Editing Apps/DysonWongTori (section)

===Fukushima (2016)===
====Setup====

[http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016, AJ, 152, id. 35, 31 pp.)] has also used ''zonal toroidal harmonics'' to examine the gravitational field external to ring-like objects. But, while continuing to associate the parameter, <math>~a</math>, with the equatorial-plane radius of the ''central ring'', and to  reference the same polar <math>~(\theta)</math> and azimuthal <math>~(\psi)</math> angles as Wong (1973), Fukushima uses a meridional-plane radial coordinate, <math>~u</math>, that, in practice, maps to [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] coordinate, <math>~\eta</math>, and 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's (1958)] parameter, <math>~c</math> as,

<div align="center">
<math>~ \cosh\eta  \leftrightarrow u \leftrightarrow \frac{1}{2}\biggl[\frac{1}{c} + c \biggr] \, .</math>
</div>

Comparing [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)] Figure 1 &#8212; digitally reproduced, immediately below &#8212; with [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)] Figure 60 reveals the following parameter notation relationships:
<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>
Given that MacMillan's parameter, <math>~c = \rho_1/\rho_2</math>, it is clear that [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)] "radial" coordinate, <math>~u</math>, can also be defined via the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u = \frac{1}{2} \biggl[\frac{1}{c} + c\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[\frac{\rho_2}{\rho_1} + \frac{\rho_1}{\rho_2} \biggr] = \frac{1}{2} \biggl[\frac{p}{q} + \frac{q}{p} \biggr] \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)], Eq. (19)
  </td>
</tr>
</table>
This is the expression for the coordinate, <math>~u</math>, that appears as equation (19) in [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] and is recorded in the legend of his Figure 1.

<table border="1" cellpadding="5" align="center" width="675px">
<tr><td align="center" width="475px" colspan="2">
Figure 1 extracted from [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016)]'''<br />
"''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects'''"<br />
Astronomical Journal, vol. 152, id. 35, 31 pp. &copy; [https://aas.org/ AAS]
</td></tr>
<tr>
  <th align="center">Reproduced without Modification</th>
  <th align="center">Modified (red font)</th>
</tr>
<tr>
<td>
[[File:FukushimaFig1.png|325px|center|To be inserted: Fig. 1 from Fukushima (2016)]]
</td>
<td>
[[File:ModifiedFukushimaFig1.png|325px|center|To be inserted: Modified Fig. 1 from Fukushima (2016)]]
</td>
</tr>
</table>

Fukushima also introduces a set of coordinate-dependent functions &#8212; <math>~\nu(u), s(\theta), c(\theta),</math> and <math>D(u,\theta) \equiv (u - c)</math> &#8212; that, in effect, allow his derivations to be presented in a relatively compact form.   [[#Table1|Table 1, below]], details the relationships between this set of functions and the coordinates/functions used by Wong as well as the coordinates/functions preferred by [http://adsabs.harvard.edu/abs/1953mtp..book.....M Morse &amp; Feshbach (1953)] that we have adopted in an [[2DStructure/ToroidalCoordinates#Toroidal_Coordinates|associated discussion]].  Then, for example, the mapping between Fukushima's adopted set of meridional-plane coordinates and ''cylindrical'' coordinates <math>~(\varpi,z)</math> is:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi}{a} = \frac{\nu}{D}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{z}{a} = \frac{s}{D} \, .</math>
  </td>
</tr>
</table>
</div>

[As is emphasized by the manner in which we have modified his Figure 1 &#8212; see the righthand panel of the composite image, immediately above &#8212; we will use <math>~\varpi</math> instead of <math>~R</math> to represent the cylindrical radial coordinate throughout our discussion of [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)] work.]  Using toroidal coordinates, a meridional-plane circle describing a cross-section through the torus is defined by setting <math>~u</math> = constant.  The relevant relation can be obtained by combining this pair of expressions to eliminate <math>~\theta</math>.  From the first, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\nu}{(u - \cos\theta)}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow
u - \cos\theta
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\nu}{\varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow
\cos^2\theta
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( u - \frac{a\nu}{\varpi^2} \biggr)^2\, .</math>
  </td>
</tr>
</table>
</div>
Using the second to replace <math>~D</math> in the first, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi}{a} \biggl( \frac{a}{z}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\nu}{\sin\theta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow
\sin\theta
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{z\nu}{\varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow
\cos^2\theta
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1- \frac{z^2\nu^2}{\varpi^2} \, .</math>
  </td>
</tr>
</table>
</div>
Together, this pair of relations implies,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl( u - \frac{a\nu}{\varpi} \biggr)^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1- \frac{z^2\nu^2}{\varpi^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow
\biggl( u - \frac{a\nu}{\varpi} \biggr)^2 + \frac{z^2\nu^2}{\varpi^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow
\frac{\varpi^2}{\nu^2}\biggl( u - \frac{a\nu}{\varpi} \biggr)^2 + z^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\varpi^2}{\nu^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\varpi^2}{\nu^2} - \biggl( \frac{\varpi u}{\nu} -a \biggr)^2
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\varpi^2}{\nu^2}\biggl[ 1 -  u^2 \biggr] +  \frac{2a\varpi u}{\nu} - a^2 \, .
</math>
  </td>
</tr>
</table>
</div>
Now, given that (see [[#Table1|Table 1]]), <math>~\nu^2 = (u^2-1)</math>, this last expression can be rewritten as,

