Numerical Studies of the Gravitational Problem of N-Bodies

Abstract

The dynamical evolution of stellar systems containing a few hundred stars can be studied by numerically integrating the equations of motion of all the stars as an N-body problem. From a methodic point of view, these numerical N-body experiments belong to two branches of dynamical astronomy: celestial mechanics and stellar dynamics. As in celestial mechanics, the basic procedure is the computation of individual orbits of mass points in the exact gravitational field of all the other bodies. However, these individual orbits are not the information at which we aim in stellar dynamics. Hence, we finally deduce from all the individual orbits the desired statistical information about the evolution of the system as a whole (‘macroscopic properties’). Of course, the individual orbits also give valuable insight into the ‘microscopic’ behaviour of a stellar system.

What can we hope to learn from N-body experiments ? There are two basic applications of the results: First, we can test the statistical theories of stellar dynamics. For that purpose, the physical situation should be chosen as simple as possible in order to test the basic assumptions and predictions of the theories. For example, we should study isolated spherical clusters of stars of equal mass. Only if the theories can successfully describe the results obtained from such simple experiments, should we introduce additional complications like different masses of the stars, external fields, etc. Second, we can simulate the dynamical evolution of real stellar systems. At present, we are able to handle systems of up to N=500 stars in the numerical experiments. This number N is typical for many open star clusters and clusters of galaxies. Hence, we may compare the N-body experiments directly with these types of astronomical objects without any further theory, provided that the models include all the physically relevant effects. For example, for open cluster we have to include the actual spectrum of stellar masses, perhaps a mass loss due to the internal evolution of the stars, the tidal field of the Galaxy, gravitational shocks by passing HI-clouds, etc..

The N-body experiments have the advantage that they are, as far as possible, free from mathematical assumptions which are not physically inherent in the problem. In contrast, all the presently available statistical theories of stellar dynamics have to introduce such additional assumptions. However, there also exists an hitherto unsolved fundamental problem in the interpretation of N-body experiments: The solutions of the general N-body problem are in most cases highly unstable. Due to this basic physical instability, we cannot trace the true individual orbits over long periods of time by any numerical technique. However, although the individual orbits of the stars are not fully reliable, it is generally argued, and supported by the comparison of different experimental results, that the derived statistical properties of the dynamical evolution of the whole stellar system are not biased by this microscopic instability.

In numerical experiments for clusters containing up to 500 stars, the following general results have been found: (1) The central density in the cluster increases with time. (2) A large halo of stars is formed in the outermost regions of a cluster. (3) The rate of dynamical evolution is strongly increased by unequal masses of the stars. (4) The most massive stars segregate towards the center of the cluster. (5) In the core, one or a few binaries are formed which absorb most of the binding energy of the cluster. (6) During close encounters, some stars gain enough energy for escape. The escapers are produced suddenly, preferentially in the core of the cluster, and not by a slow diffusion process. The rate of escape depends only weakly on the mass of a star. (7) The mean velocity of the stars in the cluster decreases with the distance from the center. (8) In the core of a cluster, the velocity distribution is isotropic. (9) In the outer parts of the cluster, the motion of the stars is primarily in the radial direction and hence strongly non-isotropic. (10) No equipartion of the kinetic energy takes place among stars of different mass. (11) The virial theorem is, on the average, nicely fulfilled for bound clusters. (12) A slow rotation does not significantly affect the dynamical evolution as outlined above. (13) The rate of escape is drastically increased by the tidal field of the Galaxy. For a typical open cluster (N= 500 stars, total mass of 250 solar masses, median radius in projection of about 1 parsec), the numerical experiments predict a total lifetime of about 5 × 10⁸ yrs. (14) A mass loss of massive stars has only a minor effect on the dynamical evolution of open clusters.

For a detailed discussion of the concepts and of the results of numerical experiments on the gravitational problem of N-bodies, we refer to the general references quoted below.