Apps/Wong1973Potential - Revision history

Jet53man at 02:56, 13 March 2023

2023-03-13T02:56:34Z

← Older revision		Revision as of 22:56, 12 March 2023
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	{{ Math/EQ_Toroidal05 }}		{{ Math/EQ_Toroidal05 }}

	we will adopt the associations:   <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math>, <math>~\nu \rightarrow (n - \tfrac{1}{2})</math>, <math>~\rho \rightarrow 0</math>, and <math>~\sigma \rightarrow ( n - \tfrac{1}{2})</math>. As a result, [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erdélyi's (1953)] expression becomes,		we will adopt the associations:   <math>z \rightarrow t</math>, <math>\mu \rightarrow 2</math>, <math>\nu \rightarrow (n - \tfrac{1}{2})</math>, <math>\rho \rightarrow 0</math>, and <math>\sigma \rightarrow ( n - \tfrac{1}{2})</math>. As a result, [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erdélyi's (1953)] expression becomes,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>
	\int_a^b\biggl[ -4(1 - t^2)^{-1} \biggr] Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) ~dt		\int_a^b\biggl[ -4(1 - t^2)^{-1} \biggr] Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) ~dt
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\biggl[		\biggl[
	(n - \tfrac{1}{2} ) Q_{n - \frac{1}{2}}^2(t) ~ X_{n - \frac{3}{2}}(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) ~X_{n - \frac{1}{2}}(t)		(n - \tfrac{1}{2} ) Q_{n - \frac{1}{2}}^2(t) ~ X_{n - \frac{3}{2}}(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) ~X_{n - \frac{1}{2}}(t)
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	{{ Math/EQ_Toroidal04 }}		{{ Math/EQ_Toroidal04 }}

	which means, after making the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, that,		which means, after making the associations, <math>z \rightarrow t</math>, <math>\mu \rightarrow 0</math> and <math>\nu \rightarrow (n-\tfrac{1}{2})</math>, that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~(n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math>		<math>(n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~2n t X_{n-\frac{1}{2}}(t) - (n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t)</math>		<math>2n t X_{n-\frac{1}{2}}(t) - (n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t)</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~(n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t) </math>		<math>\Rightarrow ~~~(n-\tfrac{1}{2})X_{n - \frac{3}{2}}(t) </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math>		<math>2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t)</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>
	\int_a^b\biggl[ \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt		\int_a^b\biggl[ \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{1}{4} \biggl\{		\frac{1}{4} \biggl\{
	\biggl[ 2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) \biggr] Q_{n - \frac{1}{2}}^2(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) ~X_{n - \frac{1}{2}}(t)		\biggl[ 2n t X_{n-\frac{1}{2}}(t) - (n+\tfrac{1}{2})X_{n+\frac{1}{2}}(t) \biggr] Q_{n - \frac{1}{2}}^2(t) - (n + \tfrac{3}{2}) Q_{n - \frac{3}{2}}^2(t) ~X_{n - \frac{1}{2}}(t)
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{1}{4} \biggl\{		\frac{1}{4} \biggl\{
	\biggl[ 2n t Q_{n - \frac{1}{2}}^2(t)		\biggl[ 2n t Q_{n - \frac{1}{2}}^2(t)
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	</tr>		</tr>
	</table>		</table>
	Returning to the same ''recurrence'' "Key Equation," but this time adopting the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, we can write,		Returning to the same ''recurrence'' "Key Equation," but this time adopting the associations, <math>z \rightarrow t</math>, <math>\mu \rightarrow 2</math> and <math>\nu \rightarrow (n-\tfrac{1}{2})</math>, we can write,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t)</math>		<math>(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	2n t Q^2_{n - \frac{1}{2}}(t) - (n + \tfrac{3}{2}) Q^2_{n - \frac{3}{2}} (t) \, ,		2n t Q^2_{n - \frac{1}{2}}(t) - (n + \tfrac{3}{2}) Q^2_{n - \frac{3}{2}} (t) \, ,
	</math>		</math>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>
	\int_a^b\biggl[ \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt		\int_a^b\biggl[ \frac{Q_{n - \frac{1}{2}}^2(t) ~X_{n - \frac{1}{2}}(t) }{(t^2-1)}\biggr]~dt
	</math>		</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	\frac{1}{4} \biggl\{		\frac{1}{4} \biggl\{
	(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ X_{n - \frac{1}{2}}(t)		(n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(t) ~ X_{n - \frac{1}{2}}(t)

Jet53man: /* Thin Ring Approximation */

2023-03-13T02:42:16Z

Thin Ring Approximation

Show changes

Jet53man: /* Contribution from Individual Modes */

2023-03-13T02:31:02Z

Contribution from Individual Modes

← Older revision		Revision as of 22:31, 12 March 2023
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	====Contribution from Individual Modes====		====Contribution from Individual Modes====

