Editing SSC/Stability/MurphyFiedler85

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=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)].

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

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

{{Math/EQ_RadialPulsation01}}
</div>

whose solution identifies eigenvectors that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.  In shifting ''from'' this more general LAWE expression to the so-called ''polytropic'' LAWE &#8212; as [[#Overview|presented above]] &#8212; the functions that quantify the structure of the underlying equilibrium configuration, <math>~\rho_0(r_0)</math>, <math>~P_0(r_0)</math>, and <math>~g_0(r_0)</math>, are re-expressed in terms of the polytropic function, <math>~\theta(\xi)</math>  [or, instead, &#x03A6;(&eta;)] and its derivative, and the dimensional Lagrangian radial coordinate, <math>~r_0</math>, is abandoned in favor the dimensionless Lagrangian radial coordinate, <math>~\xi</math> (or, instead, &eta;), that is familiarly associated with a chosen polytropic index.  In order to avoid confusion that might be associated with switching from one polytropic function to another at the core-envelope interface, here we have chosen to stick with the single Lagrangian radial coordinate, <math>~r_0</math>, throughout the configuration.

==Foundation==
Assuming that the underlying equilibrium structure is that of a [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|bipolytrope having <math>~(n_c, n_e) = (1, 5)</math>]], it makes sense to adopt the [[SSC/Structure/BiPolytropes/Analytic15#Normalization|normalizations used when defining the equilibrium structure]], namely,
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\rho_0}{\rho_c}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{(K_c/G)^{1/2}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{K_c\rho_c^{2}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{M(r_0)}{\rho_c (K_c/G)^{3/2}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~H^*</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{H}{K_c\rho_c}</math>
  </td>

  <td align="center">. &nbsp;&nbsp;&nbsp;</td>

  <td align="right" colspan="3">
&nbsp;
  </td>
</tr>

</table>
</div>

We [[SSC/Stability/Polytropes#Groundwork|note as well]] that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM(r_0)}{r_0^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
G \biggl[ M_r^* \rho_c \biggl( \frac{K_c}{G}\biggr)^{3 / 2} \biggr] \biggl[ r^*\biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr]^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{ M_r^*}{(r^*)^2}\biggl[  G\rho_c \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] 
\, .
</math>
  </td>
</tr>
</table>
Hence, multiplying the LAWE through by <math>~(K_c/G)</math> gives,
<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}{dr*^2} + \biggl[\frac{4}{r^*} -\biggl( \frac{K_c}{G} \biggr)^{1 / 2}\biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr*} 
+ \biggl( \frac{K_c}{G} \biggr)\biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ \frac{4}{r^*} -\biggl( \frac{K_c}{G} \biggr)^{1 / 2}\biggl(\frac{\rho_c \rho^*}{P^* K_c \rho_c^2}\biggr)\frac{ M_r^*}{(r^*)^2}\biggl[  G\rho_c \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] \biggr\} \frac{dx}{dr*} 
+ \biggl( \frac{K_c}{G} \biggr)\biggl(\frac{\rho^*\rho_c}{\gamma_\mathrm{g} P^* K_c \rho_c^2} \biggr)\biggl\{ \omega^2 + (4 - 3\gamma_\mathrm{g})\frac{1}{r^*} \biggl(\frac{G}{K_c}\biggr)^{1 / 2}\frac{ M_r^*}{(r^*)^2}\biggl[  G\rho_c \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr]\biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ \frac{4}{r^*} -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)^2}\biggr\} \frac{dx}{dr*} 
+ \biggl( \frac{1}{\gamma_\mathrm{g}G\rho_c} \biggr)\biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \omega^2 + (4 - 3\gamma_\mathrm{g})\frac{1}{r^*} \frac{ M_r^*}{(r^*)^2}\biggl[  G\rho_c  \biggr]\biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ \frac{4}{r^*} -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)^2}\biggr\} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{\omega^2}{\gamma_\mathrm{g} G\rho_c} + \biggl(\frac{4}{\gamma_\mathrm{g}} - 3\biggr)\frac{1}{r^*} \frac{ M_r^*}{(r^*)^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}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, .
</math>
  </td>
</tr>
</table>

This is the form of the LAWE that we will integrate from the center of the configuration to its surface <math>~(r^* = R^*)</math> in order to identify various eigenvectors that are associated with radial oscillations in <math>~(n_c, n_e) = (1, 5)</math> bipolytropes.  Before performing the numerical integrations, we need only specify the underlying dimensionless structural functions, <math>~\rho^*(r^*)</math>, <math>~P^*(r^*)</math>, and <math>~M_r^*(r^*)</math>, throughout the underlying equilibrium configuration.

