Editing Apps/RotatingPolytropes (section)

===Solutions===

In determining the equilibrium configuration's axisymmetric density distribution, <math>~\rho(r,\theta)</math>, Chandrasekhar 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>

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  <td align="right">
<math>~\Rightarrow ~~~ P = K \rho^{1+1/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<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]].)  He then argues that, for slowly rotating configurations, the function, <math>~\Theta(\xi,\mu)</math> can be effectively expressed as a small perturbation (in the two-dimensional meridional plane), <math>~v\Psi(\xi,\mu) \ll 1</math>, added to the radially dependent "Emden's function," <math>~\theta(\xi)</math>, that defines the structure of non-rotating polytropic configurations; that is,
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<tr>
  <td align="right">
<math>~\Theta(\xi,\mu)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\theta(\xi) + v \Psi(\xi,\mu) + \mathcal{O}(v^2) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (13)
  </td>
</tr>
</table>

where,
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  <td align="right">
<math>~v</math>
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<math>~\equiv</math>
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<math>~\frac{\omega_0^2}{2\pi G\rho_c} \ll 1 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (10)
  </td>
</tr>
</table>

He deduces that, to lowest order in a [https://en.wikipedia.org/wiki/Legendre_polynomials#Rodrigues'_formula_and_other_explicit_formulas Legendre polynomial series],
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  <td align="right">
<math>~\Psi</math>
  </td>
  <td align="center">
<math>~=</math>
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<math>~
\psi_0(\xi) - \frac{5}{6} ~ \frac{\xi_1^2}{3\psi_2(\xi_1) + \xi_1 \psi_2^'(\xi_1)} \psi_2(\xi) P_2(\mu) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eq. (36)
  </td>
</tr>
</table>
where:  <math>~P_2(\mu) = \tfrac{1}{2}(3\mu^2 - 1)</math>; <math>~\xi_1</math> is "<font color="darkgreen">the first zero of the Emden's function with index <math>~n</math></font>"; and <math>~\psi_0</math> and <math>~\psi_2</math> satisfy the 2<sup>nd</sup>-order ODEs,
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  <td align="right">
<math>~\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\psi_0}{d\xi} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- n\theta^{n-1} \psi_0 + 1 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\psi_2}{d\xi} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( - n\theta^{n-1} + \frac{6}{\xi^2} \biggr) \psi_2 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eqs. (37<sub>1</sub> &amp; 37<sub>2</sub>)
  </td>
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</table>

Realizing that "<font color="darkgreen">the boundary <math>~\xi_0</math> is given by <math>~\Theta = 0</math></font>," Chandrasekhar deduced as well that,
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  <td align="right">
<math>~\xi_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 + \frac{v}{|\theta_1^'|} \biggl[
\psi_0(\xi_1) - \frac{5}{6} ~ \frac{\xi_1^2 \psi_2(\xi_1) P_2(\mu)}{3\psi_2(\xi_1) + \xi_1 \psi_2^'(\xi_1)}  \biggr] \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eq. (38)
  </td>
</tr>
</table>