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User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-22T00:00:25Z

Tohline: /* Radial Dependance of the Strong Nuclear Force */

__FORCETOC__

=Radial Dependence of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"QGP (quark-gluon plasma) is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities."
</li>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:  
<ul>
<li>
"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

From the arXiv preprint of a review article by [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force A. Deur, S. J. Brodsky, & G. F. de Téramond (2020)] titled, ''The QCD Running Coupling'':
<ul>
<li>
(middle of p. 10) "We illustrate this behavior" — that is, "… the scale dependence of the coupling" — "for the coupling that arises in the static case of heavy sources and which provides a simple physical picture. Historically, and in the case of linear theories with massless force carriers, a force coupling constant is a universal coefficient that links the force to the 'charges' of two bodies (''e.g.,'' the electric charge for electricity or the mass for gravity) divided by the distance dependence <math>1/r^2</math>."
</li>
<li>
(middle of p. 10, continued) "In QFT (quantum field theory) … for weak enough forces, the first Born approximation dominates higher order contributions and the <math>1/q^2</math> propagator in momentum yields the familiar <math>1/r^2</math> factor in coordinate space. However, higher orders do contribute and deviations from the <math>1/r^2</math> law thus occur. This extra r-dependence is folded in the coupling which then acquires a scale dependence."
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

==Insert Dependence on (Energy) Density==

The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential. The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime. Generally speaking, a QGP is achieved in a very high energy-density environment.

We can mimic this behavior in our modified cosmology by assuming that the coefficient on the <math>1/r</math> term in the gravitational acceleration varies with the energy-density of the fluid. (More simply, let's have it vary with the ''mass''-dentiy.) We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>. Let's try …
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>\ddot{R}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>- \frac{GM}{R^2} \biggl\{ 1 + \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \frac{R}{a_T} \biggr\} \, .</math>
</td>
</tr>
</table>
Note that the <math>~R-</math>dependent potential from which this expression for the acceleration is derived is,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>~\Phi(R)</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>+ \frac{GM}{R} - \frac{GM_r}{a_T} \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \ln \biggl(\frac{R}{a_T}\biggr) \, .</math>
</td>
</tr>
</table>

This expression for the gravitational acceleration has the desired properties:
<ul>
<li>
In the early universe, when <math>\rho/\rho_H \gg 1</math>, the density-dependent coefficient of the second (confining) term goes to zero; we have an ''asymptotically free regime'' in which a Coulomb-like potential dominates throughout the universe.
</li>
<li>
As the universe expands, the density will steadily drop. For <math>\rho/\rho_H \ll 1</math>, the density-dependent coefficient of the confining term approaches unity and we retrieve our originally proposed, modified cosmology; that is, the potential is dominated by a logarithmic term for all distances greater than <math>~a_T</math>.
</li>
</ul>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-21T22:14:22Z

Tohline: /* Insert Dependence on (Energy) Density */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"QGP (quark-gluon plasma) is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities."
</li>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:  
<ul>
<li>
"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

From the arXiv preprint of a review article by [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force A. Deur, S. J. Brodsky, & G. F. de Téramond (2020)] titled, ''The QCD Running Coupling'':
<ul>
<li>
(middle of p. 10) "We illustrate this behavior" — that is, "… the scale dependence of the coupling" — "for the coupling that arises in the static case of heavy sources and which provides a simple physical picture. Historically, and in the case of linear theories with massless force carriers, a force coupling constant is a universal coefficient that links the force to the 'charges' of two bodies (''e.g.,'' the electric charge for electricity or the mass for gravity) divided by the distance dependence <math>1/r^2</math>."
</li>
<li>
(middle of p. 10, continued) "In QFT (quantum field theory) … for weak enough forces, the first Born approximation dominates higher order contributions and the <math>1/q^2</math> propagator in momentum yields the familiar <math>1/r^2</math> factor in coordinate space. However, higher orders do contribute and deviations from the <math>1/r^2</math> law thus occur. This extra r-dependence is folded in the coupling which then acquires a scale dependence."
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

==Insert Dependence on (Energy) Density==

The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential. The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime. Generally speaking, a QGP is achieved in a very high energy-density environment.

We can mimic this behavior in our modified cosmology by assuming that the coefficient on the <math>1/r</math> term in the gravitational acceleration varies with the energy-density of the fluid. (More simply, let's have it vary with the ''mass''-dentiy.) We want to kill off the <math>1/r</math> term when the density climbs above some threshold, <math>\rho_H</math>. Let's try …
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>\ddot{R}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>- \frac{GM}{R^2} \biggl\{ 1 + \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \frac{R}{a_T} \biggr\} \, .</math>
</td>
</tr>
</table>
Note that the <math>~R-</math>dependent potential from which this expression for the acceleration is derived is,
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>~\Phi(R)</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>+ \frac{GM}{R} - \frac{GM_r}{a_T} \biggl[\frac{\rho}{\rho_H} - 1\biggr]^{-2} \ln \biggl(\frac{R}{a_T}\biggr) \, .</math>
</td>
</tr>
</table>

This expression for the gravitational acceleration has the desired properties:
<ul>
<li>
In the early universe, when <math>\rho/\rho_H \gg 1</math>, the density-dependent coefficient of the second (confining) term goes to zero; we have an ''asymptotically free regime'' in which a Coulomb-like potential dominates throughout the universe.
</li>
<li>
As the universe expands, the density will steadily drop. For <math>\rho/\rho_H \ll 1</math>, the density-dependent coefficient of the confining term approaches unity and we retrieve our originally proposed, modified cosmology; that is, the potential is dominated by a logarithmic term for all distances greater than <math>~a_T</math>.
</li>
</ul>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-21T17:20:47Z

Tohline: /* Cosmologies */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"QGP (quark-gluon plasma) is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities."
</li>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:  
<ul>
<li>
"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

From the arXiv preprint of a review article by [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force A. Deur, S. J. Brodsky, & G. F. de Téramond (2020)] titled, ''The QCD Running Coupling'':
<ul>
<li>
(middle of p. 10) "We illustrate this behavior" — that is, "… the scale dependence of the coupling" — "for the coupling that arises in the static case of heavy sources and which provides a simple physical picture. Historically, and in the case of linear theories with massless force carriers, a force coupling constant is a universal coefficient that links the force to the 'charges' of two bodies (''e.g.,'' the electric charge for electricity or the mass for gravity) divided by the distance dependence <math>1/r^2</math>."
</li>
<li>
(middle of p. 10, continued) "In QFT (quantum field theory) … for weak enough forces, the first Born approximation dominates higher order contributions and the <math>1/q^2</math> propagator in momentum yields the familiar <math>1/r^2</math> factor in coordinate space. However, higher orders do contribute and deviations from the <math>1/r^2</math> law thus occur. This extra r-dependence is folded in the coupling which then acquires a scale dependence."
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

==Insert Dependence on (Energy) Density==

The QGP is a regime where the interaction between quarks and gluons is dominated by the ''Coulomb-like'' term in the interaction potential. The particles interact with one another as though they are not confined; this is the so-called ''asymptotically free'' regime. Generally speaking, a QGP is achieved in a very high energy-density environment.

