Editing Appendix/Ramblings/BiPolytropeStability

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=Ramblings: Marginally Unstable Bipolytropes=

Our aim is to determine whether or not there is a relationship between (1) equilibrium models at turning points along bipolytrope sequences and (2) bipolytropic models that are marginally (dynamically) unstable toward collapse (or dynamical expansion).

==Overview==
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  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined]]Figure 1: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
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  <td align="center" colspan="1">
[[File:MassVsRadiusCombined02.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
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<table border="0" cellpadding="8" align="right">
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  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 1: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
  </th>
</tr>
<tr>
  <td align="center" colspan="1">
[[File:DFBsequenceB.png|300px|Equilibrium sequences of Pressure-Truncated Polytropes]]
  </td>
</tr>
</table>
We expect the  content of this chapter &#8212; which examines the relative stability of bipolytropes &#8212; to parallel in many ways the content of an [[SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|accompanying chapter in which we have successfully analyzed the relative stability of pressure-truncated polytopes]].  Figure 1, shown here on the right, has been copied from [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|a closely related discussion]].  The curves show the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range <math>1 \le n \le 6</math>.  ([[SSC/Stability/InstabilityOnsetOverview#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Another version of this figure]] includes the isothermal sequence.)  On each sequence for which <math>~n \ge 3</math>, the green filled circle identifies the model with the largest mass.  We have shown ''analytically'' that the oscillation frequency of the fundamental-mode of radial oscillation is precisely zero<sup>&dagger;</sup> for each one of these maximum-mass models.  As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.

----

<sup>&dagger;</sup>In each case, the fundamental-mode oscillation frequency is precisely zero if, and only if, the adiabatic index governing expansions/contractions is related to the underlying ''structural'' polytropic index via the relation, <math>~\gamma_g = (n + 1)/n</math>, and if a constant surface-pressure boundary condition is imposed.

----


In another [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|accompanying chapter]], we have used purely analytic techniques to construct equilibrium sequences of spherically symmetric bipolytropes that have, <math>~(n_c,n_e) = (5,1)</math>.   For a given choice of <math>~\mu_e/\mu_c</math> &#8212; the ratio of the mean-molecular weight of envelope material to the mean-molecular weight of material in the core &#8212; a physically relevant sequence of models can be constructed by steadily increasing the value of the dimensionless radius at the core/envelope interface, <math>~\xi_i</math>, from zero to infinity.  Figure 2, which has been copied from this separate chapter, shows how the fractional core mass, <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, varies with the fractional core radius, <math>q \equiv r_\mathrm{core}/R</math>, along sequences having six different values of <math>~\mu_e/\mu_c</math>: 1 (blue diamonds), &frac12; (red squares), 0.345 (dark purple crosses), &#8531; (pink triangles), 0.309 (light green dashes),  and &frac14; (purple asterisks).  Along each of the model sequences, points marked by solid-colored circles correspond to models whose interface parameter, <math>~\xi_i</math>, has one of three values:  0.5 (green circles), 1 (dark blue circles), or 3 (orange circles).

When modeling bipolytropes, the default expectation is that an increase in <math>\xi_i</math> along a given sequence will correspond to an increase in the relative size &#8212; both the radius and the mass &#8212; of the core. As Figure 2 illustrates, this expectation is realized along the sequences marked by blue diamonds (<math>~\mu_e/\mu_c = 1</math>) and by red squares (<math>~\mu_e/\mu_c = </math>&frac12;).  But the behavior is different along the other four illustrated sequences.  For sufficiently large <math>~\xi_i</math>, the relative radius of the core begins to decrease.  Furthermore, along sequences for which <math>~\mu_e/\mu_c < \tfrac{1}{3}</math>, eventually the fractional mass of the core reaches a maximum and, thereafter, decreases even as the value of <math>~\xi_i</math> continues to increase.  (Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence|yet another accompanying chapter]].)  

<font color="red">'''The principal question is:'''</font>  ''Along bipolytropic sequences, are maximum-mass models associated with the onset of dynamical instabilities?''

