Vertical structure of an astrophysical disk

The vertical structure of a thin disk is set by the balance among vertical gravity, pressure, self-gravity, magnetic forces, and motions of gas and solids.

The disk scale height is the characteristic vertical thickness over which density or pressure decreases. Vertical hydrostatic balance in a thin Keplerian gas disk gives $H\simeq c_s/{\Omega }_K$.

The vertical epicyclic frequency is the frequency of a small vertical oscillation about a circular orbit. In an axisymmetric potential, ${\Omega }_z^2={\partial }_z^2\Phi$ at the midplane; in a spherical potential it equals the circular orbital frequency.

Dust settling is the vertical drift of solid particles toward a disk's midplane under gravity and drag force. In the strong-drag limit, a particle with aerodynamic stopping time $\tau$ settles through a linear vertical gravitational field at rate ${\Omega }_z^2\tau$.

In homologous vertical motion, every height scales by one time-dependent thickness $H(t)$, so $z=H(t)\zeta$ and the vertical velocity is $u_z=\dot{H}\zeta$. A fixed dimensionless density profile then expands and contracts without changing shape.

The vertical breathing mode is a coherent oscillation of a disk's thickness. For an adiabatic gas with pressure proportional to $H^{1-\gamma }$ in a harmonic vertical potential, ${\omega }_H=\sqrt{\gamma +1}{\Omega }_z$.