Editing 2DStructure/ToroidalCoordinates (section)

==Preface==
<table border="1" width="90%" align="center" cellpadding="15">
<tr><td align="left">

<table border="0" cellpadding="8" align="center" width="60%">
<tr><td align="center">
<font size="+1"><b>Abstract</b></font><br />Joel E. Tohline (September, 2017)
</td></tr>
</table>
<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="darkgreen">
"The important question we have tried to clarify concerns the possibility of converting the remaining double integral &hellip; into a line integral &hellip; this question remains open."</font>
</td></tr>
<tr><td align="right">
&#8212; Drawn from &sect;6 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)] 
</td></tr></table>

It appears as though we have found an answer to this question posed by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)].  The detailed derivations and associated scratch-work that support the ''summary'' discussion of this chapter can be found under the [[Appendix/Ramblings/ToroidalCoordinates|Appendix/Ramblings category]] of this H_Book and in our accompanying discussion of [[Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori|Dyson-Wong toroids]].

<table border="0" align="right"><tr><td align="center">
[[File:TThinRing72cropped.png|225px|center|Gravitational Potential surface for infinitesimally thin hoop]]
</td></tr>
<tr><td align="center">Infinitesimally Thin Hoop</td></tr></table>
As has been known for approximately a century, at any meridional-plane coordinate location, <math>~(\varpi,z)</math>, the gravitational potential due to an axisymmetric, infinitesimally thin ring (TR) of radius, <math>~a</math>,  and total mass, <math>~M</math>, is given by the "key expression,"
<table border="0" align="center"><tr><td align="center">{{ Math/EQ TRApproximation }}</td></tr></table>
and <math>~K(k)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind].  Making the coordinate-name substitutions, <math>~(\varpi,z) \rightarrow (R_*,z_*)</math>, to match much of this chapter's variable notation, we have alternatively,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(R_*,Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-  \biggl[ \frac{GM}{\pi} \biggr] \biggl[\frac{k}{(R_*a)^{1 / 2}}\biggr] K(k)  \, .</math>
  </td>
</tr>
</table>
A number of research groups (e.g., [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)], [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, ''et al.'' (2012)], [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)]) have recognized that the gravitational potential due to ''any'' axisymmetric configuration of finite extent can be obtained by integrating over the meridional-plane surface area that is occupied by the configuration, while weighting the volume density, <math>~\rho(\varpi,z)</math>, according to the prescription,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(R_* \varpi)^{1 / 2}} \biggr] K(\mu) \rho(\varpi, z) 2\pi \varpi d\varpi dz \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4R_*\varpi}{(R_* + \varpi)^2 + (Z_* - z)^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
We, like [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)], have wondered whether there is a possibility of converting this double integral into a single (line) integral.  This is a particularly challenging task when this expression for the gravitational potential is couched in terms of cylindrical coordinates because the modulus of the elliptic integral is explicitly a function of both <math>~\varpi</math> and <math>~z</math>.

We have recently realized that if a switch is made from cylindrical coordinates to a toroidal coordinate system, <math>~(\eta,\theta)</math>, that is defined such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = \frac{R_* \sinh\eta}{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~(Z_* - z) = \frac{R_* \sin\theta}{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
</tr>
</table>
</div>
then the expression for the modulus of the elliptic integral becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \biggl[1 + \frac{1}{\tanh\eta}  \biggr]^{-1} \, ,
</math>
  </td>
</tr>
</table>
which is a function of only one coordinate &#8212; the "radial" coordinate, <math>~\eta</math> &#8212; and the integral expression for the gravitational potential becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*, Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ 2^{3 / 2} G R_*^{2}
\int\limits_{\eta_\mathrm{min}}^{\eta_\mathrm{max}} \frac{K(\mu) \sinh \eta ~d\eta}{( \sinh \eta +\cosh \eta )^{1 / 2}}  
\int\limits_{\theta_\mathrm{min}(\eta)}^{\theta_\mathrm{max}(\eta)}  \rho(\eta, \theta) 
\biggl[ \frac{d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>
If the configuration's density is constant, then, as is [[#Integrate|shown below]], the integral over the angular coordinate variable, <math>~\theta</math>, can be completed analytically.  Hence, the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced a problem of carrying out a single, line integration.  We specifically illustrate how this approach can be used to evaluate the gravitational potential of a uniform-density torus with a circular cross-section and aspect ratio, <math>~R/d = 3</math>.

