Editing Apps/Wong1973Potential (section)

====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>.


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<tr><th align="center">Figure 1</th></tr>
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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
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[[File:Wong1973Fig6.png|350px|To be inserted:  Fig. 6 from Wong (1973)]]
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[[File:Wong1973Fig7.png|400px|To be inserted:  Fig. 7 from Wong (1973)]]
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&nbsp; &nbsp; &nbsp;[[File:Wong73OurFig6b.png|300px|Our line plot to compare with Wong's Figure 6]]
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[[File:Wong111N4.png|300px|Our composite to compare with Wong's Figure 7]]&nbsp; &nbsp; &nbsp; &nbsp;
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'''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.
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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, 
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<math>D_0 \equiv \frac{2^{3/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] \, ,</math>
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and the pair of parameters,
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<math>B_n(\cosh\eta_0)</math>
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<math>\equiv</math>
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<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>
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<math>C_n(\cosh\eta_0)</math>
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<math>\equiv</math>
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<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>
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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,
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<math>\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)</math>
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<math>=</math>
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<math>
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) A_n(\cosh\eta) \, ,
</math>
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where,
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<math>A_n(\cosh\eta)\biggr|_\mathrm{interior}</math>
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<math>=</math>
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<math>B_n(\cosh\eta_0)Q_{n-\frac{1}{2}}(\cosh\eta) - Q^2_{n-\frac{1}{2}}(\cosh\eta)</math>
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<tr><td colspan="3" align="center">and</td></tr>

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<math>A_n(\cosh\eta)\biggr|_\mathrm{exterior}</math>
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<math>=</math>
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<math>C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, .</math>
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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>.  

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  <th align="center" colspan="2">Figure 2:<br />3D Animated Depictions of the Warped Surface of<br />Wong's Toroidal Potential
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  <td align="center" bgcolor="#D0FFFF">[[File:MovieWongComposite.gif|center|300px|3D Depiction of Wong's Toroidal Potential Well]]
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  <td align="center" bgcolor="#D0FFFF">[[File:Wong73VaryingRoverd.gif|center|300px|3D Depiction of Wong's Toroidal Potential Well]]
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'''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>. 
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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>