Editing SSC/Stability/MurphyFiedler85 (section)

=Review of the BiPolytrope Stability Analysis by Murphy &amp; Fiedler (1985b)=

==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''
</td></tr></table>


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>

==Aside Regarding Convectively Unstable Core==
It is worth highlighting that, in their effort to determine the eigenvectors associated with radial pulsations in <math>~(n_c, n_e) = (1, 5)</math> bipolytropes, [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] assumed that fluid elements ''throughout the entire spherical configuration'' expand and contract along <math>~\gamma_g = 5/3</math> adiabats.  Referencing separately the ''structural'' polytropic index of the core and of the envelope of the equilibrium bipolytropic models, we see that,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~< </math>
  </td>
  <td align="left">
<math>~\frac{n_c+1}{n_c} = 2 \, ,</math>
  </td>
</tr>
</table>
while,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~></math>
  </td>
  <td align="left">
<math>~\frac{n_e+1}{n_e} = \frac{6}{5} \, .</math>
  </td>
</tr>
</table>

According to the so-called ''Schwarzschild criterion'' &#8212; see, for example, our [[2DStructure/AxisymmetricInstabilities#Modeling_Implications_and_Advice|accompanying discussion titled, ''Axisymmetric Instabilities to Avoid'']] &#8212; it therefore seems that the core of each of their equilibrium models should have been convectively unstable.  [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] did not comment on the impact that the presence of a convective core should have had on their radial pulsation analysis.

==More Detailed Setup==
Here we describe in more detail the steps that [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] employed in order to numerically determine the radial-oscillation eigenvectors of <math>~(n_c, n_e) = (1, 5)</math> bipolytropic spheres.

===Core Layers With n = 1===
For n = 1 structures the LAWE is,

<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[ 4 - 2 Q_1 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr)  \frac{\xi^2}{\theta} 
- \alpha_\mathrm{core} Q_1\biggr]  \frac{x}{\xi^2} 
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{d\ln\theta}{d\ln\xi} \, .</math>
  </td>
</tr>
</table>

Given that, for <math>~n = 1</math> polytropic structures,
<div align="center">
<math>
\theta(\xi) = \frac{\sin\xi}{\xi}
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
\frac{d\theta}{d\xi} = \biggl[ \frac{\cos\xi}{\xi}- \frac{\sin\xi}{\xi^2}\biggr] 
</math>
</div>
we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\xi^2}{\sin\xi} \biggl[ \frac{\cos\xi}{\xi}- \frac{\sin\xi}{\xi^2}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 -  \xi\cot\xi \, .
</math>
  </td>
</tr>
</table>

Hence, the governing LAWE for the core is,

<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[ 4 - 2 ( 1 -  \xi\cot\xi ) \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr)  \frac{\xi^3}{\sin\xi} 
- \alpha_\mathrm{core} ( 1 -  \xi\cot\xi )\biggr]  \frac{x}{\xi^2} 
</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[ 1 +  \xi\cot\xi  \biggr] \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr)  \frac{\xi^3}{\sin\xi} 
- \alpha_\mathrm{core} ( 1 -  \xi\cot\xi )\biggr]  \frac{x}{\xi^2} \, .
</math>
  </td>
</tr>
</table>
This can be rewritten as,

<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} +  \frac{2}{\xi} \biggl[ 1 +  \xi\cot\xi  \biggr]\frac{dx}{d\xi}  
+ \biggl[ \biggl( \frac{\sigma_c^2}{3\gamma_\mathrm{core} } \biggr)  \frac{\xi}{\sin\xi} 
+ \frac{2 \alpha_\mathrm{core} ( \xi\cos\xi - \sin\xi) }{\xi^2 \sin\xi} \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} +  \frac{2}{\xi} \biggl[ 1 +  \xi\cot\xi  \biggr]\frac{dx}{d\xi}  
+ \biggl[ \frac{\gamma_g}{\gamma_\mathrm{core}}\biggl( \omega_k^2 \theta_c \biggr)  \frac{\xi}{\sin\xi} 
+ \frac{2 \alpha_\mathrm{core} ( \xi\cos\xi - \sin\xi) }{\xi^2 \sin\xi} \biggr]  x \, ,
</math>
  </td>
</tr>
</table>
which matches the expression presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] (see middle of the left column on p. 223 of their article) if we set <math>~\theta_c = 1</math> and <math>~\gamma_g/\gamma_\mathrm{core} = 1</math>.  This LAWE also appears in our [[SSC/Stability/n1PolytropeLAWE#MurphyFiedler1985b|separate discussion of radial oscillations in n = 1 polytropic spheres]].

