User:Tohline/AxisymmetricConfigurations/SolvingPE - Revision history

Tohline: /* Part III */

2018-08-10T22:22:34Z

Part III

Tohline: /* Part III */

2018-08-10T22:20:10Z

Part III

← Older revision		Revision as of 22:20, 10 August 2018
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	===Part III===		===Part III===
	{{Template:LSU_CT99CommonTheme3 }}		{{Template:LSU_CT99CommonTheme3 }}


	<table border="1" align="center" cellpadding="10" width="85%">		<table border="1" align="center" cellpadding="10" width="85%">
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	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Using<br /> Toroidal Coordinates<br /> to Determine the<br /> Gravitational<br /> Potential]]		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Using<br /> Toroidal Coordinates<br /> to Determine the<br /> Gravitational<br /> Potential]]
	\|}		\|}
	Referencing [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] developed an integral expression for the Coulomb (equivalently, gravitational) potential whose foundation is the single summation, Green's function expression for the reciprocal distance between two points in toroidal coordinates. For axisymmetric mass distributions<sup>&Dagger;</sup>, Wong obtained,		{{Template:LSU_CT99CommonTheme3A }}

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

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Phi(\eta,\theta)\biggr\|_\mathrm{axisym}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2}~~
	~~\sum\limits^\infty_{n=0} \epsilon_n~~
	~~\int d\eta^' ~\sinh\eta^'~P^0_{n-1 / 2}(\cosh\eta_<) ~Q^0_~~{~~n-1 / 2}(\cosh\eta_>)~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\times \int d\theta^' \biggl\~~{ ~~\frac{ ~\cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\}~~
	~~\rho(\eta^',\theta^') \, ,~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~[http~~:~~//adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.55)~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~where, <math>~P^m_{n-1 / 2~~}~~, Q^m_{n-1 / 2~~}</math> are ''Associated Legendre Functions'' of the first and second kind of order, <math>~m</math>, and half-integer degree, <math>~n - 1/2</math> (toroidal functions). Recognizing that the <math>~m=n=0</math> toroidal function of the second kind can be expressed in terms of the complete elliptic integral of the first kind via the relation,
	~~<div align="center">~~
	~~<math>~Q_{-1 / 2}(\Chi) = \mu K(\mu)\, ,</math>     where,     <math>~\Chi \equiv \biggl[ \frac{2}{\mu^2} - 1\biggr] \, ,</math><p></p>~~
	[https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false Abramowitz & Stegun (1995)], p. 337, eq. (8.13.3)
	~~</div>~~

	~~we see that ''Version 3'' of our integral expression for the ''Gravitational Potential of an Axisymmetric Mass Distribution'' may be rewritten as,~~

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

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Phi(\eta,\theta)\biggr\|_\mathrm{axisym}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~ - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2}~~
	~~\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi) \rho(\eta^',\theta^') \, ,~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)], p. 88, Eqs. (31) & (32a)~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~<math>\mathrm{where:}~~~\Chi = \frac{[\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta)]}{[\sinh\eta \cdot \sinh\eta^']} \, .</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	Even though these two expressions for the axisymmetric potential are written entirely in terms of identically defined toroidal coordinate systems, they appear to be very different from one another. In the accompanying chapter titled, ''[[User:Tohline/2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Using Toroidal Coordinates to Determine the Gravitational Potential'']], we detail step-by-step how the first of these expressions can be transformed into the second, thereby proving that they are indeed perfectly equivalent integral expressions for the gravitational potential of axisymmetric systems.

	~~<div align="center">------------------------------</div>~~

	<sup>&Dagger;</sup>[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] actually derived a toroidal-coordinate-based integral expression for the Coulomb (equivalently, gravitational) potential that is applicable to all charge (mass) distributions, irrespective of geometric symmetries. The expression highlighted here — that is relevant to axisymmetric mass distributions — is a special case of this more general expression.

