Editing Appendix/Ramblings/BiPolytropeStability (section)

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

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>
But this surface condition is not applicable to bipolytropes.  Instead, let's return to 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>


Let's apply these relations to the core and envelope, separately.


==Envelope Layers With n = 5==
The LAWE for n = 5 structures is, then, 
<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).

==Core Layers With n = 1==
And 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]].

==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 Numerical Integration==

Let's try to integrate this bipolytrope's LAWE from the center, outward, using as a guideline an [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|accompanying ''Numerical Integration'' outline]].  Generally, for any polytropic index, the relevant LAWE can be written in the form,

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

<tr>
  <td align="right">
<math>~\theta_i {x_i''}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\mathcal{A} \biggr] \frac{x_i'}{\xi_i}  
- \frac{(n+1)}{6} \biggl[ \mathcal{B} \biggr]  x_i </math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
4\theta_i - (n+1)\xi_i (- \theta^')_i 
=
\theta_i [ 4 - (n+1)Q_i] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\sigma_c^2}{\gamma_g}  - 2\alpha \biggl(- \frac{3\theta^'}{\xi} \biggr)_i 
=
\mathfrak{F} + 2\alpha \biggl[ 1 - \biggl(- \frac{3\theta^'}{\xi} \biggr)_i  \biggr]
=
\mathfrak{F} + 2\alpha \biggl[ 1 - \frac{3\theta_i}{\xi_i^2} \cdot Q_i  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\mathfrak{F}
</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 2\alpha\biggr] 
=
\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 2\biggl(3 - \frac{4}{\gamma_g} \biggr) \biggr] 
=
\biggl[ \frac{(8 + \sigma_c^2)}{\gamma_g}  - 6\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\Rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
\sigma_c^2   =
\gamma_g (\mathfrak{F} + 6) -8
\, .
</math>
  </td>
</tr>
</table>

This leads to a discrete, finite-difference representation of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_+ \biggl[2\theta_i + \frac{\delta\xi}{\xi_i} \cdot \mathcal{A}\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_- \biggl[\frac{\delta\xi }{\xi_i} \cdot \mathcal{A} - 2\theta_i\biggr] 
+  x_i\biggl\{4\theta_i - \frac{(\delta\xi)^2(n+1)}{3}\cdot \mathcal{B} \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>

This provides an approximate expression for <math>~x_+ \equiv x_{i+1}</math>, given the values of <math>~x_- \equiv x_{i-1}</math> and <math>~x_i</math>; this works for all zones, <math>~i = 3 \rightarrow N</math>  as long as the center of the configuration is denoted by the grid index, <math>~i=1</math>.  Note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta\xi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\xi_\mathrm{max}}{(N - 1)} </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~\xi_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(i-1)\delta\xi \, . </math>
  </td>
</tr>
</table>

In order to kick-start the integration, we will set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then we will 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 will 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>

===Integration Through the n = 1 Core===

For an <math>~n = 1</math> core, we have,
<div align="center">
<math>
\theta_i = \frac{\sin\xi_i}{\xi_i}
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
Q_i = 1 -  \xi_i \cot\xi_i \, .
</math>
</div>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sin\xi_i}{\xi_i} \biggl[ 4 - 2(1 -  \xi_i \cot\xi_i) \biggr] 
=
\frac{2\sin\xi_i}{\xi_i} \biggl[ 1 + \xi_i \cot\xi_i \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{F}_\mathrm{core} + 2\alpha_\mathrm{core} \biggl[ 1 - \frac{3\theta_i}{\xi_i^2} \cdot Q_i  \biggr]
=
\mathfrak{F}_\mathrm{core} + 2\alpha_\mathrm{core} \biggl[ 1 - \frac{3\sin\xi_i}{\xi_i^3} \biggl( 1 - \xi_i \cot\xi_i \biggr)\biggr] \, .
</math>
  </td>
</tr>
</table>

So, first we choose a value of <math>~\sigma_c^2</math> and <math>~\gamma_c</math>, which means,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\mathfrak{F}_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(8 + \sigma_c^2)}{\gamma_c}  - 6\biggr] 
</math>
  </td>
</tr>
</table>

