We explain the relationship between the full phase-space volume occupied by a group of orbits and the area in a surface of section filled by these orbits, and give the physical interpretation of area in the surface of section. These results are illustrated with a simple physical example.
In order to interpret the results of complex realistic star cluster simulations, which rely on many simplifying approximations and assumptions, it is essential to study the behavior of even more idealized models, which can highlight the essential physical effects and are amenable to more exact methods. With this aim, we present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, with values of N ranging from 256 to 16384. This allows us to extrapolate the main features of the evolution to systems comparable in particle number with globular clusters. In this range, we find that the steady-state `deuterium main sequence' is characterized by a ratio of the core radius to half-mass radius that follows qualitatively the analytical estimate by Vesperini & Chernoff (1994), although the N dependence is steeper than expected. Interestingly, for an initial binary fraction f greater than 10%, the binary heating in the core during the post collapse phase almost saturates (becoming nearly independent of f), and so little variation in the structural properties is observed. Thus, although we observe a significantly lower binary abundance in the core with respect to the Fokker-Planck simulations by Gao et al. (1991), this is of little dynamical consequence. At variance with the study of Gao et al. (1991), we see no sign of gravothermal oscillations before 150 halfmass relaxation times. At later times, however, oscillations become prominent. We demonstrate the gravothermal nature of these oscillations.
The GRAPE-4, the world's fastest computer in 1995-1997, has produced some major scientific results through a wide diversity of large-scale simulations in astrophysics. Applications have included planetary formation, the evolution of star clusters and galactic nuclei, and the formation of galaxies and clusters of galaxies.
Starting with equations given by Hadjidemetriou it is possible to present the equations for the osculating elements of the two-body problem with monotonically decreasing gravitational ‘constant’ G and to derive equations for the secular variation of the mean motion and the mean longitude. The results contain as a special case the equations derived by Vinti; moreover the results produce more precise estimates. Using the theory of asymptotic approximations, expressions are given for the behaviour of the true longitude; if G behaves like a singular Jeans–Eddington function we find a remarkable simplification of the results. Using observational data, some cosmological implications are discussed.
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