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

Tohline: /* Part III */

2018-08-10T22:18:05Z

Part III

← Older revision		Revision as of 22:18, 10 August 2018
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	===Part III===		===Part III===
	Now, beginning with Version 2 of our expression for the ''Gravitational Potential of Axisymmetric Mass Distributions'', let's also map the (unprimed) cylindrical coordinate pair, <math>~(\varpi, z)</math>, to the same (but, unprimed) toroidal coordinate system, <math>~(\eta,\theta)</math>, and place the toroidal coordinate system's ''anchor ring'' in the equatorial plane of the cylindrical-coordinate system such that, <math>~(\varpi_a,z_a) = (a,0)</math>. This gives, what we will refer to as the,		{{Template:LSU_CT99CommonTheme3 }}

	~~<table border="0" cellpadding="5" align="center">~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~<font color="#770000">'''Gravitational Potential of an Axisymmetric Mass Distribution (Version 3)'''</font>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Phi(\varpi,z)\biggr\|_\mathrm{axisym}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~- 2G a^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta} \biggr]^{1 / 2} \iint\limits_\mathrm{config}~~
	~~\biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} \mu K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' \, ,</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~<math>\mathrm{where:}~~~\mu^2 = \frac{ 2 \sinh\eta^'\cdot \sinh\eta}{ \sinh\eta^'\cdot \sinh\eta + \cosh\eta^'\cdot\cosh\eta -\cos(\theta^' - \theta) } \, .</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~

	~~<!--~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\mu^2</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\mu^2 =~~
	~~\frac{ 4 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')}~~
	~~\biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)} \biggr]~~
	~~\biggl\{ \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)}+ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 +~~
	~~\biggl[\frac{ \sin\theta}{(\cosh\eta - \cos\theta)} - \frac{ \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 \biggr\}^{-1}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{ 2 \sinh\eta^'\cdot \sinh\eta}{ \sinh\eta^'\cdot \sinh\eta + \cosh\eta^'\cdot\cosh\eta -\cos(\theta^' - \theta) } \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~

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

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\mu^2</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{ 2 \sinh\eta^'\cdot \sinh\eta}{ \sinh\eta^'\cdot \sinh\eta + \cosh\eta^'\cdot\cosh\eta -\cos(\theta^' - \theta) } \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~-->~~

	<table border="1" align="center" cellpadding="10" width="85%">		<table border="1" align="center" cellpadding="10" width="85%">

Tohline: /* Part II */

2018-08-10T22:14:18Z

Part II

← Older revision		Revision as of 22:14, 10 August 2018
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	{{ Template:LSU_CT99CommonTheme2 }}		{{ Template:LSU_CT99CommonTheme2 }}


	<table border="1" align="center" cellpadding="10" width="85%">		<table border="1" align="center" cellpadding="10" width="85%">
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	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Attempt at<br />Simplification]]		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|Attempt at<br />Simplification]]
	\|}		\|}
	Referencing (equivalently, Version 1 of) the [[#Part_1\|above-identified integral expression]] for the ''Gravitational Potential of an Axisymmetric Mass Distribution,'' [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)] offer the following assessment in §6 of their paper:		{{ Template:LSU_CT99CommonTheme2A }}
	~~<table border="0" cellpadding="3" align="center" width="60%">~~
	~~<tr><td align="left">~~
	~~<font color="darkgreen">~~
	~~"The important question we have tried to clarify concerns the possibility of converting the remaining double integral … into a line integral … this question remains open."</font>~~
	~~</td></tr>~~
	~~</table>~~
	We also have wondered whether there is a possibility of converting the double integral in this Key Equation into a single (line) integral. This is a particularly challenging task when, as is the case with ''Version 1'' of the expression, the integrand is couched in terms of cylindrical coordinates because the modulus of the elliptic integral is explicitly a function of both <math>~\varpi^'</math> and <math>~z^'</math>.

