Editing Apps/RotatingPolytropes (section)

===Choices and Adaptations===

Among the set of [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|simple rotation profiles]] that have been adopted by various research groups over the years, [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)] and [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)] all choose what would generally be considered the simplest, which is the assumption of uniform rotation <math>~(\dot\varphi = \omega_0)</math>.  This means that,

<div align="center">
<math>~j^2 = r^4 \sin^4\theta \omega_0^2 \, .</math>
</div>

And in place of the co-latitude, <math>~\theta</math>, they all adopt the coordinate,
<div align="center">
<math>~\mu \equiv \cos\theta ~~~\Rightarrow ~~~ \frac{\partial}{\partial\mu} = - \frac{\partial}{\sin\theta \partial\theta} \, .</math>
</div>
As a result, the set of three remaining scalar governing equations becomes,

<table border="1" cellpadding="10" align="center" width="40%"><tr><td align="center">
<font color="#770000">'''Poisson Equation'''</font><br />

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

<tr>
  <td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2} \frac{\partial }{\partial \mu}\biggl[ (1-\mu^2) \frac{\partial \Phi}{\partial\mu}\biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (4')<br />
[https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)], p. 553, Eq. (2.1)
  </td>
</tr>
</table>

<font color="#770000">'''The Two Relevant Components of the Euler Equation'''</font>
<br />

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\rho \omega_0^2 r (1-\mu^2)
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
+ \biggl[ \frac{\partial P}{\partial r} +  \rho \frac{\partial \Phi }{\partial r} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\rho \omega_0^2 r^2 \mu
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{\partial P}{\partial\mu} +  \rho\frac{\partial \Phi}{\partial\mu}  \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="4">
[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], top of p. 126<br />
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (3')<br />
[https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)], p. 553, Eqs. (2.2) &amp; (2.3)
  </td>
</tr>
<tr>
  <td align="left" colspan="4">
NOTES:  
* In place of <math>~P</math>, Milne uses <math>~W</math>;
* For the gravitational potential, Milne and Chandrasekhar both adopt the convention, <math>~V \equiv -\Phi </math>, while James adopts the notation, <math>~\Psi \equiv - \Phi</math>;
* James includes terms with azimuthal derivatives in his equation set; these terms are set to zero (as reflected here) when seeking axisymmetric structures.
  </td>
</tr>
</table>
</td></tr></table>