Editing ProjectsUnderway/CoreCollapseSupernovae (section)

===Nonrotating, Oblate-Spheroidal Collapse===

The analogous collapse from rest of a nonrotating, uniform-density, oblate spheroid with equatorial radius, <math>~\varpi_\mathrm{eq}(t)</math>, and polar radius, <math>~Z_p(t)</math>, is governed by the equations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{\partial\Phi}{\partial \varpi} \biggr|_{\varpi_{eq}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d^2 Z_\mathrm{p}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\frac{\partial\Phi}{\partial Z} \biggr|_{Z_{p}} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, to within an additive constant,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\varpi,Z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G \rho [ A_1(e) \varpi^2 + A_3(e) Z^2] \, .
</math>
  </td>
</tr>
</table>
</div>

We should clarify and emphasize that this expression for the time-dependent gravitational potential has been written in terms of the (time-varying) eccentricity of the spheroid, <math>~e</math>, as measured in the meridional plane.  Specifically, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl( 1 - \frac{Z_p^2}{\varpi_\mathrm{eq}^2} \biggr)^{1/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~~ Z_p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\varpi_\mathrm{eq} \biggl( 1 - e^2 \biggr)^{1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
and, as is derived in our accompanying discussion of the [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Oblate_Spheroids_.28a1_.3D_a2_.3E_a3.29|properties of homogeneous ellipsoids]],

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_1(e)
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{1}{e^2} \biggl[  \frac{\sin^{-1}e}{e} - (1-e^2)^{1/2} \biggr] (1-e^2)^{1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_3(e)
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{2}{e^2} \biggl[  (1-e^2)^{-1/2} - \frac{\sin^{-1}e}{e} \biggr] (1-e^2)^{1/2} \, .
</math>
  </td>
</tr>

</table>
Hence,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~\nabla\Phi
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~2\pi G \rho \biggl[ \hat{e}_\varpi A_1(e) + \hat{e}_Z A_3(e) Z \biggr] \, ,
</math>
  </td>
</tr>

</table>
and the pair of governing dynamical equations become,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\pi G \rho A_1(e) \varpi_\mathrm{eq} = - \frac{3}{2} \biggl[ \frac{GM}{\varpi_\mathrm{eq} Z_\mathrm{p}} \biggr] A_1(e) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d^2 Z_\mathrm{p}}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\pi G \rho A_3(e) Z_\mathrm{p} = - \frac{3}{2} \biggl[ \frac{GM}{\varpi_\mathrm{eq}^2} \biggr] A_3(e) \, ,
</math>
  </td>
</tr>
</table>
</div>
where we have used the relation that is valid for uniform-density, oblate spheoids,
<div align="center">
<math>
~\rho = \frac{3M}{4\pi \varpi_\mathrm{eq}^2 Z_\mathrm{p}} \, .
</math>
</div>