Editing SSC/Stability/BiPolytrope00 (section)

==Eigenvector==
===Core Segment===

<div align="center" id="Table1">
<table border="1" cellpadding="8">
<tr>
  <th align="center" colspan="3"><font size="+1">Table 1:</font>&nbsp; Analytically Specifiable Core Eigenvectors</th>
</tr>
<tr>
  <td align="center" colspan="1">
Mode
  </td>
  <td align="center" colspan="1">
Core Eigenfunction<br />
<br />
<math>
g^2 \equiv 1  + \biggl(\frac{\rho_e}{\rho_c}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( 1-q \biggr) + 
\frac{\rho_e}{\rho_c} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] 
</math>
  </td>
  <td align="center">Core Eigenfrequency <br />
<math>~\frac{3\omega_\mathrm{core}^2}{2\pi \gamma_c G \rho_c} = 2[\alpha_c + j(2j+5)]</math>
  </td>
</tr>
<tr>
<td align="center">
<math>~j=0 </math>
</td>
<td align="left">
<math>~x_\mathrm{core} = a_0 </math>
</td>
<td align="center">
<math>~6-8/\gamma_c</math>
</td>
</tr>

<tr>
<td align="center">
<math>~j=1 </math>
</td>
<td align="left">
<math>~x_\mathrm{core} = a_0 \biggl[ 1 - \frac{7}{5}\biggr(\frac{\xi^2}{g^2}\biggr) \biggr]</math>
</td>
<td align="center">
<math>~20-8/\gamma_c</math>
</td>
</tr>

<tr>
<td align="center">
<math>~j=2 </math>
</td>
<td align="left">
<math>~x_\mathrm{core} = a_0 \biggl[ 1 - \frac{18}{5}\biggr(\frac{\xi^2}{g^2}\biggr) + \frac{99}{35}\biggr(\frac{\xi^2}{g^2}\biggr)^2 \biggr]</math>
</td>
<td align="center">
<math>~42-8/\gamma_c</math>
</td>
</tr>
</table>
</div>

===Envelope Segment===

<div align="center" id="Table2">
<table border="1" cellpadding="8">
<tr>
  <th align="center" colspan="3"><font size="+1">Table 2:</font>&nbsp; Analytically Specifiable Envelope Eigenvectors</th>
</tr>
<tr>
  <td align="center" colspan="1">
Mode
  </td>
  <td align="center" colspan="1">
Envelope Eigenfunction<br />
<br />
<math>~c_0 \equiv -1 \pm \sqrt{1+\alpha_e}</math>
  </td>
  <td align="center">Envelope Eigenfrequency<br />
<math>~\frac{3\omega_\mathrm{env}^2}{2\pi \gamma_e G \rho_e} = 3[\alpha_e + c_0(2\ell+1) + \ell(3\ell+5)]</math><br /><br /><math>~=2\alpha_e + (c_0+3\ell)(c_0+3\ell + 5)</math>
  </td>
</tr>
<tr>
<td align="center">
<math>~\ell=0 </math>
</td>
<td align="left">
<math>~x_\mathrm{env} = b_0 \xi^{c_0}</math>
</td>
<td align="center">
<math>~3[\alpha_e + c_0]</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\ell=1 </math>
</td>
<td align="left">
<math>~x_\mathrm{env} = b_0 \xi^{c_0}\biggl\{1 + \biggl[  
\frac{c_0(c_0+5)-(c_0+3)(c_0+8)}{(c_0+3)(c_0+5) - \alpha_e}
\biggr](q\xi)^3  \biggr\}</math>
</td>
<td align="center">
<math>~3[\alpha_e + 3c_0 +8]</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\ell=2 </math>
</td>
<td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\mathrm{env} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~b_0 \xi^{c_0}\biggl\{1 + \biggl[ \frac{c_0(c_0+5)-(c_0+6)(c_0+11)}{(c_0+3)(c_0+5) - \alpha_e}\biggr](q\xi)^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \frac{c_0(c_0+5)-(c_0+6)(c_0+11)}{(c_0+3)(c_0+5) - \alpha_e}\biggr]\biggl[ \frac{(c_0+3)(c_0+8)-(c_0+6)(c_0+11)}{(c_0+6)(c_0+8) - \alpha_e}\biggr](q\xi)^6 
\biggr\}
</math>
  </td>
</tr>
</table>


</td>
<td align="center">
<math>~3[\alpha_e + 5c_0 +22]</math>
</td>
</tr>
</table>
</div>

===Piecing Together===
Here we illustrate how the two segments of the eigenfunction can be successfully pieced together for the specific case of <math>(\ell,j) = (2,1)</math>.

