Statistical mechanics and equilibrium sequences of ellipticals

Abstract

Elliptical galaxies are expected to have undergone incomplete violent relaxation. Here incomplete relaxation is regarded as a process producing a metastable, long-lived state which is stabilized by the approximate conservation of some set of global quantities in addition to the total energy and the number of particles. The final state corresponds to a maximum of the classical (Boltzmann) entropy provided that the proper phase space partition and set of constraints are chosen. Here we explore two different ways of implementing these ideas. First we derive the expression for the distribution function under the assumption that only one additional quantity, Q, is conserved. An ansatz for the approximately conserved quantity Q is given on the basis of the dynamical selection criterion formulated by us elsewhere which originally had led us to study a two-parameter sequence of quasi-spherical models (⁠|$\textit f_\infty$|⁠) later found to provide excellent comparison with observations. Thus a specific example is worked out in detail and the corresponding distribution function is found to be functionally different from |$\textit f_\infty$| but still consistent with the R1/4 law of ellipticals. The actual conservation of Q is not rigorously proved, but it is demonstrated by an extensive study of numerical experiments of dissipationless galaxy formation. Alternatively, we consider the possibility that the a priori probabilities of microstates are not equal. Under the physical constraint of the conservation of the angular momentum distribution NJ for large values of J2, we derive an equation for the maximum entropy distribution function. In the limit of small values of the binding energy the solution of this equation is shown to reduce to the distribution function |$\textit f_\infty$| originally considered by us.