Editing Appendix/Ramblings/PatrickMotl (section)

===Eigenvectors of Marginally Unstable Models===

In preparation for our examination of the relative stability of bipolytropic structures having <math>~(n_c, n_e) = (5, 1)</math> &#8212; via numerical integration of the Linear-Adiabatic Wave Equation (LAWE) &#8212; we have demonstrated that we understand technically how to solve this type of eigenvalue problem by quantitatively reproducing related analyses that have been previously published by other groups.

====Good Comparisons With Previously Published Studies====

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<table border="1" align="right" cellpadding="3"><tr><td align="center" bgcolor="black"><font size="-1" color="white">Schwarzschild (1941)</font><br />[[File:Schwarzschild1941movie.gif|150px|Eigenfunctions for Standard Model]]</td></tr><tr><td align="center" bgcolor="black"><font size="-1" color="white">Taff &amp; Van Horn (1974)</font><br />[[File:TaffVanHorn1974Fundamental.gif|150px|Fundamental mode animation]]</td></tr><tr><td align="center" bgcolor="black"><font size="-1" color="white">Murphy &amp; Fiedler (1985b)</font><br />[[File:MF85Figure3.png|150px|Figure 3 (Model 17) from Murphy &amp; Fiedler (1985b)]]</td></tr></table>
[http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild (1941)] numerically integrated the LAWE for ''isolated'' <math>~n=3</math> polytropic spheres to find eigenvectors (''i.e.,'' the spatially discrete eigenfunction and corresponding eigenfrequency) for five separate oscillation modes (the fundamental mode, plus the 1<sup>st</sup>, 2<sup>nd</sup>, 3<sup>rd</sup>, and 4<sup>th</sup> overtones) for models having four different adopted adiabatic indexes <math>~\gamma_g = \tfrac{4}{3}, \tfrac{10}{7}, \tfrac{20}{13}, \tfrac{5}{3})</math>.  In an [[SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres|accompanying chapter of this H_Book]], we demonstrate that we have been able to reproduce in detail Schwarzschild's results for the specific case of <math>~\gamma_g = \tfrac{20}{13}</math>.
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[http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff &amp; Van Horn (1974)] examined radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal during the oscillations. For models having nine different truncation radii &#8212; chosen to straddle the position along the equilibrium sequence where the marginally unstable model was expected to arise &#8212; they determined and published the fundamental-mode eigenvalues.  For three of these models they also determined and published eigenvalues for the first harmonic mode of oscillation; the radial eigenfunctions associated with both the fundamental mode and the first harmonic mode of these three models also has been displayed in their Figure 1.  In a separate [[SSC/Stability/Isothermal#From_the_Analysis_of_Taff_and_Van_Horn_.281974.29|accompanying discussion]], we demonstrate that we have been able to reproduce in detail the subset of eigenfunctions and associated eigenvalues that have been previously published by [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff &amp; Van Horn (1974)].
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In their published study of bipolytropes having <math>~(n_c, n_e) = (1,5)</math> with <math>~\mu_e/\mu_c = 1</math>, [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985b)] integrated a coupled pair of LAWEs &#8212; one for the core and another for the envelope &#8212; to determine the eigenfunctions and corresponding eigenvalues of various radial modes of oscillation in more than a dozen different equilibrium models, assuming that during the oscillations, <math>~\gamma_g = 5/3</math> throughout both the core and the envelope.  In an accompanying chapter of this H_Book titled, ''[[SSC/Stability/MurphyFiedler85#Review_of_the_BiPolytrope_Stability_Analysis_by_Murphy_.26_Fiedler_.281985b.29|Review of the BiPolytrope Stability Analysis by Murphy &amp; Fiedler (1985b)]],'' we show that we have been able to duplicate in quantitative detail the eigenvectors associated with their equilibrium Models 10 and 17.
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<span id="Fig6">Building upon this set of successful</span> comparisons with stability analyses published by other groups, we have carried out numerical integrations of the relevant LAWE to identify the eigenvectors associated with the fundamental-mode of radial oscillation in pressure-truncated, <math>~n = 5</math> polytropic configurations.  Details of this analysis are provided in yet [[SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|another chapter of this H_Book]].  The following animation sequence illustrates the results of this analysis.  As far as we have been able to determine, an analysis of this type has not previously been conducted for pressure-truncated, <math>~n = 5</math> polytropes.

