Editing Apps/MaclaurinSpheroidSequence (section)

===n' = 0 Configurations===

{{ EH85 }} sought to find bifurcation points along the Maclaurin spheroid sequence where the associated, deformed equilibrium configuration has the same radial distribution of specific angular momentum &#8212; as a function of the integrated mass fraction &#8212; as does a uniformly rotating, uniform density sphere.  That is, inside the integral that defines the centrifugal potential, <math>\Psi</math>, they set,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>h(\varpi) = \varpi^2 \dot\varphi(\varpi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2M}\biggl\{ 1 - [1 - m(\varpi) ]^{2 / 3} \biggr\} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Stoeckly65 }}, &sect;II.c, eq. (12)<br />
{{ EH85 }}, &sect;2.1 (p. 290), Eq. (1)
  </td>
</tr>
</table>

where, the mass fraction,
<div align="center">
<math>m(\varpi) \equiv \frac{M_\varpi(\varpi)}{M} \, .</math>
</div>

From our example set of familiar [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profiles]], these might reasonably be referred to as [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_(n'_=_0)|<math>n' = 0</math> configurations]].  Instead, {{ EH85 }} label their deformed equilibrium configurations as follows:  "The Maclaurin spheroidal sequence bifurcates into a ''concave hamburger like'' configuration and reaches &#8212; as originally discovered and labeled by {{ MPT77 }} &#8212; the ''Maclaurin toroidal'' sequence."