Higher-order moment models of dense stellar systems: applications to the modelling of the stellar velocity distribution function

Dense stellar systems such as globular clusters, galactic nuclei and nuclear star clusters are ideal loci to study stellar dynamics due to the very high densities reached, usually a million times higher than in the solar neighbourhood; they are unique laboratories to study processes related to relaxation. There are a number of different techniques to model the global evolution of such a system. We can roughly separate these approaches into two major groups: the particle-based models, such as direct N-body and Monte Carlo models, and the statistical models, in which we describe a system of a very large number of stars through a one-particle phase-space distribution function. In this approach we assume that relaxation is the result of a large number of two-body gravitational encounters with a net local effect. We present two moment models that are based on the collisional Boltzmann equation. By taking moments of the Boltzmann equation one obtains an infinite set of differential moment equations where the equation for the moment of order n contains moments of order n + 1. In our models we assume spherical symmetry but we do not require dynamical equilibrium. We truncate the infinite set of moment equations at order n = 4 for the first model and at order n = 5 for the second model. The collisional terms on the right-hand side of the moment equations account for two-body relaxation and are computed by means of the Rosenbluth potentials. We complete the set of moment equations with closure relations which constrain the degree of anisotropy of our model by expressing moments of order n + 1 by moments of order n. The accuracy of this approach relies on the number of moments included from the infinite series. Since both models include fourth-order moments we can study mechanisms in more detail that increase or decrease the number of high-velocity stars. The resulting model allows us to derive a velocity distribution function, with unprecedented accuracy, compared to previous moment models.