Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section A / A2 / Solution

Write the proposed Jeans mass as $M_J=K{v}^a{G}^b{P}^c$, with dimensionless $K$. The dimensions of speed, the gravitational constant and pressure areEquating the exponents of mass, length and time givesThus $a=4$, $b=-3/2$ and $c=-1/2$, so the pressure-confined Jeans mass scales asA larger speed of sound gives stronger support against gravitational collapse; stronger confinement at fixed speed of sound lowers the characteristic unstable mass. Dimensional analysis determines the scaling, not $K$, which requires a particular cloud model and instability criterion.