==Profile==

Referencing the relevant [[SSC/Structure/BiPolytropes/Analytic15#Profile|derived bipolytropic model profile]], we should incorporate the following relations:

<div align="center">
<table border="1" cellpadding="6">
<tr>
  <td align="center" rowspan="2">
Variable
  </td>
  <td align="center" rowspan="2">
Throughout the Core<br>
<math>0 \le r^* \le \frac{\xi_i}{\sqrt{2\pi}}</math>
  </td>
  <td align="center" rowspan="2">
Throughout the Envelope<sup>&dagger;</sup><br>
<math>\frac{\xi_i}{\sqrt{2\pi}} \le r^* \le \frac{\xi_i e^{2(\pi - \Delta_i)}}{\sqrt{2\pi}}</math>
  </td>

  <td align="center" colspan="3">
Plotted Profiles
  </td>
</tr>

<tr>
  <td align="center">
<math>\xi_i = 0.5</math>
  </td>
  <td align="center">
<math>\xi_i = 1.0</math>
  </td>
  <td align="center">
<math>\xi_i = 3.0</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>\xi = \sqrt{2\pi}~r^*</math>
  </td>
  <td align="center">
<math>\eta = \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2}~r^*</math>
  </td>

  <td align="center" colspan="3">
&nbsp;
  </td>
</tr>

<tr>
  <td align="center">
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>\frac{\sin\xi}{\xi}</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i [\phi(\eta)]^5</math>
  </td>

  <td align="center">
<!--  [[File:PlotDensity_xi_0.5.jpg|thumb|75px]] -->
[[Image:DenXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:DenXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>~P^*</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^2</math>
  </td>
  <td align="center">
<math>\theta^{2}_i [\phi(\eta)]^{6}</math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:PresXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:PresXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="center">
<math>~M_r^*</math>
  </td>
  <td align="center">
<math>\biggl( \frac{2}{\pi}\biggr)^{1/2} (\sin\xi - \xi\cos\xi)</math>
  </td>
  <td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{2\cdot 3^3 }{\pi} \biggr)^{1/2} \theta_i 
\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) </math>
  </td>

  <td align="center">
<!-- [[File:PlotPressure_xi_0.5.jpg|thumb|75px]] -->
[[Image:MassXi05.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi10.jpg|thumb|75px]]
  </td>
  <td align="center">
[[Image:MassXi30.jpg|thumb|75px]]
  </td>
</tr>

<tr>
  <td align="left" colspan="6">
<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution|presented in Step 6 of our accompanying discussion]].
  </td>
</tr>
</table>
</div>


Throughout the core we therefore have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\xi}{\sin\xi} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sqrt{2\pi}}{\xi}\biggl( \frac{2}{\pi}\biggr)^{1/2} (\sin\xi - \xi\cos\xi) = \frac{2\sin\xi}{\xi} (1 - \xi\cot\xi) \, .</math>
  </td>
</tr>
</table>

And, throughout the envelope we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho^*}{P^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i [\phi(\eta)]^5
\biggl\{\theta^{2}_i [\phi(\eta)]^{6}\biggr\}^{-1}
=
\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) }
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_r^*}{r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{2\cdot 3^3 }{\pi} \biggr)^{1/2} \theta_i \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)
\biggl\{ \frac{1}{\eta} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \biggr\}
=
6\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr)
</math>
  </td>
</tr>
</table>


<table border="1" cellpadding="8" align="center" width="85%"><tr><th align="center" id="LaterReference">
For Later Reference
</th></tr>
<tr><td align="left">

