Editing Apps/Wong1973Potential (section)

===Evaluation===

====Composite Gravitational Potential====
In order to illustrate how the potential behaves both inside and outside (in the immediate vicinity) of a uniform-density torus, {{ Wong73 }} set <math>R/d = \cosh\eta_0 = 3</math> and evaluated the pair of boxed-in expressions for <math>\Phi_\mathrm{W}(\eta,\theta)</math> given above, over most of the cylindrical-coordinate range, <math>0 \le \varpi/a \le 2.0</math> and <math>0 \le z/a \le 1.0</math>.  (He certainly truncated the series summation at a finite number of terms, but his article does not state how many terms he included.)  The meridional-plane contour plot that resulted from this evaluation appears as Figure 7 in {{ Wong73 }} and has been reprinted here, unaltered, in the top-right panel of our Figure 1.  His Figure 6 &#8212; reprinted here, unaltered, in the top-left panel of our Figure 1 &#8212; shows how the (absolute value of the) dimensionless potential varies with <math>\varpi/a</math> at eight different heights above the equatorial plane; specifically, as labeled, for <math>z/a = 0.0, 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.2</math>.


<table border="1" align="center" cellpadding="8" width="65%">
<tr><th align="center">Figure 1</th></tr>
<tr><td align="center">

<table border="0" align="center" cellpadding="1"><tr><td align="center" colspan="2">
Figure 6 (left) &amp; Figure 7 (right) extracted without modification from pp. 296 &amp; 297, respectively, of {{ Wong73 }}<p></p>
"''Toroidal and Spherical Bubble Nuclei''"<p></p>
Annals of Physics, vol. 77, pp. 279-353 &copy; Elsevier Science
</td></tr>
<tr><td align="center">
[[File:Wong1973Fig6.png|350px|To be inserted:  Fig. 6 from Wong (1973)]]
</td>
<td align="center" width="50%">
[[File:Wong1973Fig7.png|400px|To be inserted:  Fig. 7 from Wong (1973)]]
</td></tr></table>

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

<table border="0" align="center" width="100%"><tr><td align="center">
&nbsp; &nbsp; &nbsp;[[File:Wong73OurFig6b.png|300px|Our line plot to compare with Wong's Figure 6]]
</td>
<td align="center">
[[File:Wong111N4.png|300px|Our composite to compare with Wong's Figure 7]]&nbsp; &nbsp; &nbsp; &nbsp;
</td>
</tr>
</table>

</td></tr>
<tr><td align="left" colspan="2">
'''Figure Caption:'''  Plots/images showing how the Coulomb (gravitational) potential varies across the meridional plane for a uniformly charged (uniform-density) torus with an aspect ratio, <math>R/d = \cosh\eta_0 = 3</math>.  (Top) Figures 6 &amp; 7 extracted without modification from {{ Wong73 }}.  (Bottom) Corresponding line- and contour-plots resulting from our numerical evaluation of Wong's analytically specified potential, <math>\Phi_\mathrm{w}</math>.   

'''Contour Plots (right column):''' &nbsp;The toroidal coordinate system's ''anchor ring'' is located at position <math>(\varpi/a, z/a) = (1.0,0.0)</math>, as explicitly identified by the axis label (top panel), and as marked by a small purple circular marker (bottom panel).  The circular cross-section of the torus is identified by the thick-black semicircle (top panel) and by the black circle (bottom panel).   The center of this toroidal cross-section lies just outside the location of the ''anchor ring'' at the radial-coordinate position, <math>R/a = [1 - (R/d)^{-2}]^{- 1 / 2} = 3/2^{3 / 2} </math>; this position is marked by a small white circular marker in the bottom panel.  Notice that the absolute potential minimum is positioned just ''inside'' the location of the ''anchor ring''.

