Two Limits

There are 2 limits we identify, the first is the sparse ambient limit in which we treat the ambient as a collection of parcels of gas that are attracted to the BE sphere only, and the 2nd is the dense ambient limit where we treat the ambient as a collapsing uniform sphere.

Sparse Cases

For a planar HSE atmosphere, with differential equation,

\frac{dP}{dz} = -\rho g,

the solution is an exponential with given scale height,

H=\frac{K_B T}{m_H g}

where

g = GM/r^2

Assuming the force is given by,

F = \frac{G M_{BE}M_{atm}(t)}{(R_{BE} + H)^2}

and Matm is the mass of the accumulated atmosphere, gravitationally attracted to the BE sphere through freefall,

Matm= \frac{4}{3} \pi R_{ff}^3 \rho_{amb}

(where rff is found by inverting the tff for r for 2 point masses)we get the expression for P = F/A at the surface (after plugging in the variables),

P(R_{BE},t) = P_{ext} + P(t) = P_{ext} + \frac{8}{3 \pi^2} \frac{G^2 M_{BE}^2 \rho_{amb}} {R_{BE}^2}\frac{1}{(R_{BE}+H)^2}t^2

Using this expression gives the following results,

Vertical lines indicate when a sink formed for the corresponding run. The horizontal line is the initial external pressure on the BE sphere, which since the sphere is a critical BE sphere, equals the critical pressure.

We see that as eta = rho(Rbe)/rho(amb) increases (i.e. the ambient gets sparser), it takes longer for the pressure to grow at the surface of the sphere, inducing collapse.

Plotting denser ambient mediums skews the graph, as they increase much more rapidly on the y-axis.

Here is a table of the runs summary,

Dense Case

The idea is that we have an initial uniform sphere (the ambient) that is collapsing because it is jeans unstable (this is the case for the Matched, 1/3, 1/10 cases). After some time t, a shell of the uniform sphere has fell onto the BE sphere material. We can assume it piles up into a thin shell (negligible thickness) at r = Rbe. Then we can calculate the pressure from this shell as

P = F/A = \frac{GM_{be}M_{shell}(t)}{4 \pi R_{be}^2}

To get the mass of the shell we take the difference between the masses of the original uniform sphere and the shell at some later time t.