Tori and the Kerr Metric

The initial stationary, axisymmetric analytic torus solution is presented in HSWa. For convenience, the essential results are repeated here.

The initial state for the simulations is a stationary, axisymmetric solution ($\partial_t = \partial_\phi = 0$) to equations (5)--(7) with no internal poloidal motion, i.e. $U^r=U^\theta=0$. The combined centrifugal and gravitational accelerations (which together make up an effective potential) are balanced by pressure gradients, keeping the disk in equilibrium. To develop this solution, define the specific angular momentum (l) and angular velocity ($\Omega$), as

$$U_\mu = U_t\,\left(1,0,0,-l\right)$$ (8)

$$U^\mu = U^t\,\left(1,0,0,\Omega\right).$$ (9)

Applying the orthogonality condition $U^\mu\,U_\mu= -1$ leads directly to the following expression for U_t

$$U_t = -{\left( \Vert g^{t t} - 2\,l\,g^{t \phi}+ l^2\,g^{\phi \phi}\Vert\right)}^{-1/2}$$ (10)

which is related to the binding energy, e_bind = -U_t. Here, we note that equation (94a) of HSWa should be extended to allow a more general disk outer boundary, which is specified by the binding energy at the surface e_surf = -U_tlim. The more general form of equation (94a) is

$$\epsilon = {1 \over\Gamma}\,\left({U_{t_{lim}} \over U_t} - 1\right).$$ (11)

For a constant entropy adiabatic gas the pressure is given by $P =\rho\,\epsilon\,(\Gamma - 1) = K\,\rho^\Gamma$, and density is given by $\rho={\left[{\epsilon\,(\Gamma - 1) / K}\right]}^{1/(\Gamma - 1)}$. These relations completely specify the initial equilibrium torus.

A particular constant angular momentum thick disk solution is specified by choosing the angular momentum l, the binding energy at the surface of the torus (as above), and the entropy parameter K. In keeping with HSWb, K=0.01 is fixed for all simulations, so only the specific angular momentum l, surface binding energy e_surf, and the Kerr parameter a are varied.
 

2002-06-05