Editing Apps/DysonWongTori (section)

==Thin Ring Approximation==
===MacMillan (1930)===
====Derivation of the Potential====
In &sect;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'', W. D. MacMillan (1958; originally, 1930)] derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>; as shown, immediately below, the hoop is labeled, <math>~H</math>, in his Figures 60 and 61.

<table border="0" cellpadding="8" align="center"><tr><td align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center" colspan="2">
'''Figures 60 &amp; 61 extracted without modification from, respectively, p. 195 &amp; 196 of [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 MacMillan (1958)]'''<p></p>
''The Theory of the Potential'', New York: McGraw-Hill
</td></tr>
<tr>
<td>
[[File:MacMillanFigure60.png|460px|center|MacMillan (1958, ''The Theory of the Potential'', New York: McGraw-Hill)]]
</td>
<td>
[[File:MacMillanFigure61.png|460px|center|MacMillan (1958, ''The Theory of the Potential'', New York: McGraw-Hill)]]
</tr></table>
</td></tr></table>

In setting up this problem, MacMillan (1958) says (verbatim text is typeset in a dark green font), <font color="darkgreen">Let <math>~P</math> be any point in space not in <math>~H</math>.  From <math>~P</math> drop the perpendicular <math>~PQ = z</math> to the plane of the hoop.  Draw the diameter of the circle <math>~BOA</math> which, extended, passes through <math>~Q</math>. Let <math>~m</math> be any point on the circle, and draw</font>
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Pm = \rho \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~PA = \rho_1 \, ,</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~PB = \rho_2 \, .</math>
  </td>
</tr>
</table>
</div>

<font color="darkgreen">Evidently <math>~\rho_1</math> and <math>~\rho_2</math> are the minimum and maximum values of <math>~\rho</math> as the point <math>~m</math> runs around the circle.</font> <font color="darkgreen">If the angle <math>~mOA</math> is represented by <math>~2\omega</math>, the arc element is <math>~ds = 2ad\omega</math>, and</font> &#8212; after multiplying MacMillan's &sect;102, equation (1) through by (conventionally, the negative of) the gravitational constant, <math>~G</math> &#8212; <font color="darkgreen">the expression for the gravitational potential is</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 2Ga\sigma \int_0^\pi \frac{d\omega}{\rho}  = - \biggl( \frac{GM}{\pi} \biggr) \int_0^\pi \frac{d\omega}{\rho} \, ,</math>
  </td>
</tr>
</table>
</div>

where, <math>~\sigma</math> is the (uniform) linear mass density around the hoop, hence, the total mass of the hoop is <math>~M = 2\pi a \sigma</math>.

Referring further to MacMillan's Figure 60 &#8212; digitally reproduced, above &#8212; if the lengths <math>~mQ</math> and <math>~OQ</math> are represented by <math>~h</math> and <math>~\varpi</math>, respectively, then
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi-a)^2 + z^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi+a)^2 + z^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z^2 + h^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z^2 + \varpi^2 + a^2 - 2a\varpi \cos(2\omega) \, .</math>
  </td>
</tr>
</table>
</div>
Following [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 MacMillan (1958)] (p. 196), the expression for <math>~\rho^2</math> can further be written,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\varpi^2 + a^2 + z^2)(\cos^2\omega + \sin^2\omega) - 2a\varpi (\cos^2\omega - \sin^2\omega)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[(\varpi-a)^2 + z^2]\cos^2\omega + [(\varpi+a)^2 + z^2]\sin^2\omega
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho_1^2 \cos^2\omega + \rho_2^2\sin^2\omega \, .
</math>
  </td>
</tr>
</table>
</div>
<span id="DeupreeReference">Hence, the expression for the potential becomes,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{2GM}{\pi} \biggr) \int_0^{\pi/2} \frac{d\omega}{[\rho_1^2 \cos^2\omega + \rho_2^2\sin^2\omega]^{1 / 2}} \, .</math>
  </td>
</tr>
</table>
</div>