We can mimic this behavior in our modified cosmology by assuming that the coefficient on the potential's logarithmic term varies with the energy-density of the fluid. (More simply, let's have it vary with the ''mass''-dentiy.) We want to "kill off" the logarithmic term when the density climbs above some threshold, <math>\rho_H</math>. Let's try …
<table border="0" align="center" cellpadding="5">
<tr>
<td align="right"><math>\Phi</math></td>
<td align="center"><math>=</math></td>
<td align="left">
- \frac{GM_r}{r}
</td>
</tr>
</table>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-20T15:37:09Z

Tohline: /* Tidbits */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"QGP (quark-gluon plasma) is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities."
</li>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:  
<ul>
<li>
"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

From the arXiv preprint of a review article by [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force A. Deur, S. J. Brodsky, & G. F. de Téramond (2020)] titled, ''The QCD Running Coupling'':
<ul>
<li>
(middle of p. 10) "We illustrate this behavior" — that is, "… the scale dependence of the coupling" — "for the coupling that arises in the static case of heavy sources and which provides a simple physical picture. Historically, and in the case of linear theories with massless force carriers, a force coupling constant is a universal coefficient that links the force to the 'charges' of two bodies (''e.g.,'' the electric charge for electricity or the mass for gravity) divided by the distance dependence <math>1/r^2</math>."
</li>
<li>
(middle of p. 10, continued) "In QFT (quantum field theory) … for weak enough forces, the first Born approximation dominates higher order contributions and the <math>1/q^2</math> propagator in momentum yields the familiar <math>1/r^2</math> factor in coordinate space. However, higher orders do contribute and deviations from the <math>1/r^2</math> law thus occur. This extra r-dependence is folded in the coupling which then acquires a scale dependence."
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-20T15:18:30Z

Tohline: /* Wikipedia as a Resource */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"QGP (quark-gluon plasma) is a state of matter in which the elementary particles that make up the hadrons of baryonic matter are freed of their strong attraction for one another under extremely high energy densities."
</li>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:  
<ul>
<li>
"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-19T15:14:13Z

Tohline: /* Wikipedia as a Resource */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

[https://en.wikipedia.org/wiki/Strong_interaction Strong interaction]:  
<ul>
<li>
"Unlike all other forces … the strong force does not diminish in strength with increasing distance between pairs of quarks. After a limiting distance (about the size of a hadron) has been reached, it remains at a strength of about 10,000 newtons, no matter how much farther the distance between the quarks." Hence, the effective potential has a term that is linear in r.
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-19T15:03:18Z

Tohline: /* Wikipedia as a Resource */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

[https://en.wikipedia.org/wiki/Color_confinement Color confinement]: 
<ul>
<li>
"[This] phenomenon can be understood qualitatively by noting that the force-carrying [bosonic] gluons of QCD have color charge [as well as do the fermionic quarks], unlike the photons of QED. Whereas the electric field between electrically charged particles decreases rapidly as those particles are separated, the gluon field between a pair of color charges forms a narrow flux tube (or string) between them. Because of this behavior of the gluon field, the strong force between the particles is constant regardless of their separation."
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-19T14:47:31Z

Tohline: /* Wikipedia as a Resource */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]:  
<ul>
<li>
"In normal matter quarks are ''confined''; in the QGP (quark-gluon plasma) quarks are ''deconfined''."
</li>
<li>
"In classical QCD quarks are the ''fermionic'' components of hadrons (mesons and baryons) while the gluons are considered the boson components of such particles. The gluons are the force carriers, or bosons, of the QCD color force, while the quarks by themselves are their fermionic matter counterparts."
<ul>
<li>Electrons (spin 1/2 particles) and (as a composite particle) protons are fermions; they obey Fermi-Dirac statistics.</li>
<li>According to the Standard Model of Particle Physics, photons (spin 1 particles) are one of only 5 elementary bosons; they obey Bose-Einstein statistics.</li>
</ul>
</li>
</ul>

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/StrongNuclearForce

2021-07-19T14:29:30Z

Tohline: /* Radial Dependance of the Strong Nuclear Force */

__FORCETOC__

=Radial Dependance of the Strong Nuclear Force=

{{LSU_HBook_header}}

==Wikipedia as a Resource==
[https://en.wikipedia.org/wiki/Quark–gluon_plasma QGP]

==Tidbits==
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]:
<ul>
<li>
From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, ,
</math>
</td>
</tr>
</table>
</div>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.

This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
</li>
<ul>
<li>
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
</li>
<li>
This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
</li>
</ul>
<li>
At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~V(r)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, ,
</math>
</td>
</tr>
</table>
</div>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.

This is a semi-classical approximation, but it is good enough that [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka] used it briefly in his book.
</li>
<li>
The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power.

Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
</li>
</ul>

==Pointers from Richard Imlay ''circa'' 1983==
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a radially dependent (e.g., logarithmic ) potential, he pointed me to the following:
* Quigg, C. & Rosner, J. L. [http://adsabs.harvard.edu/abs/1977PhLB...71..153Q (1977), Physics Letters, 71B, pp. 153-157], ''Quarkonium level spacings''
* Tuts, P. Michael [https://books.google.com/books/about/Proceedings_of_the_1983_International_Sy.html?id=ktnvAAAAMAAJ (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), pp. 284-327], ''Experimental results in heavy quarkonia''

<div align="center">
<table border="0" align="center" cellpadding="3" width="85%">
<tr>
<th align="center" colspan="2">Figure 1 from Tuts (1983)</th>
</tr>
<tr>
<td align="center" rowspan="2">
[[File:ImlayFig1 cropped.png|center|300px|Figure 1 from Tuts (1983)]]
</td>
<td align="left">
References Cited in Figure Caption:
<ol start="46">
<li>[<font color="orange">'''curve 3'''</font>] E. Eichten, E. ''et al.'' [http://adsabs.harvard.edu/abs/1980PhRvD..21..203E (1980), Phys. Rev. D21, pp. 203-233], ''Charmonium: Comparison with experiment''</li>
<li>[<font color="orange">'''curve 2'''</font>] W. Buchmuller, G. Grunberg, & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1980PhRvL..45..103B (1980), PRL, 45, pp. 103-106], ''Regge slope and the Λ parameter in quantum chromodynamics: An empirical approach via quarkonia''; <br />
W. Buchmuller & S.-H. H. Tye [http://adsabs.harvard.edu/abs/1981PhRvD..24..132B (1981), Phys. Rev. D24, pp. 132-156], ''Quarkonia and quantum chromodynamics''</li>
<li>[<font color="orange">'''curve 1'''</font>] A. Martin [http://adsabs.harvard.edu/abs/1980PhLB...93..338M (1980), Physics Letters B93, pp. 338-342]; ''A fit of upsilon and charmonium spectra''<br />
A. Martin [http://adsabs.harvard.edu/abs/1981PhLB..100..511M (1981) Physics Letters B100, pp. 511-514], ''A simultaneous fit of <math>~b\bar{b}</math>, <math>~c\bar{c}</math>, <math>~s\bar{s}</math> (bcs Pairs) and <math>~c\bar{s}</math> spectra''</li>
<li>[<font color="orange">'''curve 4'''</font>] G. Bhanot & S. Rudaz [http://adsabs.harvard.edu/abs/1978PhLB...78..119B (1978), Physics Letters B78, pp. 119-124], ''A new potential for quarkonium''</li>
</ol>
</td>
</tr>
<tr>
<td align="center"> </td>
</tr>
</table>
</div>