Note that we ''could have'' rewritten the governing LAWE throughout the core 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}{dr*^2} + \biggl\{ 4 -2(1-\xi\cot\xi )\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \frac{\xi}{\sin\xi} \biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g}~\frac{4\pi \sin\xi}{\xi^3} (1 - \xi\cot\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}{dr*^2} + \biggl\{ 4 -2(1-\xi\cot\xi )\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \frac{2\pi \xi}{\sin\xi} \biggl\{ \frac{\sigma_c^2}{3\gamma_\mathrm{g}}  ~+~\frac{2\alpha_\mathrm{g}}{\xi^3} \biggl(\xi\cos\xi - \sin\xi \biggr) \biggr\}  x \, ;
</math>
  </td>
</tr>
</table>

and we ''could have'' rewritten the governing LAWE throughout the envelope 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}{dr*^2} + \biggl\{ 4 - 6\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) } \biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) }
\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~6\alpha_\mathrm{g}~\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr) 
\biggl[ \frac{1}{\eta} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \biggr]^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}{dr*^2} + \biggl[ 4 - 6  \biggl(-\frac{d\ln\phi}{d\ln\eta} \biggr)  \biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) }
\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~6\alpha_\mathrm{g}~\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{ \theta_i}{\eta} \biggl(- \frac{d\phi}{d\eta} \biggr) 
\biggl(\frac{2\pi}{3}\biggr) \biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl[ 4 - 6  \biggl(-\frac{d\ln\phi}{d\ln\eta} \biggr)  \biggr] \frac{1}{r^*} \frac{dx}{dr*} 
+ \frac{2\pi}{3}\biggl( \frac{\mu_e}{\mu_c} \biggr)^2\frac{ 1}{\eta^2} 
\biggl\{ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \biggl( \frac{\sigma_c^2}{\gamma_\mathrm{g}}\biggr) \frac{\eta^2}{\theta_i \phi(\eta) }  ~-~6\alpha_\mathrm{g}~\biggl(- \frac{d\ln \phi}{d\ln\eta} \biggr)  
\biggr\}  x \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

==Model 10==
As we have [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29|reviewed in an accompanying discussion]], equilibrium Model 10 from [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy &amp; Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)] is defined by setting <math>~(\xi_i, m) = (2.5646, 1)</math>.  Drawing directly from [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29|our reproduction of their Table 1]], we see that a few relevant structural parameters of Model 10 are, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~6.5252876</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{r_i}{R} = \frac{\xi_i}{\xi_s}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.39302482</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\rho_c}{\bar\rho} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~34.346</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{env}}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5.89 \times 10^{-4}</math>
  </td>
</tr>
</table>
Here we list a few other model parameter values that will aid in our attempt to correctly integrate the LAWE to find various radial oscillation eigenvectors.
<table border="1" cellpadding="5" align="center">

