Editing SSC/Stability/MurphyFiedler85 (section)

==Overview==
In the stability analysis presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)], the relevant polytropic indexes are, <math>~(n_c, n_e) = (1,5)</math>.  Structural properties of the underlying equilibrium models have been reviewed in [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|our accompanying discussion]].

The ''Linear Adiabatic Wave Equation'' (LAWE) that is relevant to polytropic spheres may be written as,
<div align="center">
{{ Math/EQ_RadialPulsation02 }}
</div>

<table border="1" align="center" width="85%" cellpadding="10"><tr><td align="left">
See also &hellip;
* Accompanying chapter showing [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|derivation]] and overlap with [[SSC/Perturbations#Classic_Papers_that_Derive_.26_Use_this_Relation|multiple classic papers]]:
** [https://archive.org/details/TheInternalConstitutionOfTheStars A. S. Eddington (1926)], especially equation (127.6) on p. 188 &#8212; ''The Internal Constitution of Stars''
** [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux &amp; C. L. Pekeris (1941, ApJ, 94, 124)] &#8212; ''Radial Pulsations of Stars''
** [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941, ApJ, 94, 245)] &#8212; ''Overtone Pulsations for the Standard Model''
** [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1966, Annual Reviews of Astronomy &amp; Astrophysics, 4, 353)] &#8212; ''Pulsation Theory''
** [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974, Reports on Progress in Physics, 37, 563)] &#8212; ''Pulsating Stars''
* Accompanying chapter detailing [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|specific application to polytropes]] along with a couple of additional key references:
** [http://adsabs.harvard.edu/abs/1966ApJ...143..535H M. Hurley, P. H. Roberts, &amp; K. Wright (1966, ApJ, 143, 535)] &#8212; ''The Oscillations of Gas Spheres''
** [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] &#8212; ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models''
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As we have [[SSC/Stability/Polytropes#Boundary_Conditions|detailed separately]], the boundary condition at the center of a polytropic configuration is,
<div align="center">
<math>~\frac{dx}{d\xi} \biggr|_{\xi=0} = 0 \, ;</math>
</div>
and the boundary condition at the surface of an isolated polytropic configuration is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln\xi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \alpha +  \frac{\omega^2}{\gamma_g }  \biggl( \frac{1}{4\pi G \rho_c } \biggr) \frac{\xi}{(-\theta^')} </math>
&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~\xi = \xi_s \, .</math>
  </td>
</tr>
</table>


[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] apparently decided that they could not simply integrate the above-presented ''polytropic'' LAWE from the center of the configuration to its surface because the underlying bipolytropic equilibrium structure of the envelope and the core are defined by two different polytropic indexes.  Instead, they separated the problem into two pieces &#8212; integrating the relevant ''core'' LAWE from the center to the core-envelope interface, then integrating the relevant ''envelope'' LAWE from that interface to the surface &#8212; being careful to properly ''match'' the two solutions at the interface.

They also realized that the above-specified surface boundary condition is not applicable to bipolytropes.  Instead, they used what we will refer to as the [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations|original, more general expression of the surface boundary condition]]:

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

<tr>
  <td align="right">
<math>~ \frac{d\ln x}{d\ln\xi}\biggr|_s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \alpha + \frac{\omega^2 R^3}{\gamma_g GM_\mathrm{tot}} \, .</math>
  </td>
</tr>
</table>


<table border="1" align="center" width="85%" cellpadding="10"><tr><td align="left">
Utilizing an [[SSC/Stability/Polytropes#Groundwork|accompanying discussion]], let's examine the frequency normalization used by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] (see the top of the left-hand column on p. 223):
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Omega^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\omega^2 \biggl[ \frac{R^3}{GM_\mathrm{tot}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\omega^2 \biggl[ \frac{3}{4\pi G \bar\rho} \biggr]
=
\omega^2 \biggl[ \frac{3}{4\pi G \rho_c} \biggr] \frac{\rho_c}{\bar\rho}
=
\frac{3\omega^2}{(n_c+1)} \biggl[ \frac{(n_c+1)}{4\pi G \rho_c} \biggr] \frac{\rho_c}{\bar\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3\omega^2}{(n_c+1)} \biggl[ \frac{a_n^2\rho_c}{P_c} \cdot \theta_c \biggr] \frac{\rho_c}{\bar\rho}
=
\frac{3\gamma}{(n_c+1)} \frac{\rho_c}{\bar\rho} \biggl[ \frac{a_n^2\rho_c}{P_c} \cdot \frac{\omega^2 \theta_c}{\gamma} \biggr] \, .
</math>
  </td>
</tr>
</table>

For a given radial quantum number, <math>~k</math>, the factor inside the square brackets in this last expression is what [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] refer to as <math>~\omega^2_k \theta_c</math>.  Keep in mind, as well, that, in the notation we are using,
<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>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \sigma_c^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\bar\rho}{\rho_c}\biggr) \Omega^2 
=
\frac{6\gamma}{(n_c+1)} \biggl[ \frac{a_n^2\rho_c}{P_c} \cdot \frac{\omega^2 \theta_c}{\gamma} \biggr] 
=
\frac{6\gamma}{(n_c+1)} \biggl[ \omega_k^2 \theta_c \biggr] \, .
</math>
  </td>
</tr>
</table>
This also means that the surface boundary condition may be rewritten as,

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

<tr>
  <td align="right">
<math>~ \frac{d\ln x}{d\ln\xi}\biggr|_s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Omega^2}{\gamma_g } - \alpha  \, .</math>
  </td>
</tr>
</table>

</td></tr></table>