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=Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions=

You have arrived at this page from our [[User:Tohline/H_BookTiledMenu#Axisymmetric_Equilibrium_Structures|''Tiled Menu'']] by clicking on the chapter title that is also referenced in the panel of the following table that is colored light blue.  You may proceed directly to that chapter by clicking (again) on the same chapter title, as it appears in the table.  However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected.  The common thread is the "Key Equation" presented in the top panel of the table.

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==Synopses==

The gravitational potential (both inside and outside) of any axisymmetric mass distribution may be determined from the integral expression,
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{{ User:Tohline/Math/EQ_CT99Axisymmetric }}
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where, <math>~K(\mu)</math> is the complete elliptic integral of the first kind.  This "Key Equation" may be straightforwardly obtained, for example, by combining Eqs. (31), (32b), and (24) from [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)] and recognizing that the relevant differential area, <math>~d\sigma^' = \varpi^' d\varpi^' dz^' \int_0^{2\pi} d\varphi = 2\pi\varpi^'~d\varpi^' dz^'</math>; see also, [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)], [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)], and [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)].


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! style="height: 150px; width: 150px; background-color:whiteF;" |[[User:Tohline/Apps/DysonWongTori#Thin_Ring_Approximation|Dyson-Wong<br /> Tori]]<br />(Thin Ring Approximation)
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In &sect;102 of a book titled, [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 ''The Theory of the Potential'', W. D. MacMillan (1958; originally, 1930)] derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>.  Throughout our related discussions, we generally will refer to this "Key Equation" from MacMillan as providing an expression for the,
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<tr><td align="center" colspan="1"><font color="#770000">'''Gravitational Potential in the Thin Ring (TR) Approximation'''</font></td></tr>
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{{ User:Tohline/Math/EQ TRApproximation }}
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See also, [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)], &sect;III.4, Exercise (4).  As is reviewed in the chapter of our H_Book titled, ''Dyson-Wong Tori,'' a number of research groups over the years have re-derived this "thin ring" approximation in the context of their search for effective and insightful ways to determine the gravitational potential of axisymmetric systems.
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! style="height: 150px; width: 150px; background-color:#D0FFFF;" |[[User:Tohline/AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically|Solving the<br /> Poisson Equation]]
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[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] have argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' &#8212; with <math>~\delta M(\varpi^', z^')</math>  being the appropriate differential mass contributed by each ring element &#8212; that are positioned at various meridional coordinate locations throughout the mass distribution.  According to Stahler's derivation, for example (see his equation 11 and the explanatory text that follows it), the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is:
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<math>~\delta\Phi_g(\varpi,z)</math>
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<math>~=</math>
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<math>~
- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}
\times K\biggl\{ \biggl[ \frac{4\varpi^' \varpi}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .
</math>
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It is easy to see that Stahler's expression for each ''thin ring'' contribution is a generalization of the "Key Equation" expression for <math>~\Phi_\mathrm{TR}</math> given above:  The "TR" expression assumes that the ring cuts through the  meridional plane at <math>~(\varpi^', z^') = (a, 0)</math>, while Stahler's expression works for individual rings that cut through at any coordinate location.  Given that, in cylindrical coordinates, the differential mass element is, <math>~\delta M = 2\pi \rho(\varpi^', z^') \varpi^' d\varpi^' dz</math>, it is at the same time easy to see that Stahler's expression for <math>~\delta \Phi_g</math> is identical to the integrand of the "Key Equation" that appears in the top panel of this synopsis table. It is therefore clear that Deupree and, separately, Stahler were developing algorithms to numerically evaluate the gravitational potential of systems with axisymmetric mass distributions that are identical to the &hellip; well before Cohl &amp; Tohline (1999) derived the generalized "Key Equation" highlighted above.
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! style="height: 150px; width: 150px; background-color:whiteF;" |[[User:Tohline/2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|Using<br /> Toroidal Coordinates<br /> to Determine the<br /> Gravitational<br /> Potential]]
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When written in toroidal coordinates, the "Key Equation" that gives the gravitational potential of systems that have an axisymmetric mass distribution becomes,

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<math>~\Phi(\eta,\theta)\biggr|_\mathrm{axisym}</math>
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<math>~=</math>
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<math>~ - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} 
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi) \rho(\eta^',\theta^') \, ,
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[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999)], p. 88, Eqs. (31) &amp; (32a)
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<math>\mathrm{where:}~~~\Chi \equiv [\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta)]/[\sinh\eta \cdot \sinh\eta^'] \, ,</math>
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and, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are ''Associated Legendre Functions'' of the first and second kind with degree <math>~n - \tfrac{1}{2}</math> and order <math>~m</math> (toroidal harmonics).  Taking a very different approach, Wong (1973) has determined that an alternate form of this expression is,

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<math>~\Phi(\eta,\theta)\biggr|_\mathrm{axisym}</math>
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<math>~=</math>
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<math>~
- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n 
\int d\eta^' ~\sinh\eta^'~P^0_{n-1 / 2}(\cosh\eta_<) ~Q^0_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl\{ \frac{ ~\cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\}
\rho(\eta^',\theta^') \, .
</math>
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[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], p. 293, Eq. (2.55)
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In this chapter of our H_Book, we demonstrate in detail how one of these integral expressions can be transformed into the other, thereby proving that they provide identical results.
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! style="height: 150px; width: 150px; background-color:whiteF;" |[[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|Wong's <br />(1973) <br >Analytic Potential]]
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[[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|Wong's (1973) Analytic Potential]]
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[[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|Trova, Hur&eacute; &amp; Hersant (2012)]]
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=Related Discussions=


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