	It is instructive to examine how large a contribution to the ''composite'' potential, <math>~\Phi_\mathrm{W}</math>, is made by each term in the series summation. Letting <math>~\Phi_{\mathrm{W}0}</math> represent the dimensionless amplitude of the zeroth-order "mode", we have,		It is instructive to examine how large a contribution to the ''composite'' potential, <math>\Phi_\mathrm{W}</math>, is made by each term in the series summation. Letting <math>\Phi_{\mathrm{W}0}</math> represent the dimensionless amplitude of the zeroth-order "mode", we have,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Phi_{\mathrm{W}0} (\eta,\theta) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr\|_{n=0}</math>		<math>\Phi_{\mathrm{W}0} (\eta,\theta) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr\|_{n=0}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	-D_0		-D_0
	(\cosh\eta - \cos\theta)^{1 / 2} ~A_0(\cosh\eta) \, .		(\cosh\eta - \cos\theta)^{1 / 2} ~A_0(\cosh\eta) \, .
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	</tr>		</tr>
	</table>		</table>
	The 2D contour plot generated by this function is displayed — along with the label, n = 0 — in the upper-right corner of the central panel of Figure 3. For comparison and reference, the ''composite'' 2D contour plot that appears in the bottom-right panel of Figure 1 has been redisplayed in the upper-left corner of the same Figure 3 panel. Immediately we appreciate that the zeroth-order contribution to the potential, <math>~\Phi_{\mathrm{W}0}</math>, on its own plays a dominant role in determining the overall structure of the composite potential well. Notice that this component depends on the ''polar angle'', <math>~\theta</math>, only through the coefficient, <math>~(\cosh\eta - \cos\theta)^{1 / 2}</math>, which it shares in common with all other terms in the series.		The 2D contour plot generated by this function is displayed — along with the label, n = 0 — in the upper-right corner of the central panel of Figure 3. For comparison and reference, the ''composite'' 2D contour plot that appears in the bottom-right panel of Figure 1 has been redisplayed in the upper-left corner of the same Figure 3 panel. Immediately we appreciate that the zeroth-order contribution to the potential, <math>\Phi_{\mathrm{W}0}</math>, on its own plays a dominant role in determining the overall structure of the composite potential well. Notice that this component depends on the ''polar angle'', <math>\theta</math>, only through the coefficient, <math>(\cosh\eta - \cos\theta)^{1 / 2}</math>, which it shares in common with all other terms in the series.


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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Phi_{\mathrm{W}1}(\eta,\theta) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr\|_{n=1}</math>		<math>\Phi_{\mathrm{W}1}(\eta,\theta) \equiv \biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)\biggr\|_{n=1}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	-2D_0		-2D_0
	(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(\theta) A_1(\cosh\eta) \, .		(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(\theta) A_1(\cosh\eta) \, .
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	</tr>		</tr>
	</table>		</table>
	The 2D contour plot generated by this function is displayed — along with the label, n = 1 — in the left-hand column of the central panel of Figure 3. By analogy with Figure 2, above, a three-dimensional ''animated'' depiction of this same potential contour surface is displayed in the left-hand panel of Figure 3. Note, however, that the overall amplitude has been multiplied by a factor of two — that is, the warped 3D surface has been generated by the function, <math>~2\Phi_{\mathrm{W}1}</math> — in order to more vividly illustrate the structural modification to the potential that arises when this term is added to the leading, zeroth-order term.		The 2D contour plot generated by this function is displayed — along with the label, n = 1 — in the left-hand column of the central panel of Figure 3. By analogy with Figure 2, above, a three-dimensional ''animated'' depiction of this same potential contour surface is displayed in the left-hand panel of Figure 3. Note, however, that the overall amplitude has been multiplied by a factor of two — that is, the warped 3D surface has been generated by the function, <math>2\Phi_{\mathrm{W}1}</math> — in order to more vividly illustrate the structural modification to the potential that arises when this term is added to the leading, zeroth-order term.

