Editing SSC/Stability/BiPolytropes/Pt2

=Review of the Analysis by Murphy &amp; Fiedler (1985b)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes|Part I: &nbsp; The Search]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt2|Part II:&nbsp; Review of MF85b]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Stability/BiPolytropes/Pt3|III:&nbsp; (5,1) Radial Oscillations]]<br />&nbsp;</td>
  <td align="center" bgcolor="lightblue"><br />[[SSC/Stability/BiPolytropes/Pt4|IV:&nbsp; Reconciliation]]<br />&nbsp;</td>
</tr>
<tr>
  <td align="left" width="100%" colspan="4">
&nbsp; &nbsp; These four chapters, labeled Parts I - IV, are segments of the much longer chapter titled, [[SSC/Stability/BiPolytropes/PlannedApproach|SSC/Stability/BiPolytropes/PlannedApproach]].  An [[SSC/Stability/BiPolytropes/Index|accompanying organizational index]] has helped us write this chapter succinctly.
  </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>

<div id="SurfaceCondition">
<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 {{ MF85b }} &#8212; 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 {{ MF85b }} 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>
</div>

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

==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 {{ LW58 }}, 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.  

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">'''Reaffirmation:'''</font> &nbsp;  In our [[SSC/Perturbations#The_Eigenvalue_Problem|introductory discussion]] of the eigenvalue problem, we adopted the following expression for the time-dependent pressure,

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

<tr>
  <td align="right">
<math>P(m,t)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>P_0(m) + P_1(m,t) = P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \, .</math>
  </td>
</tr>
</table>
In this expression, <math>P_0(m)</math> is the function that details how the unperturbed pressure varies with Lagrangian mass shell <math>(m)</math>, and <math>P_1 = \delta P(m) e^{i\omega t}</math> traces the variation of the pressure away from its equilibrium value at each mass shell. The time-dependent and spatially dependent behavior of <math>P_1</math> has been separated, with <math>\delta P</math> carrying information about the function's spatial dependence.  Furthermore, we have adopted the shorthand notation,
<div align="center"><math>p(m) \equiv \frac{\delta P(m)}{P_0(m)} \, .</math></div>
We can just as well use <math>r_0(m)</math> to tag the (initial, unperturbed location of the) Lagrangian mass shells, in which case we write,
<div align="center"><math>p(r_0) = \frac{\delta P(r_0)}{P_0(r_0)} \, ,</math> &nbsp; &nbsp; and, similarly, &nbsp; &nbsp;<math>d(r_0)= \frac{\delta \rho(r_0)}{\rho_0(r_0)} \, ,</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>x(r_0)= \frac{\delta r(r_0)}{r_0} \, .</math></div>

These three spatially dependent quantities &#8212; <math>p, d,</math> and <math>x</math> &#8212; are related to one another via the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|set of linearized governing relations]], namely,

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
p = \gamma_\mathrm{g} d  \, .
</math>
</td></tr>
</table>
</div>
Using the third of these expressions to replace <math>d</math> in favor of <math>p</math> in the first expression, we find that,
<table border="0" align="center">

<tr>
  <td align="right">
<math>r_0 \frac{dx}{dr_0}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>-3x - \frac{p}{\gamma_g}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~p</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>-\gamma_g x \biggl[3 +  \frac{d \ln x}{d \ln r_0} \biggr] \, ,</math>
  </td>
</tr>
</table>
which is identical to the pressure-perturbation expression used by {{ LW58 }} and referenced above.  As they state, the function, <math>p = \delta P/P_0</math>, should be continuous across the core-envelope interface.
</td></tr></table>
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), {{ LW58 }} 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 {{ MF85b }} 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 {{ LW58 }}.

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