As [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 MacMillan (1958)] argues, <font color="darkgreen">&hellip; this expression shows that <math>~\Phi</math> is symmetric in <math>~\rho_1</math> and <math>~\rho_2</math>, for if <math>~\omega</math> is replaced by <math>~(\tfrac{\pi}{2} - \psi)</math> it becomes</font>

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

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{2GM}{\pi} \biggr) \int_0^{\pi/2} \frac{d\psi}{[\rho_1^2 \sin^2\psi + \rho_2^2\cos^2\psi]^{1 / 2}} \, ,</math>
  </td>
</tr>
</table>
</div>
<font color="darkgreen">and therefore</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\rho_1, \rho_2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Phi(\rho_2, \rho_1) \, .</math>
  </td>
</tr>
</table>
</div>
<font color="darkgreen">Along the axis of the hoop, <math>~\rho_1 = \rho_2</math>, and if <math>~\rho_a</math> is their common value, it is seen at once that the value of the potential along this axis <math>~\Phi_a</math> is,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_a</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{GM}{\rho_a} \, .</math>
  </td>
</tr>
</table>
</div>

Furthermore, according to [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 MacMillan (1958)], <font color="darkgreen">&hellip; the function <math>~\Phi(\rho_1, \rho_2)</math> is homogeneous of degree "- 1" in <math>~\rho_1</math> and <math>~\rho_2</math>.  Therefore, <math>~\rho_1 \Phi</math> is homogeneous of degree zero and depends only upon the ratio <math>~\rho_1/\rho_2</math>.</font>  
With this in mind, let's rewrite the expression for the potential in the form,

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

<tr>
  <td align="right">
<math>~\rho_1 \Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{2GM}{\pi}  \int_0^{\pi/2} \biggl[\sin^2\psi + \biggl( \frac{\rho_2^2}{\rho_1^2} \biggr)\cos^2\psi \biggr]^{-1 / 2} d\psi </math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{2GM}{\pi} \biggl( \frac{\rho_1}{\rho_2} \biggr) \int_0^{\pi/2} \biggl[\biggl( \frac{\rho_1^2}{\rho_2^2} \biggr)\sin^2\psi + \biggl( 1 - \sin^2\psi \biggr)\biggr]^{-1 / 2} d\psi </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{2GM}{\pi} \biggl( \frac{\rho_1}{\rho_2} \biggr) \int_0^{\pi/2} \biggl[1 - \biggl(1 - \frac{\rho_1^2}{\rho_2^2} \biggr)\sin^2\psi \biggr]^{-1 / 2} d\psi \, .</math>
  </td>
</tr>
</table>
</div>

In addition to the hoop, <math>~H</math>, Figure 61 in &sect;102 of MacMillan (1958) &#8212; digitally reproduced, above &#8212; displays a curve in the meridional plane of the hoop for which the ratio,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho_1}{\rho_2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c \, ,</math>
  </td>
</tr>
</table>
</div>
where <math>~c</math> is a constant.  As [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 MacMillan (1958)] argues, the displayed curve is a circle because this equation <font color="darkgreen">is the equation of a circle in bipolar coordinates; and this circle &hellip; divides the line <math>~BCAD</math> harmonically, since by</font> this last equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{AC}{BC} = \frac{AD}{BD}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c \, .</math>
  </td>
</tr>
</table>
</div>
<span id="RingPotential">It is clear, therefore, that at every point along this meridional circle, the potential is given by the expression,</span>

<table border="1" cellpadding="8" align="center" width="65%">
<tr>
  <th align="center">Gravitational Potential of a Thin Ring</th>
</tr>
<tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{2GMc}{\pi \rho_1} \int_0^{\pi/2} \frac{d\psi}{ \sqrt{1 - k^2 \sin^2\psi }} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl[ \frac{2GMc}{\pi \rho_1} \biggr] K(k^2)  \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[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 W. D. MacMillan (1958)], &sect;102, Eq. (5);<p></p>
see also [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)], &sect;III.4, Exercise (4)
  </td>
</tr>
</table>
where, <math>~K(k^2)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind] for the ''parameter'',
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1-c^2 \, .</math>
  </td>
</tr>
</table>
</td></tr></table>