Additionally, my handwritten notes, ''circa'' 1983, point to:
* S. M. Alladin & K. S. V. S. Narasimhan [http://adsabs.harvard.edu/abs/1982PhR....92..339A (1982), Physics Reports, 92 (#6), pp. 339 - 397], ''Gravitational interactions between galaxies'' — in 2018, this does not now seem relevant.
* J. Gasser & H. Leutwyler [http://adsabs.harvard.edu/abs/1982PhR....87...77G (1982), Physics Reports, 87, Issue 3, pp. 77 - 169], ''Quark masses''
* G. Altarelli [http://adsabs.harvard.edu/abs/1982PhR....81....1A (1982), Physics Reports, 81, Issue 1, pp. 1 - 129]. ''Partons in quantum chromodynamics''

=Cosmologies=

==Standard Presentation==
<ul>
<li>
Derivation of the [[User:Tohline/SSC/FreeFall#Relationship_to_Relativistic_Cosmologies|Friedmann Equations]] in the context of our discussion of ''Newtonian'' free-fall collapse.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Newtonian Description of Pressure-Free Collapse
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
</td>
</tr>

<tr>
<td align="right">
where,     <math>~k(R_i,v_i)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
<li>
[http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies; see also, chapter 29 of [https://www.scribd.com/doc/301615425/An-Introduction-to-Modern-Astrophysics Carroll & Ostlie (2007, 2<sup>nd</sup> Edition)]. Their equations (2) and (3) are written in the following table — with factors of <math>~c^2</math> inserted to explicitly clarify how the dimensional units are the same for every term in each equation.

<table border="1" cellpadding="10" align="center">
<tr><th align="center">
Friedmann equations:<br />
''Field equations of GR applied to the FRW metric''
</th></tr>
<tr><td align="left">

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\frac{\ddot{a}}{a}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3}\pi G \biggl[\rho + \frac{3p}{c^2} \biggr] + \frac{\Lambda c^2}{3} \, .</math>
</td>
</tr>
</table>
</div>

</td></tr>
</table>

</li>
</ul>

==ASTR4422 Class Notes==
Homework set #3 that was assigned to my ASTR4422 class in the spring of 2005 explored how solutions to the ''Newtonian'' free-fall collapse problem can be mapped directly to cosmological models of the expanding universe. The stated objective was to match the "closed universe," <math>~\Omega_0 = 2</math> model presented in Figure 27.4 (p. 1230) of the 1<sup>st</sup> edition of Carroll & Ostlie. (In the spring of 2009, this was assignment #5, and the aim was to match Figure 29.5 from the 2<sup>nd</sup> edition of Carroll & Ostlie.)

In the free-fall model, the collapse starts from rest at initial radius and density, <math>~r_0</math> and <math>~\rho_0</math>, respectively, in which case — see, for example, our [[User:Tohline/SSC/FreeFall#RoleOfIntegrationConstant|discussion of the role of the integration constant]] —
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~k_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \, .</math>
</td>
</tr>
</table>
</div>
Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H^2 = \biggl( \frac{\dot{R}}{R} \biggr)^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho - \frac{2G}{r_i} \biggl[ \frac{4}{3} \pi \rho_i r_i^3 \biggr] \frac{1}{R^2}</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \frac{\rho}{\rho_i} - \frac{r_i^2}{R^2} \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \biggl(\frac{r_i}{R}\biggr)^3 - \biggl(\frac{r_i}{R}\biggr)^2 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] \, .</math>
</td>
</tr>
</table>
Now, adopting the terminologies, <math>~\Omega \equiv \rho/\rho_\mathrm{crit}</math> and, for any <math>~H</math>, <math>~\rho_\mathrm{crit} \equiv 3H^2/(8\pi G) ~~\Rightarrow ~~ H^2 = 8\pi G \rho/(3\Omega)</math>, we have,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{8\pi G \rho}{3\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta - \sec^4\zeta \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~\frac{1}{\Omega}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\rho_i}{\rho} \biggl[ \sec^6\zeta - \sec^4\zeta \biggr] = 1 - \cos^2\zeta \, .</math>
</td>
</tr>
</table>
Hence, if in the present epoch [denoted by subscript 0], <math>~\Omega = \Omega_0 = 2</math> (as in the Carroll & Ostlie figure that we're trying to match), then in our "free-fall" model, the present epoch occurs at the dimensionless time given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~1 - \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \cos^2\zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{2}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \zeta_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{\pi}{4} \, .</math>
</td>
</tr>
</table>
</div>
This, in turn, implies that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~H_0^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ \sec^6\zeta_0 - \sec^4\zeta_0 \biggr] </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{8}{3}\pi G \rho_i \biggl[ 2^3 - 2^2\biggr] = \frac{32}{3}\pi G \rho_i </math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{1}{\tau_\mathrm{ff}^2} \biggl[\frac{3\pi}{32G\rho_i}\biggr] \frac{32}{3}\pi G \rho_i = \biggl(\frac{\pi}{\tau_\mathrm{ff}} \biggr)^2 \, .</math>
</td>
</tr>
</table>

As our [[User:Tohline/SSC/FreeFall#Parametric|parametric solution of the Newtonian free-fall problem details]], quite generally we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~t</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\tau_\mathrm{ff}}{\pi} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2}{\pi} \biggl[\frac{3\pi}{32G\rho_i} \biggr]^{1 / 2} \biggl[ \zeta + \frac{1}{2}\sin(2\zeta)\biggr]</math>
</td>
</tr>
</table>
</div>

==With Logarithmic Potential Included==
Let's return to the ''Newtonian'' expression for the acceleration equation and replace the time-dependent density, <math>~\rho</math>, with the time-independent mass, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\ddot{R}}{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{4}{3} ~\pi G\rho = - \frac{GM_R}{R^3} </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \, .</math>
</td>
</tr>
</table>
</div>
This is the form of the equation that has been integrated analytically in our [[User:Tohline/SSC/FreeFall#Single_Particle_in_a_Point-Mass_Potential|separate discussion of Newtonian free-fall collapse]]. Now, in our [http://adsabs.harvard.edu/abs/1983IAUS..100..205T published speculation about a modified force-law to explain flat rotation curves], we proposed (see that publication's equation 1) a gravitational acceleration of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{GM_R}{R^2} \biggl[1 + \frac{R}{a_\mathrm{T}}\biggr] \, .</math>
</td>
</tr>
</table>
</div>
This was intended to represent the modified gravitational acceleration felt by a (massless) test particle moving outside of a point-mass, <math>~M_R</math>. When considering a position ''inside'' of a spherical mass distribution whose radius, <math>~R_2 > R</math>, the first term remains the same because material outside of the location, <math>~R</math>, does not exert a net gravitational acceleration. But the second term cannot be treated that way. Following our [[User:Tohline/DarkMatter/UniformSphere#General_Derivation_from_Notes_Dated_29_November_1982|separate discussion of a 1/r force law]], we propose the following acceleration due to such an extended mass source:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{G}{R^2} \biggl[\frac{4}{3}\pi \rho R^3\biggr]
- \frac{G}{a_T} \biggl[ \frac{4}{3}\pi\rho R_2\biggr] R \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>
Furthermore, let's equate <math>~R_2</math> with the "size of the universe," namely, <math>~ct</math>; and let's again define the mass inside of the Lagrangian <math>~R</math> as <math>~M_R</math>. Then we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\ddot{R}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{R_2}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{GM_R}{R^2}
- \frac{GM_R }{R^2} \biggl( \frac{ct}{a_T}\biggr) \biggl\{
1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R}{ct} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
\biggr\} \, .
</math>
</td>
</tr>
</table>
</div>

