Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

===Fundamental Modes===

We decided to examine, first, whether any model along each sequence marks a transition from dynamically stable to dynamically unstable configurations.  We accomplished this by setting <math>~\sigma_c^2</math> = 0, then integrating the relevant LAWE from the center toward the surface for many different ''guesses'' of the core-envelope interface radius until an eigenfunction with no radial nodes &#8212; ''i.e.,'' an eigenfunction associated with the fundamental mode of radial oscillation &#8212; was found whose behavior at the surface matched with high precision the physically desired surface boundary condition.  We were successful in this endeavor.  A marginally unstable model was identified on each of the six separate equilibrium sequences.  

====Equilibrium Properties of Marginally Unstable Models====
Table 2 summarizes some of the equilibrium properties of these six models.  For example, the second column of the table gives the value of the core-envelope interface radius, <math>~\xi_i</math>, associated with each marginally unstable model.  The table also lists:  the value of the model's dimensionless radius, <math>~R^*_\mathrm{surf}</math>, the key structural parameters, <math>~q</math> &amp; <math>~\nu</math>, and the central-to-mean density associated with each model; and in each case the dimensionless thermal energy <math>~(\mathfrak{s})</math> and dimensionless gravitational potential energy <math>~(\mathfrak{w})</math> associated, separately, with the core and the envelope.  Note that, once the pair of parameters, <math>~(\mu_e/\mu_c, \xi_i)</math>, has been specified, we can legitimately assign high-precision values to all of the other model parameters because they are [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|analytically prescribed]].

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="10">
'''Table 2:'''  Properties of Marginally Unstable Bipolytropes Having<br /><br /><math>~(n_c, n_e) = (5, 1)</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math><br /><br />Determined from Integration of the LAWE
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~R^*_\mathrm{surf}</math></td>
  <td align="center"><math>~q \equiv \frac{r_\mathrm{core}}{R_\mathrm{surf}}</math></td>
  <td align="center"><math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{env}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">1.6686460157</td>
  <td align="right">2.139737</td>
  <td align="center">0.53885819</td>
  <td align="center">0.497747626</td>
  <td align="center">8.51704656</td>
  <td align="center">3.021916335</td>
  <td align="center">-3.356583022</td>
  <td align="center">1.47780476</td>
  <td align="center">-5.642859167</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.27925811317</td>
  <td align="right">5.146499</td>
  <td align="center">0.306021732</td>
  <td align="center">0.401776274</td>
  <td align="center">63.29514949</td>
  <td align="center">4.241287117</td>
  <td align="center">-6.074241035</td>
  <td align="center">4.284931508</td>
  <td align="center">-10.97819621</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.560146865247</td>
  <td align="right">9.554041</td>
  <td align="center">0.185160563</td>
  <td align="center">0.234302525</td>
  <td align="center">209.7739052</td>
  <td align="center">4.639705843</td>
  <td align="center">-7.125754184</td>
  <td align="center">11.72861751</td>
  <td align="center">-25.61089252</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.582007485476</td>
  <td align="right">10.120558</td>
  <td align="center">0.176288391</td>
  <td align="center">0.218241608</td>
  <td align="center">230.4125398</td>
  <td align="center">4.667042505</td>
  <td align="center">-7.200966267</td>
  <td align="center">13.15887139</td>
  <td align="center">-28.45086152</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.6274239687695</td>
  <td align="center">11.464303</td>
  <td align="center">0.158362807</td>
  <td align="center">0.184796947</td>
  <td align="center">279.0788798</td>
  <td align="center">4.722277318</td>
  <td align="center">-7.354156963</td>
  <td align="center">17.1374434</td>
  <td align="center">-36.36528446</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
  <td align="left">2.7357711469398</td>
  <td align="center">15.895632</td>
  <td align="center">0.118924863</td>
  <td align="center">0.11071211</td>
  <td align="center">430.0444648</td>
  <td align="center">4.84592201</td>
  <td align="center">-7.70305421</td>
  <td align="center">37.84289623</td>
  <td align="center">-77.67458196</td>
</tr>
</table>
</div>

<!--
OLD VERSION; IGNORE!