Note:&nbsp; As we have [[Apps/DysonWongTori#Wong_.281973.2C_1974.29|reviewed separately]], it appears as though over forty years ago [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973)] successfully employed toroidal coordinates to derive a closed-form expression for the potential (inside as well as outside) of a uniform-density torus with a circular cross-section; see, for example, his Figure 7.  That is to say, he was even able to carry out the final line integral in closed form.  We have not (yet) been able to demonstrate that our expression for the potential of a circular-cross-section torus is identical to his.
</td></tr></table>

<!-- SLIGHTLY MORE DETAILED ABSTRACT
<table border="1" width="90%" align="center" cellpadding="15">
<tr><td align="left">

<table border="0" cellpadding="8" align="center" width="60%">
<tr><td align="center">
<font size="+2"><b>Abstract</b></font><p></p>Joel E. Tohline (September, 2017)
</td></tr>
</table>
<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="darkgreen">
"The important question we have tried to clarify concerns the possibility of converting the remaining double integral &hellip; into a line integral &hellip; this question remains open."</font>
</td></tr>
<tr><td align="right">
&#8212; Drawn from &sect;6 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)] 
</td></tr></table>

It appears as though we have found an answer to this question posed by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)].  The detailed derivations and associated scratch-work that support the ''summary'' discussion of this chapter can be found under the [[Appendix/Ramblings/ToroidalCoordinates|Appendix/Ramblings category]] of this H_Book and in our accompanying discussion of [[Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori|Dyson-Wong toroids]].

[[File:MacMillanFigure61.png|250px|right|MacMillan (1958, ''The Theory of the Potential'', New York: McGraw-Hill)]]As has been known for approximately a century, at any meridional-plane coordinate location, <math>~(R_*, Z_*)</math>, the gravitational potential due to an infinitesimally thin, axisymmetric "hoop" of radius, <math>~a</math>,  and total mass, <math>~M</math>, is given by the expression,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl[ \frac{2GM}{\pi \rho_2} \biggr] K(k^2)  =
-  \biggl[ \frac{GM}{\pi} \biggr] \biggl[\frac{k}{(R_*a)^{1 / 2}}\biggr] K(k^2)  \, ,</math>
  </td>
</tr>
</table>
where, the pair of (squared) reference distances, 
<table border="0" cellpadding="5" align="center">
<tr>
<td align="right"><math>~\rho_1^2 \equiv [(R_* - a)^2 + Z_*^2] \, ,</math></td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<td align="right"><math>~\rho_2^2 \equiv [(R_* + a)^2 + Z_*^2] \, ,</math></td>
</tr>
</table>
and, <math>~K(k^2)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind] for the modulus, 
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~k^2 = 1 - \biggl(\frac{\rho_1^2}{\rho_2^2}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4R_*a}{(R_*+a)^2 + Z_*^2} \, .</math>
  </td>
</tr>
</table>
A number of research groups (e.g., [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)], [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, ''et al.'' (2012)], [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)]) have recognized that the gravitational potential due to ''any'' axisymmetric configuration of finite extent &#8212; as opposed to being infinitesimally thin &#8212; can be obtained by integrating over the meridional-plane surface area that is occupied by the configuration, while weighting the surface density, <math>~\sigma(\varpi,z)</math>, according to the prescription,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G}{R_*^{1 / 2}} \iint\limits_\mathrm{config} \varpi^{1 / 2} \mu K(\mu^2) \sigma(\varpi, z) d\varpi dz =
- \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(R_* \varpi)^{1 / 2}} \biggr] K(\mu^2) \sigma(\varpi, z) 2\pi \varpi d\varpi dz \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4R_*\varpi}{(R_* + \varpi)^2 + (Z_* - z)^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
We, like [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)], have wondered whether there is a possibility of converting this double integral into a single (line) integral.  This is a particularly challenging task when the integrals are couched in terms of cylindrical coordinates because the modulus of the elliptic integral is explicitly a function of both <math>~\varpi</math> and <math>~z</math>.