=Potentially Useful References=
* Wikipedia — [https://en.wikipedia.org/wiki/Nuclear_binding_energy#Semiempirical_formula_for_nuclear_binding_energy Semiempirical Formula for the Nuclear Binding Energy]
* [https://books.google.com/books/about/Theoretical_Nuclear_and_Subnuclear_Physi.html?id=mfphXc8b-2IC Walecka, John Dirk], ''Theoretical Nuclear and Subnuclear Physics'', World Scientific (2004)

{{LSU_HBook_footer}}

User:Tohline/Appendix/Ramblings/RadiationHydro

2021-07-05T17:45:45Z

Tohline: /* Optically Thick Regime */

__FORCETOC__


=Radiation-Hydrodynamics=

{{LSU_HBook_header}}

==Governing Equations==
===Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields===
First, referencing §2 of [http://adsabs.harvard.edu/abs/2006ApJS..165..188H J. C. Hayes et al. (2006, ApJS, 165, 188 - 228)] — alternatively see §2.1 of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M D. C. Marcello & J. E. Tohline (2012, ApJS, 199, id. 35, 29 pp)] — we see that the set of principal governing equations that is typically used in the astrophysics community to include the effects of radiation on self-gravitating fluid flows includes the,
<div align="center">
<span id="Poisson"><font color="#770000">'''Poisson Equation'''</font></span>

{{ User:Tohline/Math/EQ_Poisson01 }}

[http://adsabs.harvard.edu/abs/2006ApJS..165..188H Hayes et al. (2006)], p. 190, Eq. (15)
</div>
the,
<div align="center">
<span id="Continuity"><font color="#770000">'''Continuity Equation'''</font></span>

{{ User:Tohline/Math/EQ_Continuity01 }}
</div>

and — ignoring magnetic fields — a modified version of the,

<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the Euler Equation,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\vec{v}}{dt}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{1}{\rho}\nabla P - \nabla \Phi + \frac{1}{\rho}\biggl(\frac{\chi}{c}\biggr) \vec{F} \, ,
</math>
</td>
</tr>
</table>
</div>
plus the following pair of additional ''energy-conservation-based'' dynamical equations:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho \frac{d}{dt} \biggl( \frac{e}{\rho}\biggr) + P\nabla \cdot \vec{v} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \biggl[ \nabla \cdot \vec{F} + \bold{P}_\mathrm{st}:\nabla{\vec{v}} + c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \biggr] \, ,
</math>
</td>
</tr>
</table>
where, in this last expression, <math>~\bold{P}_\mathrm{st}</math> is the radiation stress tensor.

===Various Realizations===
====First Law====

By combining the continuity equation with the

<div id="PGE:FirstLaw" align="center">
<font color="#770000">'''First Law of Thermodynamics'''</font>

{{User:Tohline/Math/EQ_FirstLaw01}}
</div>

we can write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho T\frac{ds}{dt}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, .
</math>
</td>
</tr>
</table>
Given that the specific internal energy <math>~(\epsilon)</math> and the internal energy density <math>~(e)</math> are related via the expression, <math>~\epsilon = e/\rho</math>, we appreciate that the first of the above-identified ''energy-conservation-based'' dynamical equations is simply a restatement of the 1<sup>st</sup> Law of Thermodynamics in the context of a physical system whose fluid elements gain or lose entropy as a result of the (radiation-transport-related) source and sink terms,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho T \frac{ds}{dt}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .</math>
</td>
</tr>
</table>

====Energy-Density of Radiation Field====
By combining the left-hand side of the second of the above-identified ''energy-conservation-based'' dynamical equations with the continuity equation, then replacing the Lagrangian (that is, the [https://en.wikipedia.org/wiki/Material_derivative ''material'']) time derivative by its Eulerian counterpart, the left-hand side can be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho}~\frac{d\rho}{dt}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{dE_\mathrm{rad}}{dt} + E_\mathrm{rad}\nabla\cdot \vec{v}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial E_\mathrm{rad}}{\partial t} + \vec{v}\cdot \nabla E_\mathrm{rad}+ E_\mathrm{rad}\nabla\cdot \vec{v}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) \, ,
</math>
</td>
</tr>
</table>
which provides an alternate form of the expression, as found for example in equation (4) of [http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello & J. E. Tohline (2012)].

====Thermodynamic Equilibrium====
In an optically thick environment that is in thermodynamic equilibrium at temperature, <math>~T</math>, the energy-density of the radiation field is,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~E_\mathrm{rad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~a_\mathrm{rad}T^4 \, ,</math>
</td>
</tr>
</table>
and each fluid element will radiate — and, hence lose some of its internal energy to the surrounding radiation field — at a rate that is governed by the integrated Planck function,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~B_p = \frac{\sigma}{\pi}T^4 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{ca_\mathrm{rad}}{4\pi} T^4 \, ,</math>
</td>
</tr>
</table>
where, <math>~\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math>, is the Stefan-Boltzmann constant, and the ''radiation constant'' — which is included in an [[User:Tohline/Appendix/Variables_templates|associated appendix]] among our list of key physical constants — is,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
{{ User:Tohline/Math/C_RadiationConstant }}
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{8\pi^5}{15}\frac{k^4}{(hc)^3} \, .</math>
</td>
</tr>
</table>
Also under these conditions, it can be shown that — see, for example, discussion associated with equations (12) and (18) in [http://adsabs.harvard.edu/abs/2012ApJS..199...35M Marcello & J. E. Tohline (2012)] —

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~ \bold{P}_\mathrm{st} :\nabla{\vec{v}}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, ,</math>
</td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\vec{F}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~- \frac{1}{3}\biggl(\frac{c}{\chi}\biggr) \nabla E_\mathrm{rad} \, ,</math>
</td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl(\frac{\chi}{c}\biggr) \vec{F}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~-\nabla P_\mathrm{rad} \, ,</math>
</td>
</tr>
</table>
where we have recognized that the radiation pressure,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~P_\mathrm{rad} = \frac{1}{3}E_\mathrm{rad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{1}{3}a_\mathrm{rad}T^4 \, .</math>
</td>
</tr>
</table>

