Editing ProjectsUnderway/CoreCollapseSupernovae (section)

===Nod to Lynden-Bell's Early Contributions===
[[Image:LyndenBell1964.png|300px|right|Lynden-Bell (1964, ApJ, 139, 1195)]]I want to begin this section by paying tribute to [http://en.wikipedia.org/wiki/Donald_Lynden-Bell Donald Lynden-Bell] who, in 1962 ([http://dx.doi.org.libezp.lib.lsu.edu/10.1017/S0305004100040767 Mathematical Proceedings of the Cambridge Philosophical Society, vol. 58, pp. 709-711]), was the first to appreciate the relatively simple behavior that should be exhibited by the free-fall collapse of a uniformly rotating, uniform-density spheroid.  In an article less than two pages in length, Lynden-Bell first noted that the governing dynamical equations (written in cylindrical coordinates) take the form,
<div align="center">
<table border="0", cellpadding="5">

<tr>
  <td align="right">
<math>~\ddot{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ 2A_L(t) R + \frac{h_{LB}^2}{R^3} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\ddot{Z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ 2C_L(t) Z \, ,</math>
  </td>
</tr>

</table>
</div>
where the dots denote differentiation with respect to time, <math>~h_{LB}</math> is a constant and the two time-dependent coefficients, <math>~A(t)</math> and <math>~C(t)</math>, come from the gravitational potential that, according to Lyttleton (1953), has the form,
<div align="center">
<table border="0", cellpadding="5">

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ 2A_L(t) R^2 ~-~ C_L(t) Z^2 \, .</math>
  </td>
</tr>

</table>
</div>
Then Lynden-Bell deduced that, (a) "the result of the motion is merely a change of scales"; (b) the collapsing system "remains uniform [in density], and the boundary remains spheroidal"; and (c) "the collapse &hellip; will be through a series of [uniformly rotating] spheroids."  Two years later in a separate article, [http://adsabs.harvard.edu/abs/1964ApJ...139.1195L Lynden-Bell (1964], ApJ, 139, 1195) presented results from the numerical integration of this governing set of dynamical equations (see his Figure 1, reprinted to the right, here).  The various publications by other authors who also have modeled the free-fall collapse of rotating or nonrotating spheroids in various contexts (see our discussion that follows) have not always acknowledged Lynden-Bell's pioneering analysis of this problem.