Template:LSU CT99CommonTheme1B
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 above-highlighted 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 the meridional plane 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 expression that we have identified, above, as providing (Version 1 of) the Gravitational Potential of an Axisymmetric Mass Distribution. It is therefore clear that Deupree (1974) and, separately, Stahler (1983a) 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.
Note: It appears as though both Deupree (1974) and Stahler (1983a) only adopted this approach to evaluating the gravitational potential at locations outside of an axisymmetric mass distribution, whereas Cohl & Tohline (1999) have shown that the approach applies as well for locations inside the mass distribution.