Let us consider an accretion disc around a spinning black hole. Suppose the angular momenta of the disc and black hole are misaligned. Due to Lense Thirring precession, the line of nodes of each annulus in the disc rotates in the equatorial plane of the black hole at a rate
where is the black hole's angular momentum, is the gravitation constant, is the speed of light and is the semi major axis. In an accertion disc, matter slowly flows inward toward the black hole at some velocity (see section below), so it spends a time at radius . If the precession frequency is , then the rotation angle of the line of nodes is given by
This changing nodal angle gives rise to a spiral pattern in the disc. A illustration of this pattern can be seen in the pictures to the right.
Assuming Shakura Sunyaev model for the disc, the diffusion coefficient is given by where is the Keplerian frequency, is the gravitation constant, is the mass of the central object, is the semi major axis and is the scale height. The diffusion time is given by
Dividing the semi major by the diffusion time yields the radial velocity
The scale heigh is related to the temperature through
where is the Boltzmann constant and is the mass of the gas particles. The temperature is determined by the condition of equilibrium between the heating rate due to viscosity
and cooling through blackbody emission
where is the surface mass density, is the Stefan Boltzmann constant, is the vertical optical depth and is the opacity. The surface density can be related to the accretion rate via the conservation of mass
We can now obtain an expression for the radial velocity as a function of radius
where is Planck's constant.