Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 1 / b / ii / Solution

For $P=K{\rho }^{1+1/n}$,The pseudo-enthalpy definition therefore integrates towhere the constant makes $w=0$ at the surface. Thus

Hydrostatic balance is $c_0^{2}dw/dz=-{\Omega }^{2}z-d{\Phi }_d/dz$. Differentiate it, use the Poisson equation, and set $\zeta =\Omega z/c_0$:Since the definitions in the question give $Q_0={\Omega }^2/(\pi G{\rho }_0)$,

For $n=1$, put $\lambda =2/\sqrt{Q_0}$. The solution isThe surface $z=H$ is the first zero, so, using $c_0^2=2K{\rho }_0$,