Counter-rotating disks around black holes

Abstract

The collective behavior of stars around a black hole provides great insights into the structure of galaxies. Thus, the focal point of this thesis is to study the dynamics of a cluster of stars within the radius of influence of a black hole in a galactic nucleus. Our approach to this nearly Keplerian system follows a semi-analytical treatment of the collisionless Boltzmann equation. Not interested in the fast orbital phase, we average over it and we focus our work on the study of the secular evolution of the resulting massive rings of constant semi-major axes. We further incorporate counter-rotation between stars and we divide our rings into two populations, prograde and retrograde, where different orbital semi-major axes are assigned to each population. Having the self-consistency of the problem with the absence of collisions, we represent the two populations of rings by two separate distribution functions (DFs), which satisfy two separate collionsless Boltzmann equations (CBEs) governed by two orbit-averaged Hamiltonians (ring Hamiltonians). To describe populations of rings of small eccentricities, we expand the ring Hamiltonians to fourth order in the eccentricities and we build upon Jeans’ theorem to construct statistical distribution functions representing the eccentric rings. When the dispersion in eccentricity is relatively small, these distribution functions are completely described by their centroids. The dynamics of centroids are shown to be equivalent to a two-degree-offreedom Hamiltonian system. This system turned out to be integrable due to the presence of two conserved quantities, the Hamiltonina itself, and another quantity corresponding to the total angular momentum of the two populations. Using the conservation of the angular momentum, we reduce the system into a one-degree-of-freedom system that can be studied through phase space analysis. The linear as well as the nonlinear dynamics of the system are studied, where a criterion of linear instability is derived for initially circular discs.