<div align="center">
<font color="red">'''OLD Table 2'''</font>

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="9">
Properties of Marginally Unstable Bipolytropes Having<br /><br /><math>~(n_c, n_e) = (5, 1)</math> and <math>~(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math><br /><br />Determined from Integration of the LAWE
  </th>
</tr>
<tr>
  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>~\xi_i</math></td>
  <td align="center"><math>~q \equiv \frac{r_\mathrm{core}}{R_\mathrm{surf}}</math></td>
  <td align="center"><math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{env}</math></td>
  <td align="center"><math>~\mathfrak{s}_\mathrm{core}</math></td>
  <td align="center"><math>~\mathfrak{w}_\mathrm{core}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="left">2.467359668</td>
  <td align="center">0.597684036</td>
  <td align="center">0.692367564</td>
  <td align="center">17.09749847</td>
  <td align="center">4.518031091</td>
  <td align="center">-6.79580606</td>
  <td align="center">0.857904827</td>
  <td align="center">-3.956065776</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="left">2.65925887</td>
  <td align="center">0.314759926</td>
  <td align="center">0.481811565</td>
  <td align="center">95.04044773</td>
  <td align="center">4.759771212</td>
  <td align="center">-7.459080087</td>
  <td align="center">3.20977926</td>
  <td align="center">-8.480020858</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="left">2.77457272408</td>
  <td align="center">0.183201946</td>
  <td align="center">0.25725514</td>
  <td align="center">281.7482802</td>
  <td align="center">4.887554727</td>
  <td align="center">-7.822410223</td>
  <td align="center">10.22665208</td>
  <td align="center">-22.40600339</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
  <td align="left">2.7843424754</td>
  <td align="center">0.17412463</td>
  <td align="center">0.237677946</td>
  <td align="center">305.4881577</td>
  <td align="center">4.897826446</td>
  <td align="center">-7.852004624</td>
  <td align="center">11.6114224</td>
  <td align="center">-25.16649306</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="left">2.8050375512</td>
  <td align="center">0.156021514</td>
  <td align="center">0.197918988</td>
  <td align="center">359.3276918</td>
  <td align="center">4.91930991</td>
  <td align="center">-7.914090174</td>
  <td align="center">15.49005222</td>
  <td align="center">-32.90463409</td>
</tr>
</table>
</div>

OLD VERSION; IGNORE!
-->

As was expected from our [[#What_to_Expect_for_Equilibrium_Configurations|above discussion of virial equilibrium conditions]], we found that to high precision for each of these equilibrium models,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
(\mathfrak{w}_\mathrm{core}  
~+~\mathfrak{w}_\mathrm{env})
~+~2(\mathfrak{s}_\mathrm{core}  
~+~\mathfrak{s}_\mathrm{env})  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
<span id="Figure4">However, contrary to expectations,</span> in no case did we find that <math>~\mathfrak{s}_\mathrm{core}/\mathfrak{s}_\mathrm{env} = 5</math>.  That is to say, we found that ''none'' of the models lies on the (red-dashed) curve in the <math>~q-\nu</math> parameter space that separates stable from unstable models as defined by our [[#What_to_Expect_for_Equilibrium_Configurations|above free-energy-based stability analysis]]. The left-hand panel of Figure 4 shows this (red-dashed) demarcation curve; for all intents and purposes, it is a reproduction of the right-hand panel of [[#Virial_Stability_Evaluation|Figure 3, above]] &#8212; turning-point markers have been removed to minimize clutter, the equilibrium sequences have been labeled, and the horizontal axis has been extended to unity in order to include a longer portion of the <math>~\mu_e/\mu_c = 1</math> sequence.  The orange triangular markers that appear in the right-hand panel of Figure 4 pinpoint where each of the Table 2 "marginally unstable" models resides in this <math>~q-\nu</math> plane.  Clearly, all six of the orange triangles lie well off of &#8212; and to the ''stable'' side of &#8212; the red-dashed demarcation curve.  This discrepancy, which has resulted from our use of two separate approaches to stability analysis, will be discussed further and gratifyingly resolved, below.

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

<tr>
  <th align="center">Figure 4</th>
</tr>
<tr>
  <td align="center" colspan="10">[[File:NEWCompositeDlabeled.png|800px|Marginally unstable models]]</td>
</tr>
</table>

====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.

