Editing Apps/RotatingPolytropes (section)

===Chandrasekhar's Approach===

Using the two components of the Euler equation to express spatial derivatives of the gravitational potential in terms of spatial derivatives of the gas pressure, that is, rewriting them in the form,

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

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{\partial \Phi}{\partial\mu}  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \frac{1}{\rho} ~\frac{\partial P}{\partial\mu} - \omega_0^2 r^2 \mu  
</math>
  </td>
</tr>
</table>

we can fold them into the Poisson equation to obtain a 2<sup>nd</sup>-order PDE that relates <math>~\rho(r, \theta)</math> to <math>~P(r, \theta)</math>, namely,

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

<tr>
  <td align="right">
<math>~4\pi G\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl\{ r^2 \biggl[-  \frac{1}{\rho} ~\frac{\partial P}{\partial r} +  \omega_0^2 r (1-\mu^2)\biggr]  \biggr\} 
+ \frac{1}{r^2} \frac{\partial }{\partial \mu}\biggl\{ (1-\mu^2) \biggl[ - \frac{1}{\rho} ~\frac{\partial P}{\partial\mu} - \omega_0^2 r^2 \mu  \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
+\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\omega_0^2 r^3 (1-\mu^2)\biggr]  
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \omega_0^2 r^2 \mu (1-\mu^2)   \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
+3\omega_0^2 (1-\mu^2) - \omega_0^2  (1-3\mu^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 2\omega_0^2 -4\pi G\rho</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (5)
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)] refers to this last expression as "the fundamental equation of the problem."  If, following Chandrasekhar's lead, we adopt a [[SR#Barotropic_Structure|polytropic equation of state]],

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

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K\rho^{1+1/n} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (6)
  </td>
</tr>
</table>
</div>

this "fundamental equation" can be rewritten strictly in terms of the configuration's axisymmetric density distribution, <math>~\rho(r,\theta)</math>.  Chandrasekhar first adopts a dimensionless function, <math>~\Theta(\xi, \mu)</math>, that is related to the normalized density via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Theta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_c}  \biggr)^{1/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ P = K \rho^{1+1/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c^{1+1/n} K \Theta^{n+1} \, .
</math>
  </td>
</tr>
</table>
(In doing this, Chandrasekhar is obviously adopting an a path toward solution that parallel's the familiar method used to determine the [[SSC/Structure/Polytropes#Polytropic_Spheres|structure of isolated, nonrotating polytropes]].)  Adopting the additional pair of dimensionless variables,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~r \biggl[ \frac{(n+1)K}{4\pi G} ~\rho_c^{1/n - 1} \biggr]^{-1 / 2} </math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  <td align="right">
<math>~v</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\omega_0^2}{2\pi G \rho_c} </math>
  </td>
</tr>
</table>

Chandrasekhar's "fundamental equation" becomes,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{1}{\xi^2} \frac{\partial }{\partial \xi} \biggl[\xi^2 ~\frac{\partial \Theta}{\partial \xi} \biggr]  
+ \frac{1}{\xi^2} \frac{\partial }{\partial \mu} \biggl[ (1-\mu^2) ~\frac{\partial \Theta}{\partial\mu}  \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v -\Theta^n \, .</math>
   </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (11)
  </td>
</tr>
</table>