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? 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? Possible conditions are as follows:

1. Pram < Pthermal at the BE sphere/ambient boundary. This can happen either with a low density ambient flow or a low velocity ambient flow at the boundary. If the v of the ambient is low, but rho is not, the ambient may settle on the sphere but provide little ram pressure on it. Does the material then assimilate into the sphere? Diagnostic: ram pressure at sphere's surface, tracer of ambient material inside sphere as function of time. If the reverse happened, low enough rho that Pram is sufficiently small to not force compression, but the v of low density flow carries material into the sphere. Question: What is required force to compress sphere? How does isothermal EOS interplay with this motion?

2. Internal sound waves can increase rho within a certain r beyond critical levels. Diagnostic: M(r)/MBE at different times.

The big picture for this case is that the BE sphere accretes mass of sufficiently low ram pressure. We can ultimately check with tracers how much mass is being accreted.

Case 2 - Decrease in Radius

The concept for this case is that the radius of the sphere r gets smaller while M stays largely constant. Ways I can visualize this happening are:

1. Pram pushes on sphere faster than material can be accreted. Diagnostic: Tff ambient < Accretion timescale of sphere

2. Some barrier exists at the interface preventing material from seeping into the sphere. Could some sort of energetic barrier exist from use of an isothermal EOS in these runs? Still not entirely clear on the effects of the isothermal EOS in the transport of material from the hotter ambient medium to the colder sphere..