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^2}{\nu^2}  - \varpi^2 + \frac{2a\varpi u}{\nu} - \frac{a^2u^2}{\nu^2} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow
\biggl[\varpi - \frac{au}{\nu} \biggr]^2 + z^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^2}{\nu^2} 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)], Eq. (27)
  </td>
</tr>
</table>
</div>
This last expression not only matches equation (27) in [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] but, as has been pointed out in an [[2DStructure/ToroidalCoordinates#Chosen_Test_Mass_Distribution|accompanying discussion]] of the paper by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)], its form matches the familiar algebraic expression for an off-axis circle.  The circle &#8212; which, here, is associated with a meridional cross-section of the <math>~u</math> = constant torus &#8212; has a (cross-sectional) radius,
<div align="center">
<math>~r_t = \frac{a}{\nu} \, ,</math>
</div>
and its center is shifted a distance,
<div align="center">
<math>~\varpi_t = \frac{au}{\nu}  \, ,</math>
</div>
away from the symmetry <math>~(z)</math> axis.  Via his equations (26) and (25), Fukushima (2016) labels these key geometric lengths as, respectively, <math>~R_R</math> and <math>~R_C</math>.  We should point out that the other key geometric length, <math>~a</math> &#8212; which defines the size of the ''central ring'' as depicted in Figure 1 of [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] &#8212; is related to <math>~\varpi_t</math> and <math>~r_t</math> via the expression,

<div align="center">
<math>~a^2 = \varpi_t^2 - r_t^2 \, .</math>
</div>

And it should be emphasized that <math>~a</math> is always smaller than <math>~\varpi_t = R_C</math>, except in the limit of <math>~u \rightarrow \infty \Rightarrow u/\nu = 1</math>, in which case the two lengths ''are'' the same.


<div align="center" id="Table1">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="5">TABLE 1: &nbsp;Mapping Between Coordinate-Dependent Functions</th>
</tr>
<tr>
  <td align="center">
[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)]
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)]
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
[http://adsabs.harvard.edu/abs/1953mtp..book.....M MF53]
  </td>
</tr>
<tr>
  <td align="center">
<math>~u</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~\xi_1</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~\nu(u) = (u^2 - 1)^{1 / 2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~\sinh\eta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~(\xi_1^2 - 1)^{1 / 2}</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~s(\theta) = \sin\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~ \sin\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~ (1-\xi_2^2)^{1 / 2}</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~c(\theta) = \cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~ \cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~ \xi_2</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~D(u,\theta) = u - c(\theta)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~\cosh\eta - \cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp;<math>~\leftrightarrow</math>&nbsp; &nbsp; &nbsp;
  </td>
  <td align="center">
<math>~\xi_1 - \xi_2</math>
  </td>
</tr>
</table>
</div>