Hence, the modified Euler equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho ~ \frac{d\vec{v}}{dt}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla (P+P_\mathrm{rad}) - \rho \nabla \Phi \, ,
</math>
</td>
</tr>
</table>

and the equation governing the time-dependent behavior of <math>~E_\mathrm{rad}</math> becomes,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) + \frac{1}{3}E_\mathrm{rad} \nabla \cdot \vec{v} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p \, .
</math>
</td>
</tr>
</table>

===Optically Thick Regime===
In the optically thick regime, the following conditions hold:
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~c\kappa_E E_\mathrm{rad}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~4\pi \kappa_p B_p \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~E_\mathrm{rad}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~aT^4 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\biggl(\frac{\chi}{c}\biggr) \vec{F}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~- \nabla \biggl(\frac{aT^4}{3} \biggr) \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~ \vec{\bold{P}}:\nabla{\vec{v}}</math>
</td>
<td align="center">
<math>~\rightarrow</math>
</td>
<td align="left">
<math>~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, .</math>
</td>
</tr>
</table>

Start with,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~Tds_\mathrm{rad} = dQ</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
d\biggl(\frac{E_\mathrm{rad}}{\rho} \biggr) + P_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho}~d E_\mathrm{rad} + E_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr) + P_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{\rho}~d (aT^4 ) + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4aT^3}{\rho}~dT + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4aT}{3} \biggl[ \frac{3T^2}{\rho}~dT + T^3~d\biggl( \frac{1}{\rho} \biggr) \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{4aT}{3} ~d\biggl( \frac{T^3}{\rho} \biggr)
</math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ ds_\mathrm{rad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~d\biggl( \frac{4aT^3}{3\rho} \biggr)
</math>
</td>
</tr>
</table>

Integrating then gives us,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~s_\mathrm{rad}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
~\frac{4aT^3}{3\rho} + \mathrm{const.}
</math>
</td>
</tr>
</table>

[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-136)<br />
[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. I, §9, immediately following Eq. (9.22)
</div>

This also means that,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr) + \frac{E_\mathrm{rad}}{3} \nabla\cdot\vec{v}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho} \frac{d\rho}{dt}
+ \frac{E_\mathrm{rad}}{3} \nabla\cdot\vec{v}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{dE_\mathrm{rad}}{dt} + \frac{4E_\mathrm{rad}}{3} \nabla\cdot\vec{v}
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4E_\mathrm{rad}}{3}
\biggl[ \frac{3}{4} \cdot \frac{d\ln E_\mathrm{rad}}{dt} + \nabla\cdot\vec{v} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4E_\mathrm{rad}}{3}
\biggl[ \frac{d\ln (E_\mathrm{rad})^{3/4}}{dt} + \nabla\cdot\vec{v} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4E_\mathrm{rad}}{3}
\biggl[ \frac{d\ln T^3}{dt} - \frac{d\ln\rho}{dt} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4E_\mathrm{rad}}{3}
\biggl[ \frac{d\ln (T^3/\rho)}{dt} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4aT^4}{3} \biggl( \frac{\rho}{T^3}\biggr)
\biggl[ \frac{d(T^3/\rho)}{dt} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\rho T\biggl[ \frac{ds_\mathrm{rad}}{dt} \biggr] \, .
</math>
</td>
</tr>
</table>
</div>

Hence, the equation governing the time-dependent behavior of <math>~E_\mathrm{rad}</math> becomes an expression detailing the time-dependent behavior of the specific entropy, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\rho T~\frac{ds_\mathrm{rad}}{dt} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p \, .
</math>
</td>
</tr>
</table>

[<b>[[User:Tohline/Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], §9, Eq. (9.22)
</div>

=Traditional Stellar Structure Equations=

<div align="center">
<font color="#770000">'''Hydrostatic Balance'''</font>
{{ User:Tohline/Math/EQ_SShydrostaticBalance01 }}
<br />
<font color="#770000">'''Mass Conservation'''</font>
{{ User:Tohline/Math/EQ_SSmassConservation01 }}
<br />
<font color="#770000">'''Energy Conservation'''</font>
{{ User:Tohline/Math/EQ_SSenergyConservation01 }}
<br />
<font color="#770000">'''Radiation Transport'''</font>
{{ User:Tohline/Math/EQ_SSradiationTransport01 }}
<br />

[http://adsabs.harvard.edu/abs/1958ses..book.....S M. Schwarzschild (1958)], Chapter III, §12, Eqs. (12.1), (12.2), (12.3), (12.4)<br />
[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Chapter 6, Eqs. (6-1), (6-2), (6-3a), (6-4a)<br />
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], Eqs. (1.5), (1.1), (1.54), (1.57)<br />
[<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>], Eqs. (1.2), (2.4), (4.22), (5.11)<br />
[http://adsabs.harvard.edu/abs/1998asa..book.....R W. K. Rose (1998)], Eqs. (2.27), (2.28), (2.xx), (2.80)<br />
[<b>[[User:Tohline/Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, Eqs. (2.1), (2.2), (2.18), (2.8)<br />
[http://adsabs.harvard.edu/abs/2010asph.book.....C A. R. Choudhuri (2010)], Chapter 3, Eqs. (3.2), (3.1), (3.15), (3.16)<br />
[http://adsabs.harvard.edu/abs/2016asnu.book.....M D. Maoz (2016)], §3.5, Eqs. (3.56), (3.57), (3.59), (3.58)

</div>

=Related Discussions=
* Euler equation viewed from a [[User:Tohline/PGE/RotatingFrame|rotating frame of reference]].
* An [[User:Tohline/PGE/ConservingMomentum#Euler_Equation|earlier draft of this "Euler equation" presentation]].

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-07-02T21:13:21Z

Tohline: /* Princeton IAS */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  I doubt that she knew it, but [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. [https://ui.adsabs.harvard.edu/abs/2005BAAS...37.1555S/abstract Kevin Prendergast] (Columbia University), Dr. [https://en.wikipedia.org/wiki/Paul_L._Schechter Paul Schechter] (Harvard-Smithsonian CfA), and Dr. [https://physicalsciences.uchicago.edu/news/article/richard-miller-obituary/ Richard Miller] (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. [https://en.wikipedia.org/wiki/Joel_Primack Joel Primack] (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both [https://en.wikipedia.org/wiki/Sandra_Faber Sandy Faber] and [https://en.wikipedia.org/wiki/George_Blumenthal George Blumenthal], so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
[[File:Rood1983postcard combined.png|right|thumb|150px|Postcard from Herbert Rood (1983)]]Also, in late May of 1983, I received a postcard requesting reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). The letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. [https://en.wikipedia.org/wiki/John_N._Bahcall John N. Bahcall] (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://en.wikipedia.org/wiki/Neta_Bahcall Neta Bahcall's] article in [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. [https://en.wikipedia.org/wiki/Mike_Disney Michael Disney] (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, [https://www.ifa.hawaii.edu/users/kuhn/kuhn2.html Jeff Kuhn] (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

File:Rood1983postcard combined.png

2021-07-02T21:08:29Z

Tohline:

User:Tohline/Math/EQ FirstLaw01

2021-07-01T21:33:44Z

Tohline:

<table border=0 cellpadding=2>
<tr>
<td align="right">
[[Image:LSU_Key.png|25px|link=Appendix/EquationTemplates#Principal_Governing_Equations]]
</td>
<td align="center">
<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math>
</td>
</tr>
</table>

