Modelling of dense stellar systems with central black holes

Abstract

The presence of a black hole in a stellar system can be inferred from its gravitational interaction with the surrounding stars. This detection approach is often the only viable one, particularly in the case of a black hole of intermediate mass embedded in a dense cluster of stars. Such a regime, which is key for numerous astrophysical open questions, is notoriously difficult to attack. This difficulty arises from a number of factors, from a challenging observational environment (large amounts of crowding, lack of gas to produce significant accretion signatures etc.), to degeneracies in the effects of various observable properties of globular clusters (mass-anisotropy degeneracy, or centrally concentrated clusters of stellar remnants for example). This thesis develops mathematical and statistical tools to address this dynamical inference problem.

First, we introduce a family of self-consistent equilibria for spherically symmetric, isotropic stellar systems with a central black hole. The family is defined by a truncated isothermal distribution function in phase space, suitably modified to allow for the presence of a central point mass. We compute self-consistent solutions of the Poisson equation for the mean-field potential, which we then characterise using matched asymptotic expansions over three nested regimes of the dimensionless parameter space. This approach reveals a sharp transition between equilibria dominated by the mass of the host stellar system or by the mass of the central black hole. A thermodynamic characterisation using caloric curves shows that the black hole-dominated equilibria populate a new branch, connected, via a first-order microcanonical phase transition, to the classic truncated isothermal spheres. We also provide a numerical implementation in Python (LoKi - Loaded King Models) for the computation of the intrinsic and projected properties of the models and the sampling from the distribution function.

This class of models is then extended to the case of a rigidly rotating star cluster. We define a distribution function taking the same functional form as the LoKi models, but with the relevant Jacobi integral as the argument. This breaks the spherical symmetry of the problem, and the resulting equilibria represent axisymmetric, isotropic configurations that contain a central black hole. The resulting Poisson equation is then solved via a spectral iteration method, based on the Legendre expansion of the density and the mean-field potential, and via matched asymptotic expansions. These models are then compared to previous rotating equilibria without a central black hole. First, we note the suppression of the maximum rotation strength that may be sustained. Second, we note a change in the morphology of the central region of the system to become increasingly spherical as the black hole mass increases. The transitional behaviour in the properties of the equilibria observed in the LoKi models persists in the presence of non-vanishing global angular momentum, with an additional discontinuity in the caloric curve.

We then detail a framework for fitting the model parameters when discrete single-star data in configuration and velocity space is available. Specifically, we define a Bayesian approach that allows us to work with the discrete star data directly, without binning. We test this framework on the traditional King (1966) models, which do not contain any central black hole. We also present extensions to the framework to accommodate missing data. We then illustrate the difficulties in applying this methodology to the LoKi models, where we show the requirement for a better understanding of the parameter space to make further progress.

Finally, we examine the constraining power provided by the truncation in phase space employed in the King (1966) models, in conjunction with physically motivated bounds on each parameter. We note that this combination provides a considerable reduction in the admissible parameter space, compared to the bounds on the individual parameters alone. In turn, such a reduction allows for increased computational efficiency in model fitting, whether by designing a better prior for Bayesian inference, or by limiting the calculation of the likelihood function only to the optimal portions of its domain.