===Envelope Layers With n = 5===
The LAWE for n = 5 structures is,  
<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\eta^2} + \biggl[ 4 - 6Q_5 \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 6 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{env} } \biggr)  \frac{\eta^2}{\phi} 
- \alpha_\mathrm{env} Q_5\biggr]  \frac{x}{\eta^2}
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Q_5</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{d\ln\phi}{d\ln\eta} \, .</math>
  </td>
</tr>
</table>

From our [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution|accompanying discussion of the underlying equilibrium structure of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes]], we know that,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{B_0^{-1}\sin\Delta}{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}} \, ,</math>
  </td>
</tr>

</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\phi}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{B_0^{-1}[3\cos\Delta-3\sin\Delta + 2\sin^3\Delta] }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}}  \, .
</math>
  </td>
</tr>
</table>
</div>

where <math>~A_0</math> is a "homology factor," <math>~B_0</math> is an overall scaling coefficient, and we have introduced the notation,
<div align="center">
<math>~\Delta \equiv \ln(A_0\eta)^{1/2} = \frac{1}{2} (\ln A_0 + \ln\eta) \, .</math>
</div>

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

<tr>
  <td align="right">
<math>~Q_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \eta \biggl[  \frac{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}}{B_0^{-1}\sin\Delta} \biggr] \frac{B_0^{-1}[3\cos\Delta-3\sin\Delta + 2\sin^3\Delta] }{2\eta^{3/2}(3-2\sin^2\Delta)^{3/2}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ 3\sin\Delta - 3\cos\Delta - 2\sin^3\Delta }{2 \sin\Delta (3-2\sin^2\Delta)} \, .
</math>
  </td>
</tr>
</table>

And,
<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\eta^2} ~+~ \biggl[ 4 + \frac{ 3(3\cos\Delta - 3\sin\Delta + 2\sin^3\Delta) }{ \sin\Delta (3-2\sin^2\Delta)} \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
~+~ \biggl[ \biggl( \frac{\sigma_c^2}{\gamma_\mathrm{env} } \biggr)  \frac{B_0 \eta^{1/2}(3-2\sin^2\Delta)^{1/2}}{\sin\Delta} 
~+~ \frac{ 3\alpha_\mathrm{env} (3\cos\Delta -3\sin\Delta  +  2\sin^3\Delta )}{\eta^2 \sin\Delta (3-2\sin^2\Delta)}\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\eta^2} ~+~ \biggl[ 4 ~+~ \frac{ 3(3\cos\Delta - \tfrac{3}{2}\sin\Delta - \tfrac{1}{2}\sin3\Delta) }{ \sin\Delta (2 + \cos2\Delta)} \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
~+~ \biggl[\omega^2_k \theta_c \biggl( \frac{\gamma_g}{\gamma_\mathrm{env} } \biggr)  \frac{B_0 \eta^{1/2}(2 + \cos2\Delta)^{1/2}}{\sin\Delta} 
~+~ \frac{ 3\alpha_\mathrm{env} (3\cos\Delta -\tfrac{3}{2}\sin\Delta  -  \tfrac{1}{2}\sin3\Delta )}{\eta^2 \sin\Delta (2 + \cos2\Delta)}\biggr]  x \, ,
</math>
  </td>
</tr>
</table>
which matches the expression presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] (see middle of the left column on p. 223 of their article) if we set <math>~\theta_c = 1</math> and <math>~\gamma_g/\gamma_\mathrm{env} = 1</math>.


===Surface Boundary Condition===
Next, pulling from our [[SSC/Stability/Polytropes#Boundary_Conditions|accompanying discussion of the stability of polytropes]] and an [[SSC/Structure/BiPolytropes/Analytic15#Parameter_Values|accompanying table that details the properties of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes]], the surface boundary condition is,

