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

For a circular orbit of cylindrical radius $r$ in an axisymmetric gravitational potential,A small vertical displacement satisfiesso the vertical epicyclic frequency isFor a spherically symmetric potential $\Phi (R)$, where $R=(r^2+z^2{)}^{1/2}$,Hence $\boxed{{\Omega }_z=\Omega }$. Geometrically, a slightly tilted circular orbit remains a circular orbit in a different plane, and its height completes one oscillation per revolution.

Vertical hydrostatic equilibrium givesDefine vertically integrated pressure and the density-weighted disk scale height byMultiply the hydrostatic equation by $z$ and integrate. Since $zp\to 0$ at both boundaries, integration by parts givesTherefore

Introduce the dimensionless variablesUsing $P=\Sigma H^2{\Omega }_z^2$, vertical hydrostatic equilibrium becomes the parameter-free equationwith normalizations $\int \tilde{\rho }d\tilde{z}=1$ and $\int \tilde{p}d\tilde{z}=1$. For an isothermal atmosphere, $p=c_s^2\rho$, and these conventions give the Gaussian distribution

Gas in hydrostatic balance has zero vertical velocity. A dust grain subject to linear drag with aerodynamic stopping time $\tau$ therefore obeysThis is a damped harmonic oscillator. For ${\Omega }_z\tau >1/2$, the motion is underdamped, with angular frequencyand envelope $e^{-t/(2\tau )}$. Critical damping occurs at ${\Omega }_z\tau =1/2$. For ${\Omega }_z\tau <1/2$, the motion is overdamped. In the strong-drag limit ${\Omega }_z\tau \ll 1$, a rapid transient on timescale $\tau$ leaves slow dust settling in an astrophysical disk at rate ${\Omega }_z^2\tau$; in the weak-drag limit the grain makes many damped vertical oscillations.

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.

At the equilibrium thickness $H_0$,Set $H=H_0+h$ and retain terms linear in $h$. The linearization isThus the vertical breathing mode of an astrophysical disk has angular frequency