<span id="STEP1"><font color="red"><b>STEP 1:</b></font></span>  Choose a value of the adiabatic exponent for the envelope, <math>\gamma_e</math>.  Then, the values of both <math>\alpha_e</math> and <math>c_0</math> are known as well; actually, because it is the root of a quadratic equation, <math>c_0</math> can, in general, take on one of a ''pair'' of values.  We will elaborate on this further, below.

<span id="STEP2"><font color="red"><b>STEP 2:</b></font></span>  Acknowledging that the value of <math>q</math> has yet to be determined, fix the value of the leading, overall scaling coefficient, <math>b_0</math>, such that<sup>&dagger;</sup> <math>x_\mathrm{env} = 1</math> at the interface, that is, at <math>\xi = 1</math>.  For the case of <math>\ell=2</math>, this means that, throughout the envelope, the eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x_{\ell=2} |_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\xi^{c_0}\biggl[ \frac{ 1 +  q^3 A_{2} \xi^{3} +  q^6 A_{2}B_{2}\xi^{6} }{ 1 +  q^3 A_{2}  +  q^6 A_{2}B_{2}}\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the values of the newly introduced coefficients,
<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>A_{2}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{c_0(c_0+5) - (c_0 + 6)(c_0 + 11)}{(c_0 + 3)(c_0+5) - \alpha_e}\biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_{2}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(c_0+3)(c_0+8) - (c_0 + 6)(c_0 + 11)}{(c_0 + 6)(c_0+8) - \alpha_e}\biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
are also both known.

<span id="STEP3"><font color="red"><b>STEP 3:</b></font></span>  Recognizing that this segment of the eigenfunction will only satisfy the envelope's LAWE if we restrict our discussion to equilibrium models for which <math>g^2 = \mathcal{B} = [(1+8q^3)/(1+2q^3)^{2}]</math>, we must insert this same restriction on <math>g^2</math> into the core's eigenfunction.  At the same time, we should fix the value of the leading, overall scaling coefficient, <math>a_0</math>, such that<sup>&dagger;</sup> <math>x_\mathrm{core} = 1</math> at the interface <math>(\xi = 1)</math>.  For the case of <math>j=1</math>, this means that, throughout the core, the eigenfunction is,

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

<tr>
  <td align="right">
<math>x_{j=1} |_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5(1+8q^3) -  7 (1+2q^3)^2 \xi^2}{5(1+8q^3)-7(1+2q^3)^2} \, .</math>
  </td>
</tr>
</table>
</div>


<span id="STEP4"><font color="red"><b>STEP 4:</b></font></span>  Now we need to match the two eigenfunctions at the interface.  Following the discussion in &sect;&sect;57 &amp; 58 of {{ LW58full }}, 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 = 1)</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=1}</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=1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=1}</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=1} \, .</math>
  </td>
</tr>
</table>
</div>

In the context of this interface-matching constraint, {{ LW58 }} state the following (see their equation 62.1): &nbsp; <font color="darkgreen">"In the static</font> (''i.e.,'' unperturbed equilibrium) <font color="darkgreen">model</font> &hellip; <font color="darkgreen">discontinuities in <math>\rho</math> or in <math>\gamma</math> might occur at some [radius]</font>.  <font color="darkgreen">In the first case</font> &#8212; that is, a discontinuity only in density, while <math>\gamma_e = \gamma_c</math> &#8212; the interface conditions <font color="darkgreen">imply the continuity of <math>\tfrac{1}{x} \cdot \tfrac{dx}{d\xi}</math> at that [radius].  In the second case</font> &#8212;  that is, a discontinuity in the adiabatic exponent &#8212; <font color="darkgreen">the dynamical condition may be written</font> as above.  <font color="darkgreen">This implies a discontinuity of the first derivative at any discontinuity of <math>\gamma</math>."</font>