Then, moving from the center of the configuration, outward to the interface at <math>~\xi_i = \xi_\mathrm{interface} ~~ \Rightarrow ~~\delta\xi = \xi_\mathrm{interface}/(N-1)</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
x_2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_1 \biggl[ 1 - \frac{\mathfrak{F}_\mathrm{core}  (\delta\xi)^2}{30} \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
for <math>~i = 2 \rightarrow N \, ,</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~x_{i+1} \biggl[2\theta_i + \frac{\delta\xi}{\xi_i} \cdot \mathcal{A}_\mathrm{core} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x_{i-1} \biggl[\frac{\delta\xi }{\xi_i} \cdot \mathcal{A}_\mathrm{core} - 2\theta_i\biggr] 
+  x_i\biggl\{4\theta_i - \frac{(\delta\xi)^2(n+1)}{3}\cdot \mathcal{B}_\mathrm{core} \biggr\} \, .</math>
  </td>
</tr>
</table>
At the interface &#8212; that is, when <math>~i=N</math> &#8212; the logarithmic slope of the displacement function is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{interface}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{\xi_N}{x_N} \cdot \frac{(x_{N+1} - x_{N-1})}{2\delta\xi} \, .
</math>
  </td>
</tr>
</table>

===Interface===

Keep in mind that, as has been [[SSC/Structure/BiPolytropes/Analytic15#Parameter_Values|detailed in the accompanying ''equilibrium structure'' chapter]], for <math>~(n_c, n_e) = (1, 5)</math> bipolytropes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{1}{2\pi}\biggr)^{1 / 2} \xi \, ,</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>~0 \le \xi \le \xi_\mathrm{interface} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{\mu_e}{\mu_c}\biggr)^{-1} \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \biggr] \eta \, ,</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>~\eta_\mathrm{interface} \le \eta \le \eta_s \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\eta_s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\sqrt{3}}\biggl( \frac{\mu_e}{\mu_c}\biggr)  \xi_s = 
\frac{1}{\sqrt{3}}\biggl( \frac{\mu_e}{\mu_c}\biggr) \biggl[ \xi e^{2(\pi - \Delta)} \biggr]_\mathrm{interface}
\, .
</math>
  </td>
  <td align="left" colspan="2">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  
  </td>
</tr>
</table>

We now need to determine what the slope is at the interface, viewed from the perspective of the envelope.  From [[#Interface_Conditions|above]], we deduce that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln y}{d\ln \eta} \biggr|_\mathrm{interface}</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} \cdot \frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{interface} \, .</math>
  </td>
</tr>
</table>

Hence, letting the subscript "1" denote the interface location as viewed from the envelope, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\eta_1}{y_1} \cdot \frac{(y_2 - y_0)}{ (\eta_2 - \eta_0)}</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} \cdot \frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{interface} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ y_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_2 - \frac{2 (\delta\eta) y_1}{\eta_1} \biggl\{ 3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{interface} \biggr\} \, .</math>
  </td>
</tr>
</table>

===Integration Through the n = 5 Envelope===

For an <math>~n = 5</math> envelope, we have,
<div align="center">
<math>
\phi_i = \frac{B_0^{-1}\sin\Delta}{\eta^{1/2}(3-2\sin^2\Delta)^{1/2}}
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
Q_i = - \frac{d\ln\phi}{d\ln\eta} = \frac{ 3\sin\Delta - 3\cos\Delta - 2\sin^3\Delta }{2 \sin\Delta (3-2\sin^2\Delta)}\, ,
</math>
</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>~\mathcal{A}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\phi_i [ 4 - (n+1)Q_i] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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}}\biggl\{ 4 
~-~ 6 \biggl[ \frac{ 3\sin\Delta - 3\cos\Delta - 2\sin^3\Delta }{2 \sin\Delta (3-2\sin^2\Delta)}  \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\sigma_c^2}{\gamma_e}  - 2\alpha_\mathrm{env} \biggl(- \frac{3\phi^'}{\eta} \biggr)_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\gamma_e}\biggl[ \gamma_c (\mathfrak{F}_\mathrm{core} + 6) -8 \biggr]  - 2\biggl[ 3 - \frac{4}{\gamma_e}\biggr] Q_i \biggl( \frac{3\phi_i }{\eta_i^2} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\gamma_c}{\gamma_e}\biggl[ \mathfrak{F}_\mathrm{core} + 6  -\frac{8}{\gamma_c} \biggr]  - 6\biggl[ 3 - \frac{4}{\gamma_e}\biggr] 
\frac{B_0^{-1}\sin\Delta}{\eta^{5/2}(3-2\sin^2\Delta)^{1/2}}
\biggl[ \frac{ 3\sin\Delta - 3\cos\Delta - 2\sin^3\Delta }{2 \sin\Delta (3-2\sin^2\Delta)} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\gamma_c}{\gamma_e}\biggl[ \mathfrak{F}_\mathrm{core} + 6  -\frac{8}{\gamma_c} \biggr]  - 3B_0^{-1}\biggl[ 3 - \frac{4}{\gamma_e}\biggr] 
\biggl[ \frac{ 3\sin\Delta - 3\cos\Delta - 2\sin^3\Delta }{ \eta^{5/2}(3-2\sin^2\Delta)^{3 / 2}} \biggr]
\, .
</math>
  </td>
</tr>
</table>