Letting <math>~P</math> mark the ''external'' point at which the gravitational potential is to be evaluated &#8212; see his Figure 1, a digital replica of which is shown immediately above, on the left &#8212; [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] uses <math>~q</math> to represent the shortest distance between <math>~P</math> and the central ring, and uses <math>~p</math> to represent the greatest distance between <math>~P</math> and the central ring.  It is straightforward to show that,
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 20 August 2017:  In Fukushima's equation (23), the factor of 2 appears in the denominator, rather than in the numerator of this expression.  Barring a misinterpretation on our part, the derivation presented immediately below demonstrates that this factor of 2 should appear in the numerator, as written here.]]<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[(\varpi+a)^2 + z^2\biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a\biggl[ \frac{2(u+\nu)}{D}\biggr]^{1 / 2} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="5">
[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)], Eqs. (21) &amp; (23)
  </td>
</tr>
</table>
</div>
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 20 August 2017:  In Fukushima's equation (24), the factor of 2 appears in the denominator, rather than in the numerator of this expression.  Barring a misinterpretation on our part, our independent derivation demonstrates that this factor of 2 should appear in the numerator, as written here.]]and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~q</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[(\varpi-a)^2 + z^2\biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a\biggl[ \frac{2(u-\nu)}{D}\biggr]^{1 / 2} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="5">
[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)], Eqs. (22) &amp; (24)
  </td>
</tr>
</table>
</div>
It is worth detailing how the cylindrical-coordinate expression for <math>~p</math> is converted into the stated expression in terms of Fukushima's coordinates. (Transformation of <math>~q</math> is done in an analogous fashion.)  For <math>~p</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\nu}{D} + 1\biggr)^2 + \frac{s^2}{D^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\nu}{D} + \frac{\nu^2}{D^2} +  1 + \frac{s^2}{D^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\nu}{D} + \frac{1}{D^2} \biggl[ (u^2-1) +  (u-c)^2 + s^2\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\nu}{D} + \frac{1}{D^2} \biggl[ u^2-1 +  u^2 - 2uc + c^2 + s^2\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\nu}{D} + \frac{2u}{D^2} \biggl[ u - c\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2(\nu + u)}{D} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, along the way, we have used the fact that <math>~(s^2 + c^2) = 1</math> and, again, recognized that, <math>~\nu^2 = (u^2-1)</math>.

====Uniform-Density, Circular Torus====

[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)] tackles this same problem in &sect;5.2 of his article.  The following table identifies the parameter mapping between the notation adopted by [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] and our adopted conventions; note that, when Fukushima attaches an asterisk to a coordinate or variable, it appears as a superscript whereas when we have used an asterisk, it appears as a subscript.

<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="2">Parameter Mapping</th>
</tr>
<tr>
  <td align="center">[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)]</td>
  <td align="center">[[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential|Our Analysis]]</td>
</tr>
<tr>
  <td align="center"><math>~(R, z)</math></td>
  <td align="center"><math>~(R_*, Z_*)</math></td>
</tr>
<tr>
  <td align="center"><math>~(R^*, z^*)</math></td>
  <td align="center"><math>~(\varpi, Z)</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{CU}</math></td>
  <td align="center"><math>~\varpi_t</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{RU}</math></td>
  <td align="center"><math>~r_t</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{1U}</math></td>
  <td align="center"><math>~\varpi_t - r_t</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{2U}</math></td>
  <td align="center"><math>~\varpi_t + r_t</math></td>
</tr>
</table>
Fukushima's expression for the ''external'' gravitational potential is (see his equations 138 - 142),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_U(R,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{R_\mathrm{1U}}^{R_\mathrm{2U}} R^* ~dR^* 
\int_{-z_\mathrm{U}}^{z_\mathrm{U}}\biggl[ \frac{K(m^*)}{p^*}\biggr] dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{R_\mathrm{1U}}^{R_\mathrm{2U}} R^* ~dR^* 
\int_{-z_\mathrm{U}}^{z_\mathrm{U}}\biggl[ \frac{K(m^*)}{\sqrt{(R + R^*)^2 + (z - z^*)^2 }}\biggr] dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \Phi_U(R_*,Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{\varpi_t - r_t}^{\varpi_t + r_t} \varpi ~d\varpi 
\int_{-Z_*(\varpi)}^{Z_*(\varpi)} \biggl[ \frac{K(\mu^2)}{\sqrt{(R_* + \varpi)^2 + (Z_* - Z)^2 }}\biggr] dZ
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{\varpi_t - r_t}^{\varpi_t + r_t}  \varpi ~d\varpi 
\int_{-Z_*(\varpi)}^{Z_*(\varpi)} \biggl[ \frac{\mu^2}{4R_*\varpi} \biggr]^{1 / 2} K(\mu^2) dZ
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G\rho_\mathrm{0U} }{R_*^{1 / 2}}
\int_{\varpi_t - r_t}^{\varpi_t + r_t}  
\int_{-Z_*(\varpi)}^{Z_*(\varpi)} \biggl[ \mu^2 \biggr]^{1 / 2} K(\mu^2) \varpi^{1 / 2} ~d\varpi dZ
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m^* = \mu^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{4R^*\varpi}{(R_*+\varpi)^2 + (Z_* - Z)^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
and the integration limit,
<div align="center">
<math>~Z_*(\varpi) = \sqrt{r_t^2 - (\varpi - \varpi_t)^2 } \, .</math>
</div>
This matches the [[#Our_Integral_Expressions|integral expression that we have derived and repeated, above]].