	In a directly analogous fashion, 2D contour plots showing the behavior of <math>~\Phi_{\mathrm{W}2}</math>, <math>~\Phi_{\mathrm{W}3}</math>, and <math>~\Phi_{\mathrm{W}4}</math>, have been generated and displayed — along with the corresponding labels, n = 2, 3, & 4 — in the central panel of Figure 3. A 3D animated depiction of the warped, n = 3 surface is presented in the right-hand panel of Figure 3; note that, in this instance, the overall amplitude has been multiplied by a factor of one-hundred — that is, the warped 3D surface has been generated by the function, <math>~100\Phi_{\mathrm{W}3}</math>.		In a directly analogous fashion, 2D contour plots showing the behavior of <math>\Phi_{\mathrm{W}2}</math>, <math>\Phi_{\mathrm{W}3}</math>, and <math>\Phi_{\mathrm{W}4}</math>, have been generated and displayed — along with the corresponding labels, n = 2, 3, & 4 — in the central panel of Figure 3. A 3D animated depiction of the warped, n = 3 surface is presented in the right-hand panel of Figure 3; note that, in this instance, the overall amplitude has been multiplied by a factor of one-hundred — that is, the warped 3D surface has been generated by the function, <math>100\Phi_{\mathrm{W}3}</math>.

	A similarly vivid 3D display of the structure arising from the n = 2 and n = 4 terms in the series would have required, respectively, "magnification" of <math>~\Phi_{\mathrm{W}2}</math> by approximately a factor of fifteen and "magnification" of <math>~\Phi_{\mathrm{W}4}</math> by a factor of five-hundred; successively larger magnifications would be required for terms associated with successively higher ''polar angle'' indices, n. We conclude, therefore, that a ''composite'' potential containing only the first four terms in the series represents the ''exact'' potential to an accuracy better than one percent.		A similarly vivid 3D display of the structure arising from the n = 2 and n = 4 terms in the series would have required, respectively, "magnification" of <math>\Phi_{\mathrm{W}2}</math> by approximately a factor of fifteen and "magnification" of <math>\Phi_{\mathrm{W}4}</math> by a factor of five-hundred; successively larger magnifications would be required for terms associated with successively higher ''polar angle'' indices, n. We conclude, therefore, that a ''composite'' potential containing only the first four terms in the series represents the ''exact'' potential to an accuracy better than one percent.

	====Equatorial Plane Behavior====		====Equatorial Plane Behavior====

Jet53man: /* Composite Gravitational Potential */

2023-03-13T02:27:42Z

Composite Gravitational Potential

Show changes

Jet53man: /* W */

2023-03-13T02:13:48Z

Show changes

Jet53man: /* W */

2023-03-13T02:01:24Z

Show changes

Jet53man: /* Uniform-Density Torus */

2023-03-12T22:26:27Z

Uniform-Density Torus

Show changes

Jet53man: /* Technical Background Review */

2023-03-12T22:22:02Z

Technical Background Review

← Older revision		Revision as of 18:22, 12 March 2023
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	==Technical Background Review==		==Technical Background Review==

	Drawing principally from the published works of {{ Wong73full }} and ~~[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999, ApJ, 527, 86)]~~, in an [[2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|accompanying discussion]] we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rather than, say, cartesian, spherical or cylindrical coordinates. Specifically for [[2DStructure/ToroidalGreenFunction#Axisymmetric_Systems\|axisymmetric systems]], the relevant expression derived by {{ Wong73 }} — involving a pair of integrals over the meridional-plane coordinates, <math>~(\eta,\theta)</math>, and a summation over all ''polar angle'' modes — is,		Drawing principally from the published works of {{ Wong73full }} and {{ CT99full }}, in an [[2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|accompanying discussion]] we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rather than, say, cartesian, spherical or cylindrical coordinates. Specifically for [[2DStructure/ToroidalGreenFunction#Axisymmetric_Systems\|axisymmetric systems]], the relevant expression derived by {{ Wong73 }} — involving a pair of integrals over the meridional-plane coordinates, <math>~(\eta,\theta)</math>, and a summation over all ''polar angle'' modes — is,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Phi(\eta,\theta)</math>		<math>\Phi(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2}		- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2}
	\sum\limits^\infty_{n=0} \epsilon_n		\sum\limits^\infty_{n=0} \epsilon_n
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	</tr>		</tr>
	</table>		</table>
	while the expression derived by ~~[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)]~~ — involving different integrand expressions and no summation — is,		while the expression derived by {{ CT99 }} — involving different integrand expressions and no summation — is,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Phi(\eta,\theta)</math>		<math>\Phi(\eta,\theta)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2}		<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^') \, ,		\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>		</math>
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	~~[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)]~~, p. 88, Eqs. (31) & (32a)		{{ CT99 }}, p. 88, Eqs. (31) & (32a)
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	where, <math>~a</math> is the radius of the toroidal-coordinate system's ''anchor'' ring, <math>~P^0_{n-\frac{1}{2}}</math> and <math>~Q^0_{n-\frac{1}{2}}</math> are zero-order half-integer-degree associated Legendre functions of, respectively, the first and second kind, and		where, <math>a</math> is the radius of the toroidal-coordinate system's ''anchor'' ring, <math>P^0_{n-\frac{1}{2}}</math> and <math>Q^0_{n-\frac{1}{2}}</math> are zero-order half-integer-degree associated Legendre functions of, respectively, the first and second kind, and
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	In the subsection of our [[2DStructure/ToroidalGreenFunction#Rearranging_Terms_and_Incorporating_Special-Function_Relations\|accompanying discussion]] titled, "Rearranging Terms and Incorporating Special-Function Relations," we have explicitly demonstrated that these two expressions lead to identical evaluations of the gravitational potential.		In the subsection of our [[2DStructure/ToroidalGreenFunction#Rearranging_Terms_and_Incorporating_Special-Function_Relations\|accompanying discussion]] titled, "Rearranging Terms and Incorporating Special-Function Relations," we have explicitly demonstrated that these two expressions lead to identical evaluations of the gravitational potential.

