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Astron. Astrophys. 321, 665-671 (1997)

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4. Locations and strengths of stable shocks

The resulting stable shock location for accretion flows, i.e. [FORMULA], 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 [FORMULA] the shock location [FORMULA] 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.

[FIGURE] Fig. 3a-c. Stable shock locations for accretion flows, i.e. [FORMULA], as functions of E and l. a (upper) is for [FORMULA], where the seven lines are, from the top to the bottom, for [FORMULA] [FORMULA] respectively; b (middle) is for [FORMULA] and prograde flows, the eight lines are, from the top to the bottom, for [FORMULA] respectively; c (lower) is for [FORMULA] and retrograde flows, the seven lines are, from the top to the bottom, for [FORMULA] [FORMULA] respectively.

The corresponding shock strength, defined as the ratio of Mach numbers just before and just after the shock, [FORMULA], 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.

[FIGURE] Fig. 4a-c. Strengths of the stable shocks for accretion flows as functions of E and l. a-c are for [FORMULA] and prograde flows, and [FORMULA] and retrograde flows, respectively. The lines in the three figures are, from the rightmost to the leftmost, corresponding to those from the top to the bottom in Figs. 3a, 3b, and 3c, respectively.

For the sake of completeness, we show the location of the stable shock in winds, i.e. [FORMULA], 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.

[FIGURE] Fig. 5. Stable shock locations for [FORMULA] and wind flows, i.e. [FORMULA], as functions of E and l. The four lines are, from the top to the bottom, for [FORMULA] respectively.

[FIGURE] Fig. 6. Strengths of the shocks in Fig. 5. The four lines are, from the rightmost to the leftmost, corresponding to those from the top to the bottom in Fig.  5, respectively.
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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