==Planned Approach==

<table border="0" cellpadding="5" width="25%" align="right">
<tr>
<th colspan="2" align="center">Figure 2:  Equilibrium Sequences of Bipolytropes with <math>~(n_c,n_e) = (5,1)</math></th>
</tr>
<tr>
  <td align="center" colspan="2" bgcolor="white">
[[Image:PlotSequencesBest02.png|300px|center]]
  </td>
</tr>
</table>
Ideally we would like to answer the just-stated "principal question" using purely analytic techniques.  But, to date, we have been unable to fully address the relevant issues analytically, even in what would be expected to be the simplest case: &nbsp; [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|bipolytropic models that have <math>~(n_c,n_e) = (0, 0)</math>]].  Instead, we will streamline the investigation a bit and proceed &#8212; at least initially &#8212; using a blend of techniques.  We will investigate the relative stability of bipolytropic models having <math>~(n_c,n_e) = (5,1) </math> whose ''equilibrium structures'' are completely defined analytically; then the eigenvectors describing radial modes of oscillation will be determined, one at a time, by solving the relevant LAWE(s) numerically.  We are optimistic that this can be successfully accomplished because we have had experience numerically integrating the LAWE that governs the oscillation of:
* [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Isolated n = 3 polytropes]] &#8212; including a quantitative comparison against [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild's (1941)] published work;
* [[SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres|Pressure-truncated isothermal spheres]] &#8212; including a quantitative comparison against the published analysis of [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff &amp; Van Horn (1974)]; and
* [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Pressure-truncated n = 5 polytropes]].

A key reference throughout this investigation will be the paper by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy &amp; R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)].  They studied ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models.''  Specifically, their underlying equilibrium models were bipolytropes that have <math>~(n_c,n_e) = (1, 5)</math>.  In an [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|accompanying chapter]], we describe in detail how Murphy &amp; Fiedler obtained these equilibrium bipolytropic structures and detail some of their equilibrium properties.

Here are the steps we initially plan to take:

* Governing LAWEs:
** Identify the relevant LAWEs that govern the behavior of radial oscillations in the <math>~n_c = 5</math> core and, separately, in the <math>~n_e = 1</math> envelope.  Check these LAWE specifications against the published work of [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)].
** Determine the matching conditions that must be satisfied across the core/envelope interface.  Be sure to take into account the critical interface ''jump'' conditions spelled out by [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)], as we have already discussed in the context of an [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|analysis of radial oscillations in zero-zero bipolytropes]].
* Determine what surface boundary condition should be imposed on physically relevant LAWE solutions, i.e., on the physically relevant radial-oscillation eigenvectors.
* Initial Analysis:
** Choose a maximum-mass model along the bipolytropic sequence that has, for example, <math>~\mu_e/\mu_c = 1/4</math>.  Hopefully, we will be able to identify precisely (analytically) where this maximum-mass model lies along the sequence.  <font color="red">'''Yes!'''</font>  Our [[SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass|earlier analysis]] does provide an analytic prescription of the model that sits at the maximum-mass location along the chosen sequence.
** Solve the relevant eigenvalue problem for this specific model, initially for <math>~(\gamma_c, \gamma_e) = (6/5, 2)</math> and initially for the fundamental mode of oscillation.

=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>

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

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

{{Math/EQ_RadialPulsation01}}
</div>

<!--
<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave Equation'''</font><br />

<math>
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \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 = 0 \, ,
</math>
</div>
-->

whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.  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>

==Profile==

Now, referencing the [[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>


Therefore, throughout the core 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>~\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>
In which case the governing LAWE throughout 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}{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>

Next, 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>
So, the governing LAWE throughout the envelope 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}{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>

==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>

==Try Splitting Analysis Into Separate Core and Envelope Components==
===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>

=Begin Our Analysis=
==Relevant LAWEs==

The LAWE that is relevant to polytropic spheres may be written as,
<div align="center">
{{ Math/EQ_RadialPulsation02 }}
</div>

===Core 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\xi^2} + \biggl[ 4 - 6Q_5 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ 6 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr)  \frac{\xi^2}{\theta_5} 
- \alpha_\mathrm{core} Q_5\biggr]  \frac{x}{\xi^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\theta_5}{d\ln\xi} \, .</math>
  </td>
</tr>
</table>

From our study of the [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|equilibrium structure of <math>(n_c, n_e) = (5, 1)</math> bipolytropes]], we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\theta_5 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1 / 2} \,  ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{d\theta_5}{d\xi} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\xi}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
Q_5 = - \frac{\xi}{\theta_5} \cdot \frac{d\theta_5}{d\xi}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{1 / 2} \frac{\xi^2}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\xi^2}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>

