Editing SSC/StabilityConjecture/Bipolytrope51 (section)

===Eigenfunction===

The two '''Pairing B''' models have (almost) identical fractional core masses &#8212; specifically, <math>\nu_+ = \nu_- = 0.337200</math> &#8212; and this chosen mass-fraction is just below the maximum value associated with the <math>\mu_e/\mu_c = 0.31</math> model sequence, <math>\nu_\mathrm{max} = 0.33721701</math> (see the degenerate, '''Pairing A''').  With these two mass-radius structural profiles, we are positioned to implement the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|K-BK74 conjecture]].  Letting <math>r_+(m_r)</math> represent the run of radius with mass-fraction in the "plus" model and letting <math>r_-(m_r)</math> represent the run of radius with mass-fraction in the "minus" model, the amplitude of the eigenfunction at each value of <math>m_r</math> should be very close to the value,
<table border="0" align="center">

<tr>
  <td align="right">
<math>x</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{r_+ - r_-}{(r_+ + r_-)} \, .</math>
  </td>
</tr>
</table>
Table 3 titled, "Eigenfunction," provides this eigenfunction amplitude at twenty-three different mass-fraction locations throughout our '''Pairing B''' model(s).  For example, at the interface location where <math>m_r = 0.337200</math> (for both models), our pair of models give, respectively, <math>r_- = 6.152518</math> and  <math>r_+ = 6.306954</math>; this means that, at the interface, <math>x = 0.012395</math>, as recorded in Table 3.  Similarly, at the surface we find that, <math>r_- = 80.568084</math> and <math>r_+ = 84.492486</math> which means that, <math>x = 0.023776</math>, as recorded in Table 3.
<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="6">
Table 3: &nbsp; Eigenfunction
  </th>
</tr>

<tr>
  <td align="center"><math>\frac{M_r}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>\frac{r_+ - r_-}{2(r_+ + r_-)}</math></td>
  <td align="center" bgcolor="lightgray" rowspan="13">&nbsp;</td>
  <td align="center"><math>\frac{M_r}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>\frac{r_+ - r_-}{2(r_+ + r_-)}</math></td>
  <td align="center" rowspan="13">
[[File:Eigenfunction310.png|450px|K-BK74 Method used to determine radial eigenfunction for Maximum-Mass Bipoltrope having &mu;-ratio = 0.310]]
  </td>
</tr>

<tr>
  <td align="center"><math>0.00</math></td>
  <td align="center"><math>0.0</math></td>
  <td align="center"><math>0.50</math></td>
  <td align="center"><math>0.023454</math></td>
</tr>

<tr>
  <td align="center"><math>0.05</math></td>
  <td align="center"><math>0.000562</math></td>
  <td align="center"><math>0.55</math></td>
  <td align="center"><math>0.023537</math></td>
</tr>

<tr>
  <td align="center"><math>0.10</math></td>
  <td align="center"><math>0.000780</math></td>
  <td align="center"><math>0.60</math></td>
  <td align="center"><math>0.023596</math></td>
</tr>

<tr>
  <td align="center"><math>0.15</math></td>
  <td align="center"><math>0.001035</math></td>
  <td align="center"><math>0.65</math></td>
  <td align="center"><math>0.023634</math></td>
</tr>

<tr>
  <td align="center"><math>0.20</math></td>
  <td align="center"><math>0.001464</math></td>
  <td align="center"><math>0.70</math></td>
  <td align="center"><math>0.023663</math></td>
</tr>

<tr>
  <td align="center"><math>0.25</math></td>
  <td align="center"><math>0.002123</math></td>
  <td align="center"><math>0.75</math></td>
  <td align="center"><math>0.023686</math></td>
</tr>

<tr>
  <td align="center"><math>0.30</math></td>
  <td align="center"><math>0.004103</math></td>
  <td align="center"><math>0.80</math></td>
  <td align="center"><math>0.023704</math></td>
</tr>

<tr>
  <td align="center"><math>0.35</math></td>
  <td align="center"><math>0.020430</math></td>
  <td align="center"><math>0.85</math></td>
  <td align="center"><math>0.023724</math></td>
</tr>

<tr>
  <td align="center"><math>0.3372</math></td>
  <td align="center"><math>0.012395</math></td>
  <td align="center"><math>0.90</math></td>
  <td align="center"><math>0.023740</math></td>
</tr>

<tr>
  <td align="center"><math>0.375</math></td>
  <td align="center"><math>0.012395</math></td>
  <td align="center"><math>0.95</math></td>
  <td align="center"><math>0.023754</math></td>
</tr>

<tr>
  <td align="center"><math>0.40</math></td>
  <td align="center"><math>0.022421</math></td>
  <td align="center"><math>1.00</math></td>
  <td align="center"><math>0.023776</math></td>
</tr>

<tr>
  <td align="center"><math>0.45</math></td>
  <td align="center"><math>0.023282</math></td>
  <td align="center">&nbsp;</td>
  <td align="center">&nbsp;</td>
</tr>

</table>


The (Table 3) eigenfunction that we have constructed via the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|K-BK74 conjecture]] has several notable features:
<ul>
  <li>
Moving from the center of the configuration out to the core-envelope interface, the eigenfunction exhibits a smooth, mild steady increase.
  </li>
  <li>
Moving from the interface out to the surface of the configuration, the eigenfunction is essentially constant; this means that the envelope expands/contracts homologously.
  </li>
  <li>
At the interface the core transitions to the envelope via (essentially) a step function; for the selected model sequence <math>(\mu_e/\mu_c = 0.310)</math>, the eigenfunction amplitude jumps by a factor of <math>\approx 6</math>.  It seems reasonable to suspect that the existence of, and magnitude of, this jump is related to our choice of the size of the &mu;-jump (0.310); but it also may depend on the values of the adiabatic exponents, <math>(\gamma_c, \gamma_e) = (6/5, 2)</math>.
  </li>
  <li>
Also, following the B-KB74 conjecture, the implicit assumption is that the eigen''frequency'' associated with this marginally unstable model is zero.
  </li>
</ul>

Things to do in an effort to follow up on these recognized attributes of the eigenfunction:
<ul>
  <li>
Does the behavior (mild steady increase) of the core eigenfunction resemble, in any fashion, the [[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|analytically determined eigenfunction for the marginally unstable, pressure-truncated, n = 5 Polytrope]]?
  </li>
  <li>
[[SSC/Stability/MurphyFiedler85#Interface_Conditions|Ledoux &amp; Walraven (1958)]] have examined how the LAWE needs to be modified if there is a discontinuity at a core-envelope interface.  Did they just examine step-function changes in the adiabatic index, or did they also look at jumps in &mu;?
  </li>
  <li>
In our [[SSC/Stability/BiPolytropes#Eigenfunction_Details|earlier attempt to solve the LAWE]] for 51 bipolytropes, we did not allow the possibility of a step function at the interface for the eigenfunction.  If you flip the Figure 5 and Figure 6 curves upside down, and insert a step function, you can argue that these earlier eigenfunctions resemble our new one.
  </li>
  <li>
Is it easy to show that a constant eigenfunction throughout the envelope readily satisfies the [[SSC/Stability/InstabilityOnsetOverview#Polytropic_Stability|surface boundary condition]]?
  </li>
</ul>