Editing AxisymmetricConfigurations/PoissonEq (section)

===Deupree (1974)===

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<td align="center" bgcolor="red">&nbsp; 2D &nbsp;</td>
<td align="center" bgcolor="lightgreen">&nbsp; Sph &nbsp;</td>
<td align="center" bgcolor="yellow">&nbsp; Tor &nbsp;</td>
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"The fully nonlinear, nonradial, nonadiabatic calculation of stellar oscillations has not as yet been attempted by anyone, to the best of the author's knowledge.  Only Deupree (1974b, 1975) so far seems to have taken some steps in this direction.  He has carried out numerical calculations of the nonlinear axisymmetric oscillations in the adiabatic ([http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree 1974b]) and nonadiabatic ([http://adsabs.harvard.edu/abs/1975ApJ...198..419D Deupree 1975]) approximations, with apparently encouraging results."</font>
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&#8212; Drawn from &sect;3.5 of the review article by [http://adsabs.harvard.edu/abs/1976ARA%26A..14..247C J. P. Cox (1976, ARAA, 14, 247 - 273)] 
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Apparently, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D R. G. Deupree (1974, ApJ, 194, 393 - 402)] was the first astronomer to employ self-gravitating, numerical hydrodynamic techniques to model ''nonlinear, nonradial stellar pulsations.''  He chose to carry out his simulations on a spherical coordinate mesh with his earliest simulations being restricted to the examination of 2D (axisymmetric) configurations.  The outer boundary of his computational grid was identified as (see his &sect;IIb) <font color="darkgreen">a spherical surface completely exterior to the star.</font>

====His Derived Expression for the Boundary Potential====
In designing an algorithm to solve the Poisson equation, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] considered <font color="darkgreen">the major difficulty [to be] the evaluation of the potential boundary conditions.</font>  He decided to determine the boundary potential via an evaluation of the ''integral representation'' of the Poisson equation; <font color="darkgreen">once the boundary conditions [were] specified, the potential inside the star [was] evaluated by [employing] the [http://adsabs.harvard.edu/abs/1964ApJ...139..306H Henyey method]</font> to solve the ''differential representation'' of the Poisson equation. The assumption of azimuthal symmetry meant that, for example, the density was specified at various meridional-plane locations, <math>~\rho(r,\theta)</math>, with <font color="darkgreen">each zone [being] considered as a uniform density circular ring.</font>  He argued that <font color="darkgreen">the potential of [each such] ring at any point on the spherical boundary [could] easily be evaluated by</font> treating it as an infinitesimally thin ring of mass, <math>~\delta M = 2\pi a \rho ~\delta A</math> &#8212; where, <math>~\delta A</math> <font color="darkgreen">is the cross-sectional area of the [grid] zone, and <math>~a</math> is the radius of [that] ring</font> &#8212; then drawing upon the analytic analysis described by [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)] or, equivalently, by [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; originally 1930)].  Then, <font color="darkgreen">the total potential at the boundary is the sum of the potentials from all of the [meridional-plane] zones.</font>

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Note that, over the years, a number of other research groups have adopted this same approach to evaluate the gravitational potential of axisymmetric mass distributions &#8212; that is, summing up the potential contributions due to an ensemble of "infinitesimally thin rings."  But none appear to have recognized Deupree's earlier, pioneering investigation.  See, for example, the brief reviews that we have written regarding the related investigations by:  [[#Stahler_.281983.29|Stahler (1983)]], [[Apps/DysonWongTori#Bannikova_et_al._.282011.29|Bannikova et al. (2011)]], and [[Apps/DysonWongTori#Fukushima_.282016.29|Fukushima (2016)]].
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More specifically, equation (16) from [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] states that the differential contribution to the boundary potential due to each infinitesimally thin ring is,
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<math>~\delta\Phi_B</math>
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<math>~=</math>
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<math>~
- \frac{2G(2\pi a\rho ~\delta A)}{\pi p} \int_0^{\pi/2} \frac{d\psi}{[\cos^2\psi + (q/p)^2\sin^2\psi]^{1 / 2}} \, ,
</math>
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where, <math>~q</math> is the shortest distance between the ring and the boundary point, and <math>~p</math> is the longest distance.   (Note that, in order to match our conventions, we have inserted the negative sign along with a leading factor of <math>~G</math>.)  Referencing our accompanying [[Apps/DysonWongTori#DeupreeReference|detailed analysis of the potential due to a thin ring]], and adopting the variable mappings, <math>~p \leftrightarrow \rho_1</math> and <math>~q \leftrightarrow \rho_2</math>, we see that Deupree's expression is indeed identical to the expression for the potential derived by [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)].
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<math>~\Phi</math>
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<math>~=</math>
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<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>
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Recognizing, [[Apps/DysonWongTori#RingPotential|as did MacMillan]], that the definite integral in this expression is related to the complete elliptic integral of the first kind, and introducing the ratio of lengths, <math>~c \equiv p/q</math>, Deupree's expression for the (differential contribution to the) potential can be rewritten as,