Hence, for the core the governing 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\xi^2\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \biggr\} \frac{1}{\xi} \cdot \frac{dx}{d\xi}  
+ 6 \biggl\{ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{core} } \biggr)  \xi^2\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{1 / 2} 
- ~\alpha_\mathrm{core} \cdot \frac{\xi^2}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \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\{ 2 -  \xi^2\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \biggr\} \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{\gamma_\mathrm{core} } \biggr)  \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{1 / 2} 
- ~2\alpha_\mathrm{core} \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1} \biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x}{d\xi^2} + \biggl\{ 2 -  3\xi^2\biggl[ 3 + \xi^2 \biggr]^{-1} \biggr\} \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  \biggl[ 3 + \xi^2 \biggr]^{1 / 2} 
- ~6\alpha_\mathrm{core} \biggl[ 3 + \xi^2 \biggr]^{-1} \biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(3 + \xi^2) \frac{d^2x}{d\xi^2} + \biggl\{ 2(3 + \xi^2) -  3\xi^2 \biggr\} \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl\{ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr\}  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(3 + \xi^2) \frac{d^2x}{d\xi^2} + ( 6 - \xi^2 ) \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  x \, .
</math>
  </td>
</tr>
</table>
This exactly matches our [[SSC/Stability/n5PolytropeLAWE#Specifically_for_n.3D5_Configurations|derivation performed in the context of pressure-truncated polytropes]].  When we insert the eigenfunction obtained via a ''[[SSC/Stability/n5PolytropeLAWE#Eureka_Moment|Eureka Moment]]'' on 3/6/2017,
<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>x_0\biggl[1 - \frac{\xi^2}{15}\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{dx}{d\xi}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{2x_0 \xi}{15} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{d^2x}{d\xi^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{2x_0 }{15} \, ,</math>
  </td>
</tr>
</table>
we obtain,

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

<tr>
  <td align="right">
<math>
(3 + \xi^2) \frac{d^2x}{d\xi^2} + ( 6 - \xi^2 ) \frac{2}{\xi} \cdot \frac{dx}{d\xi}  
+ \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  x 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~(3 + \xi^2) \frac{2x_0 }{15} - 2( 6 - \xi^2 ) \frac{2x_0 }{15}  
+ \frac{x_0}{15}\biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  (15 - \xi^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{x_0}{15} \biggl\{
2(3 + \xi^2)  + 4( 6 - \xi^2 )   
- \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  (15 - \xi^2) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\frac{x_0}{15} \biggl\{
30 - 2\xi^2 ~+ ~6\alpha_\mathrm{core}   (15 - \xi^2) 
~- \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
\biggr]  (15 - \xi^2) 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

The right-hand-side goes to zero if <math>\alpha_\mathrm{core} = - 1/3</math> and <math>\sigma_c = 0</math>.  Also, notice that,

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

<tr>
  <td align="right">
<math>~ \frac{d\ln x}{d\ln \xi}\biggr|_\mathrm{core} = \frac{\xi}{x}\cdot \frac{dx}{d\xi} \biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2x_0 \xi^2}{15} \biggl\{ x_0\biggl[1 - \frac{\xi^2}{15}\biggr] \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl\{ \frac{15}{2\xi^2}\biggl[\frac{15 -\xi^2}{15}\biggr] \biggr\}^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl[\frac{15 -\xi^2}{2\xi^2}\biggr]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \biggl[1 - \frac{15}{\xi^2} \biggr]^{-1} \, .</math>
  </td>
</tr>
</table>

So, with <math>~\gamma_c = 6/5</math> and <math>~\gamma_e = 2</math> we need,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\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>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3\biggl(\frac{3}{5}  -1\biggr) + \frac{6}{5}  \biggl[1 - \frac{15}{\xi_i^2} \biggr]^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{6}{5} \biggl\{  \biggl[\frac{\xi_i^2 - 15}{\xi_i^2} \biggr]^{-1} - 1 \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{6}{5} \biggl[\frac{15}{\xi_i^2 - 15} \biggr] \, .</math>
  </td>
</tr>
</table>

===Envelope 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\eta^2} + \biggl[ 4 - 2 Q_1 \biggr] \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{env} } \biggr)  \frac{\eta^2}{\theta_1} 
- \alpha_\mathrm{env} Q_1\biggr]  \frac{x}{\eta^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_1}{d\ln\eta} \, .</math>
  </td>
</tr>
</table>

As has already been pointed out, above, for n = 1 polytropic spheres, this LAWE becomes,

<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} +  \frac{2}{\eta} \biggl[ 1 +  \eta\cot\eta  \biggr]\frac{dx}{d\eta}  
+ \biggl[ \frac{\gamma_g}{\gamma_\mathrm{env}}\biggl( \omega_k^2 \theta_c \biggr)  \frac{\eta}{\sin\eta} 
+ \frac{2 \alpha_\mathrm{env} ( \eta\cos\eta - \sin\eta) }{\eta^2 \sin\eta} \biggr]  x \, .
</math>
  </td>
</tr>
</table>

