Joel2: Created page with "[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S 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 $~\delta M(\varpi^', z^')$ being the appropriate differential mass contributed by each ring e..."

2024-06-21T23:44:29Z

Created page with "[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S 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 e..."

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[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S 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:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\delta\Phi_g(\varpi,z)</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<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>
</td>
</tr>
</table>
</div>
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,
<div align="center">
<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>,
</div>
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 [[#Part_I|identified, above]], as providing (Version 1 of) the ''Gravitational Potential of an Axisymmetric Mass Distribution.'' It is therefore clear that <font color="orange">[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] were developing robust algorithms to numerically evaluate the gravitational potential of systems with axisymmetric mass distributions well before [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)] formally derived the corresponding Key integral expression</font>.

Note: It appears as though both [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] only adopted this approach to evaluating the gravitational potential at locations ''outside'' of an axisymmetric mass distribution, whereas [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)] have shown that the approach applies as well for locations ''inside'' the mass distribution.

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