Editing SSC/Stability/MurphyFiedler85 (section)

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

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

{{Math/EQ_RadialPulsation01}}
</div>

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

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

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

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

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

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

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

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

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

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

</table>
</div>

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{ M_r^*}{(r^*)^2}\biggl[  G\rho_c \biggl( \frac{K_c}{G}\biggr)^{1 / 2} \biggr] 
\, .
</math>
  </td>
</tr>
</table>
Hence, multiplying the LAWE through by <math>~(K_c/G)</math> gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl[\frac{4}{r^*} -\biggl( \frac{K_c}{G} \biggr)^{1 / 2}\biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr*} 
+ \biggl( \frac{K_c}{G} \biggr)\biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x
</math>
  </td>
</tr>

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

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

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

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

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

==Profile==

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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

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

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


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

Note that we ''could have'' rewritten the governing LAWE throughout the core as,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ 4 -2(1-\xi\cot\xi )\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \frac{\xi}{\sin\xi} \biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\alpha_\mathrm{g}~\frac{4\pi \sin\xi}{\xi^3} (1 - \xi\cot\xi) \biggr\}  x 
</math>
  </td>
</tr>

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

and we ''could have'' rewritten the governing LAWE throughout the envelope as,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{dr*^2} + \biggl\{ 4 - 6\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) } \biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{1}{\theta_i \phi(\eta) }
\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~6\alpha_\mathrm{g}~\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \theta_i \biggl(-\eta \frac{d\phi}{d\eta} \biggr) 
\biggl[ \frac{1}{\eta} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \biggr]^2 \biggr\}  x 
</math>
  </td>
</tr>

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

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

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

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

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

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

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

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

==Numerical Integration==

===General Approach===
Here, we begin by recognizing that the 2<sup>nd</sup>-order ODE that must be integrated to obtain the desired eigenvectors has the generic form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x'' + \frac{\mathcal{H}}{r^*} x' + \mathcal{K}x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

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

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

===Special Handling at the Interface===

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

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

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

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

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

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

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

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

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

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

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

==Eigenvectors==

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

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

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

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


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



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

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


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


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