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Astron. Astrophys. 343, 325-332 (1999)

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1. Introduction

Most of theoretical astronomy uses Newtonian theory of gravitation, considering that the effects of general relativity are weak for most astronomical objects. Even in situations where these effects do become important or even dominant, pseudo-Newtonian models have often been applied which aspire to mimic the corresponding relativistic situations. Pseudo-Newtonian approaches aim to save numerical work, to enable analytical solutions where fully relativistic treatment is too cumbersome, and to provide new insights into the relativity theory and its implications.

The present contribution concerns pseudo-Newtonian models developed in order to describe very compact objects which presumably reside in the nuclei of galaxies and in some X-ray binaries (e.g., Rees 1998). In these sources, a key feature is an accretion disc around a rotating black hole. The actual accretion flows are most likely non-stationary, non-axisymmetric, and described by complex local physics. For the sake of simplicity, however, a standard model of disc accretion (see Kato et al. (1998) for a textbook exposition of the accretion theory including its recent advances) provides an ingenious approximation: the standard disc is smooth, axially symmetric, geometrically thin, and characterized by three parameters. It is astrophysically realistic only in a restricted range of its parameters. The pseudo-Newtonian approach has been devised in order to introduce effects of general relativity into accretion models, especially in the case of geometrically thick discs (Paczynski & Wiita 1980). The black hole is treated there as a Newtonian body whose gravitational field is determined by the potential [FORMULA] (geometrized units with [FORMULA] will be used throughout the paper). This simple expression mimics the gravitational field of a non-rotating (Schwarzschild) black hole quite accurately, in particular, it reproduces correctly the radii of the marginally stable and the marginally bound circular orbits of free test particles, [FORMULA] and [FORMULA]. Recently, Abramowicz et al. (1996) proposed a certain rescaling of velocities in the field of [FORMULA] to obtain better agreement with corresponding relativistic values. In this way, calculations of the observed spectra of the discs could also be improved. It has been argued, however, both on theoretical grounds (Bardeen 1970) and from observations (Iwasawa et al. 1996; Karas & Kraus 1996), that the central black holes in galactic nuclei would be more likely to rotate rapidly. Thus the main feature that should be included in the pseudo-Newtonian models is the rotation of the central object. In addition, the rotation-induced dragging must be taken into account when studying non-equatorial warped discs around compact objects (Bardeen & Petterson 1975).

The pseudo-Newtonian potential for a non-rotating black hole has been used in numerous works and we recall at least a few of them. For example, Abramowicz et al. (1988) used [FORMULA] in their introductory work on slim discs. Okazaki et al. (1987) adopted this approach to describe global trapped oscillations of relativistic accretion discs. Later, Nowak & Wagoner (1991) devised another form of the potential which is better suited for their purpose because it reproduces the epicyclic frequency [FORMULA] with higher accuracy than [FORMULA]. Szuszkiewicz & Miller (1998) studied the limit-cycle behaviour of thermally unstable flows, also with the help of the pseudo-Newtonian description. On quite a different subject, Daigne & Mochkovitch (1997) applied [FORMULA] to study the runaway instability in accretion discs which could trigger gamma ray bursts. Very recently, Ruffert & Janka (1998) used [FORMULA] in a detailed study of neutron-star mergers and related production of the bursts. It is quite apparent from this short list of different astrophysical applications that the approach has its advantages and enables qualitative (and quantitative, with some caution) studies of different topics concerning black holes. Especially the problems of accretion can be treated in this manner. It is understood that quantitative results need always be checked carefully within the full relativistic framework. This is particularly true for time-dependent phenomena (like oscillations and waves in fluids) and for computations of observed spectra. Notice that a textbook overview of the whole subject can be found in Kato et al. (1998).

Now we come to the case of a rotating black hole. At least two topics of current astrophysical interest can immediately be mentioned where the frame-dragging effects around a rotating black hole are important: the problem of oscillations of relativistic accretion discs (see Wagoner 1999 for a recent review), and that of precessing discs in low-mass X-ray binaries (Stella & Vietri 1997) and black-hole binaries (Wei et al. 1998). It is apparent that the whole subject of oscillation modes in relativistic discs calls for detailed investigation (cf. also Markovic & Lamb 1998) and a suitable pseudo-Newtonian formulation can be an appropriate tool before embarking on a complete solution. However, unlike the case of stationary gaseous configurations, nonstationary phenomena require a more refined choice of the pseudo-Newtonian model because there are additional quantities apart from the location of marginal circular orbits which must be correctly modelled (e.g. [FORMULA]).

Although the ever increasing computational facilities will make practical reasons for the pseudo-Newtonian approach rather old-fashioned in near future, what still remains desirable is a simple expression which would simulate the rotating (Kerr) black hole accurately. This is however difficult to find. For example, the potential suggested by Chakrabarti & Khanna (1992) could be useful in studying thin accretion discs around rotating holes, but it applies only to the equatorial plane and its interpretation is rather unclear (free parameters are fitted in a purely pragmatic manner for a restricted set of trajectories). Another form of the pseudo-Newtonian potential has recently been proposed by Artemova et al. (1996). This pseudo-potential reproduces very well steady circular motion but it ignores all effects which in the true Kerr metric are ascribed to the Lense-Thirring precession and which make test-particle trajectories non-planar. This is also the main reason why, in contrast to the non-rotating case, pseudo-potentials for the Kerr metric have had rather restricted impact. Also, a proper understanding of these generalizations may be as difficult as employing general relativity fully. However, the motivation which stems from attempts to understand and interpret predictions of the relativity theory has not disappeared.

It is the aim of our present contribution to discuss the pseudo-Newtonian modelling of the gravitational field of a rotating (Kerr) black hole, and also to propose how to test such models. One should remember that the idea of the pseudo-Newtonian approach represents a certain mathematical model rather than a rigorously defined approximation such as e.g. the weak-field approximation of the Einstein equations. The model does not aspire to represent any gravitational theory and, in particular, the corresponding potential is not required to satisfy any field equations. (It is exactly the lack of precise definition of the approximation method which leads us to introduce a test of accuracy of the model in the present article.) This fact, however, does not diminish the practical value of Paczyski-Wiita potential and other pseudo-Newtonian models, which apply even to regions with strong gravity and capture qualitative features of motion near the horizon.

We start by writing down the spatial components of the Kerr geodesic equation in Boyer-Lindquist coordinates [FORMULA]:




M and a are parameters of the Kerr solution, and the dot denotes differentiation with respect to the affine parameter normalized so that the 4-momentum is [FORMULA].

In the next section, Newtonian equations of test-particle motion in an axially symmetric gravitational field are given. In Sect. 3, we derive a pseudo-Newtonian potential for which these equations get a form very similar to the above geodesic equation in the Kerr spacetime. 1 In Sect. 4, numerical integration is carried out for a large number of trajectories both in the Kerr and in the "pseudo-Kerr" fields. A specific criterion analogous to the method of Lyapunov coefficients is introduced and the rate of divergence of each couple of corresponding trajectories is determined numerically. The mean values of this rate obtained for various combinations of constants of motion indicate the quality (i.e. accuracy, as defined below) of the pseudo-Newtonian approximation.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999