<tr>
  <td align="center" colspan="12">
'''A Sampling of Model 10's Equilibrium Parameter Values'''<sup>&dagger;</sup>
</td>
</tr>
<tr>
  <td align="center">Grid<br />Line</td>
  <td align="center"><math>~\frac{r}{R}</math></td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\eta</math></td>
  <td align="center"><math>~\Delta</math></td>
  <td align="center"><math>~\phi</math></td>
  <td align="center"><math>~- \frac{d\phi}{d\eta}</math></td>
  <td align="center"><math>~r^*</math></td>
  <td align="center"><math>~\rho^*</math></td>
  <td align="center"><math>~P^*</math></td>
  <td align="center"><math>~M_r^*</math></td>
  <td align="center"><math>~g_0^*\equiv \frac{M_r^*}{(r^*)^2}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">25</td>
  <td align="right">0.12093071</td>
  <td align="right">0.789108</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">0.31480842</td>
  <td align="right">0.89940188</td>
  <td align="right">0.80892374</td>
  <td align="right">0.122726799</td>
  <td align="right">1.23835945</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">40</td>
  <td align="right"> 0.19651241</td>
  <td align="right">1.2823</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right"> 0.51156369</td>
  <td align="right"> 0.74761972</td>
  <td align="right"> 0.55893525</td>
  <td align="right"> 0.473819194</td>
  <td align="right"> 1.81056130</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">79</td>
  <td align="right"> 0.393025</td>
  <td align="right">2.5646</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right">&nbsp;</td>
  <td align="right"> 1.02312737</td>
  <td align="right"> 0.21270605</td>
  <td align="right"> 0.04524386</td>
  <td align="right"> 2.150231108</td>
  <td align="right"> 2.05411964</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">79</td>
  <td align="right"> 0.393025</td>
  <td align="right">&nbsp;</td>
  <td align="right">1.4806725</td>
  <td align="right">2.6746514</td>
  <td align="right">1.000000</td>
  <td align="right">1.112155</td>
  <td align="right"> 1.02312737</td>
  <td align="right"> 0.21270605</td>
  <td align="right"> 0.04524386</td>
  <td align="right"> 2.15023111</td>
  <td align="right"> 2.0541196</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">100</td>
  <td align="right"> 0.49883919</td>
  <td align="right">&nbsp;</td>
  <td align="right">1.8793151</td>
  <td align="right">2.7938569</td>
  <td align="right">0.6505914</td>
  <td align="right">0.69070815</td>
  <td align="right"> 1.2985847</td>
  <td align="right"> 0.0247926</td>
  <td align="right"> 0.0034309</td>
  <td align="right"> 2.15127319</td>
  <td align="right"> 1.2757189</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">150</td>
  <td align="right"> 0.7507782</td>
  <td align="right">&nbsp;</td>
  <td align="right">2.8284641</td>
  <td align="right">2.9982701</td>
  <td align="right">0.2149684</td>
  <td align="right">0.30495637</td>
  <td align="right"> 1.95443562</td>
  <td align="right"> 9.7646E-05</td>
  <td align="right"> 4.4649E-06</td>
  <td align="right"> 2.15149752</td>
  <td align="right">0.563246</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">199</td>
  <td align="right"> 0.9976784</td>
  <td align="right">&nbsp;</td>
  <td align="right">3.7586302</td>
  <td align="right">3.1404305</td>
  <td align="right">0.00150695</td>
  <td align="right">0.17269514</td>
  <td align="right">2.59716948</td>
  <td align="right"> 1.653E-15</td>
  <td align="right"> 5.2984E-19</td>
  <td align="right"> 2.15149876</td>
  <td align="right">0.31896316</td>
</tr>
<tr>
  <td align="left" colspan="12">
<sup>&dagger;</sup>Our chosen (uniform) grid spacing is,
<div align="center">
<math>~\frac{\delta r}{R} = \frac{1}{78}\biggl( \frac{r_i}{R} \biggr) \approx 0.00503878 \, ;</math>
</div>
as a result, the center is at zone 1, the interface is at grid line 79, and the surface is just beyond grid line 199.
  </td>
</tr>
</table>

==Numerical Integration==

===General Approach===
Here, we begin by recognizing that the 2<sup>nd</sup>-order ODE that must be integrated to obtain the desired eigenvectors has the generic form,
<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>~
x'' + \frac{\mathcal{H}}{r^*} x' + \mathcal{K}x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{dx}{dr^*}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~x''</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d^2x}{d(r^*)^2} \, .</math>
  </td>
</tr>
</table>
Adopting the same approach [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|as before when we integrated the LAWE for pressure-truncated polytropes]], we will enlist the finite-difference approximations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{x_+ - x_-}{2\delta r^*}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~x''</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{x_+ -2x_j + x_-}{(\delta r^*)^2} \, .
</math>
  </td>
</tr>
</table>

The finite-difference representation of the LAWE is, therefore,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta r^*)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{r^*} \biggl[ \frac{x_+ - x_-}{2\delta r^*} \biggr] ~-~ \mathcal{K}x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta r^*}{2r^*} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta r^*)^2\mathcal{K}x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>

In what follows we will also find it useful to rewrite <math>~\mathcal{K}</math> in the form,
<div align="center">
<math>~\mathcal{K} ~\rightarrow ~\biggl(\frac{\sigma_c^2}{\gamma_\mathrm{g}}\biggr) \mathcal{K}_1 - \alpha_\mathrm{g} \mathcal{K}_2 \, .</math>
</div>

<font color="red">'''Case A:'''</font> &nbsp; From the above [[#Foundation|''Foundation'' discussion]], the relevant coefficient expressions for ''all'' regions of the configuration are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, .
</math>
  </td>
</tr>
</table>