	Here we build upon this technical foundation and detail how ~~[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)]~~ was able to complete both integrals to derive an analytic expression for the potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>~R/d</math>. This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, is underappreciated.		Here we build upon this technical foundation and detail how {{ Wong73 }} was able to complete both integrals to derive an analytic expression for the potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>R/d</math>. This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, is underappreciated.

	==Uniform-Density Torus==		==Uniform-Density Torus==

Jet53man: /* Technical Background Review */

2023-03-12T21:51:05Z

Technical Background Review

← Older revision		Revision as of 17:51, 12 March 2023
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	==Technical Background Review==		==Technical Background Review==

	Drawing principally from the published works of ~~[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)]~~ and [http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999, ApJ, 527, 86)], in an [[2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|accompanying discussion]] we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rather than, say, cartesian, spherical or cylindrical coordinates. Specifically for [[2DStructure/ToroidalGreenFunction#Axisymmetric_Systems\|axisymmetric systems]], the relevant expression derived by ~~[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)]~~ — involving a pair of integrals over the meridional-plane coordinates, <math>~(\eta,\theta)</math>, and a summation over all ''polar angle'' modes — is,		Drawing principally from the published works of {{ Wong73full }} and [http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl & J. E. Tohline (1999, ApJ, 527, 86)], in an [[2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential\|accompanying discussion]] we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rather than, say, cartesian, spherical or cylindrical coordinates. Specifically for [[2DStructure/ToroidalGreenFunction#Axisymmetric_Systems\|axisymmetric systems]], the relevant expression derived by {{ Wong73 }} — involving a pair of integrals over the meridional-plane coordinates, <math>~(\eta,\theta)</math>, and a summation over all ''polar angle'' modes — is,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	~~[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)]~~, p. 293, Eq. (2.55)		{{ Wong73 }}, p. 293, Eq. (2.55)
	</td>		</td>
	</tr>		</tr>

Jet53man: /* See Also */

2021-09-13T21:58:22Z

← Older revision		Revision as of 17:58, 13 September 2021
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	=See Also=		=See Also=
			<ul>
			<li>Common Theme:
			<ul>
			<li>
			Menu Tile:  [[AxisymmetricConfigurations/SolvingPE\|Solving the (Axisymmetric) Poisson Equation]]
			<math>\Rightarrow</math>     Points principally to: [[AxisymmetricConfigurations/PoissonEq\|AxisymmetricConfigurations/PoissonEq]]
			</li>
			<li>
			Menu Tile:  [[2DStructure/UsingTC\|Using Toroidal Coordinates to Determine the Gravitational Potential]]
			<math>\Rightarrow</math>     Points principally to: [[2DStructure/ToroidalGreenFunction\|2DStructure/ToroidalGreenFunction]]
			</li>
			<li>
			Menu Tile:  [[2DStructure/TCsimplification\|Attempt at Simplification]]
			<math>\Rightarrow</math>     Points principally to: [[2DStructure/ToroidalCoordinates\|2DStructure/ToroidalCoordinates]]
			</li>
			<li>
			Menu Tile:  [[Apps/WongAP\|Wong's Analytic Potential (1973)]]
			<math>\Rightarrow</math>     Points principally to: [[Apps/Wong1973Potential\|Apps/Wong1973Potential]]
			</li>
			<li>
			Menu Tile:  [[Apps/DWT\|Dyson-Wong Tori]] (Thin Ring Approximation)
			<math>\Rightarrow</math>     Points principally to: [[Apps/DysonWongTori\|Apps/DysonWongTori]]
			</li>
			</ul>
			</li>
			<li>
			(Apollonian Circles) [[2DStructure/ToroidalCoordinateIntegrationLimits\|Toroidal-Coordinates Integration Limits]]
			</li>
			<li>
			[[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Configurations_and_Related_Coordinate_Systems\|Determining the Relationships between Toroidal Coordinates and other Coordinate Systems]]
			</li>
			</ul>


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