	We have realized that if we focus, instead, on ''Version 2'' of the expression and associate the meridional-plane coordinates of the ''anchor ring'' with the coordinates of the location where the potential is ''being evaluated'' — that is, if we set <math>~(\varpi_a, z_a) = (\varpi,z)</math> — the argument of the elliptic integral becomes,
	~~<div align="center">~~
	~~<math>~\mu = \biggl[\frac~~{2}{~~1+\coth\eta^' }\biggr]^{1 / 2} \, ,</math>~~
	~~</div>~~
	~~while the integral expression for the gravitational potential becomes,~~
	~~<table border="0" cellpadding="5" align="center">~~
	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Phi(\varpi,z)\biggr\|_\mathrm{axisym}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~- 2^{3/2}G \varpi^2 \iint\limits_\mathrm{config}~~
	~~\biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} \biggl[\frac{1}{1+\coth\eta^' }\biggr]^{1 / 2} K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' </math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~- 2^{3 / 2} G \varpi^{2}~~
	~~\int\limits^{\eta_\mathrm{min}}_{\eta_\mathrm{max}} \frac{K(\mu) \sinh \eta^' ~d\eta^'}{( \sinh \eta^' +\cosh \eta^' )^{1 / 2}}~~
	~~\int\limits_{\theta_\mathrm{min}(\eta)}^{\theta_\mathrm{max}(\eta)} \rho(\eta^', \theta^')~~
	~~\biggl[ \frac{d\theta^'}{(\cosh\eta^' - \cos\theta^')^{5 / 2~~}} ~~\biggr] \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~

	Notice that, by adopting this strategy, the argument of the elliptical integral is a function only of one coordinate — the toroidal coordinate system's ''radial'' coordinate, <math>~\eta^'</math>. As result, the integral over the ''angular'' coordinate, <math>~\theta^'</math>, does not involve the elliptic integral function. Then — as is shown in an accompanying chapter titled, [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|''Attempt at Simplification'']] — if the configuration's density is constant, the integral over the angular coordinate variable can be completed analytically. Hence, <font color="orange">the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced to a problem of carrying out a single, line integration. This provides an answer to the question posed by</font> [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)].
	</td>		</td>
	</tr>		</tr>

Tohline: /* Part II */

2018-08-10T22:11:29Z

Part II

← Older revision		Revision as of 22:11, 10 August 2018
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	===Part II===		===Part II===

	Suppose we rewrite (Version 1 of) the [[#Part_I\|above-highlighted Key integral expression]] such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates <math>~(\varpi^', z^')</math> to a toroidal-coordinate system <math>~(\eta^',\theta^')</math> whose ''anchor ring'' cuts through the meridional plane at the cylindrical-coordinate location, <math>~(\varpi_a,z_a)</math>. This desired mapping is handled via the pair of relations,		{{ Template:LSU_CT99CommonTheme2 }}
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\varpi^' = \frac~~{~~\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \, ,</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~      and      ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~(z^' - z_a) = \frac{\varpi_a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \, ,</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~
	~~and the corresponding expression for each differential mass element is,~~
	~~<div align="center">~~
	~~<math>~\delta M(\eta^',\theta^') = \biggl[\frac{2\pi \varpi_a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \rho(\eta^', \theta^') d\eta^' d\theta^'</math>.~~
	~~</div>~~

	~~This gives, what we will refer to as the,~~
	~~<table border="0" cellpadding="5" align="center">~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	~~<font color="#770000">'''Gravitational Potential of an Axisymmetric Mass Distribution (Version 2)'''</font>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Phi(\varpi,z)\biggr\|_\mathrm{axisym}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~- \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{\varpi^{1 / 2}} \biggr] \biggl[ \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^{- 1 / 2}K(\mu)~~
	~~\biggl[\frac{2\pi \varpi_a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \rho(\eta^', \theta^') d\eta^' d\theta^' </math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~- 2G \biggl( \frac{\varpi_a^5}{\varpi} \biggr)^{1 / 2} \iint\limits_\mathrm{config}~~
	~~\biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^~~{~~1 / 2~~} ~~\mu K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' \, ,</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~where the square of the argument of the elliptic integral is,~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\mu^2</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{ 4\varpi \varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')}\biggl\{ \biggl[ \varpi+ \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 +~~
	~~\biggl[z- z_a - \frac{\varpi_a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 \biggr\}^{-1~~} ~~\, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~


	<table border="1" align="center" cellpadding="10" width="85%">		<table border="1" align="center" cellpadding="10" width="85%">

