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

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1. Introduction

Most previous theoretical works on shocks in accretion or wind flows around black holes were only for the non-rotating black hole case, and were only in terms of the Paczynski & Wiita (1980) pseudo-Newtonian potential (see, e.g. Chakrabarti & Molteni 1995 for references). Recently, Yang & Kafatos (1995) made a fully relativistic treatment of isothermal shocks in accretion flows onto a Schwarzschild black hole. Astrophysically realistic black holes, however, are generally believed to possess considerable angular momentum (e.g. Rees 1984). Perhaps because of the complexity of the Kerr metric components, a few papers published so far in the shock study have been devoted to the Kerr black hole case: Chakrabarti (1990) dealt analytically with standing Rankine-Hugoniot shocks in rotating accretion and wind flows in Kerr geometry, but the work was rather initial, in the sense that only a few examples of the shock solutions were presented; Sponholz & Molteni (1994) gave a plenty of numerical results on shocks in a thin, adiabatic accretion disk around a rotating black hole, but using another pseudo-Newtonian potential proposed by Chakrabarti & Khanna (1992).

In the present paper we discuss the problem of shock formation in stationary, axisymmetric, adiabatic flows of perfect fluids in, or very close to, the equatorial plane of Kerr geometry. It is known that the formation of shocks is based on the existence of multi-sonic points in the flow, because the flow is supposed to be originally subsonic and terminally supersonic (when an accretion flow crosses the black hole horizon, or a wind flow goes far away), it must pass through a sonic point on each side of the shock. It is therefore necessary to introduce some background knowledge of sonic points in the following paragraphs, which is contributed by many authors, e.g. Liang & Thompson 1980; Lu 1985; Lu & Abramowicz 1988; Anderson 1989; Abramowicz & Kato 1989.

For the flow considered here, there exist two intrinsic constants of motion along a fluid world line, namely the specific total energy, [FORMULA], and the specific angular momentum, [FORMULA] where [FORMULA] 's are the four velocity components obeying the normalization condition [FORMULA] (we use the Boyer-Lindquist coordinates together with [FORMULA] units and - + + + signature), h is the specific enthalpy, i.e. [FORMULA], with [FORMULA] and [FORMULA] being the pressure, the mass-energy density and the rest mass density, respectively. Other useful relations are the conservation of rest mass along the flow (we assume a conical shape flow), [FORMULA] const., and an equation of state which is assumed to be a polytropic one, i.e. [FORMULA] where n is the constant polytropic index, and K is a measurement of the specific entropy of the flow, which is a constant in a shock-free flow, but can increase across the shock. As Chakrabarti (1989) noticed, it is wise to define an 'entropy related' mass flow rate as [FORMULA] which is a conserved quantity for a shock-free flow, but can become larger at the shock due to the generation of entropy.

Under such circumstances, it can be shown that the properties of the critical (sonic) point in the radial motion of the flow, i.e. the location of the point, [FORMULA], and the sound speed of the flow (defined as [FORMULA] at the point, [FORMULA], which is equal to the local radial three velocity of the flow measured by a corotating observer, and accordingly the value of [FORMULA] are all determined by the two physical parameters, E and l. In particular (see Fig.  1), in the parameter space spanned by E and l, there is a strictly defined region bounded by four lines: the vertical line [FORMULA] and three characteristic functional curves [FORMULA], and [FORMULA], where [FORMULA] is the Keplerian angular momentum of the fluid (Lu, Yu & Young 1995), [FORMULA] and [FORMULA] are, respectively, the maximum and the minimum values of the function [FORMULA], i.e. the values of l satisfying [FORMULA]. Only such a flow with its parameters located within this region can have two physical sonic points, the inner one [FORMULA], and the outer one [FORMULA] ; in between there is still one more, but unphysical, sonic point [FORMULA]. The two physical sonic points are corresponding to the same pair of E and l, but are distinct in the radius and in the local sound speed, thus determining two different values of [FORMULA]. The multi-sonic point region is divided by another characteristic functional curve, [FORMULA] into two parts: in region I (= Ia + Ib) only [FORMULA] is realized in a shock-free global solution (i.e. that joining the black hole horizon to the large distance), while in region II (= IIa + IIb) only [FORMULA] is; in either case the global solution corresponds always to the smaller one of the two potential values of [FORMULA], while the larger value of [FORMULA], which is connected with the unrealizable sonic point, cannot be realized in shock-free solutions.

[FIGURE] Fig. 1a-c. Energy-angular momentum parameter space of standing shock formation in adiabatic flows around a black hole. a (upper) is for a Schwarzschild black hole ([FORMULA]), b (middle) for a rapid Kerr hole ([FORMULA]) with prograde flows, and c (lower) for a rapid Kerr hole with retrograde flows. Only for an accretion flow with its parameters E and l located within region Ia (bounded by lines [FORMULA] and [FORMULA]), or a wind flow with E and l belonging to region IIa (bounded by lines [FORMULA] and [FORMULA]), there exists an unique stable shock. See text for details.

However, when shocks are taken into account it seems that an accretion flow with parameters in region I may develop a shock in its radial motion after passing through [FORMULA], the value of [FORMULA] jumps from the smaller one to the larger one at the shock, then the flow becomes supersonic again by passing through [FORMULA] ; while a wind flow belonging to region II may also have a shock, although taking exactly the opposite way: [FORMULA] a shock [FORMULA], across the shock the value of [FORMULA] increases too. Although the formation of these shocks is only a possibility for the moment, it is certain that neither accretion flows in region II, nor winds in region I could possibly produce shocks, because such processes would require the value of [FORMULA], i.e. require the entropy of the flow to decrease, violating the second law of thermodynamics.

Based on the above knowledge we go a step further in the present paper, examining the possibilities and the properties of the standing Rankine-Hugoniot shock in terms of the intrinsic physical parameters of the flow.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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