Accretion onto a stationary black hole has only been solved analytically under the assumption of spherical symmetry. Shapiro and Teukolsky (1983) give a Newtonian treatment of accretion, and Michel (1972) gives a full general relativistic one.
For a simple model of an accretion disk around a star, consider a star with luminosity L. The energy flux a distance d away is then given by
(1)
For a particle of radius at distance d in thermodynamic equilibrium at temperature T, the energy emitted (according to the Stefan-Boltzmann law) equals the energy absorbed,
(2)
where[img]http://scienceworld.wolfram.com/physics/aimg63.gif]is the Stefan-Boltzmann constant. Solving for T gives

(3)
Now, plugging inand taking
(4)
(5)
gives
(6)
For the hydrostatic law, the scale height for a cylindrical disk is



(7)
Let X and Y be the mass fractions of H and He, then and is then defined by
(8)
Hayashi (1981) uses Plugging in J s,kg,kg,,, and K, gives
(9)
Hayashi (1981) gives the empirical best fit for the surface density of an accretion disk as
(10)
then the optical path is
(11)
giving a density of





(12)
Consider motion of an annulus of gas with inner radius R and outer radius , surface density, and angular velocity , then
(13)
(14)
(von Weizsäcker 1948, Peebles 1981). Let v be the radial velocity, then the equations of continuity are
(15)
(16)
(17)
where
(18)
is the angular momentum (Hayashi 1981). Furthermore
(19)
where C is the circumference. Plugging in gives

(20)

(21)
(22)
(23)
From conservation of angular momentum,
(24)
whereis the sum of viscous torques from neighboring annuli.
Let G be the torque of an outer annulus acting on a neighboring inner one, then
(25)
whereis the kinematic viscosity. Therefore
(26)
so
(27)
Plugging (23) into (27) gives
(28)
Letting
(29)
then gives
(30)
Now, if varies as a power of R, then (30) can be solved analytically. Furthermore, if is a constant and , then


(31)
where is an arbitrary function determined by the initial conditions. Consider a ring of mass m at radius , then
(32)
Now let
(33)
(34)
then
(35)
(von Weizsäcker 1948, Peebles 1981).