Stability of circular orbits of particles moving around black holes surrounded by axially symmetric structures

The Rayleigh criterion is used to study the stability of circular orbits of particles moving around static black holes surrounded by different axially symmetric structures with reflection symmetry, such as disks, rings, and halos. We consider three models of disks, one of infinite extension and two finite, and one model of rings. The halos are represented by external quadrupole moments (either oblate or prolate). Internal quadrupole perturbations (oblate and prolate) are also considered. For this class of disks the counterrotation hypothesis implies that the stability of the disks is equivalent to the stability of test particles. The stability of Newtonian systems is also considered and compared with the equivalent relativistic situation. We find that the general relativistic dynamics favors the formation of rings.