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}

The problem

Case 1 - Increase in Mass

Case 2 - Decrease in Radius