When evaluated at the interface, the logarithmic derivatives of our pair of parameterized eigenfunction expressions are, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d\ln x_\mathrm{env}}{d\ln \xi} \biggr|_{\xi=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0 + \frac{3A_{2}\Chi +  6A_{2}B_{2} \Chi^2}{1 + A_{2}\Chi + A_{2}B_{2}\Chi^2} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{d\ln x_\mathrm{core}}{d\ln \xi} \biggr|_{\xi=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{14(1+2\Chi)^2}{7(1+2\Chi)^2 - 5(1+8\Chi)} \, ,</math>
  </td>
</tr>
</table>
</div>
where we have made the notation substitution, <math>\Chi \equiv q^3</math>.  Allowing for a step function in the adiabatic exponent at the interface, our interface-matching constraint is, therefore,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{\gamma_c}{\gamma_e} \biggl[ \frac{14(1+2\Chi)^2}{7(1+2\Chi)^2 - 5(1+8\Chi)} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
c_0 + \frac{3A_{2}\Chi +  6A_{2}B_{2} \Chi^2}{1 + A_{2}\Chi + A_{2}B_{2}\Chi^2}
- 3\biggl(\frac{\gamma_c}{\gamma_e} -1\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mathfrak{g}_0 + (\mathfrak{g}_0+3)A_{2}\Chi +  (\mathfrak{g}_0+6)A_{2}B_{2} \Chi^2}{1 + A_{2}\Chi + A_{2}B_{2}\Chi^2}
\, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>\mathfrak{g}_0 \equiv c_0 + 3\biggl(1-\frac{\gamma_c}{\gamma_e} \biggr) \, .</math>
</div>

This can be rewritten as,
<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>
[\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] [7(1+2\Chi)^2 - 5(1+8\Chi)]
- 14(\gamma_c/\gamma_e) (1+2\Chi)^2 [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] [2 - 12\Chi + 28\Chi^2 ]
- (14 + 56\Chi + 56 \Chi^2)(\gamma_c/\gamma_e) [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2[\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] 
-12\Chi [\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] 
+ 28\Chi^2 [\mathfrak{g}_0 + (\mathfrak{g}_0 + 3)A_{2}\Chi + (\mathfrak{g}_0 + 6)A_{2}B_{2} \Chi^2] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- 14(\gamma_c/\gamma_e) [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
- 56(\gamma_c/\gamma_e)\Chi  [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] 
- 56 (\gamma_c/\gamma_e)\Chi^2 [1 + A_{2}\Chi + A_{2}B_{2}\Chi^2] \, .
</math>
  </td>
</tr>
</table>
</div>

Or we have, equivalently,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a\Chi^4 + b\Chi^3 + c\Chi^2 +d\Chi +e </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>e</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2\mathfrak{g}_0 - 14(\gamma_c/\gamma_e)  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>d</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2(\mathfrak{g}_0+3)A_{2} - 12\mathfrak{g}_0 - 14(\gamma_c/\gamma_e)A_{2} - 56(\gamma_c/\gamma_e)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2[\mathfrak{g}_0 + 3 -7(\gamma_c/\gamma_e)]A_{2} - 4[14(\gamma_c/\gamma_e) + 3\mathfrak{g}_0] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2(\mathfrak{g}_0+6)A_{2}B_{2} - 12(\mathfrak{g}_0+3)A_{2} + 28\mathfrak{g}_0 - 14(\gamma_c/\gamma_e)A_{2}B_{2} - 56(\gamma_c/\gamma_e)A_{2} - 56(\gamma_c/\gamma_e)  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2[\mathfrak{g}_0 + 6 -7(\gamma_c/\gamma_e)] A_{2}B_{2} - 4[9 + 14(\gamma_c/\gamma_e) + 3\mathfrak{g}_0]A_{2} + 28[\mathfrak{g}_0 - 2(\gamma_c/\gamma_e)]  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>b</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>-12(\mathfrak{g}_0+6)A_{2}B_{2} + 28(\mathfrak{g}_0+3)A_{2} - 56(\gamma_c/\gamma_e)A_{2}B_{2} - 56(\gamma_c/\gamma_e)A_{2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 4[3\mathfrak{g}_0+18+14(\gamma_c/\gamma_e)]A_{2}B_{2} + 28[\mathfrak{g}_0+3-2(\gamma_c/\gamma_e)]A_{2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>a</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>28[\mathfrak{g}_0+6 - 2(\gamma_c/\gamma_e)]A_{2}B_{2} \, .</math>
  </td>
</tr>
</table>