<font color="red">'''Case B:'''</font> &nbsp; Alternatively, immediately following the above [[#Profile|''Profile'' discussion]], the relevant coefficient expressions for the core are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 -2(1-\xi\cot\xi)\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\xi}{ \sin\xi} \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{4\pi }{\xi^2 \sin\xi} \biggl(\sin\xi  - \xi\cos\xi \biggr) \, ;
</math>
  </td>
</tr>
</table>

while the coefficient expressions for the envelope are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{H}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4 - 6  \biggl(-\frac{d\ln\phi}{d\ln\eta} \biggr) \biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi}{3}\biggl( \frac{\mu_e}{\mu_c} \biggr)
\biggl\{ \frac{1}{\theta_i \phi(\eta) }  \biggr\} 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{12\pi}{3}\biggl( \frac{\mu_e}{\mu_c} \biggr)^2\frac{ 1}{\eta^2} 
\biggl(- \frac{d\ln \phi}{d\ln\eta} \biggr)  \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" rowspan="2">Grid<br />Line</td>
  <td align="center" rowspan ="2"><math>~\frac{r}{R}</math></td>
  <td align="center" rowspan ="2"><math>~\xi</math></td>
  <td align="center" rowspan ="2"><math>~\eta</math></td>
  <td align="center" colspan="3"><font color="red">'''Case A'''</font></td>
  <td align="center" colspan="3"><font color="red">'''Case B'''</font></td>
</tr>
<tr>
  <td align="center"><math>~\mathcal{H}</math></td>
  <td align="center"><math>~\mathcal{K}_1</math></td>
  <td align="center"><math>~\mathcal{K}_2</math></td>
  <td align="center"><math>~\mathcal{H}</math></td>
  <td align="center"><math>~\mathcal{K}_1</math></td>
  <td align="center"><math>~\mathcal{K}_2</math></td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">25</td>
  <td align="right">0.12093071</td>
  <td align="right">0.789108</td>
  <td align="right">&nbsp;</td>
  <td align="right">3.566549</td>
  <td align="right">2.328653</td>
  <td align="right">4.373676</td>
  <td align="right">3.566549</td>
  <td align="right">2.328653</td>
  <td align="right">4.373676</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">40</td>
  <td align="right">0.19651241</td>
  <td align="right">1.2823</td>
  <td align="right">&nbsp;</td>
  <td align="right">2.761112</td>
  <td align="right">2.801418</td>
  <td align="right">4.734049</td>
  <td align="right">2.761112</td>
  <td align="right">2.801418</td>
  <td align="right">4.734049</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">79</td>
  <td align="right">0.393025</td>
  <td align="right">2.5646</td>
  <td align="right">&nbsp;</td>
  <td align="right">-5.880425</td>
  <td align="right">9.846430</td>
  <td align="right">9.4387879</td>
  <td align="right">-5.880424</td>
  <td align="right">9.846430</td>
  <td align="right">9.438787</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">79</td>
  <td align="right">0.393025</td>
  <td align="right">&nbsp;</td>
  <td align="right">1.4806725</td>
  <td align="right">-5.880425</td>
  <td align="right">9.846430</td>
  <td align="right">9.4387879</td>
  <td align="right">-5.880424</td>
  <td align="right">9.846430</td>
  <td align="right">9.438787</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">100</td>
  <td align="right">0.49883919</td>
  <td align="right">&nbsp;</td>
  <td align="right">1.8793151</td>
  <td align="right">-7.971244</td>
  <td align="right">15.134659</td>
  <td align="right">7.099025</td>
  <td align="right">-7.971184</td>
  <td align="right">15.134583</td>
  <td align="right">7.098989</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">150</td>
  <td align="right">0.7507782</td>
  <td align="right">&nbsp;</td>
  <td align="right">2.8284641</td>
  <td align="right">-2.00748E+01</td>
  <td align="right">4.58038E+01</td>
  <td align="right">6.30260</td>
  <td align="right">-2.00749E+01</td>
  <td align="right">4.58041E+01</td>
  <td align="right">6.30264</td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen">199</td>
  <td align="right">0.9976784</td>
  <td align="right">&nbsp;</td>
  <td align="right">3.7586302</td>
  <td align="right">-2.58045E+03</td>
  <td align="right">6.53411E+03</td>
  <td align="right">3.83150E+02</td>
  <td align="right">-2.58041E+03</td>
  <td align="right">6.53401E+03</td>
  <td align="right">3.83144E+02</td>
</tr>
</table>

