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
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The gravitational potential (both inside and outside) of any axisymmetric mass distribution may be determined from the integral expression,
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<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>
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<math>~=</math>
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<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>
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<math>\mathrm{where:}~~~\mu \equiv \{4\varpi \varpi^' /[ (\varpi+\varpi^')^2 + (z-z^')^2]\}^{1 / 2}</math>
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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 Cohl & 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, Bannikova et al. (2011), Trova, Huré & Hersant (2012), and Fukushima (2016).
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In §102 of a book titled, 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,
| Gravitational Potential in the Thin Ring (TR) Approximation |
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<math>~\Phi_\mathrm{TR}(\varpi,z)</math>
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<math>~=</math>
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<math>~-\biggl[ \frac{2GM}{\pi } \biggr]\frac{K(k)}{\sqrt{(\varpi+a)^2 + z^2}}</math>
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<math>\mathrm{where:}~~~k \equiv \{4\varpi a/[ (\varpi+a)^2 + z^2]\}^{1 / 2}</math>
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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.
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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:
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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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Stahler's expression for each thin ring contribution is clearly the same as our "key equation" expression for <math>~\Phi_\mathrm{TR}</math> if the individual ring being considered cuts through the meridional plane at <math>~(\varpi^', z^') = (a, 0)</math>.
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Using Toroidal Coordinates to Determine the Gravitational Potential
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Wong's (1973) Analytic Potential
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Trova, Huré & Hersant (2012)
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