User:Tohline/DarkMatter/VeraRubin

2021-06-29T18:51:42Z

Tohline: /* Early Interactions with Vera Rubin */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  I doubt that she knew it, but [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. [https://ui.adsabs.harvard.edu/abs/2005BAAS...37.1555S/abstract Kevin Prendergast] (Columbia University), Dr. [https://en.wikipedia.org/wiki/Paul_L._Schechter Paul Schechter] (Harvard-Smithsonian CfA), and Dr. [https://physicalsciences.uchicago.edu/news/article/richard-miller-obituary/ Richard Miller] (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. [https://en.wikipedia.org/wiki/Joel_Primack Joel Primack] (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both [https://en.wikipedia.org/wiki/Sandra_Faber Sandy Faber] and [https://en.wikipedia.org/wiki/George_Blumenthal George Blumenthal], so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. [https://en.wikipedia.org/wiki/John_N._Bahcall John N. Bahcall] (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://en.wikipedia.org/wiki/Neta_Bahcall Neta Bahcall's] article in [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. [https://en.wikipedia.org/wiki/Mike_Disney Michael Disney] (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, [https://www.ifa.hawaii.edu/users/kuhn/kuhn2.html Jeff Kuhn] (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-29T18:49:58Z

Tohline: /* Early Interactions with Vera Rubin */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. [https://ui.adsabs.harvard.edu/abs/2005BAAS...37.1555S/abstract Kevin Prendergast] (Columbia University), Dr. [https://en.wikipedia.org/wiki/Paul_L._Schechter Paul Schechter] (Harvard-Smithsonian CfA), and Dr. [https://physicalsciences.uchicago.edu/news/article/richard-miller-obituary/ Richard Miller] (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. [https://en.wikipedia.org/wiki/Joel_Primack Joel Primack] (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both [https://en.wikipedia.org/wiki/Sandra_Faber Sandy Faber] and [https://en.wikipedia.org/wiki/George_Blumenthal George Blumenthal], so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. [https://en.wikipedia.org/wiki/John_N._Bahcall John N. Bahcall] (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://en.wikipedia.org/wiki/Neta_Bahcall Neta Bahcall's] article in [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. [https://en.wikipedia.org/wiki/Mike_Disney Michael Disney] (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, [https://www.ifa.hawaii.edu/users/kuhn/kuhn2.html Jeff Kuhn] (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T23:01:11Z

Tohline: /* Exchange of Letters with Jim Felten (1984) … and Beyond */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. [https://ui.adsabs.harvard.edu/abs/2005BAAS...37.1555S/abstract Kevin Prendergast] (Columbia University), Dr. [https://en.wikipedia.org/wiki/Paul_L._Schechter Paul Schechter] (Harvard-Smithsonian CfA), and Dr. [https://physicalsciences.uchicago.edu/news/article/richard-miller-obituary/ Richard Miller] (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. [https://en.wikipedia.org/wiki/Joel_Primack Joel Primack] (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both [https://en.wikipedia.org/wiki/Sandra_Faber Sandy Faber] and [https://en.wikipedia.org/wiki/George_Blumenthal George Blumenthal], so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. [https://en.wikipedia.org/wiki/John_N._Bahcall John N. Bahcall] (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://en.wikipedia.org/wiki/Neta_Bahcall Neta Bahcall's] article in [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. [https://en.wikipedia.org/wiki/Mike_Disney Michael Disney] (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, [https://www.ifa.hawaii.edu/users/kuhn/kuhn2.html Jeff Kuhn] (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:54:34Z

Tohline: /* Princeton IAS */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. [https://ui.adsabs.harvard.edu/abs/2005BAAS...37.1555S/abstract Kevin Prendergast] (Columbia University), Dr. [https://en.wikipedia.org/wiki/Paul_L._Schechter Paul Schechter] (Harvard-Smithsonian CfA), and Dr. [https://physicalsciences.uchicago.edu/news/article/richard-miller-obituary/ Richard Miller] (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. [https://en.wikipedia.org/wiki/Joel_Primack Joel Primack] (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both [https://en.wikipedia.org/wiki/Sandra_Faber Sandy Faber] and [https://en.wikipedia.org/wiki/George_Blumenthal George Blumenthal], so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. [https://en.wikipedia.org/wiki/John_N._Bahcall John N. Bahcall] (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://en.wikipedia.org/wiki/Neta_Bahcall Neta Bahcall's] article in [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:45:28Z

Tohline: /* Primack at UC, Santa Cruz */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. [https://ui.adsabs.harvard.edu/abs/2005BAAS...37.1555S/abstract Kevin Prendergast] (Columbia University), Dr. [https://en.wikipedia.org/wiki/Paul_L._Schechter Paul Schechter] (Harvard-Smithsonian CfA), and Dr. [https://physicalsciences.uchicago.edu/news/article/richard-miller-obituary/ Richard Miller] (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. [https://en.wikipedia.org/wiki/Joel_Primack Joel Primack] (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both [https://en.wikipedia.org/wiki/Sandra_Faber Sandy Faber] and [https://en.wikipedia.org/wiki/George_Blumenthal George Blumenthal], so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:39:41Z

Tohline: /* Neighborhood Meeting at Yale University (1979) */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. [https://ui.adsabs.harvard.edu/abs/2005BAAS...37.1555S/abstract Kevin Prendergast] (Columbia University), Dr. [https://en.wikipedia.org/wiki/Paul_L._Schechter Paul Schechter] (Harvard-Smithsonian CfA), and Dr. [https://physicalsciences.uchicago.edu/news/article/richard-miller-obituary/ Richard Miller] (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:31:26Z

Tohline: /* Neighborhood Meeting at Yale University (1979) */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. [https://en.wikipedia.org/wiki/Vera_Rubin Vera Rubin] agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. [https://en.wikipedia.org/wiki/Jeremiah_P._Ostriker Jeremiah Ostriker] (Princeton), Dr. [https://en.wikipedia.org/wiki/Alar_Toomre Alar Toomre] (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:25:41Z

Tohline: /* Princeton IAS */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

Over the subsequent couple of decades, John and Neta Bahcall visited LSU several times. As it turns out, John had graduated from high school in Shreveport, Louisiana, and he had a brother who lived in Baton Rouge.

