Joel2: Created page with "Now, beginning with Version 2 of our expression for the ''Gravitational Potential of Axisymmetric Mass Distributions'', let's also map the (unprimed) cylindrical coordinate pair, $~(\varpi, z)$ , to the same (but, unprimed) toroidal coordinate system, $~(\eta,\theta)$ , and place the toroidal coordinate system's ''anchor ring'' in the equatorial plane of the cylindrical-coordinate system such that, $~(\varpi_a,z_a) = (a,0)$ . This gives, wh..."

2024-06-21T23:47:46Z

Created page with "Now, beginning with Version 2 of our expression for the ''Gravitational Potential of Axisymmetric Mass Distributions'', let's also map the (unprimed) cylindrical coordinate pair, <math>~(\varpi, z)</math>, to the same (but, unprimed) toroidal coordinate system, <math>~(\eta,\theta)</math>, and place the toroidal coordinate system's ''anchor ring'' in the equatorial plane of the cylindrical-coordinate system such that, <math>~(\varpi_a,z_a) = (a,0)</math>. This gives, wh..."

New page

Now, beginning with Version 2 of our expression for the ''Gravitational Potential of Axisymmetric Mass Distributions'', let's also map the (unprimed) cylindrical coordinate pair, <math>~(\varpi, z)</math>, to the same (but, unprimed) toroidal coordinate system, <math>~(\eta,\theta)</math>, and place the toroidal coordinate system's ''anchor ring'' in the equatorial plane of the cylindrical-coordinate system such that, <math>~(\varpi_a,z_a) = (a,0)</math>. This gives, what we will refer to as the,

<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Gravitational Potential of an Axisymmetric Mass Distribution (Version 3)'''</font>
</td>
</tr>
<tr>
<td align="right">
<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
- 2G a^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta} \biggr]^{1 / 2} \iint\limits_\mathrm{config}
\biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} \mu K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' \, ,</math>
</td>
</tr>
<tr>
<td align="center" colspan="3">
<math>\mathrm{where:}~~~\mu^2 = \frac{ 2 \sinh\eta^'\cdot \sinh\eta}{ \sinh\eta^'\cdot \sinh\eta + \cosh\eta^'\cdot\cosh\eta -\cos(\theta^' - \theta) } \, .</math>
</td>
</tr>
</table>

Template:LSU CT99CommonTheme3 - Revision history