===Special Handling at the Center===
In order to kick-start the integration, we set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \delta\xi</math>, away from the center.  Specifically, we set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
x_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F}  (\delta\xi)^2}{60} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

===Special Handling at the Interface===

Integrating outward from the center, the ''general approach'' will work up through the determination of <math>~x_{j+1}</math> when "j+1" refers to the interface location.  In order to properly transition from the core to the envelope, we need to determine the value of the slope at this interface location.  Let's do this by setting j = i, then projecting forward to what <math>~x_+</math> ''would be'' &#8212; that is, to what the amplitude just beyond the interface ''would be'' &#8212; if the core were to be extended one more zone.  Then, the slope at the interface (as viewed from the perspective of the core) will be,

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

<tr>
  <td align="right">
<math>~x'_i\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta r^*} \biggl\{
x_+ - x_{i-1}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{x_{i-1}}{2\delta r^*} 
+
\frac{1}{2\delta r^*} \biggl\{
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta r^*} \biggl\{
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]x_{i-1} ~-~\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta r^*} \biggl\{
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~2x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]^{-1}
</math>
  </td>
</tr>
</table>

Conversely, as viewed from the ''envelope'', if we assume that we know <math>~x_i</math> and <math>~x'_i</math>, we can determine the amplitude, <math>~x_{i+1}</math>, at the first zone beyond the interface as follows:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_-</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
x_{i+1} - 2\delta r^*\cdot x'_i\biggr|_\mathrm{env}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{i+1} \biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i ~-~\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
\biggl[ x_{i+1} - 2\delta r^*\cdot x'_i\biggr|_\mathrm{env}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{i+1} \biggl[1 + \biggl( \frac{\delta r^*}{2r^*}\biggr) \mathcal{H} \biggr]
~+~
\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
x_{i+1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta r^*)^2\mathcal{K}\biggr] x_i 
~+~
\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
2\delta r^*\cdot x'_i\biggr|_\mathrm{env} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{i+1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 - \tfrac{1}{2}(\delta r^*)^2\mathcal{K}\biggr] x_i 
~+~
\biggl[ 1 - \biggl( \frac{\delta r^*}{2r^*} \biggr) \mathcal{H} \biggr]
\delta r^*\cdot x'_i\biggr|_\mathrm{env} 
</math>
  </td>
</tr>
</table>

==Eigenvectors==

Keep in mind that, for all models, we ''expect'' that, at the surface, the logarithmic derivative of each proper eigenfunction will be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Omega^2}{\gamma} - \alpha \, .</math>
  </td>
</tr>
</table>

Also, keep in mind that, for Model 10 <math>~(\xi_i = 2.5646)</math>:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r_i}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.39302482</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp;&nbsp;&nbsp;</td>
  <td align="right">
<math>~\frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~34.3460405</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
'''Our Determinations for Model 10'''
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" rowspan="2"><math>~\sigma_c^2</math></td>
  <td align="center" rowspan="2"><math>~\Omega^2 \equiv \frac{\sigma_c^2}{2} \biggl( \frac{\rho_c}{\bar\rho}\biggr)</math></td>
  <td align="center" rowspan="2"><math>~x_\mathrm{surf}</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_3</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_3</math></td>
</tr>
<tr>
  <td align="center">''expected''</td>
  <td align="center">measured</td>
</tr>
<tr>
  <td align="center">1<br /><font size="-1">(Fundamental)</font></td>
  <td align="right">0.92813095170326</td>
  <td align="right">15.93881161</td>
  <td align="right">+85.17</td>
  <td align="right">8.963286966</td>
  <td align="right">8.963085</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right">1.237156768978</td>
  <td align="right">21.24571822</td>
  <td align="right">- 610</td>
  <td align="right">12.14743093</td>
  <td align="right">12.147337</td>
  <td align="right">0.5724</td>
  <td align="right">3.05E-05</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="right">1.8656033984</td>
  <td align="right">32.0380449</td>
  <td align="right">+3225</td>
  <td align="right">18.62282676</td>
  <td align="right">18.6228</td>
  <td align="right">0.4845</td>
  <td align="right">1.35E-04</td>
  <td align="right">0.787</td>
  <td align="right">2.05E-07</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="right">2.65901504799</td>
  <td align="right">45.66331921</td>
  <td align="right">-9410</td>
  <td align="right">26.79799153</td>
  <td align="right">26.797977</td>
  <td align="right">0.4459</td>
  <td align="right">2.620E-04</td>
  <td align="right">0.7096</td>
  <td align="right">1.834E-06</td>
  <td align="center">0.8632</td>
  <td align="center">1.189E-08</td>
</tr>
<tr>
  <td align="center" colspan="12">[[File:MF85Figure2B.png|800px|Match Figure 2 from MF85]]</td>
</tr>
</table>



