Editing SSC/Stability/BiPolytrope00 (section)

===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>.  
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<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>
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<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>
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</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.
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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>