Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions
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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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This expression can be obtained, for example, by straightforwardly combining Eqs. (31), (32b), and (24) from Cohl & Tohline (1999) and recognizing that the differential meridional-plane area, <math>~d\sigma^' = \varpi^' d\varpi^' dz^' \int_0^{2\pi} d\varphi = 2\pi\varpi^'~d\varpi^' dz^'</math>.
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Dyson-Wong Tori
(Thin Ring Approximation)
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Solving the Poisson Equation 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.
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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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