Editing Appendix/Ramblings/PatrickMotl (section)

====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.

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'''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.