Thinking of a stability analysis for modified BE sphere phase of Matched case...

I am thinking about some analysis Di Francesco did and how we may adopt a similar, but potentially more correct way of doing it…

In assessing the stability of the 'modified' BE sphere in my simulation, i.e. the post-compression wave phase, where the BE sphere has seemingly re-equilibrated in a smaller volume at higher densities, I am curious if there is a better way to evaluate xi for this modified BE sphere and if it exceeds xi_crit = 6.5.

My idea was as follows… I could gather from the plots I already made the new central density of the modified sphere, and the radial extent (i.e. from flat inner core region to flat envelope region). Then, initialize a new BE sphere with this central density, and radial extent, but with different xi's. The goal would be to see if we could 'fit' the profiles. Only if the profiles were within some small margin of error, would I feel comfortable using the definition of xi and xi crit in evaluating stability as Di Francesco did…

The idea is based on this plot:

Here we see how different xi's (given same rho_c and r) are in HSE of varying slope… Steeper slopes are in unstable HSE as the top curve is the critical value of xi. We would like to try to fit one of these unstable curves to the center, over dense modified BE sphere if possible…

In attempting to do this, I am still trying to work out scaling correctly, but here is a first shot:

Obviously the pink is the data from my matched run, from the chombo that began to resemble a supercritical BE sphere… Note there is a slight bump in it still, perhaps a remnant from the compression wave phase…

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