This leads to a discrete, finite-difference representation of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_+ \biggl[2\phi_i + \frac{\delta\eta}{\eta_i} \cdot \mathcal{A}_\mathrm{env} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y_- \biggl[\frac{\delta\eta }{\eta_i} \cdot \mathcal{A}_\mathrm{env} - 2\phi_i\biggr] 
+  y_i\biggl[4\phi_i - 2(\delta\eta)^2 \cdot \mathcal{B}_\mathrm{env} \biggr] </math>
  </td>
</tr>
</table>
</div>

This provides an approximate expression for <math>~y_+ \equiv y_{i+1}</math>, given the values of <math>~y_- \equiv y_{i-1}</math> and <math>~y_i</math>; this works for all zones, <math>~i = 3 \rightarrow M</math>  as long as the interface between the core and the envelope of the configuration is denoted by the grid index, <math>~i=1</math>.  Note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta\eta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\eta_\mathrm{surf}- \eta_\mathrm{interface} }{M - 1} </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="right">
<math>~\eta_i</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{interface} + (i-1)\delta\eta \, . </math>
  </td>
</tr>
</table>

At the interface, we need special treatment in order to ensure that both the amplitude and the first derivative of the displacement function behave properly.  Specifically, when <math>~i = 1</math>, we must set, <math>~y_1 = x_N</math> and <math>~\eta_1 = (\mu_e/\mu_c)\xi_N/\sqrt{3}</math>.  Then the value of <math>~y_2</math> is obtained from the expression,

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

<tr>
  <td align="right">
<math>~y_2 \biggl[2\phi_1 + \frac{\delta\eta}{\eta_1} \cdot \mathcal{A}_\mathrm{env} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y_0 \biggl[\frac{\delta\eta }{\eta_1} \cdot \mathcal{A}_\mathrm{env} - 2\phi_1\biggr] 
+  y_1\biggl\{4\phi_1 - 2(\delta\eta)^2 \cdot \mathcal{B}_\mathrm{env} \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y_1\biggl[ 4\phi_1 - 2(\delta\eta)^2 \cdot \mathcal{B}_\mathrm{env} \biggr] 
+  
\biggl[\frac{\delta\eta }{\eta_1} \cdot \mathcal{A}_\mathrm{env} - 2\phi_1\biggr] 
\biggl\{
y_2 ~-~ \frac{2 (\delta\eta) y_1}{\eta_1} \biggl[ 3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{interface} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~y_2 \biggl[4\phi_1  \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
y_1\biggl[ 4\phi_1 - 2(\delta\eta)^2 \cdot \mathcal{B}_\mathrm{env} \biggr] 
~-~  
\frac{2 (\delta\eta) y_1}{\eta_1} \biggl[\frac{\delta\eta }{\eta_1} \cdot \mathcal{A}_\mathrm{env} - 2\phi_1\biggr] 
\biggl\{3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{interface} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{y_2}{y_1} \biggl[\phi_1  \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \phi_1 - \frac{(\delta\eta)^2}{2} \cdot \mathcal{B}_\mathrm{env} \biggr] 
~-~  
\frac{ (\delta\eta) }{2\eta_1} \biggl[\frac{\delta\eta }{\eta_1} \cdot \mathcal{A}_\mathrm{env} - 2\phi_1\biggr] 
\biggl\{3\biggl(\frac{\gamma_c}{\gamma_e}  -1\biggr) + \frac{\gamma_c}{\gamma_e} \cdot \frac{d\ln x}{d\ln\xi}\biggr|_\mathrm{interface} 
\biggr\} \, .
</math>
  </td>
</tr>
</table>