Is time to go unstable some constant fraction of the free fall time of the ambient?

Attached PDF with calculations.

Basically it does seem a pretty good approximation that the collapse time for the BE sphere is constant (t~0.2) for the non-compression wave cases. I checked this in the simulations, using the time it took between sink formation, and monotonic increase in density past original values (since in non-compression wave cases we see breathing modes). On the other hand, it seems not as straight-forward to gauge collapse time for strong compression wave cases. If I take it as beginning at the time I see monotonic increase of core densities above initial values, the compression wave cases lead to short collapse times.

No clear trend in the fraction of free-fall times is observed. You can check the pdf values in the attachment.

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