User:Tohline/DarkMatter/UniformSphere - Revision history

Tohline: /* Limiting Cases */

2018-02-24T16:25:28Z

Limiting Cases

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	===Limiting Cases===		===Limiting Cases===
	Some limiting cases are of interest for the uniform sphere, ''i.e.,'' when <math>~R_1 = 1</math>. First, notice that (Gradshteyn & Ryzhik 1965, formula 0.141-2),		Some limiting cases are of interest for the uniform sphere, ''i.e.,'' when <math>~R_1 = 0</math>. First, notice that (Gradshteyn & Ryzhik 1965, formula 0.141-2),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: Tag derivation with date of 29 November 1982

2015-03-08T21:37:10Z

Tag derivation with date of 29 November 1982

← Older revision		Revision as of 21:37, 8 March 2015
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	==Tohline ~~Circa 1983~~==		==Tohline 1982==

	===General Derivation===		===General Derivation from Notes Dated 29 November 1982===
	If the force per unit mass exerted at the position, <math>~\vec{r}</math>, from a single point mass, <math>~m</math>, is given by,		If the force per unit mass exerted at the position, <math>~\vec{r}</math>, from a single point mass, <math>~m</math>, is given by,
	<div align="center">		<div align="center">

Tohline: /* See Also */ Add link to 1963 paper by Finzi

2015-03-08T21:03:40Z

See Also: Add link to 1963 paper by Finzi

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	=See Also=		=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://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://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]

Tohline: /* See Also */ Insert link to Tinsley's 1978 notes

2015-03-07T21:05:15Z

See Also: Insert link to Tinsley's 1978 notes

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	=See Also=		=See Also=

			* [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://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://www.phys.lsu.edu/~tohline/vita/Tohline.C5.pdf Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?]

Tohline: /* See Also */

2015-03-05T20:57:11Z

Tohline: /* See Also */ Add Kuhn & Kruglyak reference; and point to ADS for official scanned copy of my 1983 paper

2015-03-05T20:20:04Z

See Also: Add Kuhn & Kruglyak reference; and point to ADS for official scanned copy of my 1983 paper

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	=See Also=		=See Also=

	* [http://~~www~~.~~phys.lsu~~.edu/~~~tohline~~/~~vita/Tohline~~.C4.~~pdf~~ Stabilizing a Cold Disk with a 1/r Force Law]		* [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://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)]


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Tohline: /* Tohline's Derivations Circa 1983 */ Finish typing in 1983 notes

2015-03-04T22:19:58Z

Tohline's Derivations Circa 1983: Finish typing in 1983 notes

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	==Tohline~~'s Derivations~~ Circa 1983==		==Tohline Circa 1983==

			===General Derivation===
	If the force per unit mass exerted at the position, <math>~\vec{r}</math>, from a single point mass, <math>~m</math>, is given by,		If the force per unit mass exerted at the position, <math>~\vec{r}</math>, from a single point mass, <math>~m</math>, is given by,
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	</div>		</div>
	where, <math>M(R_2) \equiv 4\pi \rho_0 R_2^3/3</math>. If		where, <math>~M(R_2) \equiv 4\pi \rho_0 R_2^3/3</math>. If the position, <math>~r</math>, is located interior to a uniform-density ''shell'', then <math>~r < R_1 < R_2</math> and the aggregate acceleration is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F(r)_\mathrm{shell} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{4\pi}{3} G^' \rho_0 R_2 r \biggl\{1 - \frac{R_1}{R_2}
			- 3 \sum_{n=1}^{\infty} \biggl[ \biggl( \frac{r}{R_2} \biggr)^{2n} - \frac{R_1}{R_2} \biggl( \frac{r}{R_1} \biggr)^{2n}\biggr] \biggl[(2n-1)(2n+1)(2n+3)
			\biggr]^{-1} \biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>


			If <math>~r</math> is inside a uniform-density sphere, then <math>~R_1 = 0</math> and <math>~ r < R_2</math>, so the aggregate acceleration is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F(r)_\mathrm{in} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{4\pi}{3} G^' \rho_0 R_2 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>

			===Limiting Cases===
			Some limiting cases are of interest for the uniform sphere, ''i.e.,'' when <math>~R_1 = 1</math>. First, notice that (Gradshteyn & Ryzhik 1965, formula 0.141-2),
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\sum_{n=1}^{\infty} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \frac{1}{12} \, .</math>
			</td>
			</tr>
			</table>
			</div>

			<font color="maroon" face="Arial">'''Sitting on the Surface:'''</font> Therefore, when <math>~r = R_2</math> — that is, ''on'' the surface of the uniform-density sphere,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ - \frac{3 G^' M(R_2)}{4R_2} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			So the force acts as though the mass is all concentrated at a point, ''not'' at the center of the sphere, but at a distance <math>~4/3</math> of the sphere's radius away.

			<font color="maroon" face="Arial">'''Well Inside the Surface:'''</font> When <math>~r \ll R_2</math>,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F(r)_\mathrm{in}</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~ - \frac{G^' M(R_2)}{R_2} \biggl( \frac{r}{R_2} \biggr) \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			that is, the acceleration grows linearly with <math>~r</math>, as in any harmonic potential.

			<font color="maroon" face="Arial">'''Well Outside the Sphere:'''</font> When <math>~r \gg R_2</math>,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F(r)_\mathrm{out}</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~ - \frac{G^' M(R_2)}{r} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			which is in line with the adopted point-mass specification.

