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

3. Equipotential surfaces of the marginally stable configurations orbiting Schwarzschild-de Sitter black holes

Equilibrium configurations of test perfect fluid rotating around an axis of rotation in a given spacetime are determined by the equipotential surfaces, where the gravitational and inertial forces are just compensated by the pressure gradient. (In an axially symmetric spacetime, the axis of rotation coincides with the axis of symmetry of the spacetime, while in a spherically symmetric spacetime the axis of rotation can be any radial line; usually, the coordinate system is chosen so that the rotation axis corresponds to .)

The equipotential surfaces can be closed or open. Moreover, there is a special class of critical, self-crossing surfaces (with a cusp), which can be either closed or open. The closed equipotential surfaces determine stationary equilibrium configurations. The fluid can fill any closed surface-at the surface of the equilibrium configuration pressure vanish, but its gradient is non-zero (Kozlowski et al. 1978). On the other hand, the open equipotential surfaces are important in dynamical situations, e.g., in modeling jets (Lynden-Bell 1969; Blandford 1987). The critical, self-crossing closed equipotential surfaces are important in the theory of thick accretion disks, because accretion onto the black hole through the cusp of the equipotential surface located in the equatorial plane is possible due to the Paczynski mechanism.

According to Paczynski, the accretion into the black hole is driven through the vicinity of the cusp due to a little overcoming of the critical equipotential surface, , by the surface of the disk. The accretion is thus driven by a violation of the hydrostatic equilibrium, rather than by viscosity of the accreting matter (Kozlowski et al. 1978).

It is well known that all characteristic properties of the equipotential surfaces for a general rotation law are reflected by the equipotential surfaces of the simplest configurations with uniform distribution of the angular momentum density - see Jaroszynski et al. (1980). Moreover, these configurations are very important astrophysically, being marginally stable (Seguin 1975). Under the condition

holding in the rotating fluid, a simple relation for the equipotential surfaces follows from Eq. (10):

with being determined by , and the metric coefficients only.

The equipotential surfaces are described by the formula , which can be given by the differential equation

which for the configurations with reduces to

The influence of a non-zero cosmological constant on character of the equipotential surfaces of the marginally stable configurations rotating around a black hole will be examined in the simplest case of Schwarzschild-de Sitter spacetimes corresponding to a repulsive cosmological constant, . (For completeness, we briefly discuss the case of Schwarzschild-anti-de Sitter spacetimes corresponding to an attractive cosmological constant, .)

In the standard Schwarzschild coordinates, the non-zero metric coefficients of the Schwarzschild-(anti)-de Sitter spacetimes are

Here, the radial coordinate r is expressed in units of the mass parameter M, and the dimensionless cosmological constant parameter

is introduced. It should be stressed that a static region exists in the Schwarzschild-de Sitter spacetimes with subcritical values of

of course, the equilibrium configurations are possible only in these spacetimes. Now, the equipotential surfaces are given by the formulae

and

for these relations reduce to the well known Schwarzschild formulae (Jaroszynski et al. 1980).

The best insight into the nature of the configurations can be obtained by the examination of the behavior of the potential in the equatorial plane (). There are two reality conditions of :

The first condition is identical with the condition for the static regions (located between the black-hole and cosmological horizons); the second condition can be expressed in the form

The function is the effective potential of the photon geodesic motion; recall that corresponds to the definition of the impact parameter for photon's geodesic motion - see Stuchlík & Hledík (1999). Further, the condition of the local extrema of the potential is identical with the condition of vanishing of the pressure gradient (, ). Since at the equatorial plane there is independently of the , and

we arrive at the condition

The extrema of correspond to the points, where the fluid moves along a circular geodesic, since corresponds to the distribution of the angular momentum density of the circular geodesic orbits. Clearly,

where

is the specific energy of the circular geodesics. (Recall that the specific energy of circular geodesics corresponds to the local extrema of the effective potential of the geodesic motion (Stuchlík & Hledík 1999).) The most important properties of the potential are determined by its behavior at the equatorial plane, and, especially, by the properties of the functions , and . Discussion of these properties enables us to give a classification of the Schwarzschild-(anti)-de Sitter spacetimes according to the properties of the equipotential surfaces of test perfect fluid. We shall separate the discussion to the case of the Schwarzschild-de Sitter (), and Schwarzschild-anti-de Sitter spacetimes (). For the pure Schwarzschild spacetime () the analysis can be found in (Kozlowski et al. 1978).

