Black hole discs and spheres in galactic nuclei - exploring the landscape of vector resonant relaxation equilibria

Vector resonant relaxation (VRR) is known to be the fastest gravitational process that shapes the geometry of stellar orbits in nuclear star clusters. This leads to the realignment of the orbital planes on the corresponding VRR time-scale t(VRR) of a few million years, while the eccentricity e and semimajor axis a of the individual orbits are approximately conserved. The distribution of orbital inclinations reaches an internal equilibrium characterized by two conserved quantities, the total potential energy among stellar orbits, E-tot, and the total angular momentum, L-tot. On time-scales longer than t(VRR), the eccentricities and semimajor axes change slowly, and the distribution of orbital inclinations are expected to evolve through a series of VRR equilibria. Using a Monte Carlo Markov Chain method, we determine the equilibrium distribution of orbital inclinations in the microcanonical ensemble with fixed E-tot and L-tot for isolated nuclear star clusters with a power-law distribution of a, e, and m, where m is the stellar mass. We explore the possible equilibria for nine representative E-tot-L-tot pairs that cover the possible parameter space. For all cases, the equilibria show anisotropic mass segregation, where the distribution of more massive objects is more flattened than that for lighter objects. Given that stellar black holes are more massive than the average main-sequence stars, these findings suggest that black holes reside in disc-like structures within nuclear star clusters for a wide range of initial conditions.

Automated summary

Vector resonant relaxation (VRR) is the fastest gravitational process that shapes the stellar orbits in nuclear star clusters. It leads to the realignment of orbital planes on a time-scale of a few million years, while the eccentricity and semimajor axis of the individual orbits remain conserved. The distribution of orbital inclinations reaches an equilibrium characterized by the total potential energy and total angular momentum. Using a Monte Carlo Markov Chain method, the equilibrium distribution of orbital inclinations in isolated nuclear star clusters with a power-law distribution of orbital parameters is determined. The equilibria show anisotropic mass segregation, with more massive objects being more flattened.