Jet53man: Created page with "Referencing (equivalently, Version 1 of) the above-identified integral expression for the ''Gravitational Potential of an Axisymmetric Mass Distribution,'' [http:/..."

2021-09-10T17:42:51Z

Created page with "Referencing (equivalently, Version 1 of) the above-identified integral expression for the ''Gravitational Potential of an Axisymmetric Mass Distribution,'' [http:/..."

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Referencing (equivalently, Version 1 of) the [[#Part_1|above-identified integral expression]] for the ''Gravitational Potential of an Axisymmetric Mass Distribution,'' [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)] offer the following assessment in §6 of their paper:
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"The important question we have tried to clarify concerns the possibility of converting the remaining double integral … into a line integral … this question remains open."</font>
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We also have wondered whether there is a possibility of converting the double integral in this Key Equation into a single (line) integral. This is a particularly challenging task when, as is the case with ''Version 1'' of the expression, the integrand 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 realized that if we focus, instead, on ''Version 2'' of the expression and associate the meridional-plane coordinates of the ''anchor ring'' with the coordinates of the location where the potential is ''being evaluated'' — that is, if we set <math>~(\varpi_a, z_a) = (\varpi,z)</math> — the argument of the elliptic integral becomes,
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<math>~\mu = \biggl[\frac{2}{1+\coth\eta^' }\biggr]^{1 / 2} \, ,</math>
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while the integral expression for the gravitational potential becomes,
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<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>
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<math>~=</math>
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<math>~
- 2^{3/2}G \varpi^2 \iint\limits_\mathrm{config}
\biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} \biggl[\frac{1}{1+\coth\eta^' }\biggr]^{1 / 2} K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' </math>
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<math>~=</math>
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<math>~- 2^{3 / 2} G \varpi^{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>
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Notice that, by adopting this strategy, the argument of the elliptical integral is a function only of one coordinate — the toroidal coordinate system's ''radial'' coordinate, <math>~\eta^'</math>. As result, the integral over the ''angular'' coordinate, <math>~\theta^'</math>, does not involve the elliptic integral function. Then — as is shown in an accompanying chapter titled, [[2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|''Attempt at Simplification'']] — if the configuration's density is constant, the integral over the angular coordinate variable can be completed analytically. Hence, <font color="orange">the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced to a problem of carrying out a single, line integration. This provides an answer to the question posed by</font> [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)].

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Jet53man: Created page with "Referencing (equivalently, Version 1 of) the above-identified integral expression for the ''Gravitational Potential of an Axisymmetric Mass Distribution,'' [http:/..."