The physically relevant (real) root of this quartic equation in <math>\Chi</math> &#8212; see our [[SSC/Stability/BiPolytrope00Details#Quartic|accompanying detailed presentation]] &#8212; gives us the ''specific'' value of the dimensionless interface location, <math>q</math>, for which the values of the two eigenfunctions match at the interface, and for which the first derivatives of the two eigenfunctions are discontinuous by the properly prescribed amount at the interface.

===Example Solutions===

<div align="center" id="Table3">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="6"><font size="+1">Table 3:</font>&nbsp; Example Analytic Model Parameters for <math>~(\ell,j) = (2,1)</math><br />NOTE:  &nbsp;<math>\mathfrak{F}_\mathrm{core} = 14</math></th></tr>
<tr>
  <td align="center">Eigenfunction</td>
  <td align="center"><math>~\gamma_c ~(n_c)</math></td>
  <td align="center"><math>~\gamma_e</math></td>
  <td align="center"><math>~q</math></td>
  <td align="center"><math>~\frac{\rho_e}{\rho_c}</math></td>
  <td align="center"><math>~\sigma_c^2</math></td>
</tr>
<tr>
  <td>[[File:A21.png|center|thumb|100px|Model A21]]
  <td align="center">&nbsp;<math>~\frac{5}{3} ~~\biggl(\frac{3}{2} \biggr)</math></td>
  <td align="center">1.1340607</td>
  <td align="center">0.6684554</td>
  <td align="center">0.3739731</td>
  <td align="center">25.333333</td>
</tr>
<tr>
  <td>[[File:B21.png|center|thumb|100px|Model B21]]
  <td align="center">&nbsp;<math>~\frac{4}{3} ~~\biggl( 3 \biggr)</math></td>
  <td align="center">1.0263212</td>
  <td align="center">0.6385711</td>
  <td align="center">0.3424445</td>
  <td align="center">18.666666</td>
</tr>
<tr>
  <td>[[File:C21.png|center|thumb|100px|Model C21]]
  <td align="center">&nbsp;<math>~\frac{6}{5} ~~\biggl( 5 \biggr)</math></td>
  <td align="center">1.0028319</td>
  <td align="center">0.6187646</td>
  <td align="center">0.3214875</td>
  <td align="center">16.000000</td>
</tr>
</table>
</div>


It appears as though the eigenvectors (eigenfunction and eigenfrequency) of other radial oscillation modes can be identified by holding all other parameters fixed but changing the value of the quantum number, <math>~\ell</math>, in the [[#OtherSigmas|expression provided below]].  Picking the configuration identified as model '''C21''' in [[#Table3|Table 3]], for example, in addition to the parameter values provided in the table we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha_e = 3 - \frac{4}{\gamma_e} = -0.9887044</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~c_0 = \sqrt{1+\alpha_e} - 1 = -0.8937192 \, ,</math>
  </td>
</tr>
</table>
</div>

so we expect the variation in (the square of) the eigenfrequency, <math>~\sigma_c</math>, with <math>~\ell</math> to be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_c^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~12\biggl[\frac{c_0^2 + c_0(2\ell+3) + \ell(3\ell+5) }{(3-\alpha_e) } \biggr]\frac{\rho_e}{\rho_c} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.9671938[c_0^2 + c_0(2\ell+3) + \ell(3\ell+5) ] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.9671938[-1.8824236 +(3.2125616)\ell + 3\ell^2 ] \, .</math>
  </td>
</tr>
</table>