Tohline: /* Part I */

2018-08-10T22:08:12Z

Part I

← Older revision		Revision as of 22:08, 10 August 2018
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	! style="height: 150px; width: 150px; background-color:#D0FFFF;" \|[[User:Tohline/AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically\|Solving the<br /> Poisson Equation]]		! style="height: 150px; width: 150px; background-color:#D0FFFF;" \|[[User:Tohline/AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically\|Solving the<br /> Poisson Equation]]
	\|}		\|}
	[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] have argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — with <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring element — that are positioned at various meridional coordinate locations throughout the mass distribution. According to Stahler's derivation, for example (see his equation 11 and the explanatory text that follows it), the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is:		{{ Template:LSU_CT99CommonTheme1B }}
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\delta\Phi_g(\varpi,z)</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~- \biggl[\frac{2G}~~{~~\pi }\biggr] \frac~~{~~\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}~~
	~~\times K\biggl\{ \biggl[ \frac{4\varpi^' \varpi}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~
	~~Stahler's expression for each ''thin ring'' contribution is a generalization of the above-highlighted Key Equation expression for <math>~\Phi_\mathrm{TR~~}</math>: The "TR" expression assumes that the ring cuts through the meridional plane at <math>~(\varpi^', z^') = (a, 0)</math>, while Stahler's expression works for individual rings that cut through the meridional plane at any coordinate location. Given that, in cylindrical coordinates, the differential mass element is,
	~~<div align="center">~~
	~~<math>~\delta M = \rho(\varpi^', z^') \varpi^' d\varpi^' dz^' \int_0^{2\pi~~}~~d\varphi = 2\pi \rho(\varpi^', z^') \varpi^' d\varpi^' dz^'</math>,~~
	~~</div>~~
	it is easy to see that Stahler's expression for <math>~\delta \Phi_g</math> is identical to the integrand of the expression that we have [[#Part_I\|identified, above]], as providing (Version 1 of) the ''Gravitational Potential of an Axisymmetric Mass Distribution.'' It is therefore clear that <font color="orange">[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] were developing robust algorithms to numerically evaluate the gravitational potential of systems with axisymmetric mass distributions well before [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)] formally derived the corresponding Key integral expression</font>.

	Note: It appears as though both [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] only adopted this approach to evaluating the gravitational potential at locations ''outside'' of an axisymmetric mass distribution, whereas [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)] have shown that the approach applies as well for locations ''inside'' the mass distribution.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Part I */

2018-08-10T22:05:44Z

Part I

← Older revision		Revision as of 22:05, 10 August 2018
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	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/Apps/DysonWongTori#Thin_Ring_Approximation\|Dyson-Wong<br /> Tori]]<br />(Thin Ring Approximation)		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/Apps/DysonWongTori#Thin_Ring_Approximation\|Dyson-Wong<br /> Tori]]<br />(Thin Ring Approximation)
	\|}		\|}
	In §102 of a book titled, [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential ''The Theory of the Potential'', MacMillan (1958; originally, 1930)] derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>. Throughout our related discussions, we generally will refer to this additional Key Equation from MacMillan as providing an expression for the,		{{ Template:LSU_CT99CommonTheme1Awhite }}
	~~<table border="0" cellpadding="5" align="center">~~
	~~<tr><td align="center" colspan="1"><font color="#770000">'''Gravitational Potential in the Thin Ring (TR) Approximation'''</font></td></tr>~~
	~~<tr>~~
	~~<td align="center">~~
	{{ ~~User~~:~~Tohline/Math/EQ TRApproximation~~ }}
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	See also, §III.4, Exercise (4) in [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)]. As is reviewed in an accompanying chapter titled, ''[[User:Tohline/Apps/DysonWongTori#Thin_Ring_Approximation\|Dyson-Wong Tori]],'' a number of research groups over the years have re-derived this "thin ring" approximation in the context of their search for effective and insightful ways to determine the gravitational potential of axisymmetric systems.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Part I */

2018-08-10T22:01:08Z

Part I

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	===Part I===		===Part I===
	~~The gravitational potential (both inside and outside) of any axisymmetric mass distribution may be determined from the following integral expression that we will refer to as the,~~		{{Template:LSU_CT99CommonTheme1 }}
	~~<div align="center">~~
	~~<font color="#770000">'''Gravitational Potential of an Axisymmetric Mass Distribution (Version 1)'''</font>~~
	{{ ~~User~~:~~Tohline/Math/EQ_CT99Axisymmetric~~ }}
	~~</div>~~
	and, <math>~K(\mu)</math> is the complete elliptic integral of the first kind. This Key Equation<sup>&dagger;</sup> may be straightforwardly obtained, for example, by combining Eqs. (31), (32b), and (24) from [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)]; see also, [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)], [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)], and [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)].