'''Line Plots (left column):''' &nbsp;The curve labeled <math>z/a = 0.0</math> shows how the (absolute value of the) dimensionless potential, <math>|a\Phi_\mathrm{W}/(GM)|</math>, varies with radius, <math>\varpi/a</math>, in the equatorial plane; this corresponds to variation along a horizontal line in the mid-plane of either contour plot (right column).  Each of the other curves displays variations along a separate horizontal line in the contour plot, drawn at the specified distance, <math>z/a</math>, above (or below) the equatorial plane.  The horizontal lines corresponding to three of these curves &#8212; specifically, at <math>z/a = 0.0, 0.2, 0.3</math> &#8212; cut through the torus and, as a result, sample the behavior of the ''interior'' as well as the ''exterior'' potential.  In the bottom-left plot, a pair of black circular markers identifies where the horizontal sampling curve intersects the surface of the torus, so the portion of each curve that samples the ''interior'' potential lies between the pair of markers; in the top-left plot, Wong uses a dashed curve segment to identify the analogous ''interior'' region.
</td></tr></table>


We have also evaluated the Wong-derived potential function, <math>\Phi_\mathrm{W}(\eta,\theta)</math>, both inside and outside (in the immediate vicinity) of a uniform-density torus whose aspect ratio is, <math>~R/d = \cosh\eta_0 = 3</math>.  (As is [[#Contribution_from_Individual_Modes|discussed further, below]], we included only the terms, n = 0, 1, 2, &amp; 3, in the series summation.)  A colored contour plot resulting from our evaluation is displayed in the bottom-right panel of our Figure 1; the height and width of this rectangular image have been linearly scaled to lengths that allow straightforward &#8212; although regrettably not a quantitative &#8212; comparison with the {{ Wong73 }} published Figure 7.   The colored line plot in the bottom-left panel of Figure 1 shows, for our model, how the (absolute value of the) dimensionless potential varies with <math>\varpi/a</math> at eight different heights above the equatorial plane, as indicated by the inset legend of that plot.  The Figure 1 caption highlights some specific points of comparison between Wong's displayed results and our independent evaluation of his model.  Our line plot and plotted contours match quite well the corresponding figures presented by {{ Wong73 }}.

<span id="D0andCn">In designing a numerical algorithm</span> to evaluate <math>\Phi_\mathrm{W}</math>, we first followed {{ Wong73 }} lead and rewrote the interior/exterior expressions in a more compact form.  After defining the leading amplitude coefficient, 
<div align="center">
<math>D_0 \equiv \frac{2^{3/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \, ,</math>
</div>
and the pair of parameters,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>(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}) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>C_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>(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_{n - \frac{1}{2}}(\cosh \eta_0)~Q^2_{n + \frac{1}{2}}(\cosh \eta_0) \, ,
</math>
  </td>
</tr>
</table>

each of which is only a function of the axis ratio of the torus (via the parameter <math>\eta_0</math>) and the ''polar angle'' index, <math>n</math>, the (dimensionless) expression for the potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) A_n(\cosh\eta) \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A_n(\cosh\eta)\biggr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>B_n(\cosh\eta_0)Q_{n-\frac{1}{2}}(\cosh\eta) - Q^2_{n-\frac{1}{2}}(\cosh\eta)</math>
  </td>
</tr>

<tr><td colspan="3" align="center">and</td></tr>

<tr>
  <td align="right">
<math>A_n(\cosh\eta)\biggr|_\mathrm{exterior}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, .</math>
  </td>
</tr>
</table>


Next, in FORTRAN, we wrote a group of functions/subroutines that allowed us to evaluate the toroidal functions, <math>P^0_{n-\frac{1}{2}}</math> and <math>Q^m_{n-\frac{1}{2}}</math>, individually for m = 0, 1, &amp; 2 and for all desired ''polar angle'' index values; a (double-precision version of a) set of ''Numerical Recipes'' algorithms was used to evaluate complete elliptic integrals of the first and second kind.  An [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|accompanying appendix]] provides details regarding some key tests that we conducted in order to demonstrate the accuracy of these various functions/subroutines.  Then, as has already been stated, the (composite) meridional-plane contour diagram displayed in the bottom-right panel of Figure 1 was generated by setting <math>R/d = \cosh\eta_0 = 3</math> and <math>\mathrm{nmax} = 3</math>.  