<div align="center" id="Table4">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="6"><font size="+1">Table 4:</font>&nbsp; Additional Hypothesized Oscillation Modes for Model C21</th>
</tr>
<tr>
  <td align="center"><math>~\ell = 0</math><p></p><math>~\sigma_c^2 = -1.821</math></td>
  <td align="center"><math>~\ell = 1</math><p></p><math>~\sigma_c^2 = +4.188</math></td>
  <td align="center"><math>~\ell = 2</math><p></p><math>~\sigma_c^2 = +16</math></td>
  <td align="center"><math>~\ell = 3</math><p></p><math>~\sigma_c^2 = +33.615</math></td>
  <td align="center"><math>~\ell = 4</math><p></p><math>~\sigma_c^2 = +57.033</math></td>
  <td align="center"><math>~\ell = 5</math><p></p><math>~\sigma_c^2 = +86.255</math></td>
</tr>
<tr>
  <td align="center">[[File:C01hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C11hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C21hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C31hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C41hypothetical.png|150 px|center]]</td>
  <td align="center">[[File:C51hypothetical.png|150 px|center]]</td>
</tr>
</table>
</div>

Each of the six plots displayed in [[#Table4|Table 4]] (click on a panel in order to view a larger image) was [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface.  At the interface: &nbsp; the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4|according to the above "Step 4" discussion]].  Each of the graphically illustrated Table 4 eigenfunctions has been scaled in such a way that the central value is unity; note that the panel labeled <math>(\ell=2; \sigma_c^2 = +16)</math> displays an eigenfunction that is identical to the ''analytically defined'' eigenfunction displayed as Model C21 in [[#Table3|Table 3]], but it has been rescaled &#8212; and by necessity inverted &#8212; to provide a central value of unity.


{{ SGFworkInProgress }}

===Old, Incorrect Solutions===
As is shown by the plot displayed in the right-hand panel of Figure 1, we have found different values of <math>q</math> for each choice (STEP 1) of <math>\gamma_e</math> (or, equivalently, choice of <math>\alpha_e</math>).  In this plot we have purposely flipped the horizontal axis so that the extreme left <math>(\alpha_e = +3)</math> represents an incompressible <math>(n = 0)</math> envelope, while the extreme right represents an isothermal <math>(\gamma_e = 1)</math> envelope. 
<!--
The green and orange curves, respectively, show as well how the corresponding model parameters, <math>\nu</math> and <math>\rho_e/\rho_c</math>, vary with <math>\alpha_e</math>.  The right-hand panel displays one example of an <math>(\ell, j) = (2,1)</math> eigenfunction that simultaneously satisfies the LAWE of the core and the LAWE of the envelope, and matches smoothly at the interface.  This ''particular'' plotted solution corresponds to the case of <math>\alpha_e = -0.35</math>, for which:  <math>\gamma_e =1.1940299</math>; <math>n_e = 5.1538462</math>; <math>q = 0.7943853</math>; <math>\nu =0.6675302</math>; and <math>\rho_e/\rho_c =0.5006468</math>.  
-->

<div align="center" id="Figure1">
<table border="1" cellpadding="8">
<tr><th align="center" colspan="4">Figure 1</th></tr>
<tr>
  <td align="center" colspan="3"><math>\alpha_e = -0.35 \, ;~~~c_0 = \sqrt{1+\alpha_e} - 1</math></td>
  <td align="center" rowspan="18">

<table border="0" width="100%">
<tr><td align="center">
[[File:Quartic21Solution02Corrected.png|500px|quartic solution]]
</td></tr>
<tr><td align="left">''Directly Above:'' Plot shows for ''which'' equilibrium bipolytropic configurations with <math>(n_c, n_e) = (0,0)</math> we are able to construct analytically prescribed eigenvectors for the radial oscillation mode, <math>(\ell, j) = (2,1)</math>.  The top (blue), middle (green), and bottom (orange) curves show how <math>q</math>, <math>\nu</math>, and <math>\rho_e/\rho_c</math> vary with the specified value of the envelope's adiabatic exponent over the full, physically reasonable range of the parameter, <math>-1 \le \alpha_e \le 3</math>.  For the upper portion of each curve (dark blue, dark green, dark orange), the parameter, <math>c_0</math>, is taken to be the "plus" root of its defining quadratic equation; the "minus" root defines <math>c_0</math> along the lower portion of each curve (light blue, light green, light orange).  

