Editing SSC/Stability/n1PolytropeLAWE (section)

=Radial Oscillations of n = 1 Polytropic Spheres (Pt 1)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/n1PolytropeLAWE|Part I: &nbsp; Search for Analytic Solutions]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/n1PolytropeLAWE/Pt2|Part II:&nbsp; New Ideas]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/n1PolytropeLAWE/Pt3|Part III:&nbsp; What About Bipolytropes?]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Stability/n1PolytropeLAWE/Pt4|Part IV:&nbsp; Most General Structural Solution]]<br />&nbsp;</td>
</tr>
</table>


As far as we have been able to ascertain, the first technical examination of radial oscillation modes in <math>n=1</math> polytropes was performed &#8212; using numerical techniques &#8212; in 1951 by L. D. Chatterji; at the time, he was in the Mathematics Department of Allahabad University.  His two papers on this topic were published in, what  is now referred to as, the ''Proceedings of the Indian National Science Academy'' (PINSA).  The citations that immediately follow this opening paragraph provide inks to both of these papers by Chatterji, but the links may be insecure.  (Citations/links to articles that provide analyses of models having other polytropic indexes are provided at the [[#See_Also|bottom of this chapter]].)  Apparently [https://www.springer.com/journal/43538 Springer] is archiving recent PINSA volumes, but their holdings do not date back as early as 1951.

* {{ Chatterji51full }}, ''Radial Oscillations of a Gaseous Star of Polytropic Index I'' 
* {{ Chatterji52full }}, ''Anharmonic Pulsations of a Polytropic Model of Index Unity''

A detailed review of {{ Chatterji51hereafter }} is provided in [[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2#Setup|an accompanying discussion]].

==Groundwork==

In an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. Because this widely used form of the radial pulsation equation is not dimensionless but, rather, has units of inverse length-squared, we have found it useful to also recast it in the following [[SSC/Perturbations#Dimensionless_Expression|dimensionless form]]:
<div align="center">
<math>
\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0} - \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} + \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{P_0}{P_c}\biggr)^{-1} \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr]  x = 0 ,
</math><br />
</div>
where,
<div align="center">
<math>~g_\mathrm{SSC} \equiv \frac{P_c}{R\rho_c} \, ,</math> &nbsp; &nbsp; &nbsp; and  &nbsp; &nbsp; &nbsp; <math>~\tau_\mathrm{SSC} \equiv \biggl[\frac{R^2 \rho_c}{P_c}\biggr]^{1/2} \, .</math>
</div>

In a [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|separate discussion]], we showed that specifically for isolated, ''polytropic'' configurations, this linear adiabatic wave equation (LAWE) can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4 - (n+1)V(\xi)}{\xi} \biggr] \frac{dx}{d\xi} + 
\biggl[\frac{\omega^2}{\gamma_g \theta} \biggl(\frac{n+1 }{4\pi G \rho_c} \biggr) - 
\biggl(3-\frac{4}{\gamma_g}\biggr)  \cdot \frac{(n+1)V(x)}{\xi^2} \biggr]  x </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d\xi^2} + \biggl[\frac{4}{\xi} - \frac{(n+1)}{\theta} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] \frac{dx}{d\xi} + 
\frac{(n+1)}{\theta}\biggl[\frac{\sigma_c^2}{6\gamma_g } - 
\frac{\alpha}{\xi} \biggl(- \frac{d\theta}{d\xi}\biggr)\biggr]  x \, ,</math>
  </td>
</tr>
</table>
</div>
where we have adopted the dimensionless frequency notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_c^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3\omega^2}{2\pi G \rho_c}  \, .</math>
  </td>
</tr>
</table>
</div>

Here we focus on an analysis of the ''specific'' case of isolated, <math>~n=1</math> polytropic configurations, whose unperturbed equilibrium structure can be prescribed in terms of analytic functions.  Our hope &#8212; as yet unfulfilled &#8212; is that we can discover an analytically prescribed eigenvector solution to the governing LAWE.