<span id="CylindricalLocation">The ''parameter'', <math>~k^2</math>, always lies between zero and unity.  For later reference, we note that,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \biggl[ \frac{2GMc}{\pi \rho_1} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2GM}{\pi \rho_2} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2GM}{\pi } \biggr]\frac{1}{\sqrt{(\varpi+a)^2 + z^2}} \, ,</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~k^2 = 1-c^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - \frac{\rho_1^2}{\rho_2^2} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{[(\varpi+a)^2 + z^2] - [(\varpi-a)^2 + z^2]}{(\varpi+a)^2 + z^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4a\varpi }{(\varpi+a)^2 + z^2} \, .
</math>
  </td>
</tr>
</table>
</div>
<span id="TRApproximation">This gives us what we will henceforth refer to as the,</span>
<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">
{{ Math/EQ TRApproximation }}
  </td>
</tr>
</table>
Notice that in finalizing the [https://en.wikipedia.org/wiki/Elliptic_integral#Notational_variants precise ''notational form''] of this "key equation," we have chosen to rewrite the argument of the complete elliptic integral of the first kind in terms of the ''elliptic modulus'', <math>~k</math>, rather than in terms of the ''parameter'' <math>~k^2</math>.

====Some Geometric Relations====
<table border="0" cellpadding="8" align="right"><tr><td align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center" colspan="1">
'''Modification of MacMillan's Figure 61'''
</td></tr>
<tr>
<td>
[[File:ModifiedMacMillan61.png|460px|center|Modification of MacMillan's Figure 61]]
</tr></table>
</td></tr></table>
Throughout his derivation, [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 MacMillan (1958)] uses the parameter, <math>~a</math>, to represent the radius of the circular "hoop" &#8212; that is, the distance from the center of the hoop to either point <math>~A</math> or point <math>~B</math> as marked in both Figure 60 and Figure 61.  In the diagram presented here, on the right, we have modified his Figure 61 (modifications are in red) to explicitly identify two additional lengths that will come into play when we reference toroidal coordinates, below: &nbsp; The parameter, <math>~R</math>, identifies the distance from the center of the hoop to the center of the meridional-plane circle; and the parameter, <math>~d</math>, identifies the radius of this meridional-plane circle.  (Note that the distance between point <math>~A</math> and the center of the meridional-plane circle is <math>~R-a</math>.)  Given that the meridional-plane circle has been drawn in such a way that the ratio, <math>~\rho_1/\rho_2 = c</math>, at all points <math>~P</math> along the circle, a useful relationship can be derived between the three parameters, <math>~R, d</math> and <math>~a</math> as follows.  

If <math>~P</math> is moved around the circle to align with point <math>~D</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_1 = \rho_2 c = d + (R-a) \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and, &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\rho_2 = 2a + (R-a) + d </math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
<math>~\Rightarrow ~~~ 
c = \frac{d + (R-a)}{2a + (R-a) + d} \, .
</math>
  </td>
</tr>
</table>
</div>

Similarly, if <math>~P</math> is moved around the circle to align with point <math>~C</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_1 = \rho_2 c = d - (R-a) \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and, &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\rho_2 = 2a - [ d - (R-a)]</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
<math>~\Rightarrow ~~~ 
c = \frac{d - (R-a)}{ 2a - [ d - (R-a)]} \, .
</math>
  </td>
</tr>
</table>
</div>
Equating these two expressions for <math>~c</math> then gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d + R-a}{a + R + d}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d - R + a}{ a  -d + R}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
(d + R-a)(a  -d + R )
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(d - R + a)(a + R + d )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
[R - (a- d) ][R + (a  -d) ]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[(a + d) - R ][(a + d) + R ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
R^2 - (a- d)^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(a + d)^2 - R^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~
2R^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(a + d)^2 + (a- d)^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2a^2 + 2d^2 \, ,
</math>
  </td>
</tr>
</table>
</div>
<span id="TorusGeometry">or, finally,</span>
<table border="1" cellpadding="8" align="center" width="25%">
<tr>
  <th align="center">Geometric Relationship</th>
</tr>
<tr>
  <td align="center">
<math>~a^2 = R^2 - d^2 \, .</math>
  </td>
</tr>
</table>
Similarly, it can be shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \biggl(\frac{R-d}{R+d}\biggr)^{1 / 2}  \biggr]\biggl[1 + \biggl(\frac{R-d}{R+d}\biggr)^{1 / 2}   \biggr]^{- 1 }</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{d} \biggl[ 1 - \sqrt{1 - \frac{d^2}{R^2}} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Thus, the ''aspect ratio'',
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R}{d}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1+c^2}{2c} = \frac{1}{2}\biggl[ \frac{\rho_2}{\rho_1} + \frac{\rho_1}{\rho_2} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

