Joel2: Created page with "Beginning with ''his'' integral expression for $\Phi(\eta,\theta)|_\mathrm{axisym}$ , {{ Wong73 }} 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..."

2024-06-21T23:49:29Z

Created page with "Beginning with ''his'' integral expression for <math>\Phi(\eta,\theta)|_\mathrm{axisym}</math>, {{ Wong73 }} 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..."

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Beginning with ''his'' integral expression for <math>\Phi(\eta,\theta)|_\mathrm{axisym}</math>, {{ Wong73 }} 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, [[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:

<table border="0" align="center" cellpadding="8">
<tr><td align="center">'''Exterior Solution:'''  <math>~\eta_0 \ge \eta</math></td></tr>
<tr><td align="left">

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

<tr>
<td align="right">
<math>~\Phi_\mathrm{W}(\eta,\theta)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2}
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times P_{n-1 / 2}(\cosh\eta)
\biggl[
(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0)
- (n - \tfrac{3}{2})Q^2_{n + \frac{1}{2}}(\cosh \eta_0) ~ Q_{n - \frac{1}{2}}(\cosh \eta_0)
\biggr] \, .
</math>
</td>
</tr>
</table>

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

<table border="0" align="center" cellpadding="8">
<tr><td align="center">'''Interior Solution:'''  <math>~\eta \ge \eta_0 </math></td></tr>
<tr><td align="left">

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

<tr>
<td align="right">
<math>~\Phi_\mathrm{W}(\eta,\theta)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- \frac{2\sqrt{2}~ GM}{3\pi^2 a} \biggl[ \frac{\sinh^3\eta_0}{ \cosh{\eta_0}} \biggr] (\cosh\eta - \cos\theta)^{1 / 2}
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
 
</td>
<td align="left">
<math>~
\times \biggl\{
Q_{n-1 / 2}(\cosh\eta)
\biggl[
(n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0})
- (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0})
\biggr] - Q_{n - \frac{1}{2}}^2(\cosh\eta) \biggr\} \, .
</math>
</td>
</tr>
</table>

The following pair of animated images result from [[Apps/Wong1973Potential#Evaluation|our numerical evaluation]] of this pair of expressions for <math>~\Phi_\mathrm{W}</math> (including the first four, and most dominant, terms in the series summation) for the case of: (left) <math>~R/d = 3</math>, which is the aspect ratio Wong chose to illustrate in his publication; and (right) tori having a variety of different aspect ratios over the range, <math>~1.8 \le R/d \le 8</math>.

<table border="1" cellpadding="10" align="center" width="65%">
<tr>
<td align="center" bgcolor="#D0FFFF">[[File:MovieWongComposite.gif|center|250px|3D Depiction of Wong's Toroidal Potential Well]]
</td>
<td align="center" bgcolor="#D0FFFF">[[File:Wong73VaryingRoverd.gif|center|250px|3D Depiction of Wong's Toroidal Potential Well]]
</td>
</tr>
</table>

</td></tr>
</table>
Does this expression for the potential behave as we expect in the "thin ring" approximation? On p. 295 of {{ Wong73 }}, we find the following statement:
<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="darkgreen">
"For the case of a very thin ring (i.e., <math>~\eta_0 \rightarrow \infty</math>), the exterior solution has contributions mostly from the first term in the expansion of the series …"</font>
</td></tr>
</table>
Using the notation, <math>\Phi_\mathrm{W0}</math>, to represent the leading-order term in the expression for the ''exterior'' potential, we have (see the [[Apps/Wong1973Potential#Thin_Ring_Approximation|accompanying chapter]] for details),


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

<tr>
<td align="right">
<math>~\Phi_{\mathrm{W}0} (\eta,\theta)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~-\biggl( \frac{GM}{a} \biggr)
F(\cosh\eta_0)\cdot
[\cosh\eta - \cos\theta]^{1 / 2}
P_{-\frac{1}{2}}(\cosh\eta)
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \frac{2}{\pi}\biggl( \frac{GM}{a} \biggr)
F(\cosh\eta_0)\cdot
\biggl[ \frac{2a^2}{(\varpi + a)^2 + z^2} \biggr]^{1 / 2} K(k) \, ,
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right">
<math>~k</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl[ \frac{2}{1+\coth\eta}\biggr]^{1 / 2}
= \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + z^2} \biggr]^{1 / 2} \, ,
</math>
      and,
</td>
</tr>

<tr>
<td align="right">
<math>~F(\cosh\eta_0)</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{2^{1 / 2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \biggl[ Q_{+\frac{1}{2}}(\cosh \eta_0) Q_{- \frac{1}{2}}^2(\cosh \eta_0)
+ 3 Q_{- \frac{1}{2}}(\cosh \eta_0)~Q^2_{+ \frac{1}{2}}(\cosh \eta_0)\biggr] \, .
</math>
</td>
</tr>
</table>
In our [[Apps/Wong1973Potential#Thin_Ring_Approximation|accompanying discussion]] we show that,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~F(\cosh\eta_0)\biggr|_{\eta_0\rightarrow \infty}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl\{ \frac{2^{1 / 2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \biggl[ \biggl( \frac{3 \pi^2}{2} \biggr) \frac{1}{\cosh^2\eta_0} \biggr] \biggr\}_{\eta_0\rightarrow \infty}
= \frac{1}{\sqrt{2}} \, .
</math>
</td>
</tr>
</table>

Hence, we have,

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

<tr>
<td align="right">
<math>~\Phi_{\mathrm{W}0} (\eta,\theta)\biggr|_{\eta_0 \rightarrow \infty}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \biggl[ \frac{2GM}{\pi} \biggr]
\frac{K(k)}{\sqrt{(\varpi + a)^2 + z^2}} \, ,
</math>
</td>
</tr>
</table>
which precisely matches the above-referenced [[#Chapter_Synopses|''Gravitational Potential in the Thin Ring (TR) Approximation.'']]

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