Astron. Astrophys. 363, 425-439 (2000)

1. Introduction

Recent observations give strong evidence that the energy sources of quasars and active galactic nuclei are accretion disks around central massive black holes (Abramowicz & Percival 1997; Blandford 1990). Similar, scaled down, accretion disks appear in some extraordinary galactic binary systems containing a black hole (or a neutron star). In the accretion disks, the potential gravitational energy of matter orbiting the central black hole is liberated and transferred into heat, due to viscous stresses acting against shearing motion, and radiated (at least partly) away. During the process, angular momentum of the accreting matter has to be transported outwards.

It is well known that at low accretion rates the pressure is negligible, and the accretion disk is geometrically thin. Its basic properties are determined by the circular geodesic motion in the black-hole background. The radius of the marginally stable circular orbit represents the inner edge of the Keplerian disks, since matter, following quasikeplerian orbits down to , falls freely into the black hole from this radius (Novikov & Thorne 1973; Stoeger 1976). At high accretion rates, the pressure is relevant, and the accretion disk must be geometrically thick. Its basic properties are determined by equipotential surfaces of test perfect fluid (i.e., perfect fluid that does not alter the black-hole geometry) rotating in the black-hole background. The accretion is possible, if a toroidal equilibrium configuration of the test fluid containing a critical, self-crossing equipotential surface can exist in the background. The cusp of this equipotential surface corresponds to the inner edge of the disk, and the accretion inflow of matter into the black hole is possible due to a mechanical non-equilibrium process, i.e., because of matter slightly overcoming the critical equipotential surface. The pressure gradients push the inner edge of the thick disks under the radius (Kozlowski et al. 1978; Abramowicz et al. 1978).

The simplest, but quite illustrative case of the equipotential surfaces of the test fluid can be constructed for the configurations with uniform distribution of the angular momentum density. This case is fully governed by the geometry of the spacetime, however, it contains all the characteristic features of more complex cases of the rotation of the fluid (Jaroszynski et al. 1980). Moreover, this case is also very important physically since it corresponds to marginally stable equilibrium configurations (Seguin 1975).

The equipotential surfaces were analyzed for both Schwarzschild and Kerr black-hole spacetimes. The critical closed surfaces with a cusp can exist for angular momentum density higher (lower) than the one corresponding to the marginally stable (bound) circular geodesic, and the location of the cusp shifts from to the radius of the marginally bound geodesic orbit . The cusp close to the horizon enables the inflow of matter into the black hole. However, the character of the equilibrium configurations does not allow outflow of matter and transfer of the angular momentum for disks around isolated black holes. In binary systems the outflow is possible through the Lagrange point L2 - see, e.g., Novikov & Thorne (1973).

Very recently, a wide variety of cosmological observations (measurements of the present value of the Hubble parameter, details of the anisotropy of the cosmic relic radiation, statistics of gravitational lensing of quasars, and high-redshift supernovae) suggest a non-zero, repulsive cosmological constant (Krauss & Turner 1995; Ostriker & Steinhardt 1995; Krauss 1998). Therefore, it is interesting to clarify the influence of the repulsive cosmological constant on astrophysically relevant properties of black-hole spacetimes.

Here, we shall show that in the field of black-hole spacetimes with a repulsive cosmological constant the outflow of matter from the accretion disk is possible, because equipotential surfaces with an outer cusp in vicinity of the so called static radius can exist (beside the critical surfaces with the inner cusp nearby the horizon), if the mass of the black hole is small enough to admit existence of the stable circular geodesics (Stuchlík & Hledík 1999). Moreover, if the uniform angular momentum density of the equilibrium configuration corresponds to the marginally bound orbit of the background, the critical equipotential surface has both the inner and outer cusps. In this situation, any mechanical non-equilibrium in the thick disk leads to both inflow into the black hole, and outflow from the disk near the static radius.

The plan of this paper is following. In Sect. 2, the basic formulae for the equilibrium configurations of test perfect fluid in a given stationary and axially symmetric background are summarized, following the papers of Abramowicz and coworkers (Kozlowski et al. 1978; Abramowicz et al. 1978; Jaroszynski et al. 1980). In Sect. 3, the equipotential surfaces of the marginally stable configurations (having a uniform distribution of angular momentum density) of the test perfect fluid are determined for the Schwarzschild-de Sitter black-hole spacetimes. For completeness, we include also discussion of the case of the Schwarzschild-anti-de Sitter spacetimes with an attractive cosmological constant. In Sect. 4, some concluding remarks are presented, and astrophysical consequences of the presented results are pointed out. We shall use the geometric system of units (), if not stated otherwise.

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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