<div align="center">
'''Figure 6'''<br />
[[File:N5Truncated2.gif|500px|Fundamental-mode eigenvectors for pressure-truncated n = 5 polytropes]]
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In a [[SSC/Stability/n5PolytropeLAWE#Search_for_Analytic_Solutions_to_the_LAWE|subsection of this separate chapter]], we have also shown that, at the maximum-mass turning point along the pressure-truncated <math>~n=5</math> equilibrium sequence &#8212; identified by the green circular marker in the left-hand panel of this animation &#8212; the fundamental-mode eigenfrequency is precisely zero and the associated eigenfunction is described exactly by the formula for a parabola.

====Our Numerical Analysis of Bipolytropes Having (n<sub>c</sub>, n<sub>e</sub>) = (5, 1)====
In an [[SSC/Stability/BiPolytropes#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|accompanying discussion]] we have shown that we can integrate the linear adiabatic wave equation (LAWE) &#8212; that is, we effectively have been able to solve the eigenvalue problem &#8212; to obtain the eigenvector associated with marginally unstable models along equilibrium sequences for bipolytropes having <math>~(n_c, n_e) = (5, 1)</math>.  The marginally unstable models that have been identified via this more rigorous approach &#8212; see the orange triangles in the following figure &#8212; fall at different points along each equilibrium sequence than the points that were identified via our free-energy analysis &#8212; see the red-dashed demarcation curve.  Assuming that the algorithm that we developed to integrate the LAWE was basically error-free, we trust the model identifications generated via this more rigorous technique.  

Our analysis indicates that, along the <math>~\mu_e/\mu_c = 1</math> equilibrium sequence, the core-envelope interface of the marginally unstable model is located at <math>~\xi_i = 1.6686460157</math>; in the following figure, the red arrow points to this location along that sequence.  In the brief table that accompanies the figure, we have listed values for the dimensionless specific entropy in the core and, separately, in the envelope, along with the interface location, <math>~\xi_i</math>.


<table border="0" align="center" width="80%"><tr><td align="center">
<table border="1" align="right" cellpadding="8">
<tr><td align="center">'''Figure 8'''</td></tr>
<tr><td align="center"><math>~\gamma_c = 6/5</math> &nbsp; &nbsp; and &nbsp; &nbsp;  <math>~\gamma_e = 2</math><br />&nbsp;<br />[[File:Entropy04Annotated.png|350px|Entropy distribution]]</td></tr></table>
<table border="1" align="left" cellpadding="8" width="40%">
<tr><td align="center" colspan="4">'''Figure 7'''</td></tr>
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  <td align="center" colspan="4" width="80%"><b>Initial Model Parameters<br />for<br />LAWE-Determined<br />Marginally Unstable Model</b></td>
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  <td align="center"><math>~\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" width="25%"><math>~\xi_i</math></td>
  <td align="center" width="25%"><math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math></td>
  <td align="center" width="25%"><math>~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math></td>
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  <td align="center">1</td>
  <td align="center">1.6686460157</td>
  <td align="center">8.04719</td>
  <td align="center">1.31310</td>
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  <td align="center" colspan="4" width="100%">[[File:MotlLAWEdetermination02.png|350px|LAWE determination of marginally unstable model]]</td>
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  <td align="left" colspan="4" width="100%">For more details, see the accompanying discussion titled, ''[[SSC/Stability/BiPolytropes#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|Eigenvectors from Solution of LAWE]]''</td>
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