3.1. Schwarzschild-de Sitter black holes

If , the function diverges at the black-hole horizon, , and the cosmological horizon, , determined by equality in the condition (23). The horizons are given by the relations

where

The radii of the horizons are illustrated in Fig. 1. The local minimum of is located at , independently of y, and determines the unstable photon circular geodesic with the impact parameter

Fig. 1. Characteristic radii of the Schwarzschild-de Sitter spacetimes as functions of the parameter y. The black hole () and cosmological () horizons are given by bold solid lines, the static radius () by bold dotted line, the radii of marginally stable orbits ( and ) by thin dashed lines, and marginally bound orbits ( and ) by thin solid lines.

The function , determining the Keplerian (geodesic) circular orbits, has a zero point at the so called static radius given by

and it is not well defined at , being negative there. At the static radius (unstable) stationary equilibrium of test particles is possible because the gravitational attraction of the black hole is just compensated by the cosmological repulsion there.

The function diverges at the black-hole horizon: ; at the cosmological horizon, there is . Since

the local extrema of are given by the condition

determining the marginally stable circular geodesics. The local maximum of gives the critical value of the parameter y admitting stable circular orbits

If , there exists an inner (outer) marginally stable circular geodesic at , see Fig. 1. The angular momentum density of the marginally stable orbits , and , is simultaneously determined by Eqs. (27) and (36)-see Fig. 2. The specific energy of these orbits , and , is simultaneously determined by Eqs. (29) and (36)-see Fig. 3. There is other special value of y, corresponding to the situation, where the value of the minimum of equals to the maximum of . We denote this value . It can be found that

Fig. 2. The angular momentum density of the marginally stable ( and , solid line) and marginally bound (, dashed line) orbits as functions of the parameter y of the Schwarzschild-de Sitter spacetimes. Note that , .

Fig. 3. The specific energy of the marginally stable ( and , solid line) and marginally bound (, bold dotted line) orbits as functions of the parameter y of the Schwarzschild-de Sitter spacetimes. Note that , .

In the Schwarzschild-de Sitter spacetimes, there is another important class of circular geodesics-namely the marginally bound orbits. These orbits exist in the Schwarzschild-de Sitter spacetimes admitting existence of the stable circular orbits, i.e., spacetimes with . In these spacetimes, there exists an inner, (outer, ), marginally bound orbit close to the black-hole horizon (static radius). These orbits are defined by the condition

and are determined by an appropriate numerical procedure (see Fig. 1-Fig. 3). In the Schwarzschild spacetime () the marginally bound orbit is located at , and -it is because the effective potential of the geodesic motion at independently of the angular momentum density in the Schwarzschild spacetime. In the Schwarzschild-de Sitter spacetimes with the marginally bound circular orbits are not defined because only unstable circular orbits exist in these spacetimes; particles from them can always escape to infinity.

We can distinguish four qualitatively different cases of the behavior of the functions , which give four classes of the Schwarzschild-de Sitter black holes with different character of the equipotential surfaces of the rotating perfect fluid. These four classes are defined according to values of the cosmological parameter y in the following way:

• (A) ,

• (B) ,

• (C) ,

• (D) .

For these classes, the typical behavior of the functions , , with y fixed, is given in Figs. 4a-d. For completeness, the corresponding value of is exhibited in these figures. Note that the descending parts of the curve (with y fixed) correspond to the unstable circular geodesics, while its growing part (if it exists) corresponds to the stable circular geodesics. The extrema of , if they exist, have an important role: the minimum , at , determines the inner marginally stable circular geodesic, while the maximum , at , determines the outer marginally stable circular geodesic.

Fig. 4a-d. Behavior of the functions and in the four qualitatively different cases determining the four classes of the Schwarzschild-de Sitter spacetimes with different properties of the equipotential surfaces (both r and are given in units of M). Figures a-d reflect subsequently the cases , , , and . In the shaded region, the equipotential surfaces are not defined in the equatorial plane of the spacetime, defined by the axis of rotation of the perfect fluid. The descending parts of the function determine the cusps, while the growing parts determine central rings of the equilibrium configurations. The dotted line () determines the impact parameter of the photon circular geodesic at .

