Disk Properties and the Kerr Parameter

One of the goals of H91 was to study the effect of the proximity of the cusp in the relativistic potential to the inner edge of the torus, and how accretion into the black hole affects mode growth. Thus the majority of the H91 runs dealt with tori with angular momentum above the marginally bound value, that is the smallest specific angular momentum for which a particle orbit has an inner turning point. In H91, an initial torus configuration was specified by choosing the angular momentum and the surface binding energy. Here, the structural properties of the initial torus are also influenced by the Kerr parameter. We therefore need to understand the interdependence of l, U_tlim, and a in order to specify initial configurations for the simulations.

To help guide the choice of l, it is useful to consider the quantity l_mb that gives the specific angular momentum for a test particle on a marginally bound orbit. Particles on retrograde orbits require larger absolute values of the specific angular momentum to remain out of the hole than do particles on prograde orbits. This can be seen in Figure 1, which plots the dependence of l_mb on the Kerr parameter a; the relation l_mb(a) can be found, for instance, in Frolov and Novikov (1998). The quantity l_mb also helps distinguish families of tori that have stable orbits from those that do not. Tori with l = l_mb are referred to as marginal tori from hereon.

Figure 1: Plot of l_mb(a), the angular momentum of a marginally bound orbit as a function of black hole angular momentum.

The PPI affects slender, intermediate, and wide tori in different ways, so the choice of torus dimensions is important. Given a value for l, the choice of surface binding energy, -U_tlim, sets the overall size of the disk. Figure 2 illustrates how different choices of -U_tlim determine the width of the torus, here shown for l = 4.5 and a range of choices for a. This illustration reinforces the notion that the torus is created by ``filling" a local minimum in the energy surface; it also shows that an inner bound for a torus may not exist as a is made increasingly negative, and the reason for this is best seen in the next figure.

Figure 2: Plots of energy function e = -U_t for l=4.5. The choice of -U_tlim determines the width of the disk. Here, the relevant portion of the energy surface (in the equatorial plane) is shown for a =-0.8 (left), a =0.0 (middle), a =1.0 (right), along with four reference lines for -U_tlim=0.970, 0.974, 0.978, and 0.982. Notice how -U_tlim=0.982 has no inner bound for a=-0.8, and -U_tlim=0.970 has no solution for a=1.0.

The role of the Kerr parameter a on the disk configuration is illustrated by Figure 3, which shows equipotentials for disks with l=4.5, and the same values of a as Figure 2. The equipotentials are defined by the Boyer energy parameter $\Phi=\log(-{U_t})$. This figure presents polar ``slices'' through the axisymmetric disk. The set of level contours is the same in each plot, so the effect of a is readily apparent: the equipotentials shrink in the prograde case, and expand in the retrograde case, a result consistent with what was shown in Figure 2 for the equatorial plane. So, for a given choice of surface binding energy, a retrograde torus would be larger, and a prograde torus would be more compact, than the Schwarzschild case.

In H91 the value of l=4.5 was chosen since it corresponded to a bound torus in the Schwarzschild metric. Figures 1 and 2 hint that this choice of l may not correspond to bound tori for all values of a down to a = -1. Figure 3 further reinforces this observation, and also provides a clearer picture: many of the contours for the a = -0.8 case are open onto the black hole, and the viable range of parameters for which a bound torus can be constructed is growing smaller. For choices of a below a = -0.8 virtually all contours are open onto the hole, so an equilibrium initial state cannot be constructed.

Figure 3: Equipotentials for l=4.5 disks. The left panel is for a retrograde torus, with a = -0.8, the middle panel is for a = 0, the right is for a prograde torus, with a = 1.0. These choices of l and a are the basis for torus models A3 and B3 described in §4.

Figure 4: Equipotentials for marginal and slightly below marginal prograde tori with a = -0.5. The right panel shows the equipotentials for the case where l = l_mb, and the left for l = l_mb-0.04. Note how the potential cusp at r = 4.95M in the right panel opens up in the left panel.

The transition from closed to open contours can best be seen in Figure 4, which shows a plot of equipotentials for l at and just below the marginally bound value for a disk orbiting a Kerr hole with a=-0.5 (the point in Figure 1 where the l_mb curve passes through l=4.5). The effect near the cusp is striking: a small reduction in l below the marginally bound value opens a pathway to the black hole in the equatorial plane, so that a disk whose structural parameters give it an inner bound near r = 4.95 M would not be expected to remain in equilibrium, and any small departure from equilibrium would trigger accretion. The opening of the potential cusp can also be accomplished by choosing a < -0.5 for an l = 4.5 torus, which explains the shape of the contours in the left panel of Figure 3.

Finally the analytic thick disk solution is characterized by a number of derived parameters. One is the location of the pressure maximum, r_Pmax, which is obtained by root-finding using the condition $\partial_r \epsilon(r,\pi/2)=0$ since the pressure and energy maxima are coincident. The angular frequency at the pressure maximum, $\Omega_{P_{max}} = (g^{t \phi} - l g^{\phi \phi})/(g^{tt} - l g^{t \phi})$, and the orbital period at the pressure maximum, $T_{orb} =2\,\pi/\Omega_{P_{max}}$. Time in the simulations will be reported in terms of this orbital period. The boundaries of the disk in the equatorial plane are r_in and r_out. Since r_in is an input parameter (through the choice of U_tlim), a simple plot of $\rho(r,\pi/2,\phi)$ helps locate r_out, which is in turn used to specify the required outer boundary of the radial grid (usually set at about $r_{max} \approx 2\,r_{out}$).

2002-06-05