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

<tr>
  <td align="right">
<math>~ \frac{d\ln x}{d\ln\eta}\biggr|_s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl(\frac{\gamma_g}{\gamma_\mathrm{env}}\biggr) \alpha + \frac{\omega^2 R^3}{\gamma_\mathrm{env} GM_\mathrm{tot}} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
\frac{d\ln x}{d\ln\eta}\biggr|_s + \biggl(\frac{\gamma_g}{\gamma_\mathrm{env}}\biggr) \alpha 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\omega^2 (R_s^*)^3}{\gamma_\mathrm{env} GM^*_\mathrm{tot}} \biggl( \frac{K_c}{G}\biggr)^{3 / 2}\biggl( \frac{K_c}{G}\biggr)^{-3 / 2} \frac{1}{\rho_0}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\omega^2 }{\gamma_\mathrm{env} G\rho_0 } \biggl[ (2\pi)^{-1/2} \xi_i e^{2(\pi - \Delta_i)} \biggr]^3 
\biggl[ \biggl( \frac{3}{2\pi} \biggr)^{1/2} \sin\xi_i \biggl( \frac{3}{\sin^2\Delta_i} - 2 \biggr)^{1/2} e^{(\pi - \Delta_i)} \biggr]^{-1} \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\omega^2 }{\gamma_\mathrm{env}(2\pi G\rho_0)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{1}{\sqrt{3}} \biggl[  \frac{\xi_i^2}{\theta_i}  \biggr] 
\biggl( \frac{3}{\sin^2\Delta_i} - 2 \biggr)^{-1 / 2}  e^{5(\pi - \Delta_i)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\omega^2 }{\gamma_\mathrm{env}(2\pi G\rho_0)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{e^{5\pi}}{\sqrt{3}} \biggl[  \frac{\xi_i^2}{\theta_i}  \biggr] 
\xi_i^{1 / 2}B\theta_i  (\xi_i A)^{-5/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\omega^2 }{\gamma_\mathrm{env}(2\pi G\rho_0)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{B e^{5\pi}}{\sqrt{3} ~A^{5 / 2}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\omega_k^2 \theta_c}{(n_c+1)} \biggl( \frac{\mu_e}{\mu_c}\biggr) \frac{B e^{5\pi}}{\sqrt{3} ~A^{5 / 2}} \, .
</math>
  </td>
</tr>
</table>

After acknowledging that, in their specific stability analysis, <math>~\theta_c = 1</math>, <math>~n_c = 1</math>, and <math>~\mu_e/\mu_c = 1</math>, this right-hand-side expression matches the equivalent term published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] (see the bottom of the left-hand column on p. 223).

===Interface Conditions===
Here, we will simply copy the discussion already provided in the context of our attempt to analyze the stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes; specifically, we will draw from [[SSC/Stability/BiPolytrope00#Piecing_Together|<font color="red">'''STEP 4:'''</font> in the ''Piecing Together'' subsection]].  Following the discussion in &sect;&sect;57 &amp; 58 of [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)], the proper treatment is to ensure that fractional perturbation in the gas pressure (see their equation 57.31),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\delta P}{P}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma x \biggl( 3 + \frac{d\ln x}{d\ln \xi} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>

is continuous across the interface.  That is to say, at the interface <math>~(\xi = \xi_i)</math>, we need to enforce the relation,
<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>~\biggl[ \gamma_c x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \gamma_e x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=\xi_i}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_e \biggl[ \frac{\gamma_c}{\gamma_e} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr) - \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr)\biggr]_{\xi=\xi_i}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=\xi_i}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} \, .</math>
  </td>
</tr>
</table>
</div>
In the context of this interface-matching constraint (see their equation 62.1), [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)] state the following: &nbsp; <font color="darkgreen"><b>In the static</b></font> (''i.e.,'' unperturbed equilibrium) <font color="darkgreen"><b>model</b></font> &hellip; <font color="darkgreen"><b>discontinuities in <math>~\rho</math> or in <math>~\gamma</math> might occur at some [radius]</b></font>.  <font color="darkgreen"><b>In the first case</b></font> &#8212; that is, a discontinuity only in density, while <math>~\gamma_e = \gamma_c</math> &#8212; the interface conditions <font color="darkgreen"><b>imply the continuity of <math>~\tfrac{1}{x} \cdot \tfrac{dx}{d\xi}</math> at that [radius].  In the second case</b></font> &#8212;  that is, a discontinuity in the adiabatic exponent &#8212; <font color="darkgreen"><b>the dynamical condition may be written</b></font> as above.  <font color="darkgreen"><b>This implies a discontinuity of the first derivative at any discontinuity of <math>~\gamma</math></b></font>.

The algorithm that [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] used to "<font color="#007700">&hellip; [integrate] through each zone &hellip;</font>" was designed "<font color="#007700">&hellip; with continuity in <math>~x</math> and <math>~dx/d\xi</math> being imposed at the interface &hellip;</font>"  Given that they set <math>~\gamma_c = \gamma_e = 5/3</math>, their interface matching condition is consistent with the one prescribed by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)].

==Our Confession==

When we tried to integrate the governing LAWEs in the piecemeal fashion described by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] &#8212; as we have just detailed &#8212; we initially failed to match their published eigenvector solutions.  In retrospect, it appears as though we did not correctly implement the interface-matching conditions.  In an effort to diagnose this problem, we backed up to a more generalized prescription of the LAWE that allowed us to smoothly integrate a ''single'' equation from the center to the surface of the configuration without having to mess with interface-matching conditions.  In what follows, we describe this alternate approach.  This approach has allowed us to derive radial-oscillation eigenvectors that match in detail the results published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)].