Jeans Analysis

A uniform cloud of gas will collapse under gravity if its gravitational timescale (the freefall time, t_{ff}) is shorter than its thermal timescale (sound crossing time, t_{sc}). That is, if t_{ff} \propto 1/(G\rho)^{1/2} is less than t_{sc}=R/C_s, where R is the radius of the cloud and Cs is its sound speed, a cloud is unstable to collapse. Setting these equal and solving for r gives an estimate of the Jeans length,

\lambda_J \propto (Cs^2/G\rho)^{1/2}

By substituting for rho, we can turn this into the more physically intuitive condition:

\lambda_J \propto (Cs^2/(GM/r^3))^{1/2}

Thus, for a given cloud of radius r, sound speed Cs, and mass M, we can compute a Jeans length, which is either less than r or greater than r. For a \lambda_J < r, the cloud is gravitationally unstable. For the opposite, the cloud is gravitationally stable.

Given the formulation of \lambda_J (T, M, r) we can easily see how the Jeans length depends on the various quantities of interest for the BE problem. For instance, consider we hold M and T fixed and vary r. We see that decreasing r leads to a decrease in \lambda_J, which corresponds to making the sphere LESS stable. If we instead hold T and r fixed, but vary M, we see that an increase in M has the same effect. Thus we can think of the collapse in the various cases as occurring from either situation, either the r of the BE sphere shrinks or the M of the sphere increases, both of which can trigger collapse. We seem to be seeing different effects happening in the different simulations.

The problem

Now, given it is likely that a combination of these effects are happening, how do we decipher the main contributor to collapse? If one happens, does the other always happen as well? To begin, let's first ponder the reasons one would happen over the other, and potential diagnostics of the differing mechanisms. Note that the sphere we are dealing with is a critical BE sphere, one with M = M_J and Pext = Pcrit.

Case 1 - Increase in Mass

Collapse is triggered by this case when the mass is assimilated into the sphere faster than the ram pressure squeezes the sphere into a smaller radial volume, thereby reducing the Jeans length. But how would this happen?

1. Pram < Pthermal at the BE sphere/ambient boundary.

Case 2 - Decrease in Radius