===Bannikova et al. (2011) -- Thin Ring===
In a paper titled, ''Gravitational Potential of a Homogeneous Circular Torus: a New Approach'', [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B E. Y. Bannikova, V. G. Vakulik &amp; V. M. Shulga (2011, MNRAS, 411, 557 - 564)] begin by reviewing what the expression is for an infinitesimally thin, axisymmetric hoop.  Specifically, referencing the ''central ring'' &#8212; see their equations (1) - (3) &#8212; they state that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_c(\varpi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_c}{\pi a} \biggl[ \frac{am}{\varpi} \biggr]^{1 / 2} K(m) \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K(m) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int_0^{\pi/2} \frac{d\beta}{\sqrt{ 1 - m\sin^2\beta}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~m </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4a \varpi}{(\varpi + a)^2 + z^2} \, .
</math>
  </td>
</tr>
</table>
</div>
Noting that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{GM_c}{\pi a} \biggl[ \frac{am}{\varpi} \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{GM_c}{\pi }\biggr] \biggl[ \frac{m}{a\varpi} \biggr]^{1 / 2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2GM_c}{\pi }\biggr] \frac{1}{\sqrt{(\varpi + a)^2 + z^2}} \, .
</math>
  </td>
</tr>
</table>
</div>
We see that, aside from the adopted sign convention, the [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)]  expression for <math>~\Phi_c</math> exactly matches the expression for the "[[#RingPotential|Gravitational Potential of a Thin Ring]]" that is obtained from [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 MacMillan's (1958)] derivation.

===Fukushima (2016) -- Thin Ring===
As is discussed in considerable depth, [[#Fukushima_.282016.29|below]], [http://adsabs.harvard.edu/abs/2016AJ....152...35F Toshio Fukushima (2016, AJ, 152, id. 35, 31 pp.)] has used ''zonal toroidal harmonics'' to examine the gravitational field external to ring-like objects.  In the first segment of &sect;3 of his paper, Fukushima introduces a potential function, <math>~\Phi_P</math>, that, in his words, "is a special solution of simplified Poisson's equation being valid in the whole space."  From his equations (21), (62), (63), and (64), we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\frac{GM}{\sqrt{(\varpi + a)^2 + z^2 }} \biggl[ \frac{2}{\pi} \cdot K(m)\biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K(m)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\int_0^{\pi/2} \frac{d\varphi}{\sqrt{ 1 - m\sin^2\varphi}}
</math>
  </td>
</tr>
</table>
</div>
and the parameter,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\nu}{u + \nu} \, .
</math>
  </td>
</tr>
</table>
</div>
Given that (see, for example, Fukushima's equations 19 and 7, in conjunction with his Figure 1),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2u = \biggl(\frac{p}{q} + \frac{q}{p}\biggr) = \frac{p^2 + q^2}{pq} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~2\nu = 2(u^2 - 1)^{1 / 2} = 2 \biggl[ \frac{(p^2 + q^2)^2 - 4p^2q^2}{4p^2q^2} \biggr]^{1 / 2} = \frac{p^2 - q^2}{pq} \, ,</math>
  </td>
</tr>
</table>
</div>
we recognize that the parameter, <math>~m</math>, can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\nu \biggl[2u + 2\nu \biggr]^{-1} = \frac{2(p^2 - q^2)}{pq} \biggl[\frac{2p^2}{pq}\biggr]^{-1} = 1 - \frac{q^2}{p^2} \, .
</math>
  </td>
</tr>
</table>
</div>
Acknowledging furthermore &#8212; see our more extensive [[#Fukushima_.282016.29|discussion, below]] &#8212; the parameter mapping,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \rho_2 \biggr]_\mathrm{MacMillan} \leftrightarrow \biggl[ p \biggr]_\mathrm{Fukushima}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\biggl[ \rho_1 \biggr]_\mathrm{MacMillan} \leftrightarrow \biggl[ q \biggr]_\mathrm{Fukushima}</math>
  </td>
</tr>
</table>
it is clear that Fukushima's parameter, <math>~m</math>, is identical to MacMillan's modulus, <math>~k^2</math>, and that Fukushima's expression for <math>~\Phi_P</math> is identical to the expression for the "[[#RingPotential|Gravitational Potential of a Thin Ring]]" that we have presented above.  Indeed, immediately preceding his equation (62), Fukushima explicitly acknowledges that his expression for <math>~\Phi_P</math> has been drawn from [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)].