We have recently realized that if a switch is made from cylindrical coordinates to a toroidal coordinate system, <math>~(\eta,\theta)</math>, that is defined such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi = \frac{R_* \sinh\eta}{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~(Z_*-z) = \frac{R_* \sin\theta}{(\cosh\eta - \cos\theta)} \, ,</math>
  </td>
</tr>
</table>
</div>
then the expression for the modulus of the elliptic integral becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \biggl[1 + \frac{1}{\tanh\eta}  \biggr]^{-1} \, ,
</math>
  </td>
</tr>
</table>
which is a function of only one coordinate &#8212; the "radial" coordinate, <math>~\eta</math> &#8212; and the integral expression for the gravitational potential becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*, Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ 2^{3 / 2} G R_*^{2}
\int\limits_{\eta_\mathrm{min}}^{\eta_\mathrm{max}} \frac{K(\mu^2) \sinh \eta ~d\eta}{( \sinh \eta +\cosh \eta )^{1 / 2}}  
\int\limits_{\theta_\mathrm{min}(\eta)}^{\theta_\mathrm{max}(\eta)}  \rho(\eta, \theta) 
\biggl[ \frac{d\theta}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>
If the configuration's surface density is constant, then, as is shown below, the integral over the angular coordinate variable, <math>~\theta</math>, can be completed analytically.  Hence, the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced a problem of carrying out a single, line integration.  We specifically illustrate how this approach can be used to evaluate the gravitational potential of a uniform-density torus with a circular cross-section and aspect ratio, <math>~R/d = 3</math>.
</td></tr></table>

-->

{| class="Chap25A" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 150px; width: 150px;" |[[H_BookTiledMenu#Axisymmetric_Equilibrium_Structures|Exploring the Use of<br />Toroidal Coordinates<br />to Determine the<br />Gravitational<br /> Potential]]
|}
As I have studied the structure and analyzed the stability of (both self-gravitating and non-self-gravitating) toroidal configurations over the years, I have often wondered whether it might be useful to examine such systems mathematically using a toroidal &#8212; or at least a toroidal-like &#8212; coordinate system.  Is it possible, for example, to build an equilibrium torus for which the density distribution is one-dimensional as viewed from a well-chosen toroidal-like system of coordinates?

I should begin by clarifying my terminology.  In volume II (p. 666) of their treatise on [[Appendix/References#MF53|''Methods of Theoretical Physics'', Morse &amp; Feshbach (1953; hereafter MF53)]] define an orthogonal toroidal coordinate system in which the Laplacian is separable.<sup>1</sup>  (See details, below.)  It is only this system that I will refer to as ''the'' toroidal coordinate system; all other functions that trace out toroidal surfaces but that don't conform precisely to Morse &amp; Feshbach's coordinate system will be referred to as ''toroidal-like.'' 

I became particularly interested in this idea while working with Howard Cohl (when he was an LSU graduate student).  Howie's dissertation research uncovered a ''Compact Cylindrical Greens Function'' technique for evaluating Newtonian potentials of rotationally flattened (especially axisymmetric) configurations.<sup>2,3</sup>  The technique involves a multipole expansion in terms of half-integer-degree Legendre functions of the <math>2^\mathrm{nd}</math> kind &#8212; see [http://dlmf.nist.gov/14.19 NIST digital library discussion] &#8212; where, we have discovered, the argument of this special function is a function only of the ''radial'' coordinate of Morse &amp; Feshbach's orthogonal toroidal coordinate system &#8212; see more on this, [[#Connection_With_the_Physical_Problem|below]].