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.

Chapter Synopses

Part I

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 (Version 1)

CT99 v1

	<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>

and, <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); 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, §III.4, Exercise (4) in Kellogg (1929). 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>~ - \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 = \rho(\varpi^', z^') \varpi^' d\varpi^' dz^' \int_0^{2\pi}d\varphi = 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.

Part II

Suppose that we rewrite the above integral expression (CT99.v1) such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates <math>~(\varpi^', z^')</math> to a toroidal-coordinate system <math>~(\eta^',\theta^')</math> whose anchor ring cuts through the meridional plane at the cylindrical-coordinate location, <math>~(\varpi_a,z_a)</math>. The desired mapping is handled via the pair of relations,

<math>~\varpi^' = \frac{\varpi_a \sinh\eta}{(\cosh\eta - \cos\theta)} \, ,</math>

and

<math>~(z^' - z_a) = \frac{\varpi_a \sin\theta}{(\cosh\eta - \cos\theta)} \, ,</math>

and the corresponding expression for each differential mass element is,

<math>~\delta M(\eta^',\theta^') = \biggl[\frac{2\pi \varpi_a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \rho(\varpi^', z^') d\eta^' d\theta^'</math>.

	<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>

Attempt at Simplification

Referencing the above-identified integral expression for the Gravitational Potential of an Axisymmetric Mass Distribution, Trova, Huré & Hersant (2012) offer the following assessment in §6 of their paper:

"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."

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 written above, 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, instead of using the cylindrical-coordinate pair <math>~(\varpi^',z^')</math> to identify the meridional-plane location of each mass element, we switch to a toroidal coordinate system <math>~(\eta,\theta)</math> that is defined such that its anchor ring cuts through the meridional plane at the coordinate location, <math>~(\varpi,z)</math> — that is, at the meridional-plane location where the potential is being evaluated — then the appropriate transformation of the "primed" coordinates is,

<math>~\varpi^' = \frac{\varpi \sinh\eta}{(\cosh\eta - \cos\theta)} \, ,</math>

and

<math>~(z^' - z) = \frac{\varpi \sin\theta}{(\cosh\eta - \cos\theta)} \, ,</math>

and the expression for the modulus of the elliptic integral becomes, <math>~\mu^2 = 2[1+1/\tanh\eta]^{-1}</math>. In doing this, the modulus becomes a function of only one coordinate — the "radial" coordinate, <math>~\eta</math> — and the integral expression for the gravitational potential becomes,

<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>

<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>

If the configuration's density is constant, then — as is shown in an accompanying chapter titled, Attempt at Simplification — the integral over the angular coordinate variable, <math>~\theta</math>, can be completed analytically. Hence, 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.

Part III

Using Toroidal Coordinates to Determine the Gravitational Potential

Referencing [MF53], Wong (1973) developed an integral expression for the Coulomb (equivalently, gravitational) potential whose foundation is the single summation, Green's function expression for the reciprocal distance between two points in toroidal coordinates. For axisymmetric mass distributions, Wong obtained,

<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_>) </math>
		<math>~ \times \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)

where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are Associated Legendre Functions of the first and second kind of order, <math>~m</math>, and half-integer degree, <math>~n - 1/2</math> (toroidal functions). Alternatively, if our "Key" integral expression for the Gravitational Potential of an Axisymmetric Mass Distribution is directly mapped from cylindrical to toroidal coordinates, we obtain,

<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>

These appear to be very different expressions. In the accompanying chapter titled, Using Toroidal Coordinates to Determine the Gravitational Potential, we detail step-by-step how the first of these expressions can be transformed into the second, thereby proving that they are indeed perfectly equivalent integral expressions for the gravitational potential of axisymmetric systems.

Wong's (1973) Analytic Potential

Wong's (1973) Analytic Potential

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