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<math>~\delta\Phi_B</math>
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<math>~=</math>
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<math>~-  \frac{2G (2\pi a\rho ~\delta A) c}{\pi p} \int_0^{\pi/2} \frac{d\psi}{ \sqrt{1 - k^2 \sin^2\psi }} </math>
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&nbsp;
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<math>~=</math>
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<math>~-  \biggl[ \frac{2G (2\pi a\rho ~\delta A) c}{\pi p} \biggr] K(k)  \, ,</math>
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where, <math>~K(k)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind] for the modulus, <math>~k = \sqrt{1-c^2}</math>.

====Comparison With Other Related Derivations====
Now, given that Deupree chose to construct and evolve his models using a spherical coordinate system, he would have specified the relevant lengths, <math>~p</math> and <math>~q</math>, and each ring's differential cross-section, <math>~\delta A</math>, in terms of spherical coordinates.  In an effort to more clearly illustrate the connection between Deupree's expression for the (differential contribution to the) boundary potential and the expression for the boundary potential that we have [[#For_Axisymmetric_Systems|derived above for axisymmetric systems]], we will insert expressions for these terms that apply, instead, to a cylindrical-coordinate mesh.

Following the same line of reasoning as has been presented in our [[Apps/DysonWongTori#CylindricalLocation|accompanying discussion of MacMillan's work]], if the meridional-plane locations of the infinitesimally thin ring and the desired point on the boundary are, respectively, <math>~(\varpi^',z^')</math> and <math>~(\varpi,z)</math>, we see that,
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<math>~k = \sqrt{1 - c^2}</math>
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<math>~=</math>
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<math>~ \biggl[\frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2 }\biggr]^{1 / 2} \, ,</math>
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and,
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<math>~\frac{c}{p}</math>
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<math>~=</math>
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<math>~\biggl[(\varpi + \varpi^')^2 + (z - z^')^2  \biggr]^{- 1 / 2} = \frac{k}{\sqrt{4\varpi \varpi^'}} \, .</math>
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Hence &#8212; after acknowledging that, in cylindrical coordinates, the radius of each "infinitesimally thin ring" is, <math>~a = \varpi^'</math>, and the differential cross-section of each ring is, <math>~\delta A = \delta\varpi^' \delta z^'</math> &#8212; Deupree's expression for the (differential contribution to the) potential may be rewritten as,

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<math>~\delta\Phi_B(\varpi,z)</math>
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<math>~=</math>
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<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr]  K(k) </math>
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&nbsp;
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<math>~=</math>
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<math>~- \frac{2G (\delta M) }{\pi } \cdot \frac{K(k) }{ \sqrt{(\varpi + a)^2 + (z - z^')^2 }}  \, ,  </math>
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or it may be rewritten as,

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<math>~\delta\Phi_B(\varpi,z)</math>
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<math>~=</math>
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<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr]  K(k) </math>
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&nbsp;
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<math>~=</math>
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<math>~- \frac{2G }{\sqrt{\varpi}} \biggl[ \delta\varpi^' \delta z^' \rho(\varpi^', z^') \sqrt{\varpi^'} k K(k) \biggr] \, .</math>
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Notice that the first of these two rewritten expressions aligns perfectly with our "[[Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
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{{ Math/EQ TRApproximation }}
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<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png|225px|link=Apps/DysonWongTori#ThinRingContours]]</td>
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(See our [[Apps/DysonWongTori#ThinRingContours|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.)  Next, referring back to the expression that was [[#For_Axisymmetric_Systems|derived above for axisymmetric systems from a toroidal-function-based Green's function]], namely,

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<math>~ \Phi_B(\varpi,z)\biggr|_{2D}</math>
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<math>~=</math>
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<math>~ 
-\frac{2G}{\sqrt{\varpi}} \int\limits_{\varpi^'} \int\limits_{z^'} d\varpi^' dz^' \rho(\varpi^',z^') \sqrt{\varpi^'}  
\mu K(\mu) \, , 
</math>
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[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl &amp; J. E. Tohline (1999)], p. 89, Eqs. (31) &amp; (32)
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where,
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<math>~\mu </math>
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<math>~=</math>
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<math>~
\biggl[ \frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
\, ,
</math>
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we see that the second of these rewritten expressions for Deupree's <math>~\delta\Phi_B</math> aligns perfectly with our derived expression for the differential contribution to the potential of any axisymmetric mass distribution.  It is therefore fair to say that the expression that Deupree used to determine the gravitational potential along the boundary of his modeled configurations is derivable from a 3D Green's function that is written in terms of ''toroidal functions''.