Editing Appendix/Ramblings/PatrickMotl (section)

===Initial Comments===
First, I need to make sure that I understand the new plots that you have provided.  For discussion purposes, I'll focus on your [http://www.patrickmotl.net/simulations/single_stars/n5_3.25_constp_m12/plots/plots.html plots for n = 5 polytrope &xi; = 3.25 and constant pressure truncation mark 12].  It appears to me that a key plot/animation is the "x ''versus'' m" plot shown in the upper-right corner of this web page; on your web page this plot/animation is labeled, "<math>~(R - R_0)/R_0</math> vs Mass."  I presume that this shows the time-evolution of the radial eigenfunction; am I correct?  Immediately below, I have reproduced one frame from this movie.

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[[File:Motl n5 eigenfunction3.25.png|350px|Patrick's Eigenfunction]]
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''Above:'' &nbsp;A frame showing radial eigenfunction from Motl's nonlinear simulation of model having initial <math>~\tilde\xi = 3.25</math>.  ''Below-right:'' &nbsp;A frame from Tohline's "[[SSC/Stability/n5PolytropeLAWE#Numerical_Integration_of_LAWE|Composite Display 1]]," showing eigenfunction from Tohline's solution of the LAWE for the same initial model.
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[[File:MotlN5movieFrameM26.png|600px|pressure-truncated n = 5 eigenvector]]
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We expect Motl's eigenfunction to lie right on top of the '''blue segment''' of Tohline's eigenfunction.  The good news is that, at least at first glance, the two eigenfunctions look quite similar.  But we need to generate new plots having the same axes.  For example, Tohline's eigenfunction plot is x versus R<sub>0</sub> whereas Motl's plot shows x versus m.  Note that the vertical amplitude (x) can be rescaled to an arbitrary value at the center.

What about the eigenfrequency?  Well, for the model with an initial value of <math>~\tilde\xi = 3.25</math>, Tohline's solution of the LAWE gives <math>~\sigma_c^2 = - 0.039629</math>.  What value is obtained from Motl's simulation?  Well &hellip; I don't quite know how to interpret Motl's plot of <math>~|F(\rho)|</math> versus <math>~f/f_0</math>.  But Motl's plot of <math>~\rho</math> versus <math>~t/T_0</math> ought to tell us something.  Because <math>~\sigma_c^2</math> is negative, this model is dynamically unstable and it should collapse on the timescale of  a ''single'' oscillation frequency <math>~\sigma_c = \sqrt{0.039629} \approx 0.200</math>.  If Motl generates a semi-log plot &#8212; that is <math>~\ln\rho</math> versus <math>~t/T_0</math>, he should see a line with slope given by this value of <math>~\sigma_c</math>. &nbsp; &nbsp; <font color="red">NOTE:  The relationship between <math>~\sigma_c</math> and <math>~\omega</math> is given in the following subsection.</font>

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