Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section B / 5 / c

The particle count is the enclosed mass divided by the particle mass $m$:Integrating the isothermal Lane-Emden equation giveswhere regularity makes the lower endpoint vanish. ThusThe prefactor is a mass divided by a mass, and the remaining factor is dimensionless. In particular, at fixed $K$ the dependence on central mass density is $N\propto {\rho }_c^{-1/2}f(\xi )$, with $f(\xi )={\xi }^2{\Psi }^{\prime }(\xi )$. Both changes must be retained when comparing equilibria at fixed $N$.