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

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4. Testing approximative equations

As mentioned above, the pseudo-Newtonian potential has been used frequently in various situations in which general relativistic effects on the motion of material (either test particles or fluids) are essential but exact calculations are too difficult. It is however impossible to estimate, a priori , the error which is introduced by replacing the original system, described in the framework of general relativity, by a corresponding pseudo-Newtonian system. The plausibility of a particular form of the simulating potential can be verified by solving analogous situations within the exact theory. For example, several specific questions in the astrophysics of accretion discs had been first analysed using the pseudo-Newtonian theory, and the results were only later supported by more complicated calculations within the Schwarzschild spacetime (with identical local physics of the fluid). Indeed, it is quite trivial to recall that the above-mentioned standard model of thin discs was formulated within the Newtonian (Shakura & Sunyaev 1973), pseudo-Newtonian (Paczynski & Wiita 1980), and also relativistic frameworks (Novikov & Thorne 1973). A specific response of relativistic accretion discs to oscillations (Kato & Fukue 1980) was also explored with a modified Newtonian potential (Nowak & Wagoner 1991) before further steps were carried out (Perez et al. 1997). These examples indicate that the pseudo-Newtonian model is appropriate for investigating the motion of material around black holes.

It is quite straightforward to check whether the pseudo-Newtonian potential is suitable for treating the motion of test particles and fluids when one deals with a spherically symmetric system. What appears more difficult is to develop an analogous pseudo-Kerr theory, describing the relativistic effects near a rotating compact object. Now we will propose a systematic approach to estimate the quality of a particular model by computing a sufficiently large number of trajectories with different initial conditions and comparing results with exact calculations of geodesic motion. The test we suggest determines the rate of divergence of trajectories (as given by approximative versus exact equations). We propose that this type of check should be carried out for each particular set of approximative equations before they are applied to astrophysical situations. (Until now, only rather restricted tests on a relatively small number of trajectories have been applied. As a consequence, one cannot be sure about predictions based on approximative equations of such models because there is little or no control of their overall precision.) For illustration, we then submit the approximative equations of Sect. 3 to our test.

4.1. Criterion of accuracy of approximative solutions

Our criterion is motivated by the definition of the Lyapunov characteristic coefficients which have been introduced in order to characterize chaotic systems (see, e.g., Chapt. 5 of Lichtenberg & Lieberman (1983)). The most important features of the Lyapunov coefficients relevant for the present work are summarized in the Appendix.

We will now introduce a parameter which is analogous to the maximum Lyapunov characteristic exponent [FORMULA]. One should however realize the basic difference between the standard formulation of the problem of chaotic system and our present situation. While [FORMULA] characterizes the rate of divergence of two neighbouring (initially close to each other) trajectories, in the present work one always starts with exactly identical initial conditions. Trajectories then separate because the motion of the first test particle is determined by the geodesic equation in the Kerr spacetime, while the other (fictitious) particle moves according to approximative equations. The geodesic equation could be integrated analytically but not the approximative equations, so we have to resort to numerical solution. Two points should be mentioned: (i) as the test particles move in a stationary system with preferred time coordinate (t), the separation of trajectories in the phase space is calculated in the [FORMULA] slice; (ii) periodic rescalings of the separation w to zero length keep the two trajectories close to each other during their evolution.

In analogy with standard definition (A.2), we introduce a critical parameter

[EQUATION]

with [FORMULA]

Now:

  • (i) two particles start with identical initial conditions; the first one evolves according to geodesic equations (1)-(3), while the second one according to approximative equations (Eqs. (20)-(22), for example);

  • (ii) [FORMULA] is determined as a time interval during which the two trajectories remain close to each other and the initial separation [FORMULA] is an arbitrary small number (typically, [FORMULA] is of the order of the Keplerian orbital period at the initial radius);

  • (iii) Eq. (35) is evaluated and it is determined, numerically, whether convergence has been reached for large n with a pre-determined accuracy. The final value, [FORMULA], characterizes the rate of divergence of the two trajectories with the same initial conditions: [FORMULA] corresponds to separation increasing exponentially while [FORMULA] corresponds to a polynomial (i.e. much slower) increase of the separation.

In order to estimate the accuracy of the approximative equations, one needs to follow a large number of trajectories with randomly chosen initial conditions. By averaging over [FORMULA] corresponding to different initial conditions, one obtains a value which shows whether the approximative equations describe, on the whole , the motion of test matter with good precision. Regions where the mean value is positive, [FORMULA], indicate that the approximation is not acceptable. The process is described in more detail below where we calculate [FORMULA] for Eqs. (11)-(13) and (20)-(22) as examples.

