Editing Appendix/Ramblings/PatrickMotl (section)

==October 07 - 08 (between Joel &amp; Patrick)==

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Patrick,

Please look at figure 6 and the lines of text that immediately precede and that immediately follow this figure on the following web page:
https://tohline.education/SelfGravitatingFluids/index.php/Appendix/Ramblings/PatrickMotl

I put this material together some months ago, but had since forgotten about it.  It contains quantitative results that will significantly strengthen any presentation that you make to show the results of your student's 3D simulations.  I have carried out a numerical, linear-stability analysis of models along the pressure-truncated, n = 5 model sequence.  Specifically, I have numerically determined the eigenfunction and the eigenfrequency (squared) of the fundamental mode of oscillation for 16 different models falling between \xi_i = 0.75 and \xi_i = 5.0.  You should plot omega-squared versus \xi_i from this linear-stability analysis, then plot your oscillation frequencies on the same graph to see if they lie along the curve that you get from my analysis.  Also, can you actually generate a plot of the radial eigenfunction of the vibration modes that develop in your simulations?  You could directly compare these eigenfunction plots against my Figure-6 eigenfunction plots.  [I'll be happy to explain this in more detail via a zoom session, if that would help.]

Seeing how your results compare to my linear analysis **quantitatively** will be a lot of fun!!  We can then march boldly toward a similar analysis of bipolytropes!
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Hi Joel,

So, measuring the frequencies is relatively straightforward. 

For the eigenfunction, do you know off the top of your head or a reference for rewriting variation in r over r in terms of variation of rho over rho?
I am trying to think of something I can measure from the simulations.
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Patrick,

This question also popped up in my mind while lying in bed last night.  That is, I recognized that the variation in r is a **lagrangian** measurement and you are not set up to make that measurement easily.  So you probably need the density variation.

I am quite sure that the answer is buried in the following chapter of my on-line book:
https://tohline.education/SelfGravitatingFluids/index.php/SSC/Perturbations#Summary_Set_of_Linearized_Equations
For example, this specific link will take you to a subsection of the chapter that gives the relationship between p (fractional pressure variation), d (density), and x (lagrangian radial location) via the three key linearized **differential** equations.  I am going to look for a better answer to your question, though.
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