	~~<div align="center">------------------------------</div>~~

	<sup>&dagger;</sup>Building upon a previously little-known ''[[User:Tohline/Appendix/Ramblings/CCGF#Compact_Cylindrical_Green_Function_.28CCGF.29\|Compact Cylindrical Green's Function]]'' expansion, [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)] derived an integral expression for the gravitational potential that is applicable to all mass distributions, irrespective of geometric symmetries. The Key Equation highlighted here — that is relevant to axisymmetric mass distributions — is a special case of this more general expression.


	<table border="1" align="center" cellpadding="10" width="85%">		<table border="1" align="center" cellpadding="10" width="85%">

Tohline: /* Part III */

2018-08-10T20:55:01Z

Part III

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	! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|Wong's <br />(1973) <br >Analytic Potential]]		! style="height: 150px; width: 150px; background-color:whiteF;" \|[[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|Wong's <br />(1973) <br >Analytic Potential]]
	\|}		\|}
	Beginning with ''his'' integral expression for <math>~\Phi(\eta,\theta)\|_\mathrm{axisym}</math>, [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] was able to complete the integrals in ''both'' coordinate directions and obtain an analytic expression for the potential both inside and outside of a uniformly charged (equivalently, uniform-density), circular torus. This is a remarkable result that has been largely unnoticed and unappreciated by the astrophysics community. We detail how he accomplished this task in an accompanying chapter titled, [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|''Wong's (1973) Analytic Potential'']].		Beginning with ''his'' integral expression for <math>~\Phi(\eta,\theta)\|_\mathrm{axisym}</math>, [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] was able to complete the integrals in ''both'' coordinate directions and obtain an analytic expression for the potential both inside and outside of a uniformly charged (equivalently, uniform-density), circular torus. <font color="orange">This is a remarkable result that has been largely unnoticed and unappreciated by the astrophysics community.</font> We detail how he accomplished this task in an accompanying chapter titled, [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|''Wong's (1973) Analytic Potential'']].

	If a torus has a major radius, <math>~R</math>, and cross-sectional radius, <math>~d</math>, Wong realized that every point on the surface of the torus will have the same toroidal-coordinate radius, <math>~\eta_0 = \cosh^{-1}(R/d)</math>, if the ''anchor ring'' of the selected toroidal coordinate system has a radius, <math>~a = \sqrt{R^2 - d^2}</math>. His derived expressions for the potential — one, outside, and the other, inside the torus — are:		If a torus has a major radius, <math>~R</math>, and cross-sectional radius, <math>~d</math>, Wong realized that every point on the surface of the torus will have the same toroidal-coordinate radius, <math>~\eta_0 = \cosh^{-1}(R/d)</math>, if the ''anchor ring'' of the selected toroidal coordinate system has a radius, <math>~a = \sqrt{R^2 - d^2}</math>. His derived expressions for the potential — one, outside, and the other, inside the torus — are:

Tohline: /* Part II */

2018-08-10T20:53:50Z

Part II

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	</table>		</table>

	Notice that, by adopting this strategy, the argument of the elliptical integral is a function only of one coordinate — the toroidal coordinate system's ''radial'' coordinate, <math>~\eta^'</math>. As result, the integral over the ''angular'' coordinate, <math>~\theta^'</math>, does not involve the elliptic integral function. Then — as is shown in an accompanying chapter titled, [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|''Attempt at Simplification'']] — if the configuration's density is constant, the integral over the angular coordinate variable can be completed analytically. Hence, the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced to a problem of carrying out a single, line integration. This provides an answer to the question posed by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)].		Notice that, by adopting this strategy, the argument of the elliptical integral is a function only of one coordinate — the toroidal coordinate system's ''radial'' coordinate, <math>~\eta^'</math>. As result, the integral over the ''angular'' coordinate, <math>~\theta^'</math>, does not involve the elliptic integral function. Then — as is shown in an accompanying chapter titled, [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|''Attempt at Simplification'']] — if the configuration's density is constant, the integral over the angular coordinate variable can be completed analytically. Hence, <font color="orange">the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced to a problem of carrying out a single, line integration. This provides an answer to the question posed by</font> [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)].
	</td>		</td>
	</tr>		</tr>