<div id="ThinRingContours">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="4">Graphical Depiction in the Meridional Plane of the Gravitational Potential of a Thin, Axisymmetric Ring</th>
</tr>
<tr>
  <td align="center" rowspan="5" bgcolor="#D0FFFF">
[[File:TThinRing72cropped.png|380px|Our Thin Ring equipotential surface]]
  </td>
<td align="center" colspan="3">
Extracted without modification from [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016)]'''<p></p>
"''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects'''"<p></p>
Astronomical Journal, vol. 152, id. 35, 31 pp. &copy; [https://aas.org/ AAS]
</td>
</tr>
<tr>
  <td align="center">Fukushima's Figure 4</td>
  <td align="center">Fukushima's Figure 5</td>
  <td align="center">Fukushima's Figure 6</td>
</tr>
<tr>
  <td align="center">
[[File:Fukushima2016Fig4.png|225px|To be inserted: Fig. 4 from Fukushima (2016)]]
  </td>
  <td align="center">
[[File:Fukushima2016Fig5.png|225px|To be inserted: Fig. 5 from Fukushima (2016)]]
  </td>
  <td align="center">
[[File:Fukushima2016Fig6.png|225px|To be inserted: Fig. 6 from Fukushima (2016)]]
  </td>
</tr>
<tr>
  <td align="center" align="center" colspan="3">
Our Effort to Reproduce Fukushima's Figures
  </td>
</tr>
<tr>
  <td align="center" bgcolor="#D0FFFF">
[[File:FlatColorContoursCropped.png|225px|Our Thin Ring equipotential surface]]
  </td>
  <td align="center">
[[File:OurFig5Bhalf.png|225px|Our Thin Ring equipotential surface]]
  </td>
  <td align="center">
[[File:OurFig6Chalf.png|225px|Our Thin Ring equipotential surface]]
  </td>
</tr>
</table>
</div>


Figures 4, 5, &amp; 6 of [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] display, with quantitative accuracy, the behavior of the potential function, <math>~\Phi_P/|\Phi_0|</math>, where <math>~\Phi_0</math> is a normalization factor.  We have extracted digital copies of these three figures and have displayed them, without modification, in the above figure ensemble.  We have evaluated the expression for the potential of an infinitesimally thin ring as derived by MacMillan (1958) and, in the same figure ensemble, have displayed the results in a manner that facilitates comparison with Fukushima's published results; our displayed results incorporate the normalization, <math>~\Phi_0 = GM/(2\pi a)</math>, so, given the high degree of quantitative overlap, we presume that this is the normalization that was adopted by Fukushima.  In the left-most panel of our figure ensemble, we have displayed the absolute value of this same two-dimensional, normalized potential function in the form of a warped surface; this has been done largely for visual effect.  We should point out that our pair of multi-colored contour plots &#8212; both the warped surface and its flat projection onto the meridional plane &#8212; cover an area that extends from the cylindrical-coordinate axes out to the boundaries, <math>~(R/a, Z/a) = (\tfrac{3}{2}, \pm\tfrac{3}{4})</math>, whereas Fukushima's contour plot (his Figure 4) extends twice as far in both directions.