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An accretion disk is an orbiting flow in which stresses permit matter to move inward by transporting angular momentum to other matter or an outflow. Simply losing heat is insufficient for gas that retains too much angular momentum to reach the center. The stresses also dissipate orbital energy, potentially producing strong radiation. A spreading ring illustrates the exchange: some material can accrete while a smaller fraction moves outward carrying the required angular momentum.^[ref]

An alpha disk uses a dimensionless parameter $\alpha$ to represent unresolved transport by an effective kinematic viscosity, commonly $\nu =\alpha c_{s}H$. With gas-supported hydrostatic equilibrium, this gives $\nu \sim \alpha c_s^2/\Omega$ and a viscous timescale much longer than an orbital time when $\alpha (H/r{)}^2\ll 1$. The prescription makes disk evolution calculable without resolving every turbulent motion. A value of $\alpha$ is a closure assumption, rather than an identification of the physical instability producing the stress.^[ref]

The disk scale height measures the vertical spread of matter or pressure. For an isothermal, non-self-gravitating thin disk, $\rho (z)={\rho }_0\exp [-z^2/(2H^2)]$ and $H=c_s/{\Omega }_z$. Stronger vertical gravity confines the same gas more tightly; a higher sound speed makes it thicker. Here $H$ is a standard deviation, not a sharp surface; a polytropic surface uses a different convention.^[ref]

For a locally plane-parallel thin disk, assume a polytropic equation of state $P=K{\rho }^{1+1/n}$ with $K>0$ and polytropic index $n>0$. Let ${\rho }_0$ be the midplane mass density and $c_0^2=(1+1/n)K{\rho }_0^{1/n}$ the midplane squared sound speed. Take the external vertical gravitational acceleration to be $-{\Omega }^{2}z$, as for a local Keplerian disk.

The specific enthalpy is $\mathcal{H}=(n+1)K{\rho }^{1/n}$. Define $w=\mathcal{H}/c_0^2=n(\rho /{\rho }_0{)}^{1/n}$ and $\zeta =\Omega z/c_0$. Hydrostatic equilibrium gives $d\mathcal{H}/dz=-{\Omega }^{2}z-d{\Phi }_d/dz$, while the plane-parallel Poisson equation gives $d^2{\Phi }_d/d{z}^2=4\pi G\rho$. Combining them yieldsThis $Q_0$ is a midplane-density parameter; it is not the surface-density Toomre parameter. Midplane symmetry imposes $w(0)=n$ and $w^{\prime }(0)=0$. The first positive zero of $w$ defines the model's surface, where the mass density vanishes.

In the negligible-self-gravity limit $Q_0\to \infty$, $w=n-{\zeta }^2/2$ and the half-thickness is $\sqrt{2n}{c}_0/\Omega$. This finite polytropic surface differs from an isothermal Gaussian atmosphere with no sharp edge.^[ref]

The Shakura--Sunyaev thin disk combines a thin disk, large optical depth, efficient local radiative cooling and an alpha disk stress prescription. At each radius, orbital-energy dissipation is balanced by radiation escaping vertically. The model predicts a radial sequence of effective temperatures and therefore a broad spectrum. It is useful when vertical adjustment and cooling are faster than radial evolution; strong inward heat transport, winds or large thickness require a different approximation.^[ref]

Thermal stability of an accretion disk asks whether a small temperature disturbance decays while the local surface density is effectively fixed. Write net heating as $Q^{+}-Q^{-}$; with positive effective heat capacity, stability requires ${\partial }_T(Q^{+}-Q^{-})<0$ at equilibrium. A hotter patch must gain a restoring excess of cooling. The comparison belongs on the thermal timescale of an accretion disk, before slower radial redistribution changes the column. Its outcome depends on the stress and opacity prescriptions. The standard total-pressure alpha disk closure predicts instability in a radiation-supported, electron-scattering regime.^[ref]

The vertical structure of an astrophysical disk describes how mass density, pressure and temperature vary away from its midplane. For a thin, centrally dominated gas layer, hydrostatic equilibrium gives $dp/dz=-\rho {\Omega }_z^{2}z$. An isothermal ideal gas has $p=c_s^2\rho$, producing a Gaussian density profile with disk scale height $H=c_s/{\Omega }_z$. Disk self-gravity or magnetic support changes that balance.^[ref]

Viscous evolution of an accretion disk follows the redistribution of surface density as internal stresses exchange angular momentum between rings. For an axisymmetric Keplerian disk without mass loss, combining mass and angular-momentum conservation gives ${\partial }_t\Sigma =(3/r){\partial }_r[r^{1/2}{\partial }_r(\nu \Sigma r^{1/2})]$. Here $\nu$ is an effective kinematic viscosity. The equation describes both inward accretion and outward spreading; the sign of local mass flow follows from the radial gradient, rather than viscosity alone.^[ref]

The viscous torque in an accretion disk is the rate of outward angular momentum transport by an internal shear stress. For an axisymmetric thin disk rotating with angular velocity $\Omega (r)$, a local Newtonian viscous stress has $T_{r\varphi }=\rho \nu rd\Omega /dr$. Integrating that stress over a cylinder and multiplying by the lever arm givesHere $\bar{\nu }$ is the density-weighted mean kinematic viscosity, with units of length squared per time. The vertically integrated quantity is $\bar{\nu }\Sigma$. For decreasing $\Omega$, the sign convention makes $\mathcal{G}>0$, describing outward angular momentum transport. More general internal stresses can be represented by an effective $\bar{\nu }$ through the same integrated torque relation.^[ref]