<table border="1" cellpadding="10" align="center" width="65%">
<tr>
  <th align="center" colspan="2">Figure 2:<br />3D Animated Depictions of the Warped Surface of<br />Wong's Toroidal Potential
  </th>
</tr>
<tr>
  <td align="center" bgcolor="#D0FFFF">[[File:MovieWongComposite.gif|center|300px|3D Depiction of Wong's Toroidal Potential Well]]
  </td>
  <td align="center" bgcolor="#D0FFFF">[[File:Wong73VaryingRoverd.gif|center|300px|3D Depiction of Wong's Toroidal Potential Well]]
  </td>
</tr>
<tr>
  <td align="left" colspan="2">
'''Figure Caption:''' &nbsp;3D animated depictions of the warped potential surface associated with various uniform-density  tori.   (''left panel'')  The torus has the same axis ratio <math>(R/d = 3)</math> as was highlighted by {{ Wong73 }}; in an effort to better illustrate key features of the warped surface, different frames of the animation present the surface as viewed from different lines of sight.  (''right panel'') The warped potential surfaces of tori having nine different axes are depicted, as viewed from the same line of sight; specifically, the tori have <math>R/d = 1.8, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0</math>. 
  </td>
</tr>
</table>


A three-dimensional ''animated'' depiction of this same potential contour surface is displayed in the left panel of Figure 2.  In an effort to better illustrate key features of the warped surface, different frames of the animation present the same surface as viewed from different lines of sight.

Elements of the accompanying 3D animated depiction:
<ul>
<li>The pair of (cylindrical-coordinate based) meridional-plane axes are (red) <math>\varpi/R_0</math> and (green) <math>Z/R_0</math>, where <math>R_0</math> is the distance in the equatorial plane from the symmetry axis to the center of the circular cross-section of the torus.</li>
<li>The scalar value of the potential is plotted along the third (blue) axis; the surface, as a whole, has been shifted vertically in order to place the potential minimum at zero.</li>
<li>The pink, translucent cylinder identifies the meridional-plane location of the surface of the torus; because the model being illustrated has an axis ratio, <math>R_0/d = \cosh\eta_0 =3</math>, the radius of the pink cylinder is <math>d/R_0 = 1/3</math>.  This same cylindrical surface of the torus is identified by the thick black circle in the 2D ''projection'' of the warped contour surface that is displayed in the bottom-right panel of our Figure 1.  A thin, pink vertical rod identifies the center of the circular cross-section of the (pink) torus; it intersects the red, radial-coordinate axis at the location that is identified in the bottom-right panel of Figure 1 by a small white circular marker.</li>
<li>As in the bottom-right panel of Figure 1, a small spherical purple marker identifies the radial location of the ''anchor ring'' associated with the adopted toroidal coordinate system.</li>
</ul>

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

<tr>
  <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>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~A_0(\cosh\eta) \, .
</math>
  </td>
</tr>
</table>
The 2D contour plot generated by this function is displayed &#8212; along with the label, n = 0 &#8212; 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.