4.2. An example: test of the pseudo-Kerr equations

We submitted our approximative equations to the above-described test on [FORMULA]. Relative position of the two test particles is determined by difference equations:

[EQUATION]

with [FORMULA] The suffix "Kerr" indicates that the trajectory is a geodesic in the Kerr spacetime while the suffix "approx" corresponds to pseudo-Newtonian equations. We should stress in this place that comparing processes in different spacetimes is a serious problem in general relativity. Here, however, we do not compare two relativistic spacetimes and the situation is quite different: though a proper physical justification gives an additional appeal to any pseudo-Newtonian model and we have therefore emphasized also a physical content of our approach (in Sect. 3), in the final stage one mainly asks whether the test particles in the model field in some coordinates move along trajectories that are sufficiently close to certain geodesics of a given relativistic field in some particular coordinates. It is only necessary to define the way of correspondence of the initial conditions. In our test, we consider as counterparts the particles which start from a given position (r,[FORMULA]) with given (specific) constants of motion ([FORMULA][FORMULA]). Of course, it is possible that we would obtain a better fit with some other choice, e.g. if the "corresponding" particles had the same initial velocities with respect to some (corresponding) local frames.

We have integrated difference Eqs. (36) using the Bulirsch-Stoer scheme (Press et al. 1994). Initial conditions cover the parameter space of

[EQUATION]

Typically, we set [FORMULA]. As expected, most of the particles with [FORMULA] escape quickly to large distances [FORMULA] where both the exact and approximative equations give identical results. It is thus relevant to investigate trajectories with lower energies which often make a number of revolutions around the black hole. Some of these trajectories tend to diverge in a chaotic-type manner, as we will see below.

Each run, characterized by the value of [FORMULA], resulted in about [FORMULA] values of [FORMULA]. For each set of initial conditions (37) within the run, the separation of the two corresponding trajectories was periodically rescaled down to a pre-determined value after a fixed interval [FORMULA]. We have typically chosen [FORMULA] which is comparable to the Keplerian orbital period near the horizon. The integration was terminated when one of the following conditions had been satisfied: (i) [FORMULA] converged to a finite positive value [FORMULA] (numerically, we checked that the relative change of [FORMULA] during the last 30 rescalings did not exceed 0.5 %); (ii) the upper limit of [FORMULA] on the number of rescalings was exceeded, while [FORMULA] was oscillating or decreasing monotonically to negative values; (iii) the trajectory was captured by a black hole, [FORMULA] The case (iii) was excluded from further consideration.

Fig. 1 illustrates the evolution of [FORMULA] for two sets of initial conditions which both fall under item (i) above. Apparently, both cases shown there converge to positive [FORMULA], although the numbers of rescalings were different. In this respect, it is interesting to note that the separation of nearby geodesics in the Kerr spacetime never increases exponentially. This fact is a consequence of the existence of the fourth (Carter's) constant of motion, as discussed in Karas & Vokrouhlický (1992).

[FIGURE] Fig. 1. This graph illustrates how the critical parameter [FORMULA] oscillates at first and then settles to a positive value after [FORMULA] rescalings, indicating the chaotic-type increase of separation [FORMULA] (cf. Eq. (35)). Two different examples of trajectories are shown.

Next we illustrate the mean value [FORMULA] as a function of the constants of motion, e and l. Graphs corresponding to Eqs. (20)-(22) are shown for an extremely rotating black hole (Fig. 2a) and a non-rotating black hole (Fig. 2b). For comparison, the Keres-Israel model (11)-(13) with [FORMULA] was also included (Fig. 2c). One can locate easily the regions of positive [FORMULA]. In general these regions are bound to small e and l which correspond to trajectories that plunge close to the horizon. In the case of a non-rotating hole the graph is symmetrical about [FORMULA] because both the Schwarzschild spacetime and the adopted pseudo-Newtonian field with [FORMULA] are spherically symmetric. Distortion introduced by rotation is visible in the graphs (a) and (c). Comparing these two graphs one can verify that model (19) is superior to (10) (cf. the positive values of [FORMULA] in Fig. 2c, indicating that most of the corresponding trajectories separate rapidly from each other).

[FIGURE] Fig. 2a-c. Isocurves of the mean terminal value of the critical parameter, [FORMULA], as a function of the particles' specific energy e and their specific axial angular-momentum l. Three panels correspond to a  Eqs. (20)-(22) with [FORMULA], b  Eqs. (20)-(22) with [FORMULA], and, for comparison, c  Eqs. (11)-(13) with [FORMULA] (scalar, Keres-Israel model). Positive [FORMULA] indicates chaotic-type behaviour (see the text for details).

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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