The dynamical evolution of three star cluster models has been investigated by numerically integrating the equations of motion of all the stars as an N-body problem. The integration is performed in high order with variable and individual time steps to reduce the computing time. The results are used as experimental data for testing analytical theories of stellar dynamics. Each star cluster model contains 100 stars, divided into 4 mass groups. The mass spectrum is designed to imitate that of stars in real clusters. The stars are initially distributed independently of their mass. The initial values of the stellar positions and velocities in "model P" are generated according to the frequency distribution of PLUMMER's model. Hence model P is stationary in the sense of the collisionless continuum theory of stellar dynamics. Model P is spherically symmetrical; its density distribution corresponds to that of a polytropic gas sphere of index five, and its velocity distribution is isotropic. "Model R" differs from model P only by its differential rotation. "Model S" is an unstationary system showing rigid rotation. All the three models are treated as isolated systems. The period over which model P is integrated corresponds to 2.7*10^8 years for a typical open star cluster, or to 1.1*10^11 years for a cluster of galaxies. Models R and S are integrated over shorter time intervals. Results: For the stationary models P and R, experimental relaxation times have been obtained from the variations of the stellar energies with time. These variations are due to encounters only. As far as the rough order of magnitude is concerned, no discrepancy is found between the experimental relaxation time and CHANDRASEKHAR's theoretical prediction. A detailed comparison is hampered by the impossibility of applying the idealized theoretical definition of the relaxation time to the more realistic star cluster models. The obtained relative dependence of the relaxation time on stellar mass and on the position within the cluster roughly confirms the theoretical expectations. The experimentally obtained total number of escaping stars agrees well with theoretical predictions (e.g. according to CHANDRASEKHAR). However, contrary to theoretical expectation, the escape rate is almost independent of stellar mass. The mechanism of escape does not act like a slow diffusion process, but rather each escaper is produced suddenly by a close encounter with another star or with a small group of stars. This confirms the conclusion of a theoretical investigation of the escape mechanism performed by HENON. The experimental escape rate proves that the evaporation due to internal interactions contributes considerably to the disintegration of open star clusters. The escapers carry away an important part of the total kinetic energy of a cluster. For models P and R the virial theorem remains valid in the time average during their dynamical evolution; the mean deviation of the instantaneous ratio of the total kinetic energy to the potential energy from the predicted ratio is +/- 10 %. In agreement with the prediction of the collisionless theory, we find that the density distribution and therefore the mean gravitational field change only slightly with time in the models P and R. Significant variations occur only in the central part of the clusters (formation of a density peak) and in the outermost parts (halo of ejected stars). The stellar encounters produce a strong sedimentation of the most massive stars to the nucleus of the cluster. The velocity distribution deviates markedly from a Maxwellian one. No equipartition of the mean kinetic energy over the stars of different mass occurs; the mean stellar velocity remains almost independent of mass.
Veroeffentlichungen des Astronomischen Rechen-Instituts Heidelberg
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