Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

====Eigenfunction Details====

Here we examine some of the properties of the fundamental-mode eigenfunctions that we have found are associated with marginally unstable, <math>~(n_c, n_e) = (5,1)</math> bipolytropes.
<table border="0" align="right" width="40%">
<tr>
  <th align="center">Figure 5</th>
</tr>
<tr><td align="center">
[[File:Mod0MuRatio100.png|450px|Example eigenvector]]
</td></tr>
</table>
Consider the model on the <math>~\mu_e/\mu_c = 1</math> sequence for which <math>~\sigma_c^2=0~</math>; key properties of this specific equilibrium model are enumerated in the first row of numbers provided in [[#Equilibrium_Properties_of_Marginally_Unstable_Models|Table 2, above]].  Figure 5 shows how our numerically derived, fundamental-mode eigenfunction, <math>~x = \delta r/r_0</math>, varies with the fractional radius over the entire range, <math>~0 \le r/R \le 1</math>.  By prescription, the eigenfunction has a value of unity and a slope of zero at the center  <math>~(r/R = 0)</math>.  Integrating the LAWE outward from the center, through the model's core (blue curve segment), <math>~x</math> drops smoothly to the value <math>~x_i = 0.81437</math> at the interface <math>~(\xi_i = 1.6686460157 ~\Rightarrow~ q = r_\mathrm{core}/R_\mathrm{surf} = 0.53885819)</math>.  Our numerical integration of the LAWE showed that, at the interface, the logarithmic slope of the ''core'' (blue) segment of the eigenfunction is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i \biggr\}_\mathrm{core}
=
\biggl\{ \frac{d\ln x}{d\ln \xi}\biggr|_i \biggr\}_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 0.455872 \, .</math>
  </td>
</tr>
</table>
Next, following the [[#Interface|above discussion of matching conditions at the interface]], we determined that, from the perspective of the envelope, the slope of the eigenfunction at the interface must therefore be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i \biggr\}_\mathrm{env}
=
\biggl\{ \frac{d\ln x}{d\ln \eta}\biggr|_i \biggr\}_\mathrm{env}
</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}{d\ln \xi}\biggr|_i \biggr\}_\mathrm{core} = -1.47352 \, .</math>
  </td>
</tr>
</table>

Adopting this "env" slope along with the amplitude, <math>~x_i = 0.81437</math>, as the appropriate ''interface'' boundary conditions, we integrated the LAWE from the interface to the surface, obtaining the green-colored segment of the eigenfunction that is shown in Figure 5.  The amplitude continued to steadily decrease, reaching a value of <math>~x_s = 0.38203</math>, at the model's surface <math>~(r/R = 1)</math>.  At the surface, this ''envelope'' (green) segment of the eigenfunction exhibits a logarithmic slope that matches to eight significant digits the value that is [[#SurfaceCondition|expected from astrophysical arguments]] for this marginally unstable <math>~(\sigma_c^2=0)</math> model, namely,
<div align="center">
<math>~
\frac{d\ln x}{d\ln \eta}\biggr|_s = \biggl[ \biggl( \frac{\rho_c}{\bar\rho} \biggr)\frac{\cancelto{0}{\sigma_c^2}}{2\gamma_e} - \biggl(3 - \frac{4}{\gamma_e}\biggr)\biggr] = -1 \, .
</math>
</div>


<table border="1" cellpadding="10" width="80%" align="center"><tr><td align="left">
<font color="red">'''Key Reminder:'''</font>  We were able to find an eigenfunction whose surface boundary condition matched the desired value &#8212; in this particular case, a logarithmic slope of negative one &#8212; to this high level of precision only by iterating many times and, at each step, fine-tuning our choice of the equilibrium model's radial interface location, <math>~\xi_i</math> before performing a numerical integration of the LAWE.
</td></tr></table>


The discontinuous jump that occurs in the ''slope'' of the eigenfunction at the interface results from our assumption that the effective adiabatic index of material in the core <math>~(\gamma_c = 6/5)</math> is different from the effective adiabatic index of the envelope material <math>~(\gamma_e = 2)</math>.   In an effort to emphasize and more clearly illustrate the behavior of this fundamental-mode eigenfunction as it crosses the core/envelope interface, we have added a pair of dashed line segments to the Figure 5 plot.  The red-dashed line segment touches, and is tangent to, the blue segment of the eigenfunction at the location of the core/envelope interface; it has a slope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{dx}{d(r/R)}\biggr|_i
=
\frac{x_i}{(r_i/R)}\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i\biggr\}_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 0.455872 \biggl(\frac{ 0.81437 }{ 0.53885819 }\biggr) = - 0.68895\, .</math>
  </td>
</tr>
</table>
On the other hand, the purple-dashed line segment  touches, and is tangent to, the green segment of the eigenfunction at the location of the core/envelope interface; it has a slope,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{dx}{d(r/R)}\biggr|_i
=
\frac{x_i}{(r_i/R)}\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i\biggr\}_\mathrm{env}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 1.47352 \biggl(\frac{ 0.81437 }{ 0.53885819 }\biggr) = - 2.22691\, .</math>
  </td>
</tr>
</table>
For comparison purposes, the eigenfunction shown in Figure 5 has been presented again in Figure 6, along with several other of our numerically derived eigenfunctions, but in Figure 6 the plotted amplitude has been renormalized to give a surface value &#8212; rather than a central value &#8212; of unity.


In Figure 6 we show the behavior of the fundamental-mode eigenfunction for each of the marginally unstable models identified in Table 2.  In the top figure panel, each curve shows &#8212; on a linear-linear plot &#8212; how the amplitude varies with radius; in the bottom figure panel, the amplitude is plotted on a logarithmic scale.  On each curve, the black plus sign marks the radial location of the core-envelope interface; in the bottom panel, these markers are accompanied by the values of <math>~\xi_i</math> that are associated with each corresponding model (see also the second column of Table 2).  Each eigenfunction has been normalized such that the surface amplitude is unity.  In the top panel, the value of the central amplitude of the eigenfunction that results from this normalization is recorded near the point where each eigenfunction touches the vertical axis.   (In each case, the value provided on the plot is simply the inverse of the value of <math>~x_s</math> given in Table 3, below.)

<table border="0" align="center" cellpadding="8">
<tr>
  <th align="center">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/LinearPerturbation/FaulknerBipolytrope1.xlsx --- worksheet = Mode0Ensemble]]'''Figure 6:  Eigenfunctions Associated with the Fundamental-Mode of Radial Oscillation'''<br /> 
'''in Marginally Unstable Models having Various''' <math>~\mu_e/\mu_c</math>
  </th>
</tr>
<tr>
  <td align="center">
[[File:Mode0EigenfunctionsCombinedSmall.png|800px|Eigenfunctions for Marginally Unstable Models]]
  </td>
</tr>
</table>

Notice that, especially as they approach the surface, the "envelope" segments of these six marginally unstable eigenfunction appear to merge into the same curve, irrespective of their value of the ratio of mean molecular weights.  Note as well that the discontinuous jump that occurs in the ''slope'' of each eigenfunction at the radial location of the core/envelope interface &#8212; resulting from our choice to adopt a different adiabatic index, <math>~\gamma_g</math>, in the core from the one in the envelope &#8212; becomes less and less noticeable for smaller and smaller values of the ratio of mean molecular weights.