For Model 17 <math>~(\xi_i = 3.0713)</math>:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r_i}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.93276717</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp;&nbsp;&nbsp;</td>
  <td align="right">
<math>~\frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3.79693903</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="12">
'''Our Determinations for Model 17'''
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" rowspan="2"><math>~\sigma_c^2</math></td>
  <td align="center" rowspan="2"><math>~\Omega^2 \equiv \frac{\sigma_c^2}{2} \biggl( \frac{\rho_c}{\bar\rho}\biggr)</math></td>
  <td align="center" rowspan="2"><math>~x_\mathrm{surf}</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_3</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_3</math></td>
</tr>
<tr>
  <td align="center">''expected''</td>
  <td align="center">measured</td>
</tr>
<tr>
  <td align="center">1<br /><font size="-1">(Fundamental)</font></td>
  <td align="left">1.149837904</td>
  <td align="left">2.182932207</td>
  <td align="right">+1.275</td>
  <td align="right">0.7097593</td>
  <td align="right">0.7097550</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="left">7.34212930615</td>
  <td align="left">13.93880866</td>
  <td align="right">- 2.491</td>
  <td align="right">7.763285</td>
  <td align="right">7.763244</td>
  <td align="right">0.7215</td>
  <td align="right">0.24006</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="left">16.345072567</td>
  <td align="left">31.03062198</td>
  <td align="right">+4.33</td>
  <td align="right">18.01837</td>
  <td align="right">18.01826</td>
  <td align="right">0.5806</td>
  <td align="right">0.5027</td>
  <td align="right">0.848</td>
  <td align="right">0.0541</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="left">27.746934203</td>
  <td align="left">52.6767087</td>
  <td align="right">-9.1</td>
  <td align="right">31.0060</td>
  <td align="right">31.0058</td>
  <td align="right">0.4859</td>
  <td align="right">0.6737</td>
  <td align="right">0.7429</td>
  <td align="right">0.1974</td>
  <td align="center">0.8957</td>
  <td align="center">0.0171</td>
</tr>
<tr>
  <td align="center" colspan="12">[[File:MF85Figure3.png|800px|Match Figure 3 from MF85]]</td>
</tr>
</table>


<table border="1" align="center" cellpadding="8" >
<tr>
  <td align="center" colspan="7">
'''Numerical Values for Some Selected <math>~(n_c, n_e) = (1, 5)</math> Bipolytropes'''<br />
[to be compared with Table 1 of [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy &amp; Fiedler (1985)]]
  </td>
</tr>
<tr>
  <td align="center">MODEL</td>
  <td align="center">Source</td>
  <td align="center"><math>~\frac{r_i}{R}</math></td>
  <td align="center"><math>~\Omega_0^2</math></td>
  <td align="center"><math>~\Omega_1^2</math></td>
  <td align="center"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center"><math>~1-\frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
</tr>
<tr>
  <td align="center" rowspan="2">10</td>
  <td align="center" bgcolor="pink">MF85</td>
  <td align="left">0.393</td>
  <td align="left">15.9298</td>
  <td align="left">21.2310</td>
  <td align="left">0.573</td>
  <td align="left">1.00E-03</td>
</tr>
<tr>
  <td align="center">Here</td>
  <td align="right">0.39302</td>
  <td align="right">15.93881161</td>
  <td align="right">21.24571822</td>
  <td align="right">0.5724</td>
  <td align="left">3.05E-05</td>
</tr>
<tr>
  <td align="center" rowspan="2">17</td>
  <td align="center" bgcolor="pink">MF85</td>
  <td align="left">0.933</td>
  <td align="left">2.1827</td>
  <td align="left">13.9351</td>
  <td align="left">0.722</td>
  <td align="left">0.232</td>
</tr>
<tr>
  <td align="center">Here</td>
  <td align="left">0.93277</td>
  <td align="left">2.182932207</td>
  <td align="left">13.93880866</td>
  <td align="left">0.7215</td>
  <td align="left">0.24006</td>
</tr>
</table>

