Bardeen Petterson Effect

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

$\omega_{lt} \approx \frac{G J}{c^2 a^3}$

where $J$ is the black hole's angular momentum, $G$ is the gravitation constant, $c$ is the speed of light and $a$ is the semi major axis. In an accertion disc, matter slowly flows inward toward the black hole at some velocity $v_r$ (see section below), so it spends a time $t \approx r/v_r$ at radius $r$. If the precession frequency is $\omega_{lt}$, then the rotation angle of the line of nodes is given by

$\alpha \approx \omega_{lt} r/v_r \approx \frac{G^{6/5} J M^{1/5} \mu^{4/5} \sigma^{1/5}}{a^{8/5} c^2 k^{4/5} \dot{M}^{2/5} \alpha^{4/5} \kappa{1/5}} \approx \frac{G^{6/5} J M^{1/5} \mu^{4/5}}{a^{8/5} c^{12/5} h^{3/5} \dot{M}^{2/5} \alpha^{4/5} \kappa^{1/5}}$

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.

Radial Velocity

Assuming Shakura Sunyaev $\alpha$ model for the disc, the diffusion coefficient is given by $\nu \approx \alpha \Omega H^2$ where $\Omega \approx \sqrt{G M/a^3}$ is the Keplerian frequency, $G$ is the gravitation constant, $M$ is the mass of the central object, $a$ is the semi major axis and $H$ is the scale height. The diffusion time is given by

$t_d \approx \frac{a^2}{\nu} \approx \frac{a^2}{\alpha \Omega H^2}$

Dividing the semi major by the diffusion time yields the radial velocity

$v_r \approx \frac{a}{t_d} \approx \alpha \Omega H^2/a$

The scale heigh $H$ is related to the temperature $T$ through

$k T \approx \frac{G M \mu H^2}{a^3}$

where $k$ is the Boltzmann constant and $\mu$ is the mass of the gas particles. The temperature is determined by the condition of equilibrium between the heating rate due to viscosity

$L_h \approx \frac{\Sigma \Omega^2 a^4}{t_d}$

and cooling through blackbody emission

$L_c \approx \sigma a^2 T^4/\tau$

where $\Sigma$ is the surface mass density, $\sigma$ is the Stefan Boltzmann constant, $\tau \approx \kappa \Sigma$ is the vertical optical depth and $\kappa$ is the opacity. The surface density can be related to the accretion rate $\dot{M}$ via the conservation of mass

$\dot{M} \approx a^2 \Sigma v_r$

We can now obtain an expression for the radial velocity as a function of radius

$v_r \approx \frac{k^{4/5} \dot{M}^{2/5} \alpha^{4/5} \kappa^{1/5}}{a^{2/5} G^{1/5} M^{1/5} \mu^{4/5} \sigma^{1/5}} \approx \frac{\hbar^{3/5} c^{2/5} \dot{M}^{2/5} \alpha^{4/5} \kappa^{1/5}}{a^{2/5} G^{1/5} M^{1/5} \mu^{4/5}}$

where $\hbar$ is Planck's constant.