User:Tohline/AxisymmetricConfigurations/SolvingPE

Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions

You have arrived at this page from our 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.

Synopses

The gravitational potential (both inside and outside) of any axisymmetric mass distribution may be determined from the following "Key" integral expression that we will refer to as the,

Gravitational Potential of an Axisymmetric Mass Distribution

	<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>	<math>~=</math>	<math>~ - \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(\varpi~ \varpi^')^{1 / 2}} \biggr] K(\mu) \rho(\varpi^', z^') 2\pi \varpi^'~ d\varpi^' dz^' </math>
<math>\mathrm{where:}~~~\mu \equiv \{4\varpi \varpi^' /[ (\varpi+\varpi^')^2 + (z-z^')^2]\}^{1 / 2}</math>

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 Cohl & Tohline (1999) and recognizing that, in cylindrical coordinates, 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, Bannikova et al. (2011), Trova, Huré & Hersant (2012), and Fukushima (2016).

Dyson-Wong Tori (Thin Ring Approximation) In §102 of a book titled, The Theory of the Potential, 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 additional "Key Equation" from MacMillan as providing an expression for the, Gravitational Potential in the Thin Ring (TR) Approximation <math>~\Phi_\mathrm{TR}(\varpi,z)</math> <math>~=</math> <math>~-\biggl[ \frac{2GM}{\pi } \biggr]\frac{K(k)}{\sqrt{(\varpi+a)^2 + z^2}}</math> <math>\mathrm{where:}~~~k \equiv \{4\varpi a/[ (\varpi+a)^2 + z^2]\}^{1 / 2}</math> See also, O. D. Kellogg (1929), §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.

Solving the Poisson Equation Deupree (1974) and, separately, 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 — with <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring element — 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: <math>~\delta\Phi_g(\varpi,z)</math> <math>~=</math> <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> Stahler's expression for each thin ring contribution is a generalization of the "Key Equation" expression for <math>~\Phi_\mathrm{TR}</math>: 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 easy to see that Stahler's expression for <math>~\delta \Phi_g</math> is identical to the integrand of the "Key" expression that we have identified, above, as providing the Gravitational Potential of an Axisymmetric Mass Distribution. It is therefore clear that Deupree and, separately, Stahler were developing robust algorithms to numerically evaluate the gravitational potential of systems with axisymmetric mass distributions well before Cohl & Tohline (1999) formally derived the corresponding "Key" integral expression.

Using Toroidal Coordinates to Determine the Gravitational Potential When written in toroidal coordinates, the "Key" integral expression for the Gravitational Potential of an Axisymmetric Mass Distribution becomes, <math>~\Phi(\eta,\theta)\biggr|_\mathrm{axisym}</math> <math>~=</math> <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^') \, , </math> Cohl & Tohline (1999), p. 88, Eqs. (31) & (32a) <math>\mathrm{where:}~~~\Chi \equiv [\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta)]/[\sinh\eta \cdot \sinh\eta^'] \, ,</math> 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, Referencing [MF53], Wong (1973) states that, in toroidal coordinates, the Green's function is, <math>~\Phi(\eta,\theta)\biggr|_\mathrm{axisym}</math> <math>~=</math> <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> Wong (1973), p. 293, Eq. (2.55) 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.

Wong's (1973) Analytic Potential Wong's (1973) Analytic Potential

Trova, Huré & Hersant (2012)

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