In a [[SSC/Stability/InstabilityOnsetOverview#Configurations_Having_an_Index_Less_Than_Three|separate chapter]], we explain that the analytically defined eigenfunction that satisfies this LAWE &#8212; when <math>\omega_k^2 = 0</math> and <math>\alpha_\mathrm{env} = +1</math> &#8212; is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_P\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3b_e}{\eta^2}\biggl[ 1- \eta \cot\eta  \biggr]   \, .
</math>
  </td>
</tr>
</table>
</div>

===Summary===

For a given choice of the ''equilibrium'' model parameters, <math>\xi_i</math> and <math>\mu_e/\mu_c</math>, we can pull the parameters and profiles of the base equilibrium model from our accompanying chapter on <math>(n_c, n_e) = (5, 1)</math> bipolytropes.  [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|Note, in particular, that]]:  
<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{3}{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_i \, .</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} \frac{1}{\sqrt{2\pi}} \biggl(1 + \frac{\xi_i^2}{3}  \biggr) \biggr] \eta \, ,</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>\eta_i \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>
\eta_i + \frac{\pi}{2} + \tan^{-1} \biggl[\frac{1}{\eta_i} - \frac{\xi_i}{\sqrt{3}}   \biggr]
\, .
</math>
  </td>
  <td align="left" colspan="2">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  
  </td>
</tr>
</table>


Then, for a choice of the pair of exponents, <math>\gamma_\mathrm{core}</math> and <math>\gamma_\mathrm{env}</math>, that govern the behavior of adiabatic oscillations, the LAWE to be numerically integrated 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>
(3 + \xi^2) \frac{d^2x_\mathrm{core}}{d\xi^2} + ( 6 - \xi^2 ) \frac{2}{\xi} \cdot \frac{dx_\mathrm{core}}{d\xi}  
+ \biggl[ \biggl( \frac{\sigma_c^2}{\sqrt{3}~\gamma_\mathrm{core} } \biggr)  ( 3 + \xi^2 )^{3 / 2} 
- ~6\alpha_\mathrm{core} \biggr]  x_\mathrm{core} \, ,
</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>~0 \le \xi \le \xi_i \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x_\mathrm{env}}{d\eta^2} + \biggl[ 1 +  \eta\cot\eta  \biggr] \frac{2}{\eta} \cdot \frac{dx_\mathrm{env}}{d\eta}  
+ 2 \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_\mathrm{env} } \biggr)  \frac{\eta^3}{\sin\eta} 
- \alpha_\mathrm{env} ( 1 -  \eta\cot\eta )\biggr]  \frac{x_\mathrm{env}}{\eta^2} \, ,
</math>
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; for, 
  </td>
  <td align="left">
<math>~\eta_i \le \eta \le \eta_s \, .</math>
  </td>
</tr>
</table>

The boundary conditions at the center of the configuration are, <math>x_\mathrm{env} = 1</math>, while <math>dx_\mathrm{env}/d\xi = 0</math>.

As [[#Interface_Conditon|described above]], the two interface boundary conditions are, <math>x_\mathrm{env} = x_\mathrm{core}</math>, and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\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_\mathrm{core}}{\gamma_\mathrm{env}}  -1\biggr) + \frac{\gamma_\mathrm{core}}{\gamma_\mathrm{env}} \biggl( \frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr)_{\xi=\xi_i} \, .</math>
  </td>
</tr>
</table>

Finally, drawing from a [[SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29|related discussion of the surface boundary condition for isolated n = 3 polytropes]], at the surface we need,

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

<tr>
  <td align="right">
<math>\frac{d\ln x_\mathrm{env}}{d\ln \eta}\biggr|_\mathrm{surface}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\gamma_\mathrm{env}} \biggl( 4 - 3\gamma_\mathrm{env} + \frac{\omega^2 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>\biggl[ \frac{3\omega^2 R^3}{4\pi G \gamma_\mathrm{env} \bar\rho} - \alpha_\mathrm{env} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \biggl[ \frac{\sigma_c^2}{\gamma_\mathrm{env} } \biggl(\frac{\rho_c}{\bar\rho}\biggr) -2 \alpha_\mathrm{env} \biggr] \, ,</math>
  </td>
</tr>
</table>

where, for an <math>(n_c, n_e) = (5, 1)</math> bipolytrope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\rho_c}{\bar\rho}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s^2}{3A\theta_i^5} \, .
</math>
  </td>
</tr>
</table>

=See Also=

* [http://adsabs.harvard.edu/abs/2018Sci...362..201D K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206)], ''A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.''


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