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:21:11Z

Tohline: /* Princeton IAS */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr>
<tr><td align="right">
-- John N. Bahcall
</td></tr></table>

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:18:25Z

Tohline: /* Exchange of Letters with Jim Felten (1984) … and Beyond */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr></table>

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable and intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T22:15:27Z

Tohline: /* Exchange of Letters with Jim Felten (1984) … and Beyond */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr></table>

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In the spring of 1985, I received a package of documents from Dr. Michael Disney (Cardiff, Wales). In addition to a wonderfully worded cover letter, the package contained a couple of thick draft articles in which he explored in a number of different directions what the implications would be of a modification to Newton's Law of the type I was proposing. He had not previously seen my work on this topic, but his thoughts were so in tune with my own that I was sure he had read my mind! It was wonderful to have heard from such a kindred spirit; his (draft) writings were like poetry, to me. Toward the end of the decade, at the invitation of Mike Disney and his astronomy colleagues at University College, Cardiff, I spent ten very enjoyable intellectually stimulating days visiting Wales.
</li>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T21:40:31Z

Tohline: /* Princeton IAS */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall (also, Princeton's IAS) that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law."</font>
</td></tr></table>
Bahcall then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 kpc … Very intriguing."</font>
</td></tr></table>

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T21:36:13Z

Tohline: /* Princeton IAS */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law.."</font>
</td></tr></table>
He then asked whether or not I had considered how the standard mathematical expression of the virial theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30"</font> relative to what is observed. His reference regarding the relevant observational measurements was [https://www.annualreviews.org/doi/10.1146/annurev.aa.15.090177.002445 Neta Bahcall's article in Annual Review of Astronomy and Astrophysics (1977, Vol. 15, pp. 505 - 540)].

I wrote back to John Bahcall, stating that I agreed with his derived correction to the virial theorem but that my interpretation of the published observational results — as found for example in Neta Bahcall's review article — was different from his. It looked to me as though the modified virial theorem expression nicely explains the mass-to-light ratio measurements in rich clusters. After studying my response, Dr. Bahcall wrote back:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"… Thank you for your interesting letter on the r<sup>-1</sup> force. You are indeed right that the mass to light ratio is typically 300 for rich clusters (I misquoted Neta's article). Thus there is no inconsistency with the virial equation I derived, provided one takes [a scale length] a &sim; 10 npc … Very intriguing."</font>
</td></tr></table>

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T20:59:27Z

Tohline: /* Princeton IAS */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law.."</font>
</td></tr></table>
He then asked whether or not I had considered how the standard mathematical expression of the viral theorem would be generalized in the context of my proposed non-Newtonian gravitational attraction. He derived what he considered to be the appropriate additional term that would be required, then applied the result to the case of "rich clusters" of galaxies. HIs conclusion was that my proposed modification of Newton's Law <font color="darkgreen">"… would overestimate the missing mass [in rich clusters] by a factor of 30."</font>

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T18:31:25Z

Tohline: /* Rubin's Scientific American Article (1983) */

=Early Interactions with Vera Rubin=
<font color="red">Note from Joel E. Tohline:</font>  Whether she knew it or not, [https://en.wikipedia.org/wiki/Vera_Rubin Dr. Vera C. Rubin] was a significant influence on my early astronomy career. What follows are some highlights of my early professional interactions with her.

{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|125px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty, postdocs and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1½ days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that, at that time, the astrophysics community, worldwide, was making significant progress on a number of issues related to galaxies — both observationally and theoretically — the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin agreed to be our opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jeremiah Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner in the summer of 1982 at the IAU Symposium No. 100.

==IAU Symposium No. 100 (1982)==

An astronomical symposium sanctioned by the International Astronomical Union (IAU) titled, ''Internal Kinematics and Dynamics of Galaxies,'' was held in Besançon, France, August 9 - 13, 1982. This is the professional conference at which I presented a short "poster paper" titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T "Stabilizing a Cold Disk with a 1/r Force Law."]. It appears as a two-page article that begins on p. 105 of the published symposium proceedings.

Dr. Rubin was (again!) the lead-off speaker for this five-day symposium; accordingly, the paper that she prepared for the symposium — titled, ''Systematics of HII Rotation Curves'' — appears as the first article (pp. 3 - 8) in the proceedings. Two pages of text (pp. 9 - 10) that immediately follow her article record six questions that were asked of Dr. Rubin at the end of her presentation, along with her six responses. The sixth question was from me; here is the published record:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="maroon">
TOHLINE:   I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V<sup>2</sup>r/L of the directly measurable quantities V, r and L. This ratio V<sup>2</sup>r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should now be seriously questioned. I hope that observers who have definitive evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
</font>
</td></tr>
<tr><td align="left">
<font color="purple">
RUBIN:   While we have observed that most Sa, Sb and Sc, galaxies have flat or slightly rising rotation curves, a few have slightly falling curves. However, I know of no isolated galaxy with rotation velocities decreasing as rapidly as <math>V \propto r^{-1 / 2}</math>. The point you raise is worth keeping in mind although I believe most of us would rather alter Newtonian gravitational theory only as a last resort.
</font>
</td></tr></table>

==Rubin's Scientific American Article (1983)==

Vera Rubin published a detailed description of the observational evidence for ''Dark Matter in Spiral Galaxies'' in the [https://ui.adsabs.harvard.edu/abs/1983SciAm.248f..96R/abstract June, 1983 issue of Scientific American (pp. 96 - 108)]. An excerpt from near the bottom of p. 102 of that article reads,
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Perhaps the most radical idea for explaining the observed high rotational velocities is one advanced independently by Joel E. Tohline of Louisiana State University and M. Milgrom and J. Bekenstein of the Weizmann Institute of Science. They have proposed that at great distances the Newtonian theory of gravitation must be modified, thereby allowing rotational velocities in galaxies to remain high at such distances from the galactic center even in the absence of unseen mass."</font>
</td></tr>
<tr><td align="right">
-- Vera C. Rubin
</td></tr></table>
This nod of recognition from Dr. Rubin broadened my visibility — both professionally and among the public — more than any other single citation. For example …

===Primack at UC, Santa Cruz===
That same month (June 1983) I received a (hand-written) letter from Dr. Joel Primack (a physicist at UC, Santa Cruz) containing the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"I'm working on a review of dark matter for Annual Reviews (with Sandy Faber, George Blumenthal and Doug Lin) and would very much appreciate it if you would send me a copy of your paper on gravity weakening with distance as an explanation of constant rotation curves, referred to in <b>Vera Rubin's recent ''Scientific American'' article</b>… Please also send a copy to George Blumenthal at UC Santa Cruz. Thanks very much!"</font>
</td></tr>
<tr><td align="right">
-- Joel Primack
</td></tr></table>
Receiving this request from Joel Primack was exciting for me on two fronts: (1) Having my work cited in a high-quality review article would significantly enhance its visibility. (2) I had only recently (1978) earned my doctoral degree in astronomy from UC, Santa Cruz and had completed courses taught by both Sandy Faber and George Blumenthal, so I knew them well and was happy to hear that they had become aware of research endeavors that I was pursuing beyond the focus of my dissertation research.

