Editing Apps/DysonWongTori (section)

====Introducing Toroidal Coordinates====

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Figure 1 extracted without modification from [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973)]'''<br />
"''Toroidal and Spherical Bubble Nuclei'''"<br />
Annals of Physics, vol. 77, pp. 279 - 353 &copy; Elsevier Science
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[[File:Wong1973Figure1.png|400px|center|To be inserted: Fig. 1 from Wong (1973, Annals of Physics, 77, p. 284)]]
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973)] introduces the toroidal coordinate system <math>~(\eta, \theta, \psi)</math> as follows (direct quotes from the article are displayed here in a dark green font).  Referencing the figure  &#8212; shown here on the right &#8212; that has been extracted without modification from Wong's article, <font color="darkgreen">the surfaces of constant <math>~\eta</math> are generated by rotating a circle about the axis of symmetry, the <math>~z</math>-axis.  These surfaces are toroidal surfaces.  A toroidal surface of coordinate <math>~\eta</math> can be characterized by a "major radius" <math>~R</math> and a "minor radius" <math>~d</math>.</font> Note that in the article by Wong the three parameters, <math>~(a, R, d)</math>, represent the same geometric lengths as in our discussion, above, of [[#MacMillan_.281930.29|MacMillan's (1930) derivation]]; and Wong's ''radial coordinate'', <math>~\eta</math>, plays the same role as MacMillan's parameter, <math>~c</math> &#8212; the mathematical relationship between the two is [[#WongMacComparison|presented below]].  

<font color="darkgreen">The quantity <math>~\eta</math> varies from zero to infinity.  The larger the value of <math>~\eta</math>, the smaller is the "minor radius" <math>~d</math>; when <math>~\eta</math> approaches infinity, the two-dimensional toroidal surface degenerates into a 1-dimensional circle with a radius <math>~a</math>.</font>  It is in this limit that Wong's torus ''becomes'' the infinitesimally thin, axisymmetric ''hoop'' analyzed by MacMillan (1930). 
<!-- [Note that, otherwise, <math>~R</math> (the location of the center of the circular cross-section of the torus) does not coincide with <math>~a</math> (the location of the off-axis "origin" of the toroidal coordinate system).] -->

<font color="darkgreen">The surfaces of constant <math>~\theta</math> are spherical bowls.  The coordinate <math>~\theta</math> is defined in such a way that points above the x-y plane are characterized by positive values of <math>~\theta</math> while points below the x-y plane by negative values of <math>~\theta</math>.  Thus <math>~\theta</math> varies from <math>~- \pi</math> to <math>~+\pi</math>.</font>  As is also the case for a spherical coordinate system, <font color="darkgreen">the surfaces of constant <math>~\psi</math> are half planes through the axis of symmetry.  The coordinate <math>~\psi</math> varies from <math>~0</math> to <math>~2\pi</math>.
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Referencing the [[#TorusGeometry|above-derived ''geometric relationship'']], we see that for a toroidal surface of major radius <math>~R</math> and minor radius <math>~d</math>, the parameter <math>~a</math> is defined such that,
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<math>~a^2</math>
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<math>~\equiv</math>
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<math>~R^2 - d^2 \, .</math>
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.8)
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The corresponding "radial" coordinate location <math>~\eta_0</math> of the relevant toroidal surface is,
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<math>~\eta_0</math>
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<math>~=</math>
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<math>~\cosh^{-1}\biggl(\frac{R}{d}\biggr) \, .</math>
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.9)
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We see, therefore, that [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>, is related to Wong's "radial" coordinate via the expression,
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<math>\cosh\eta_0 = \frac{1+c^2}{2c} \, .</math>
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Alternatively, given <math>~\eta_0</math> and the value of the parameter <math>~a</math>, we have,
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<math>~R</math>
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<math>~=</math>
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<math>~a \coth\eta_0 \, ,</math>
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<math>~d</math>
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<math>~=</math>
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<math>~\frac{a}{\sinh\eta_0} \, .</math>
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eqs. (2.10) &amp; (2.11)
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Hence, the aspect ratio is,
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<math>~\frac{R}{d}</math>
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<math>~=</math>
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<math>~\cosh\eta_0 \, .</math>
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.12)
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Given the value of the scale-length, <math>~a</math>, the relationship between toroidal coordinates and Cartesian coordinates is [see equations 2.1 - 2.3 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ],
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<math>~x</math>
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<math>~=</math>
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<math>~\frac{a \sinh\eta \cos\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
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<math>~y</math>
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<math>~=</math>
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<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
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<math>~z</math>
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<math>~=</math>
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<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>
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LaTeX mathematical expressions cut-and-pasted directly from
<br />
NIST's ''Digital Library of Mathematical Functions''
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As an additional point of reference, note that according to [http://dlmf.nist.gov/14.19 &sect;14.19 of NIST's ''Digital Library of Mathematical Functions''], the relationship between Cartesian and ''toroidal'' coordinates is given by the expressions,
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<math>~x</math>
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<math>~=</math>
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<math>~\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta},</math>
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<math>~y</math>
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<math>~=</math>
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<math>~\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta},</math>
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<math>~z</math>
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<math>~=</math>
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<math>~\frac{c\sin\theta}{\cosh\eta-\cos\theta}\, .</math>
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Mapping the other direction [see equations 2.13 - 2.15 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ], we have,
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<math>~\eta</math>
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<math>~=</math>
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<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
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<math>~\cos\theta</math>
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<math>~=</math>
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<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
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<math>~\tan\psi</math>
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<math>~=</math>
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<math>~\frac{y}{x} \, ,</math>
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where,
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<math>~r_1^2 </math>
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<math>~\equiv</math>
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<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + z^2 \, ,</math>
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<math>~r_2^2 </math>
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<math>~\equiv</math>
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<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + z^2 \, ,</math>
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and <math>~\theta</math> has the same sign as <math>~z</math>. 
 