====Is There an Analytic Expression for the Eigenfunction?====
After noticing that, in Figure 6,  the ''envelope'' segments of all of the marginally unstable eigenfunctions merge into the same curve, we began to wonder whether a single expression &#8212; and, even better, an ''analytically defined'' expression &#8212; would perfectly describe the eigenfunction.  We had reason to believe that this might actually be possible because, in [[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|pressure-truncated polytropic configurations, we have derived analytic expressions for the marginally unstable, fundamental-mode eigenfunctions]] of both <math>~n = 5</math> and <math>~n=1</math> systems.

Very quickly, we convinced ourselves that a parabolic function does indeed perfectly match the "core" segment of each displayed eigenfunction.  Specifically, throughout the core <math>~(0 \le \xi \le \xi_i)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 - \frac{\xi^2}{15}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{dx_P}{d\xi}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2\xi}{15} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\frac{d\ln x_P}{d\ln \xi}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{2\xi^2}{15} \biggl[ \frac{(15 - \xi^2)}{15} \biggr]^{-1} = - \frac{2\xi^2}{(15 - \xi^2)}  \, .</math>
  </td>
</tr>
</table>


The envelope segment posed a much greater challenge.  In the context of our [[SSC/Stability/n1PolytropeLAWE#Radial_Oscillations_of_n_.3D_1_Polytropic_Spheres|discussion of ''Radial Oscillations of n = 1 Polytropic Spheres'']], and in an [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Is_There_an_Analytic_Expression_for_the_Eigenfunction.3F|accompanying ''Ramblings Appendix'' chapter]] we have detailed some trial derivations that are mostly blind alleyways.  Twice &#8212; once in [[SSC/Stability/n1PolytropeLAWE#tagJanuary2019|January, 2019]] and again (independently) in [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B|April 2019]] &#8212; we have analytically demonstrated that the following appears to work for the envelope:

Given that the ''Structural Properties'' of the envelope are described by the Lane-Emden function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a_0 \biggl[ \frac{\sin(\eta - b_0)}{\eta} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ Q \equiv - \frac{d\ln \phi}{d\ln\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \eta \cot(\eta - b_0) \biggr] \, ,</math>
  </td>
</tr>
</table>
the relevant LAWE is satisfied by the fractional displacement function,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3c_0 Q}{\eta^2} \, ,</math>
  </td>
</tr>
</table>
where, <math>~c_0</math> is an arbitrary scale factor.  

<table border="1" align="center" cellpadding="8" width="60%"><tr><td align="left">
Note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3c_0}{\eta^2}\biggl[\eta -\cot(\eta - b_0)
+\eta\cot^2(\eta - b_0) 
\biggr] 
- \frac{6c_0 Q}{\eta^3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{d\ln x_P}{d\ln\eta} = \frac{\eta}{x_P}\cdot \frac{dx_P}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\eta^3}{3c_0 Q} \biggl\{
\frac{3c_0}{\eta^2}\biggl[\eta -\cot(\eta - b_0)
+\eta\cot^2(\eta - b_0) 
\biggr] 
- \frac{6c_0 Q}{\eta^3} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{Q} \biggl\{
\biggl[\eta^2 -\eta \cot(\eta - b_0)
+\eta^2\cot^2(\eta - b_0) 
\biggr] 
- 2 Q 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{Q} 
\biggl[\eta^2 -(1-Q)
+(1-Q)^2 
\biggr] 
- 2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{Q} 
\biggl[\eta^2  - Q + Q^2\biggr] 
- 2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\eta^2}{Q} + Q - 3 \, .
</math>
  </td>
</tr>
</table>

</td></tr></table>
But, as far as we have been able to determine (as of 16 April 2019), this analytic displacement function does not match the displacement function that has been generated through numerical integration of the LAWE (see the light-green segment of the eigenfunction displayed [[#Eigenfunction_Details|above in Figure 5]]).  It remains unclear whether (a) the numerical integration is at fault, (b) we are imposing an incorrect slope at the core-envelope interface, or ( c) we are misinterpreting how to compare the two separately derived (one, numerical, and the other, analytic) envelope eigenfunctions.