<table align="center" border="1" cellpadding="10">
<tr><th align="center" colspan="3">Figure 3: &nbsp; Contribution from Various ''Polar Angle'' "Modes"</th></tr>
<tr>
  <td align="center">n = 1 (magnification 2)</td>
<td align="center" rowspan="2" bgcolor="#D0FFFF">
[[File:Montage04.png|350px|Mode Ensemble]]
</td>
  <td align="center">n = 3 (magnification 100)</td>
</tr>
<tr>
<td align="center" bgcolor="#D0FFFF">
[[File:MovieWongN2.gif|300px|Contribution to potential by mode n = 1 (magnified by 2)]]
</td>
<td align="center" bgcolor="#D0FFFF">
[[File:MovieWongN4b.gif|300px|Contribution to potential by mode n = 3 (magnified by 100)]]
</td>
</tr>
</table>


The second (n = 1) term in the series is,
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
-2D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\cos(\theta) A_1(\cosh\eta) \, .
</math>
  </td>
</tr>
</table>
The 2D contour plot generated by this function is displayed &#8212; along with the label, n = 1 &#8212; 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 &#8212; that is, the warped 3D surface has been generated by the function, <math>2\Phi_{\mathrm{W}1}</math> &#8212; 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 &#8212; along with the corresponding labels, n = 2, 3, &amp; 4 &#8212; 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 &#8212; 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.

====Equatorial Plane Behavior====
=====Interior Solution (n = 0)=====

Let's examine in more detail how the potential varies with radial position in the equatorial plane, focusing first on the ''interior'' region &#8212; that is, <math>~\infty \ge \eta \ge \eta_0</math> &#8212; and (cylindrical-coordinate-based) radial locations, <math>~\varpi < a</math>, in which case the corresponding value of the (toroidal-coordinate-based) polar angle is, <math>~\theta = \pi</math>.  In this region, the zeroth-order contribution to the potential is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Interior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{
Q_{-\frac{1}{2}}(\cosh\eta) [B_0(\cosh\eta_0)]
- Q^2_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
where, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(\cosh\eta_0) \cdot Q_{- \frac{1}{2}}^2(\cosh\eta_0) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(\cosh\eta_0) \cdot Q^2_{+ \frac{1}{2}}(\cosh\eta_0) \, .
</math>
  </td>
</tr>
</table>

Now if <math>~\cosh\eta_0 = 3</math>, as in the above example illustration, then, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[3^2-1]^{1 / 2} = 2^{3/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ D_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2^{3/2}}{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
= \frac{2^{3/2}}{3\pi^2} \biggl[ \frac{2^{9/2}}{3}\biggr]
=\biggl[\frac{2^3}{3\pi}\biggr]^{2} = 0.720506195 \, ;
</math>
  </td>
</tr>
</table>
and, drawing from the derivations and example double-precision evaluations of selected toroidal functions provided in an [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|accompanying appendix]],

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

<tr>
  <td align="right">
<math>~B_0(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(3) \cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[ 1.597386605 \cdot 1.104816977 \biggr]
+ \frac{3}{2} \biggl\{   \frac{0.8346268417 }{\sqrt{3^2-1}} \biggr[ 
-\frac{3}{2} Q^1_{+\frac{1}{2}}(3) - \frac{3}{2} Q^1_{-\frac{1}{2}}(3)
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.882409920
+ \frac{3^2}{2^2} \biggl\{   \frac{0.8346268417 }{\sqrt{3^2-1}} \biggr[ 
0.1718911443 + 0.6753219405
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.882409920
+ \frac{3^2}{2^2} \biggl\{   
\frac{1}{2^2}\biggr\} = 1.444909920 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~0.720506195 
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{ 1.444909920
Q_{-\frac{1}{2}}(\cosh\eta) - Q^2_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, .
</math>
  </td>
</tr>
</table>

=====Exterior Solution (n = 0)=====
In the region ''exterior'' to the torus, the zeroth-order contribution to the potential is,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{
P_{-\frac{1}{2}}(\cosh\eta) [C_0(\cosh\eta_0)]
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
where, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2}~Q_{+\frac{1}{2}}(\cosh \eta_0) Q_{- \frac{1}{2}}^2(\cosh \eta_0) 
+ \frac{3}{2}~ Q_{- \frac{1}{2}}(\cosh \eta_0)~Q^2_{+ \frac{1}{2}}(\cosh \eta_0) \, .
</math>
  </td>
</tr>
</table>