''Upper-left Quadrant:'' An <math>x(r_0/R)</math> plot showing the radial structure of the analytically prescribed eigenfunction for <math>\alpha_e = -0.35</math> and <math>c_0</math> (plus); its underlying, equilibrium model characteristics are identified by the black circular marker in the above plot.

''Lower-left Quadrant:'' The analytcially prescribed eigenfunction, <math>x(r_0/R)</math>,  for <math>\alpha_e = -0.9</math> and <math>c_0</math> (minus); its underlying, equilibrium model characteristics are identified by the yellow circular marker in the above plot.

<sup>&dagger;</sup>Note that, as displayed here, the sign has been flipped on both <math>x(r_0/R)</math> eigenfunctions so that, in practice, the amplitude at the interface is ''negative'' one, rather than positive one. Plotted in this way, we immediately recognize that both eigenfunctions are ''qualitatively'' similar to the <math>j = 2</math> radial oscillation eigenfunction that [[SSC/Stability/UniformDensity#Properties_of_Eigenfunction_Solutions|was derived by]] {{ Sterne37 }} in the context of isolated, homogeneous spheres.
</td></tr>
</table>

  </td>
</tr>
<tr>
  <td align="right"><math>c_0</math> (plus):</td>
  <td align="center"><math>-0.1937742</math></td>
  <td align="center" rowspan="8">
[[File:EigenfunctionP1Corrected.png|270px|quartic solution]]
  </td>
</tr>
<tr>
  <td align="right"><math>\gamma_e</math>:</td>
  <td align="center"><math>1.1940299</math></td>
</tr>
<tr>
  <td align="right"><math>n_e</math>:</td>
  <td align="center"><math>5.1538462</math></td>
</tr>
<tr>
  <td align="right"><math>q</math>:</td>
  <td align="center"><math>0.6840119</math></td>
</tr>
<tr>
  <td align="right"><math>\nu</math>:</td>
  <td align="center"><math>0.5466868</math></td>
</tr>
<tr>
  <td align="right"><math>\rho_e/\rho_c</math>:</td>
  <td align="center"><math>0.3902664</math></td>
</tr>
<tr>
  <td align="right"><math>\alpha_c</math>:</td>
  <td align="center"><math>+0.8326585</math></td>
</tr>
<tr>
  <td align="right"><math>\gamma_c</math>:</td>
  <td align="center"><math>+1.845579</math></td>
</tr>
<tr>
  <td align="center" colspan="3"><math>\alpha_e = -0.9 \, ;~~~c_0 = -\sqrt{1+\alpha_e} - 1</math></td>
</tr>
<tr>
  <td align="right"><math>c_0</math> (minus):</td>
  <td align="center"><math>- 1.3162278</math></td>
  <td align="center" rowspan="8">
[[File:EigenfunctionM1Corrected.png|270px|quartic solution]]
  </td>
</tr>
<tr>
  <td align="right"><math>\gamma_e</math>:</td>
  <td align="center"><math>1.0256410</math></td>
</tr>
<tr>
  <td align="right"><math>n_e</math>:</td>
  <td align="center"><math>39</math></td>
</tr>
<tr>
  <td align="right"><math>q</math>:</td>
  <td align="center"><math>0.5728050</math></td>
</tr>
<tr>
  <td align="right"><math>\nu</math>:</td>
  <td align="center"><math>0.4586270</math></td>
</tr>
<tr>
  <td align="right"><math>\rho_e/\rho_c</math>:</td>
  <td align="center"><math>0.2731929</math></td>
</tr>
<tr>
  <td align="right"><math>\alpha_c</math>:</td>
  <td align="center"><math>-0.9595214</math></td>
</tr>
<tr>
  <td align="right"><math>\gamma_c</math>:</td>
  <td align="center"><math>+1.0102231</math></td>
</tr>
</table>
</div>


