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

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2. Multiplicity of the shock location

The Rankine-Hugoniot shock conditions for a relativistic fluid flow are continuities for rest-mass, energy, and momentum flux densities(Landau & Lifshitz 1987):

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] 's are the energy-momentum tensor components, [FORMULA] 's are the metric components, and the square brackets denote the difference between the values of a quantity on the two sides of the shock. For flows with properties described in Sect.1 we transform Eq.(1) into the following working set of the jump conditions across the shock:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where L is the effective potential of the flow motion defined as [FORMULA] (Lu et al. 1995), [FORMULA] is the location of the shock, the subscripts - and + denote the values before and after the shock, respectively.

As stated in Sect.1, once the values of the two constants, E and l, are given, the value of [FORMULA] is accordingly obtained, so it is not another independent constant, but is the eigenvalue of the problem, [FORMULA]. A flow with E and l located within the multi-sonic point region of the parameter space (bounded by lines [FORMULA] and [FORMULA] in Fig. 1) can have two different values of [FORMULA], the smaller one, [FORMULA], is realized either in a shock-free global solution, or in the preshock part of a shock-included solution, while the larger one, [FORMULA], cannot be realized in a shock-free solution, it can be realized only in the postshock part of a shock solution. Therefore, with a suitably given pair of E and l, and two accordingly determined quantities, [FORMULA] and [FORMULA], the five Eqs. (2a-e) enable us to solve for five unknowns: [FORMULA]. The radial three-velocity measured by a corotating observer, [FORMULA], is obtained by a transformation (Lu 1986): [FORMULA], and the Mach number is defined as [FORMULA]. The shock solution of the flow, as in the shock-free case, is completely determined by the two constants of motion, E and l.

The first thing is the location of the shock. The computing results show that there exists a new strictly defined function, [FORMULA], drawn in Fig. 1 as two curves [FORMULA] and [FORMULA], 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 are four formal shock locations, denoted by [FORMULA] and [FORMULA], which are related to the three formal locations of the sonic point, [FORMULA] and [FORMULA], as

[EQUATION]

while for an accretion flow belonging to region Ib (bounded by lines [FORMULA] and [FORMULA] in Figs. 1a and 1b, and by lines [FORMULA] and [FORMULA] in Fig. 1c), or a wind flow belonging to region IIb (bounded by lines [FORMULA] and [FORMULA] in Figs. 1a and 1b, and by lines [FORMULA] and [FORMULA] in Fig. 1c), there are only two formal shock locations, denoted by [FORMULA] and [FORMULA], satisfying

[EQUATION]

An example of the multiplicity of the formal shock location, taken from region I of Fig.  1a, is shown in Fig. 2, where [FORMULA], and the corresponding [FORMULA] (hereafter [FORMULA] and the black hole's specific angular momentum a are all in units of the black hole mass), [FORMULA], it is clearly seen that for [FORMULA] there are four formal shock locations satisfying relation (3), while for [FORMULA] there are only two satisfying relation (4). These results are qualitatively similar to those for adiabatic flows around a non-rotating black hole in the context of Paczynski & Wiita (1980) potential (Chakrabarti 1989), as well as those for isothermal accretion flows in Schwarzschild geometry (Yang & Kafatos 1995). A preliminary discussion on the nature of these formal shock locations can be made by employing the boundary conditions: for an accretion flow onto a black hole both [FORMULA] and [FORMULA] can be ruled out, because the flow must be supersonic when crossing the black hole horizon and subsonic when starting at the large distance, and no sonic point exists inside [FORMULA] or outside [FORMULA] (however for accretion onto a neutron star [FORMULA] is a possible shock location, because the flow is terminally subsonic); while for a wind flow [FORMULA] is not possible either, but [FORMULA] is possible if the flow is terminally subsonic. The possibilities of shocks at [FORMULA] and [FORMULA], however, cannot be judged in this way, on which we focus our attention from now on.

[FIGURE] Fig. 2. An example of the multiplicity of the formal shock location in accretion flows, where [FORMULA] and [FORMULA] For [FORMULA] there are four formal shock locations satisfying relation (3), while for [FORMULA] there are only two satisfying relation (4).
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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