Editing SSC/Stability/BiPolytropes/HeadScratching (section)

==Attempts to Identify Marginally Unstable (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Bipolytropes==

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  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMax.xlsx --- worksheet = PlotSigma0]]Figure 3: &nbsp;Conflicting Instability Regions</th>
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  <td align="center" colspan="10">[[File:EigenvectorStability.png|300px|Conflicting Instability Regions]]</td>
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<ul>
<li>[[SSC/Stability/BiPolytropes#Virial_Analysis|Virial Analysis]]</li>
<li>Solving the Relevant LAWE:
  <ul>
  <li>[[SSC/Stability/BiPolytropes#Review_of_the_Analysis_by_Murphy_.26_Fiedler_.281985b.29|Review of the Analysis by]] {{ MF85b }}</li>
  <li>[[SSC/Stability/BiPolytropes#Radial_Oscillations_of_.28nc.2C_ne.29_.3D_.285.2C_1.29_Models|Radial Oscillations of 51 Models]] &nbsp; <math>\Leftarrow</math> &nbsp; [[SSC/Stability/BiPolytropes#Interface|Interface]] probably not handled correctly here.</li>
  <li>[[SSC/Stability/BiPolytropes#Variational_Principle|Variational Principle]]</li>
  </ul>
</li>
<li>[[SSC/StabilityConjecture/Bipolytrope51|K-BK74 Conjecture]]<br />
Through this analysis, we were quite successful at generating a reasonably shaped radial-oscillation eigenfunction for the equilibrium model that lies along the sequence at the maximum mass fraction, <math>\nu_\mathrm{max}</math>.  We obtained the eigenfunction by subtracting the structural profile of one model (immediately to the left of this maximum) from the structural profile of a separate model (immediately to the right of this maximum).  But, after doing so, we realized that the result could not be physically justified.  Although we had been careful to ensure that the core mass fraction <math>(\nu)</math> was identical in the two separate models, we had not paid attention to the ''total'' mass; it had not been held fixed.  Hence, we cannot claim to have performed a valid dynamical perturbation of models in the vicinity of <math>\nu_\mathrm{max}</math>.
<br />&nbsp;<br />

The eigenfunction that was constructed in this manner was nevertheless eye-opening!  It appears to contain a sizable step function at the interface; that is to say, it appears as though the radial-displacement function at the surface of the core is offset (discontinuously) from the radial-displacement function at the base of the envelope.  We had not allowed this to happen in our separate investigation of [[SSC/Stability/BiPolytropes#Radial_Oscillations_of_.28nc.2C_ne.29_.3D_.285.2C_1.29_Models|Radial Oscillations of 51 Models]].
</li>
</ul>

We will attempt to incorporate these new insights into our analyses that follow.
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