6 Summary and discussion

In this paper we have calculated two-body relaxation times for different mass distributions, number of particles, softenings and particle velocities. For this we launched test particles in a configuration of rigid field particles and measured the relaxation time from the deflection angles (measured from the theoretical trajectory of the same particles) and the transit times.

We first determine the range of softening values for which the error in the force calculation is dominated by noise, rather than by bias. These extend to larger values of the softening for smaller number of particles and for less centrally concentrated configurations, in good agreement with what was found by AFLB. We also find them to be somewhat larger for models with a larger cut-off radius.

We confirm that the relaxation time increases with the number of particles. Indeed a larger number of particles entails a lesser graininess and thus a smaller effect of two-body encounters. In particular for homogeneous density distributions we confirm the analytical result that the relaxation time is proportional to the number of particles. We find, however, that this dependence does not hold for all mass distributions.

We find that the relaxation time depends strongly on how centrally concentrated the mass distribution is, in the sense that more centrally concentrated configurations have considerably shorter relaxation times. This can be understood if we make an N-body realisation not by distributing particles of equal masses in such a way as to follow the density, but by distributing the particles homogeneously in space and attributing to each one of them an appropriate mass. Our effect then simply follows from the fact that the deviation of a particle trajectory by a single very massive particle is larger than that due to a sum of low mass particles of the same total mass.

This dependence of relaxation time on central concentration is strong. E.g. for a softening $\epsilon =0.01$ and a $f_{\rm v}$ of 1.5 the relaxation times of models D and H (or P) differ by roughly an order of magnitude. In order to achieve this difference by changing the number of particles one has to increase them by a factor of 10 also. I.e. in order to avoid two-body relaxation ten times more particles are necessary for a simulation of model D than for one of model H, provided one uses the same softening. The difference is even larger if the softening is chosen to be optimal in each configuration, since the optimal softenings differ by more than an order of magnitude, so an extra factor of two is introduced to the necessary particle number.

Also the dependence of the relaxation time on the number of particles changes with the central concentration of the configuration. We find a shallower dependence for our more concentrated models, the difference being more important for smaller values of the softening. For a softening of 0.01 the difference in the exponent of the power law dependence is of the order of 20%.

We find that the relaxation time increases with velocity, as expected. The reason is that the deflections in two-body encounters are larger when the relative velocity is smaller. The dependence of the relaxation time on the effective velocity is linear in a log-log plane for the larger values of the effective velocity we have considered and deviates strongly from linear for smaller velocities. In the simple analytical estimates of Sect. 2, $T_{\rm relax}$ is proportional to the third power of the velocity. Our more precise numerical estimates argue that these estimates are only about 10% off for the case of the homogeneous sphere, to which they apply.

The strong decrease of relaxation time with encounter velocities entails that two-body relaxation has little effect in simulations of "violent'' events as collapses or mergings. On the other hand it may, depending on the configuration, the number of particles and the softening, play a role in simulations of quasi-equilibrium configurations. Furthermore two-body relaxation will be less of a menace in simulations of objects with high velocity dispersions, like giant ellipticals which are largely pressure supported, than in cases with less pressure and more rotational support, like small ellipticals or discs, putting aside of course the effects of shape and rotation, which we have not addressed here.

We have also examined the dependence of the relaxation time on softening. We find that, as expected, the relaxation time increases with softening. The dependence, however, is complicated, and not given by a simple mathematical formula. Nevertheless for the not too centrally concentrated models the increase is not too large. Thus for our models H and P we increase the relaxation time by a factor of the order of 2 if we increase the softening by a factor of 10. In this we agree well with Theis (1998). The only case where the increase is more pronounced is for model D and particularly for high values of the softening. It should, however, be remembered that this is a bias dominated regime. We can thus conclude that in the noise dominated regime the increase of the relaxation time with the softening is relatively small.

Finally we compared results obtained using GRAPE-3 with those found by GRAPE-4, and found they are similar. From the above results we can deduce that the relaxation times of the two types of GRAPE systems are essentially the same. This can be understood as follows. Athanassoula et al. (1998) argued that the limited precision of GRAPE-3 does not influence the accuracy with which the force is calculated since the error in the calculation of the pairwise forces can be considered as random (cf. their Fig. 5). This is in good agreement with what was initially argued by Hernquist et al. (1993) and Makino (1994), who pointed out that the two-body relaxation dominates the error and that the effect of the error in the force calculation is practically negligible, provided this error is random.

Our results for the relaxation time are always smaller than those given by the simple analytical formula. The differences are relatively small if one uses $b_{\rm max}=R$ in the formula, but quite large if one uses $b_{\rm max}=l$. Thus our results argue strongly that the former is the right value to use, at least for collisionless simulations. In this we agree with Spitzer & Hart (1971), Farouki & Salpeter (1982), Spitzer (1987) and Theis (1998). It should also be noted that the analytical estimate obtained with $b_{\rm max}=l$ is always considerably larger that the numerical result, and thus is falsely reassuring.

By extending somewhat the standard method based on the angular deflections we obtained an estimate for the relaxation time of an N-body simulation of a Plummer sphere. Comparing it with the results found by Huang et al. (1993) we find excellent agreement. This is very interesting since Huang et al. (1993) obtain their estimate of the relaxation time directly from an N-body simulation, i.e. include collective effects. This agreement could argue that such effects are not very large, and thus gives more weight to the results obtained with our simple and straightforward method.

It is often stressed that galaxies have so many stars that their relaxation times are far longer than the age of the universe, and thus that N-body simulations extending to long periods of time should have a very large number of particles to also achieve sufficiently large relaxation times. As a counter-argument one could say that real galaxies are not only composed of individual stars, but also of star clusters and gaseous clouds, which, being considerably more massive than individual stars, will introduce two-body relaxation and change the dynamics from that of a collisionless system. This, however, is no argument in favour of N-body simulations with short relaxation times, since the deviations from the evolution of a smooth stellar fluid brought by the graininess of the N-body system need not be the same as those brought by the compact objects, star clusters or gaseous clouds. It is thus necessary in N-body simulations to strive for high relaxation times and believe the results only for times considerably shorter than that. If desired, the effects of the compact objects, star clusters or gaseous clouds can then be studied separately.

Acknowledgements

We would like to thank A. Bosma for many useful discussions. We would also like to thank IGRAP, the INSU/CNRS, the region PACA and the University of Aix-Marseille I for funds to develop the computing facilities used for the calculations in this paper. C. Vozikis acknowledges the European Commission ERBFMBI-CT95-0384 T.M.R. postdoctoral grant and the hospitality of the Observatoire de Marseille.