Editing SSC/Stability/BiPolytropes/PlannedApproach (section)

==Planned Approach==
<ol>
  <li>[[SSC/Stability/BiPolytropes/PlannedApproach|Earlier Planned Approach]]</li>
  <li>[[SSC/Stability/BiPolytropes/HeadScratching|Headscratching]]</li>
  <li>[[SSC/Stability/BiPolytropes/SuccinctDiscussion|Succinct Discussions]]</li>
  <li>[[SSC/Stability/BiPolytropes/51Models|51Models]]</li>
</ol>

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<th colspan="2" align="center">Figure 2:  Equilibrium Sequences of Bipolytropes <br />with <math>(n_c,n_e) = (5,1)</math> and Various <math>\mu_e/\mu_c</math></th>
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  <td align="center" colspan="2" bgcolor="white">
[[Image:TurningPoints51Bipolytropes.png|300px|center]]
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Ideally we would like to answer the just-stated "principal question" using purely analytic techniques.  But, to date, we have been unable to fully address the relevant issues analytically, even in what would be expected to be the simplest case: &nbsp; [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|bipolytropic models that have <math>(n_c,n_e) = (0, 0)</math>]].  Instead, we will streamline the investigation a bit and proceed &#8212; at least initially &#8212; using a blend of techniques.  We will investigate the relative stability of bipolytropic models having <math>(n_c,n_e) = (5,1) </math> whose ''equilibrium structures'' are completely defined analytically; then the eigenvectors describing radial modes of oscillation will be determined, one at a time, by solving the relevant LAWE(s) numerically.  We are optimistic that this can be successfully accomplished because we have had experience numerically integrating the LAWE that governs the oscillation of:
* [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|Isolated n = 3 polytropes]] &#8212; including a quantitative comparison against the published work of {{ Schwarzschild41full }};
* [[SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres|Pressure-truncated isothermal spheres]] &#8212; including a quantitative comparison against the published analysis of {{ TVH74full }}; and
* [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Pressure-truncated n = 5 polytropes]].

A key reference throughout this investigation will be the paper by {{ MF85bfull }}.  They studied ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models.''  Specifically, their underlying equilibrium models were bipolytropes that have <math>(n_c,n_e) = (1, 5)</math>.  In an [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5|accompanying chapter]], we describe in detail how {{ MF85b }} obtained these equilibrium bipolytropic structures and detail some of their equilibrium properties.

Here are the steps we initially plan to take:

* Governing LAWEs:
** Identify the relevant LAWEs that govern the behavior of radial oscillations in the <math>~n_c = 5</math> core and, separately, in the <math>~n_e = 1</math> envelope.  Check these LAWE specifications against the published work of {{ MF85b }}.
** Determine the matching conditions that must be satisfied across the core/envelope interface.  Be sure to take into account the critical interface ''jump'' conditions spelled out by {{ LW58full }}, as we have already discussed in the context of an [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope|analysis of radial oscillations in zero-zero bipolytropes]].
* Determine what surface boundary condition should be imposed on physically relevant LAWE solutions, i.e., on the physically relevant radial-oscillation eigenvectors.
* Initial Analysis:
** Choose a maximum-mass model along the bipolytropic sequence that has, for example, <math>\mu_e/\mu_c = 1/4</math>.  Hopefully, we will be able to identify precisely (analytically) where this maximum-mass model lies along the sequence.  <font color="red">'''Yes!'''</font>  Our [[SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass|earlier analysis]] does provide an analytic prescription of the model that sits at the maximum-mass location along the chosen sequence.
** Solve the relevant eigenvalue problem for this specific model, initially for <math>(\gamma_c, \gamma_e) = (6/5, 2)</math> and initially for the fundamental mode of oscillation.