Drawing again from the derivations and example double-precision evaluations of selected toroidal functions provided in an [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|accompanying appendix]], this means that, for <math>~\cosh\eta_0 = 3</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_0(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2}~Q_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2}~ Q_{- \frac{1}{2}}(3)\cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[ 0.1128885424 \cdot 1.104816977 \biggr]
+ \frac{3}{2} \biggl\{   \frac{1.311028777 }{\sqrt{3^2-1}} \biggr[ 
-\frac{3}{2} Q^1_{+\frac{1}{2}}(3) - \frac{3}{2} Q^1_{-\frac{1}{2}}(3)
\biggr] \biggr\}
</math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.0623605885
+ \frac{3^2}{2^2} \biggl\{   \frac{1.311028777 }{\sqrt{3^2-1}} \biggr[ 
0.1718911443 + 0.6753219405
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.0623605885
+ \frac{3^2}{2^2} \biggl\{  0.3926990811 \biggr\} = 0.945933521 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~0.720506195~
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{0.945933521~
P_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, .
</math>
  </td>
</tr>
</table>



<!--
=====Second Attempt=====

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM}\biggr)\Phi_\mathrm{W}(\eta,\theta)\bigr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~D_0  (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
~B_n(\cosh\eta_0)~Q_{n-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ (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}) \, ,
</math>
  </td>
</tr>
</table>

For the model illustrated above, <math>~\cosh\eta_0 = 3</math>, in which case,
<div align="center">
<math>~D_0
= \frac{2^{3/2} }{3\pi^2} \cdot  \frac{(\sinh^2\eta_0)^{3/2}}{\cosh\eta_0} 
= \frac{2^{3/2} }{3\pi^2} \cdot  \frac{(3^2-1)^{3/2}}{3} 
= \biggl(\frac{2^3 }{3\pi} \biggr)^2 = 0.720506194 \, .
</math>
</div>
And, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(3) \cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~[1.597386605] \cdot [1.104816977 ]
+ \frac{3}{2} ~ [0.8346268417] \cdot [Q^2_{+ \frac{1}{2}}(3) ]
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~Q_{n - \frac{1}{2}}^{2}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z^2-1)^{-\frac{1}{2}} \{ (n-\tfrac{3}{2}) z Q^1_{n - \frac{1}{2}}(z) - (n+\tfrac{1}{2})Q^1_{n - \frac{3}{2}}(z)\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ Q_{+\frac{1}{2}}^{2}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(3^2-1)^{-\frac{1}{2}} \biggl[ -\frac{3}{2}  Q^1_{+\frac{1}{2}}(3) - \frac{3}{2}Q^1_{- \frac{1}{2}}(3) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{3}{2^{5/2}}  \biggr)
\biggl[  Q^1_{+\frac{1}{2}}(3) + Q^1_{- \frac{1}{2}}(3) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{3}{2^{5/2}}  \biggr)
\biggl[  -0.1718911443 - 0.6753219405 \biggr] = 0.4493025877 \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~ \frac{1}{2} ~[1.597386605] \cdot [1.104816977 ]
+ \frac{3}{2} ~ [0.8346268417] \cdot [0.4493025877 ]
=
1.444909919 \, .
</math>
  </td>
</tr>
</table>

As a result, including only the zeroth-order term in the series summation gives,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM}\biggr)\Phi_\mathrm{W0}(\eta,\theta)\bigr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~D_0  (\cosh\eta - \cos\theta)^{1 / 2}  
~B_0(\cosh\eta_0)~Q_{-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~1.041066547  (\cosh\eta - \cos\theta)^{1 / 2}  ~Q_{-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~1.041066547  ~\sqrt{\frac{\cosh\eta - \cos\theta}{\cosh\eta +1}} ~K\biggl( \sqrt{\frac{1}{\cosh\eta +1}} \biggr) \, .
</math>
  </td>
</tr>
</table>

-->