Properties of the equipotential surfaces can be established easily, using the behavior of the potential in the equatorial plane. The properties of the potential are closely related to the properties of the effective potential of the geodesic motion, and at their local extrema, located at the same radii, the condition (28) is satisfied. Further, , if or . The topological properties of the equipotential surfaces can be directly inferred from the properties of the potential . The local extrema of the potential are determined by the condition

therefore, at the radii determined by the local extrema of , perfect fluid follows free, geodesic circular orbits. The maxima of the potential are determined by the descending part of , they correspond to the cusps of the equipotential surfaces, and the matter moves along an unstable geodesic orbit at the corresponding radii. The minima of the potential are determined by the rising part of , they correspond to the central rings of the equilibrium configurations, and the matter moves along a stable geodesic orbit at the corresponding radii.

Now, we give a complete survey of the behavior of the equipotential surfaces, and the related potential . We start with the astrophysically most important case.

• (A) . From Fig. 4a, we obtain nine qualitatively different cases of the behavior of the potential , and corresponding nine qualitatively different families of the equipotential surfaces, according to the values of . (In the following, we consider only. This can be done due to the symmetry of the spacetimes under consideration.)

  • (I) . Open surfaces only; no disks are possible. Surface with the outer cusp exists. (Fig. 5a)

  • (II) . An infinitesimally thin, unstable ring located at exists. An open surface with the outer cusp exists. (Fig. 5b)

  • (III) . Closed surfaces exist. Many equilibrium configurations without cusps are possible, and one with the inner cusp. An open surface with the outer cusp exists. (Fig. 5c)

  • (IV) . Many equilibrium configurations without cusps are possible. There is an equipotential surface with both the inner and outer cusps. Now, the mechanical non-equilibrium causes an inflow into the black hole, and an outflow from the disk, with the same efficiency; it is the most interesting new feature of the accretion processes caused by the presence of a repulsive cosmological constant. (Fig. 5d)

  • (V) . Equilibrium configurations are possible because closed equipotential surfaces exist. However, accretion into the black hole is impossible because the equilibrium configurations (closed surfaces) have no inner cusp; the inner cusp has an open equipotential surface. The outer cusp belongs to a closed surface, and the outflow from the disk is possible. (Fig. 5e)

  • (VI) . The potential diverges at the photon circular orbit located at , and the inner cusp disappears. The closed equipotential surfaces still exist, with the most extended one containing the outer cusp that enables outflow from the disk. (Fig. 5f)

  • (VII) . In the region defined by , the equipotential surfaces cannot reach the equatorial plane. The closed equipotential surfaces exist, one with the outer cusp. (Fig. 5g)

  • (VIII) . An infinitesimally thin, unstable ring located at exists (the center, and the outer cusp coalesce). (Fig. 5h)

  • (IX) . Open equipotential surfaces exist only. There is no cusp in this case. (Fig. 5i).

• (B) . For this special value of y (Fig. 4b), we still obtain the families of equipotential surfaces given by (A-I)-(A-V) and (A-IX). However, the case (A-VII) disappears, and the cases (A-VI) and (A-VIII) coalesce, giving the case

  • (X) . The inner cusp just disappears, while the outer cusp coalesce with the center. (Fig. 5j)

• (C) . From Fig. 4c it follows that the intervals of , and the families of equipotential surfaces (A-I)-(A-IV) remain. The following new intervals of the angular momentum density must be introduced.

  • (XI) . This case is equivalent to the case (A-V).

  • (XII) . There is the inner cusp of an open equipotential surface, but the center and the outer cusp coalesce-this corresponds to an infinitesimally thin unstable ring, located at . (Fig. 5k)

  • (XIII) . There are open surfaces only, one being with the inner cusp. (Fig. 5l)

  • (XIV) . This case corresponds to the case (A-IX).

• (D) . For this interval of y, the function is descending everywhere (see Fig. 4d). Only maxima of the potential are possible (if ), and open equipotential surfaces can exist only. Equilibrium configurations corresponding to toroidal disks are not possible. This is quite natural result, since in the spacetimes under consideration stable circular geodesics cannot exist. Now, there are only two different intervals of the parameter .

  • (XV) . This family of equipotential surfaces corresponds to the case (A-I).

  • (XVI) . This family of equipotential surfaces corresponds to the case (A-IX).

