Astronomisches Rechen-Institut Heidelberg

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ARI-Heidelberg Mitteilungen Serie A, No. 19

Author(s): von Hoerner, S.
Title: Die numerische Integration des n-Körper-Problems für Sternhaufen, II.
Source: Z. Astrophys. 57, 47-82
Year: 1963
Abstract: The 6n equations of motion of n stars in a random cluster are integrated for n=25; one example over 40 times of relaxation, a second one over 24. This covers more than the actual age of globular clusters and of clusters of galaxies, and of most of the galactic clusters. Though both examples show differences in their development their general trend of development is the same. The distributions of density, velocity and energy (and their change with time) are investigated. While their mean values stay remarkably constant, their standard deviations increase steadily with time. This might be used for age estimates.It also shows that theoretical cluster models cannot be given in general, but only as a function of age. At the end of our calculation, the density distribution spreads over 3 orders of magnitude in r, and over 9 orders in the density rho. At the center, the density builds a sharp cusp with about rho~r-2, and the outermost parts give about rho~r-4. Within a factor of six, rho~r-3 would be a rough approximation over the whole region covered; the isothermal polytrope cannot be used as an approximation at all. The velocity distribution is very different from a Maxwellian: the excess of high velocities is due to close pairs and groups at the center, and the excess of low velocities belongs to stars of high energy, lingering close to their apocenter at large r. The energy distribution forms a pointed maximum at low negative energies close to zero and drops toward large negative energies about ~(-E)-p, with p decreasing with time. No distinction between a "remaining cluster" and an "outer region" is possible, since the density drops smoothly from its central cusp to the outermost star. The supply of stars with positive energy is much less than theoretically expected, thus the disintegration of clusters by their internal exchange of energy is slow. An approximate formula for calculating the density distribution from the energy distribution is given and checked. With this formula, the partial differential equation for the time-development of the energy distribution is derived; it shows some similarity to the equation of heat conduction.
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