=Reconcile Approaches=
==Core:==
Given that, <math>~\sqrt{2\pi}~r^* = \xi</math>, lets multiply the LAWE through by <math>~(2\pi)^{-1}</math>.  This gives,

<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 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{\xi} \cdot \frac{dx}{d\xi} 
+ \frac{1}{2\pi}\biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, .
</math>
  </td>
</tr>
</table>

Specifically for the core, therefore, the finite-difference representation of the LAWE is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta \xi)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{\xi} \biggl[ \frac{x_+ - x_-}{2\delta \xi} \biggr] ~-~ \biggl[ \frac{\mathcal{K}}{2\pi} \biggr]x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta \xi}{2\xi} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \xi)^2 \biggl[ \frac{\mathcal{K}}{2\pi} \biggr] x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \xi}{2\xi}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta \xi)^2\biggl( \frac{\mathcal{K}}{2\pi} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \xi}{2\xi} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>
This also means that, as viewed from the perspective of the core, the slope at the interface is

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{dx}{d\xi}\biggr]_\mathrm{interface}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2\delta \xi} \biggl\{
\biggl[ 2 - (\delta \xi)^2 \biggl( \frac{\mathcal{K}}{2\pi} \biggr)\biggr] x_i ~-~2x_{i-1} 
\biggr\}\biggl[1 + \biggl( \frac{\delta \xi}{2\xi}\biggr) \mathcal{H} \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

==Envelope:==
Given that,
<div align="center">
<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2}~r^* = \eta \, ,</math>
</div>
let's multiply the LAWE through by <math>~(3/2\pi)( \mu_e/\mu_c)^{-2} </math>.  This gives,

<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 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{\eta} \cdot \frac{dx}{d\eta} 
+ \frac{3}{2\pi} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, .
</math>
  </td>
</tr>
</table>

Specifically for the envelope, therefore, the finite-difference representation of the LAWE is, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x_+ -2x_j + x_-}{(\delta \eta)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\mathcal{H}}{\eta} \biggl[ \frac{x_+ - x_-}{2\delta \eta} \biggr] ~-~  \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{3\mathcal{K}}{2\pi} \biggr]x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \frac{\delta \eta}{2\eta} \biggl[ x_+ - x_- \biggr]\mathcal{H} ~-~ (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}\biggl[ \frac{3\mathcal{K}}{2\pi} \biggr] x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \eta}{2\eta}\biggr) \mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2 - (\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{3\mathcal{K}}{2\pi} \biggr) \biggr] x_j ~-~\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>

This also means that, once we know the slope at the interface (see immediately below), the amplitude at the first zone outside of the interface will be given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_{i+1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 - \tfrac{1}{2}(\delta \eta)^2 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{3\mathcal{K}}{2\pi} \biggr)\biggr] x_i 
~+~
\biggl[ 1 - \biggl( \frac{\delta \eta}{2\eta} \biggr) \mathcal{H} \biggr]
\delta \eta \cdot \biggl[ \frac{dx}{d\eta} \biggr]_\mathrm{interface} \, .
</math>
  </td>
</tr>
</table>

==Interface==
If we consider only cases where <math>~\gamma_e = \gamma_c</math>, then at the interface we expect,

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

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d\ln x}{d\ln \xi} = \frac{d\ln x}{d\ln \eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r^*\frac{dx}{d r^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi \frac{dx}{d \xi} = \eta \frac{d x}{d \eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{dx}{dr^*}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2\pi)^{1 / 2}\frac{dx}{d\xi} = \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \frac{dx}{d\eta} \, .</math>
  </td>
</tr>
</table>
Switching at the interface from <math>~\xi</math> to <math>~\eta</math> therefore means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \biggl[ \frac{dx}{d\eta}\biggr]_\mathrm{interface}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{3}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl[ \frac{dx}{d\xi}\biggr]_\mathrm{interface} \, .</math>
  </td>
</tr>
</table>


=See Also=


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