Editing Apps/MaclaurinSpheroidSequence (section)

=Bifurcation Points Along Maclaurin-Spheroid Sequence=

==The Perturbed Configuration==

Referencing the [[AxisymmetricConfigurations/HSCF#Constructing_Two-Dimensional,_Axisymmetric_Structures|Hachisu Self-Consistent Field (HSCF) technique]], our objective is to solve an ''algebraic'' expression for hydrostatic balance,
<div align="center">
<math>~H + \Phi + \Psi = C_0</math> ,
</div>
in conjunction with the Poisson equation in a form that is appropriate for two-dimensional, axisymmetric systems &#8212; written in cylindrical coordinates, for example,
<div align="center">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho .
</math>
</div>
In both of these expressions, <math>\Phi</math> is the gravitational potential.  In the algebraic expression, <math>C_0</math> is a constant throughout the volume, and on the surface, of the equilibrium configuration.  Here, we seek a uniform-density (incompressible) configuration, in which [[SR#Time-Independent_Problems|the enthalpy]], <math>H = P/\rho</math>, goes to zero at all points across the surface.  And the centrifugal potential, <math>\Psi</math>, [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|is given by the expression]]

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right" width="40%"><math>\Psi</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>- ~\int \frac{h^2(\varpi)}{\varpi^3} d\varpi \, ,</math></td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ OM68 }}, &sect;IIId (p. 1084), eq. (44)<br />
{{ MPT77 }}, &sect;III (p. 590), eq. (3.4)
</td>
</tr>
</table>
where, the (cylindrical) radial distribution of the specific angular momentum,
<div align="center">
<math>h(\varpi) = \varpi^2 \dot\varphi(\varpi) \, ,</math>
</div>
is to be specified according to the physical problem in hand &#8212; usually chosen from a familiar set of "[[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profiles]]."  Therefore, across the surface of each equilibrium configuration, the algebraic expression for hydrostatic balance takes the form,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>\Phi - \int \frac{h^2(\varpi)}{\varpi^3} d\varpi</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>C_0 \, .</math></td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ EH85 }}, &sect;2.1 (p. 290), Eq. (5)
  </td>
</tr>
</table>

===Uniform Rotation===

{{ HE83 }} sought to find bifurcation points along the Maclaurin spheroid sequence where the associated, deformed equilibrium configuration is uniformly rotating.  From the set of familiar [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profiles]], therefore, they set
<div align="center">
<math>
\Psi = - \frac{1}{2} \varpi^2 \omega_0^2 \, ,
</math>
</div>
in which case, in their investigation, the condition (along the surface) for hydrostatic balance is,

<div align="center">
<math>\Phi - \frac{1}{2} \varpi^2 \omega_0^2 = C_0</math> .
</div>
Replacing <math>\varpi^2</math> with its equivalent expression in terms of oblate spheroidal coordinates gives,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>- ~\Phi + \frac{\omega_0^2}{2} a_0^2 (1 + \xi^2)(1 - \eta^2) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- ~C_0 \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ HE83 }}, &sect;A.1 (p. 587), Eq. (6)
  </td>
</tr>
</table>
which is the same as their Appendix (&sect;A.1) Eq. (6), except they chose a different sign when defining the constant, <math>C_0</math>.


===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."

==Particularly Interesting Models Along the Maclaurin Spheroid Sequence==

<table border="1" cellpadding="8" align="center" width="60%"><tr><td align="center" bgcolor="lightblue">
Go to our associated discussion of [[Appendix/Ramblings/MacSphCriticalPoints|Critical Points along the Maclaurin Spheroid Sequence]].
</td></tr></table>