Fig. 5a-h. Equipotential surfaces (meridional sections) for the marginally stable () configurations of test perfect fluid orbiting the Schwarzschild-de Sitter black-holes, and the related potential . The radial coordinate is expressed in units of M; the logarithmic scale is used, in order to cover whole the range between the inner and outer cusps. The central black hole is shaded. The sequence of figures a-l covers all the possibilities of the behavior of the equipotential surfaces for black holes in spacetimes with a repulsive cosmological constant. The sequence a-i gives successively all the possibilities for the behavior of the equipotential surfaces in the spacetimes of class A, with , which is the astrophysically most plausible class. For the spacetimes of the classes B-D, the relevant sequences of the equipotential surfaces are determined in the text. The cusps of the toroidal disks correspond to the local maxima of , the central rings correspond to their local minima. The dashed lines give asymptotics of , and determine the interval of radii where the equipotential surfaces cannot reach the equatorial plane.

Fig. 5i-l.

Values of the potential at the central ring and the cusps (provided they exist) are given in Table 1. Note that the maximum difference between the values of the potential W on the boundary and at the center of the toroidal disk in the Schwarzschild spacetime is (Abramowicz et al. 1978). Comparing this with the value from Table 1 characterizing the limiting accretion disk with , we can conclude that the presence of a repulsive cosmological constant makes the structure of the disk `smoother'.

Table 1. Radii of the inner cusp (), outer cusp (), and the central ring (), the corresponding values of the potential (, , ), and the differences (, ) for the equilibrium configurations with in the Schwarzschild-de Sitter spacetimes. (Radii and are in units of mass parameter M, while W and are in units of .)

The Schwarzschild case was discussed in (Kozlowski et al.1978) and will not be repeated here. We only mention that the critical self-crossing surface for the marginally bound configurations , while .

3.2. Schwarzschild-anti-de Sitter black holes

If , the function diverges at infinity, and at the black-hole horizon given by the relation

The local minimum of is again located at , and the impact parameter of the corresponding photon circular geodesic is given by Eq. (33). If , there is no zero point of and , . Now, Eq. (36) determines only one marginally stable circular geodesic, close to the horizon. On the other hand, in the Schwarzschild-anti-de Sitter spacetimes the notion of marginally bound circular geodesic ceases any meaning because particles from the unstable circular orbits never escape to infinity, since the effective potential diverges at infinity for each value of the angular momentum density (Stuchlík & Hledík 1999).

If , the behavior of the functions and is qualitatively the same as in the Schwarzschild case. It is illustrated in Fig. 6. The function has a minimum at corresponding to the marginally stable circular geodesic. The unstable geodesics are given by the descending part of , while the stable are given by the rising part.

Fig. 6. Behavior of the functions and for the Schwarzschild-anti-de Sitter spacetimes, given for (both r and are given in units of M). The dotted line determines , as in Fig. 4. It is qualitatively similar to the pure Schwarzschild case (), and it has the same character for all . In the shaded region, the equipotential surfaces are not defined in the equatorial plane.

Now, it is immediately clear that for all of the Schwarzschild-anti-de Sitter spacetimes we always obtain four possible cases of the behavior of the potential and four corresponding families of the equipotential surfaces; notice that as . These cases are given by the following intervals of :

• (I) . There are open equipotential surfaces only. (Fig. 7a)

• (II) . An infinitesimally thin unstable ring is located at . (Fig. 7b)

• (III) . Closed equipotential surfaces exist, one with the cusp that enables accretion from the toroidal disk into the black hole. (Fig. 7c)

• (IV) . Closed equipotential surfaces exist, but no with a cusp at the equatorial plane. In vicinity of the horizon (in region limited by radii determined by the equation ) the equipotential surfaces cannot cross the equatorial plane. (Fig. 7d)

Fig. 7a-d. Equipotential surfaces (meridional sections) for the marginally stable () configurations of test perfect fluid orbiting the Schwarzschild-anti-de Sitter black holes, and the related potential , given for . The behavior of the equipotential surfaces has the same character for all . There are four possibilities described in the text. We express the radial coordinate in units of M, and use the logarithmic scale. The central black hole is shaded. Notice the special shape of the equipotential surfaces with a cusp, resembling a falling wave. The dashed lines give asymptotics of , and determine the interval of radii where the equipotential surfaces cannot reach the equatorial plane.

Values of the potential at the cusp and the central ring (provided they exist) are given in Table 2.

Table 2. Radii of the cusp and central ring, the corresponding values of the potential and their difference for equilibrium configurations with in the Schwarzschild-anti-de Sitter spacetimes.

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
helpdesk.link@springer.de