Astrophysical disk

An astrophysical disk is a flattened rotating distribution of gas, dust, stars, or other matter whose orbital motion strongly influences its structure and evolution.

A thin disk has vertical semi-thickness $H$ much smaller than cylindrical radius $r$. Vertical gravity is then approximately linear in height and orbital dynamics is nearly planar.

The surface density of a disk is its volume density integrated vertically, $\Sigma ={\int }_{-\infty }^{\infty }\rho dz$.

An accretion disk transports mass inward and angular momentum outward through stresses, while dissipating orbital energy as heat.

Viscous disk evolution combines mass conservation with radial transport of angular momentum by stress. In a steady Keplerian disk, the sum of advective and stress-carried angular-momentum fluxes is independent of radius.

For vertically integrated kinematic viscosity $\bar{\nu }$, the viscous torque across a cylinder of radius $r$ is

A decretion disk is fed near its inner edge and transports mass outward. Rapidly rotating Be stars can sustain gaseous decretion disks whose outer material is removed by a companion, radiation, or a wind.

A Be star is a rapidly rotating B-type star showing Balmer emission from a circumstellar gaseous decretion disk.

The alpha-disk prescription writes turbulent kinematic viscosity as $\nu =\alpha c_{s}H\simeq \alpha c_s^2/\Omega$, with dimensionless $\alpha \lesssim 1$. It reflects eddies no larger than the disk thickness and no faster than the sound speed.

The Shakura--Sunyaev thin disk is a geometrically thin, optically thick, radiatively efficient accretion-disk model whose turbulent stress is parametrized by the alpha prescription.

The magnetorotational instability destabilizes a magnetized differentially rotating flow when angular velocity decreases outward. Magnetic tension couples neighboring fluid elements, allowing the inner one to lose angular momentum and fall inward while the outer one gains angular momentum and moves outward.

For a vertical magnetic field in a Keplerian disk, axisymmetric vertical MRI modes are unstable when $k^2{v}_A^2<3{\Omega }^2$. The shortest unstable wavelength is ${\lambda }_{\mathrm{crit}}=2\pi v_A/(\sqrt{3}\Omega )$.

A local thermal equilibrium is stable when a small temperature increase raises cooling faster than heating at fixed surface density. The standard total-pressure alpha prescription makes a radiation-pressure-dominated, Thomson-opacity disk thermally unstable.

Disk self-gravity can overcome pressure and rotational support, producing growing density disturbances, spiral structure, or fragmentation.

For a razor-thin isothermal gas disk,Axisymmetric disturbances are stable when $Q\ge 1$: pressure stabilizes short wavelengths, epicyclic motion stabilizes long wavelengths, and self-gravity drives intermediate wavelengths.

The shearing sheet is a local Cartesian approximation to a differentially rotating disk. It retains Coriolis force, linearized tidal gravity, and a background linear shear.

Thermal relaxation drives the local sound speed toward an equilibrium value on a cooling time $\tau$. Fast relaxation gives an isothermal response, while slow relaxation gives an adiabatic response.