Statistical Mechanics of Violent Relaxation in Stellar Systems

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Abstract
An explanation of the observed light distributions of elliptical galaxies is sought and found. The violently changing gravitational field of a newly formed galaxy is effective in changing the statistics of stellar orbits. The equilibrium distribution under this encounterless relaxation is found by use of a fourth type of statistics related to both Fermi–Dirac statistics and equipartition of energy per unit mass. In the relevant limit this becomes Maxwell's distribution but with temperature proportional to mass. The predicted light distributions are those of the modified isothermal spheres developed by Michie from considerations of stellar relaxation in globular clusters. Both these and the special case further developed by King are known to give agreement with observations of spherical systems. Application to clusters of galaxies will remove Zwicky's paradox. The theory is also developed for rotating systems where allowance must be made for anisotropy of stellar motions if the outer parts are not to be much flatter than the inner parts. The new statistics developed here should have important applications to collisionless plasmas and collisionless shocks. Kelvin's theorem is rederived for collisionless dynamics. It is suggested that the typical ‘equilibrium’ state of a stellar system may be hierarchical.