Meeting update

I have been working through details of the derivation we discussed the other day. Please see my previous post on the pressure expression and my concern there.

In the dense ambient case limit, we talked about using a spherical wave that has had time enough to fall onto the BE sphere via freefall. For this, we considered Vff=Sqrt(GM/r) (which may be off by a factor of root 2), and said v=r/t, r=vt.

I think this may be better given by the following.

Imagine the following situation, m falling down to mass Mr from initial velocity = 0. For the case of spherical collapse, m is the mass of a thin shell, attracted to all the mass Mr within the shell by Gauss's law. None of the shells cross, so Mr is constant over the collapse.

By energy conservation, Ei = Ef, or

where r0 is the initial radius material is falling from, rf is the final radius (for our case, rf=Rbe). Solving for v, we have,

For the case rf << r0, the 1/rf term dominates, and this gives the expression,

The flow near the core, at late times of spherical collapse approaches this value (in both the LP solutions and the BE collapse solns).

In looking for R(t), maybe we should write it this way?,

i.e. the velocity of the material at a distance r0 above Rbe is the freefall velocity. Rearranging and solving for r0(t) gives,

which is to say, everything within this radius had had time enough to fall onto the BE sphere..

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