Statistics – Computation
Scientific paper
Mar 1994
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994mnras.267..379v&link_type=abstract
Monthly Notices of the Royal Astronomical Society, vol. 267, no. 2, p. 379-389
Statistics
Computation
17
Distribution Functions, Elliptical Galaxies, Galactic Evolution, Galactic Structure, Gravitational Collapse, Many Body Problem, Particle Density (Concentration), Angular Momentum, Astronomical Models, Celestial Mechanics, Computational Astrophysics, Density Distribution, Dissipation, Energy Distribution
Scientific paper
A method is described for creating a distribution function for an anisotropic nearly spherical dissipationless gravitational system that originates from a cold collapse and is relaxed close to an equilibrium state. The method is based on taking advantage of several regularities that appear in the numerical data of N-body experiments. In particular, a two-dimensional histogram of the number density of particles N(E, L2) on a plane of energy E and square angular momentum L2 is constructed and the shapes of equidensity contours studied. An analytic formula is obtained that gives a very good fit to N(E,L2) throughout the domain of the histogram. The radial period Tr(E, L2) is found to depend almost entirely on the energy, being independent of angular momentum apart from a small variation of less than 1 per cent. The dependence of Tr(E,L2) on energy is close to the Keplerian case for small absolute values of energy and becomes steeper for larger values. The numerical integration of the distribution function obtained from our analytic formulae reproduces the cumulative mass as a function of the radius with an error of less than 5 per cent. The function N(E,L2) is given analytically as the sum of two terms. The two terms contain a common factor which resembles Lynden-Bell's formula for violent relaxation. One term describes the distribution of matter trapped around the circular orbits in the monopole component of the potential. This matter dominates in the core. In this term, the energy of the above circular orbits plays an important role by separating regions of positive and negative temperature. A similar separation also occurs in the other term, which describes the rest of the matter, but for different characteristic values of energy. Furthermore, the form of this term reveals a new kind of correlation of L2 with E which is different from that for circular orbits.
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