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):
where 's are the energy-momentum tensor components, '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:
where L is the effective potential of the flow motion defined as (Lu et al. 1995), 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 is accordingly obtained, so it is not another independent constant, but is the eigenvalue of the problem, . A flow with E and l located within the multi-sonic point region of the parameter space (bounded by lines and in Fig. 1) can have two different values of , the smaller one, , is realized either in a shock-free global solution, or in the preshock part of a shock-included solution, while the larger one, , 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, and , the five Eqs. (2a-e) enable us to solve for five unknowns: . The radial three-velocity measured by a corotating observer, , is obtained by a transformation (Lu 1986): , and the Mach number is defined as . 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, , drawn in Fig. 1 as two curves and , for an accretion flow with its parameters E and l located within region Ia (bounded by lines , and ), or a wind flow with E and l belonging to region IIa (bounded by lines and ), there are four formal shock locations, denoted by and , which are related to the three formal locations of the sonic point, and , as
while for an accretion flow belonging to region Ib (bounded by lines and in Figs. 1a and 1b, and by lines and in Fig. 1c), or a wind flow belonging to region IIb (bounded by lines and in Figs. 1a and 1b, and by lines and in Fig. 1c), there are only two formal shock locations, denoted by and , satisfying
An example of the multiplicity of the formal shock location, taken from region I of Fig. 1a, is shown in Fig. 2, where , and the corresponding (hereafter and the black hole's specific angular momentum a are all in units of the black hole mass), , it is clearly seen that for there are four formal shock locations satisfying relation (3), while for 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 and 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 or outside (however for accretion onto a neutron star is a possible shock location, because the flow is terminally subsonic); while for a wind flow is not possible either, but is possible if the flow is terminally subsonic. The possibilities of shocks at and , however, cannot be judged in this way, on which we focus our attention from now on.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998