<div align="center" id="WongMacComparison">
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  <th align="center">Relationship Between Wong's and MacMillan's Parameter Notation</th>
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A comparison between Wong's Figure 1 and, for example, MacMillan's Figure 61 reveals the following parameter notation relationships:
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<math>~\biggl[ \rho_2 \biggr]_\mathrm{MacMillan} \leftrightarrow \biggl[ r_1 \biggr]_\mathrm{Wong}</math>
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&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
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<math>~\biggl[ \rho_1 \biggr]_\mathrm{MacMillan} \leftrightarrow \biggl[ r_2 \biggr]_\mathrm{Wong}</math>
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Given that MacMillan's parameter, <math>~c = \rho_1/\rho_2</math>, it is clear that <math>~c</math> is related to Wong's "radial" coordinate, <math>~\eta</math> via the expression,
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<math>~\eta = \ln\biggl(\frac{r_1}{r_2}\biggr)_\mathrm{Wong}</math>
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<math>~=</math>
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<math>~\ln\biggl(\frac{\rho_2}{\rho_1}\biggr)_\mathrm{MacMillan} = - \ln(c) \, .</math>
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This also makes sense, in that,
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<math>~\cosh\eta</math>
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<math>~=</math>
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<math>~\frac{1}{2}\biggl[e^\eta + e^{-\eta}  \biggr]</math>
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<math>~=</math>
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<math>~\frac{1}{2}\biggl[\frac{1}{c} + c  \biggr]</math>
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<math>~=</math>
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<math>~\frac{1+c^2}{2c} = \frac{R}{d} \, .</math>
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[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 16 August 2017:  In equation (2.17) of his &sect;IIB &#8212; when Wong (1973) introduces the differential volume element &#8212; the variable used to represent the azimuthal coordinate angle switches from &psi; to &#x003A6;.  We will stick with the &psi; notation, here.]]<span id="volume">Drawing from equations (2.7), (2.17) and (2.18) of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], we see that the volume,</span> <math>~V</math>, of a torus that is bounded by surface <math>~\eta_s</math> is,
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<math>~\frac{V}{a^3} = \frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>
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<math>~=</math>
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<math>~\iiint\limits_{\eta_s} \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi~ = \frac{2\pi^2\cosh{\eta_s}}{\sinh^3\eta_s} \, .</math>
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If <math>~\eta_s \rightarrow \eta_0</math> then, in terms of the major and the minor radii of the torus, the volume is,
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<math>~V</math>
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<math>~=</math>
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<math>~2\pi^2 Rd^2 \, .</math>
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], Eq. (2.19)
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<span id="DensityFormula">
If such a torus has a uniform density</span>, <math>~\rho_0</math>, throughout, and a total charge (mass), <math>~q</math>, then the charge (mass) and density will be related through the toroidal-coordinate expression (see Wong's equation 2.51),
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<math>~\rho_0 = \frac{q}{V}</math>
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<math>~=</math>
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<math>~\frac{q\sinh^3\eta_0}{2\pi^2 a^3 \cosh{\eta_0}} \, .</math>
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Also, as [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] points out (see his equation 2.50), in this case the density ''distribution'' may be written as,
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<math>~\rho(\eta^', \theta^', \psi^')</math>
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<math>~=</math>
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<math>~\rho_0 \Theta(\upsilon) \, ,</math>
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where, the argument <math>~\upsilon \equiv (\eta - \eta_s)</math>, and <math>~\Theta(\upsilon)</math> is the step function defined by,
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<math>~\Theta(\upsilon)</math>
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<math>~=</math>
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<math>~0</math>
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  <td align="right">&nbsp; &nbsp;&nbsp; for &nbsp;<math>~\upsilon < 0 \, ,</math></td>
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<math>~\Theta(\upsilon)</math>
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<math>~=</math>
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<math>~1</math>
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  <td align="right">&nbsp; &nbsp;&nbsp; for &nbsp;<math>~\upsilon \ge 0 \, .</math></td>
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