===Princeton IAS===
Also in early June of 1983, I received a request for reprints of my work from Dr. Herbert Rood (Princeton's Institute for Advanced Study). I do not now have a copy of his request, but the letter that I wrote to him in response (dated 3 June 1983) contains the following text:
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="purple">"Dear Herbert:  Your recent request for reprints of my work on the non-Newtonian Force Law came as a bit of a shock. I was, until then, unaware that Vera had mentioned my work in her article — in fact, that issue of ''Scientific American'' only arrived to subscribers here at LSU the day that your request for reprints appeared in my mailbox."</font>
</td></tr></table>
Shortly thereafter I received a letter (dated 17 June 1983) from Dr. John N. Bahcall that began as follows …
<table border="0" align="center" width="60%"><tr><td align="left">
<font color="darkgreen">"Dear Joel:   Herb Rood showed me an abstract that you had written in which you consider some of the Solar system consequences of modifying the Newtonian force law.."</font>
</td></tr></table>

==Exchange of Letters with Jim Felten (1984) … and Beyond==

[[File:JamesFelten1983small.png|right|thumb|150px|Draft of James E. Felten's (1984) paper]]About half a year later, Dr. James E. Felten — at the time, a research scientist at the NASA Goddard Space Flight Center in Maryland — was discussing with Dr. Rubin the published work of Milgrom & Bekenstein and she told him that Joel Tohline "worked on 'Milgrom' ideas before Milgrom!" (See Felten's hand-written comment inside the red oval of the image shown here, on the right; click to make the thumbnail image larger.). I presume that Dr. Rubin was recalling the discussion that I had had [[#Tohline_Visits_CIW:DTM_.281980.29|with her group in early 1980]]. Dr. Felten then contacted me and we exchanged a few letters on the subject. Dr. Felten's critique of the Milgrom-Bekenstein work was published in [https://ui.adsabs.harvard.edu/abs/1984ApJ...286....3F/abstract The Astrophysical Journal (1984, 286, pp. 3-6)]; page 5 of this article includes a ''Note added in manuscript 1984 June 6'' acknowledging these discussions.

In my (old-fashioned, paper) files, I have a record of insightful written exchanges that I had with a number of researchers throughout the decade of the 80s. For example …
<ul>
<li>
In early 1986, Jeff Kuhn (Princeton, at the time) sent me a preprint of the paper he had written in collaboration with Kruglyak; he had heard of my work from Milgrom. A brief exchange of letters followed. An acknowledgement of my effort to examine the stability of cold stellar disks appears in their published paper [https://ui.adsabs.harvard.edu/abs/1987ApJ...313....1K/abstract (ApJ, 313, 1)].
</li>
</ul>

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T17:58:48Z

Tohline: /* Rubin's Scientific American Article (1983) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T17:55:57Z

Tohline: /* Rubin's Scientific American Article (1983) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T16:28:00Z

Tohline: /* Rubin's Scientific American Article (1983) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T16:26:48Z

Tohline: /* Exchange of Letters with Jim Felten (1984) … and Beyond */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T16:25:40Z

Tohline: /* Exchange of Letters with Jim Felten (1984) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T15:46:58Z

Tohline: /* Neighborhood Meeting at Yale University (1979) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T15:44:43Z

Tohline: /* Rubin's Scientific American Article (1983) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T15:35:28Z

Tohline: /* IAU Symposium No. 100 (1982) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T15:32:30Z

Tohline: /* IAU Symposium No. 100 (1982) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T15:30:07Z

Tohline: /* Tohline Visits CIW:DTM (1980) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T15:28:05Z

Tohline: /* Neighborhood Meeting at Yale University (1979) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T05:44:39Z

Tohline:

User:Tohline/DarkMatter/VeraRubin

2021-06-28T05:18:10Z

Tohline: /* IAU Symposium No. 100 (1982) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T05:12:13Z

Tohline: /* Rubin's Scientific American Article (1983) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T05:09:48Z

Tohline:

File:JamesFelten1983small.png

2021-06-28T04:50:54Z

Tohline:

User:Tohline/DarkMatter/VeraRubin

2021-06-28T04:43:35Z

Tohline: /* Rubin's Scientific American Article (1983) */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T04:17:13Z

Tohline: /* Early Interactions with Vera Rubin */

User:Tohline/DarkMatter/VeraRubin

2021-06-28T03:53:52Z

Tohline:

User:Tohline/DarkMatter/VeraRubin

2021-06-28T03:48:40Z

Tohline: /* Neighborhood Meeting at Yale University (1979) */

=Early Interactions with Vera Rubin=
{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|thumb|200px|Yale Neighborhood Meeting (1979)]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1.5 days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that the astrophysics community, worldwide, was presently making significant progress on a number of issues — both observationally and theoretically — related to galaxies, the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin was the opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jerry Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner at the IAU Symposium No. 100 in a paper titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law].

==IAU Symposium No. 100 (1982)==

==Rubin's Scientific American Article (1983)==

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

User:Tohline/DarkMatter/VeraRubin

2021-06-28T03:41:41Z

Tohline: /* Neighborhood Meeting at Yale University (1979) */

=Early Interactions with Vera Rubin=
{{LSU_HBook_header}}

==Neighborhood Meeting at Yale University (1979)==

[[File:Yale1979Meeting.jpg|right|200px|Yale 1979 Neighborhood Meeting]]For two years, beginning in the summer of 1978, I held a J. Willard Gibbs instructorship in the astronomy department at Yale University. In my first year, I was encouraged — along with another young astronomer, Dr. Carol A. Christian — to organize a so-called ''Neighborhood Meeting'' at Yale. The idea was to focus on a topic that would bring together faculty and graduate students from universities and research centers that were "within driving distance" of the Yale campus; this, and limiting the gathering to 1.5 days (just one overnight stay) would keep travel expenses to a minimum. We accepted the challenge. Given that the astrophysics community, worldwide, was presently making significant progress on a number of issues — both observationally and theoretically — related to galaxies, the topic we picked was …
<div align="center">'''Rotation:''' The Dynamical Structure of Galaxies<br />''(A Neighborhood Meeting at Yale University)''<br />Dates: 23 - 24 March 1979</div>

Dr. Vera Rubin was the opening speaker. It was an opportunity for the (> 90) attendees to hear and see — first hand from the expert — how significant the evidence was for flat rotation curves. Five speakers followed: Dr. Jerry Ostriker (Princeton), Dr. Alar Toomre (MIT), Dr. Kevin Prendergast (Columbia University), Dr. Paul Schechter (Harvard-Smithsonian CfA), and Dr. Richard Miller (Chicago).

==Tohline Visits CIW:DTM (1980)==

In early February, 1980, I visited the Carnegie Institution of Washington's Department of Terrestrial Magnetism (CIW:DTM) in Washington, DC to meet and interact with Vera Rubin and her research group. During that visit, I had the opportunity to present an informal talk in which I pitched the idea that flat rotation curves in galaxies might be explained by modifying Newton's law of gravity at large distances. This is the idea that I first presented in a formal manner at the IAU Symposium No. 100 in a paper titled, [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law].

==IAU Symposium No. 100 (1982)==

==Rubin's Scientific American Article (1983)==

=See Also=

* [http://adsabs.harvard.edu/abs/1963MNRAS.127...21F Finzi (1963)] — ''On the Validity of Newton's Law at a Long Distance''
* [http://www.phys.lsu.edu/~tohline/TinsleyNotes1978.pdf Notes from Beatrice Tinsley dated July 3, 1978]
* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]
* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]
* [http://adsabs.harvard.edu/abs/1987ApJ...313....1K Kuhn & Kruglyak (1987)] — ''Non-Newtonian forces and the invisible mass problem''
* [http://arxiv.org/abs/1404.0531 Sanders (2014)] — ''A Historical Perspective on Modified Newtonian Dynamics''

{{LSU_HBook_footer}}

File:Yale1979Meeting.jpg

2021-06-28T03:38:51Z

Tohline: