4. Locations and strengths of stable shocks
The resulting stable shock location for accretion flows, i.e. , is shown in Fig. 3 as a function of E and l. The range of E and l values in Fig. 3 is identical with region Ia in Fig. 1. It is seen from Fig. 3 that there is always an almost uniform lower limit of stable shock locations for different values of E and l: in the case of a rapid Kerr black hole with prograde flows (Fig. 3b) this limit is about three gravitational radii, which is considerably closer to the hole than that in the case of a rapid Kerr hole with retrograde flows (about twenty gravitational radii, see Fig. 3c) as well as that in the Schwarzschild black hole case (somewhere between ten and twenty gravitational radii, see Fig. 3a). On the other hand, the shock location can always extend outwards to almost the same distance for flows with the same total energy, no matter what kind the central black hole is of, for example, for the shock location extends to about a thousand gravitational radii in all the three cases shown in Figs. 3a, b, c. It is also observed from Fig. 3 that the shock location for retrograde flows around a rapid Kerr black hole (Fig. 3c) behaves similarly as that for flows around a Schwarzschild hole with higher values of l (Fig. 3a, here we mean the absolute value of l), reflecting the fact that in the former case some centrifugal force is 'spent' to fight against the frame-dragging; for prograde flows around a rapid Kerr hole (Fig. 3b, cf. also Fig. 1), however, the frame-dragging helps to strengthen the centrifugal barrier, so that only those with lower angular momenta and higher energies are allowed to go through the barrier, i.e. to form shocks. Astrophysical implications of this distinction need further studying.
The corresponding shock strength, defined as the ratio of Mach numbers just before and just after the shock, , is drawn in Fig. 4 as functions of E and l. It is seen that the strength generally increases with decreasing energy and/or decreasing angular momentum (in absolute value), and that the shock in prograde flows around a rapid Kerr black hole is stronger than that in flows around a Schwarzschild hole as well as that in retrograde flows around a rapid Kerr hole.
For the sake of completeness, we show the location of the stable shock in winds, i.e. , in Fig. 5 as functions of E and l, the range of E and l values is identical with region IIa in Fig. 1a. The corresponding shock strength is given in Fig. 6.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998