<span id="STEP5"><font color="red"><b>STEP 5:</b></font></span> Finally, for each choice of <math>\gamma_e</math> &#8212; or, alternatively, <math>\alpha_e</math> &#8212; the physically relevant value of the core's adiabatic exponent is set by demanding that the ''dimensional'' eigenfrequencies of the envelope and core precisely match.  That is, we demand that,
<table border="1" cellpadding="8" align="right">
<tr><th align="center" colspan="1">Figure 2</th></tr>
<tr>
  <td align="center">
[[File:AlphaVsAlpha21BothCorrected.png|350px|quartic solution]]
  </td>
</tr>
</table>
<div align="center">
<math>\omega^2_\mathrm{env} = \omega^2_\mathrm{core} \, .</math>
</div>
From [[#Eigenvector|above]], we know that, for the core,
<div align="center">
<math>3\omega^2_\mathrm{core}\biggr|_\mathrm{j=1} =  2\pi \gamma_c G \rho_c [ 20 - 8/\gamma_c] \, ;</math>
</div>
whereas, for the envelope,
<div align="center">
<math>3\omega^2_\mathrm{env}\biggr|_\mathrm{\ell=2} =  2\pi \gamma_e G \rho_e [ 3(\alpha_e + 5c_0 + 22)] \, .</math>
</div>
By demanding that these frequencies be identical, we conclude that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> \gamma_c </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{20} \biggl[ 8 + 3\gamma_e \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(\alpha_e + 5c_0 + 22 \biggr)\biggr] \, .</math>
  </td>
</tr>
<!--
<tr>
  <td align="right">
<math>\Rightarrow ~~~ \gamma_c </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{20}\biggl\{ 8 + \biggl(\frac{12}{3-\alpha_e}\biggr) \biggl[\alpha_e + 5(\sqrt{1+\alpha_e}-1) + 22 \biggr]\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggr\}\, .</math>
  </td>
</tr>
-->
</table>
</div>

Figure 2 shows how the required value of <math>\alpha_c</math> varies with the choice of <math>\alpha_e</math>; here, both axes have been flipped in order to run from incompressible <math>(\alpha = +3)</math> at the left/bottom, to isothermal <math>(\alpha = -1)</math> at the right/top.  For the lower portion of the curve (red circular markers), the parameter, <math>c_0</math>, is taken to be the "plus" root of its defining quadratic equation; the "minus" root defines <math>c_0</math> along the upper portion of the curve (purple circular markers).  The diagonal dashed-black line identifies where <math>\alpha_c = \alpha_e</math>; in models below and to the right of this line, the envelope is more compressible than is the core, whereas in models above and to the left of this line, the core is more compressible than the envelope.  

<!-- OMIT Evidently there is one model for which the <math>(\ell,j) = (2,1)</math> eigenvector is analytically specifiable in which the envelope and core are equally compressible; it is the model with <math>\gamma_c = \gamma_e \approx 1.13</math> that is identified by where the <math>c_0</math> (minus) segment of the curve intersects the diagonal black-dashed line.
-->

The eigenfrequency that corresponds to the ''specific'' eigenfunction that is displayed in upper-left quadrant of Figure 1 is identified by the black circular marker in Figure 2; as is indicated by the row of numbers on the left in Figure 1, this model has,
<div align="center">
<math>\gamma_c = 1.845579 </math>
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow </math>&nbsp; &nbsp; &nbsp; 
<math>\alpha_c = +0.8326535 \, . </math>
</div>
The yellow circular marker in Figure 2 identifies the model whose analytically prescribed, <math>(\ell,j) = (2,1)</math> eigenfunction is displayed in the lower-left quadrant of Figure 1; it has,
<div align="center">
<math>\gamma_c = 1.0102231 </math>
&nbsp; &nbsp; &nbsp; <math>~\Rightarrow </math>&nbsp; &nbsp; &nbsp; 
<math>\alpha_c = -0.9595214 \, . </math>
</div>