	=See Also=		=See Also=

Tohline: /* Tohline's Derivations Circa 1983 */ Add to old notes/derivation

2015-03-04T21:33:01Z

Tohline's Derivations Circa 1983: Add to old notes/derivation

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	</div>		</div>

			For a spherically symmetric mass distribution, <math>~\rho(r^')</math>, the magnitude of the force that is directed along the radial vector, <math>~\vec{r}^'</math>, and measured from the center of the mass distribution can be expressed as the following single integral over <math>~r^'</math>:
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F(r) \equiv \vec{F}\cdot \frac{\vec{r}}{r} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ -2\pi G^' \int\limits_{R_1}^{R_2} \rho(r^') (r^')^2
			\biggl[\frac{1}{r} + \frac{1}{2r^2 r^'} \biggl( r^2 - {r^'}^2 \biggr) \ln\biggl( \frac{r^' + r}{\|r^' - r\|} \biggr) \biggr] dr^'
			\, .</math>
			</td>
			</tr>
			</table>
			</div>
			This integral can be completed analytically if <math>~\rho(r^') = \rho_0</math>, that is, for a uniform-density mass distribution. Independent of whether the limits of integration, <math>~R_1</math> and <math>~R_2</math>, are less than or greater than <math>~r</math>, the integral gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F(r) </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{3G^'}{8r} \biggl( \frac{4\pi}{3}\rho_0 \biggr) \biggl\{ \biggl( R_2^3 - R_1^3 \biggr) + r^2 \biggl(R_2 - R_1\biggr)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~ +
			r^3 \biggl[ \frac{1}{2} + \frac{1}{2}\biggl( \frac{R_1}{r} \biggr)^4 - \biggl( \frac{R_1}{r} \biggr)^2\biggr]
			\ln\biggl( \frac{R_1 + r}{\|R_1 - r\|} \biggr)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~ -
			r^3 \biggl[ \frac{1}{2} + \frac{1}{2}\biggl( \frac{R_2}{r} \biggr)^4 - \biggl( \frac{R_2}{r} \biggr)^2\biggr]
			\ln\biggl( \frac{R_2 + r}{\|R_2 - r\|} \biggr)
			\biggr\} \, .</math>
			</td>
			</tr>
			</table>
			</div>

			If the position, <math>~r</math>, is located outside of a uniform-density sphere, then <math>~R_1 = 0</math> and <math>~R_2 < r</math>, so the aggregate acceleration becomes,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~F(r)_\mathrm{out} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{3G^'}{8r} \biggl( \frac{4\pi}{3}\rho_0 \biggr) \biggl\{ R_2^3 + r^2 R_2 -
			r^3 \biggl[ \frac{1}{2} + \frac{1}{2}\biggl( \frac{R_2}{r} \biggr)^4 - \biggl( \frac{R_2}{r} \biggr)^2\biggr]
			\ln\biggl( \frac{r+R_2}{r- R_2} \biggr)
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{G^' M(R_2)}{r} \biggl\{
			1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R_2}{r} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}
			\biggr\} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where, <math>M(R_2) \equiv 4\pi \rho_0 R_2^3/3</math>. If

	=See Also=		=See Also=

Tohline: /* Force Exerted by a Uniform-Density Shell or Sphere */ Begin typing in notes from 1983

2015-03-04T20:36:02Z

Force Exerted by a Uniform-Density Shell or Sphere: Begin typing in notes from 1983

← Older revision		Revision as of 20:36, 4 March 2015
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			==Tohline's Derivations Circa 1983==
			If the force per unit mass exerted at the position, <math>~\vec{r}</math>, from a single point mass, <math>~m</math>, is given by,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec{F}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \biggl( \frac{G^'m}{r} \biggr) \frac{\vec{r}}{r} \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			then the force per unit mass exerted at <math>~\vec{x}</math> by a continuous mass distribution, whose mass density is defined by the function <math>~\rho(\vec{x}^')</math>, is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec{F}(\vec{x})</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \int G^' \rho(\vec{x}^') \biggl[ \frac{\vec{x}^' - \vec{x}}{\| \vec{x}^' - \vec{x} \|^2} \biggr] d^3x^' \, .</math>
			</td>
			</tr>
			</table>
			</div>
			This central force can also be expressed in terms of the gradient of a scalar potential, <math>~\Phi(\vec{x})</math>, specifically,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\vec{F}(\vec{x})</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \vec\nabla\Phi(\vec{x}) \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Phi(\vec{x}) </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \int G^' \rho(\vec{x}^') \ln \| \vec{x}^' - \vec{x} \| d^3x^' \, .</math>
			</td>
			</tr>
			</table>
			</div>

	=See Also=		=See Also=

Tohline: Begin chapter documenting derivations from early 1980s

2015-03-04T20:16:29Z

Begin chapter documenting derivations from early 1980s

New page

=Force Exerted by a Uniform-Density Shell or Sphere=
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=See Also=

* [http://www.phys.lsu.edu/~tohline/vita/Tohline.C4.pdf 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?]

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← Older revision		Revision as of 20:57, 5 March 2015
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	* [http://adsabs.harvard.edu/abs/1983IAUS..100..205T Stabilizing a Cold Disk with a 1/r Force Law]		* [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://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)]		* [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''


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