Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 3 / e / Solution

Let $\zeta =z/H(t)$ label a fluid element and use the homologous vertical motion of an astrophysical diskBecause $D\zeta /Dt=0$ and $\nabla \cdot \mathbf{u}=\dot{H}/H$, this ansatz satisfies mass conservation. The pressure equation for an adiabatic process giveshenceThe vertical acceleration is $D{u}_z/Dt=\ddot{H}\zeta$. Using the dimensionless hydrostatic profiles from part c, the pressure force is $(P/\Sigma H)\zeta$, and the vertical momentum equation reduces towhere $C$ is constant.