Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 2 / a

For $|z|\ll r$, the gravitational potential has the Taylor expansionHence the vertical gravity is $-{\partial }_z\Phi \simeq -{\Omega }_K^{2}z$, where ${\Omega }_K^2=G{M}_{\star }/r^3$. Vertical hydrostatic equilibrium balances a pressure gradient of order $P/H$ against $\rho {\Omega }_K^{2}H$. Since $c_s^2\sim P/\rho$,The thin disk condition $H/r\ll 1$ is therefore equivalent to a highly supersonic orbital speed.

Put $K(r)=e^{s(r)}$. The perfect gas relation in the question becomes the polytropic equation of stateAt fixed $r$, vertical hydrostatic balance isAfter one integration,Defining the midplane adiabatic sound speed by $c_{s0}^2=\gamma P_0/{\rho }_0=\gamma K{\rho }_0^{\gamma -1}$ givesThis is a vertically truncated polytropic atmosphere; the physical perfect-gas case has $\gamma >1$.

At the midplane, radial hydrostatic equilibrium givesIf $d{P}_0/dr<0$, then $\Omega <{\Omega }_K$: the pressure gradient provides part of the inward force and the gas is Sub-Keplerian. For radial pressure scale comparable to $r$,Using $H\sim c_s/{\Omega }_K$ and ${\Omega }_K-\Omega \simeq ({\Omega }_K^2-{\Omega }^2)/(2{\Omega }_K)$,