The distribution of stars around a massive central black hole in a spherical stellar system. I - Results for test stars with a unique mass and radius

Abstract

view Abstract Citations (10) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The distribution of stars around a massive central black hole in a spherical stellar system. I. Results for test stars with a unique mass and radius. Ipser, J. R. Abstract The Fokker-Planck equation in phase space is solved in an approximate way for the steady- state star distribution around a massive central black hole in a spherical stellar system. Our approach differs from previous ones, which have solved diffusion equations in energy (E)- angular-momentum (J) space, equations whose equivalence to the fundamental Fokker-Planck equation has not been proved for the present problem. The hole strongly influences the dynamics at r <% = 2GMH!Vc2 (MH is the hole's mass, Vc is the rms velocity in the stellar system's core), and our approximate solution there differs significantly from previous ones based on diffusion in E-J space. For r <% r /lO and rant (rorit is the minimum radius at which stars with low J have a good chance to scatter to large J and thereby avoid being consumed by the hole), we obtain a stellar density n(r) const. x r - a, with a 5/4 rather than the value 7/4 obtained in other approaches. Conversely, for r /lO <% r <% , where our results become less solidly grounded as MH approaches the largest values envisaged, we obtain a density enhancement that is steeper than that of other approaches. (Like others, we presently neglect the contribution of the stars to the gravitational field.) Roughly, we find n(r) const. x exp (3r /2r) for r > r /3 if the surrounding cluster is isothermal; and n(r) grows faster than r -2, but not faster than , for r /lO < r < r /3. The discrepancy remains unresolved. The question is raised whether it can possibly be traced to an inequivalence of the basic equations (Fokker-Planck equation in phase space versus diffusion equation in E-J space) underlying the different approaches; and/or to inequivalent applications of the boundary conditions of the problem. Analyses suggesting that the latter may be at least partially involved in a resolution are provided: evidence is presented that over a range of smaller radii, our density profile would steepen toward something resembling an a = 7/4 power law if we were to handle the "loss-cone" boundary condition governing consumption of low-J stars by the hole in the (unacceptable, we claim) way it is handled by previous calculations yielding a = 7/4. In addition, it is shown that an energy-conservation argument which previously suggested a = 7/4 suggests a < 7/4 when loss-cone consumption is accounted for. The argument suggests a 5/4 if the time scales for energy diffusion and "loss-cone diffusion" are nearly equal. Our analyses allow the possibility that a 7/4 for rant < r < r /lO, but rant < r /lO only for the largest values of MH envisaged for globular clusters and galactic nuclei. Our density profile implies a stellar consumption rate, when MH is large, that exceeds by a large amount the rates calculated previously, but the gravity of the stars would flatten our profiles and reduce our rates somewhat. Subject headings: black holes - clusters: globular - stars: stellar dynamics Publication: The Astrophysical Journal Pub Date: June 1978 DOI: 10.1086/156216 Bibcode: 1978ApJ...222..976I Keywords: Black Holes (Astronomy); Globular Clusters; Star Distribution; Stellar Systems; Angular Momentum; Fokker-Planck Equation; Galactic Nuclei; Gravitational Collapse; Iterative Solution; Quasars; Steady State; Stellar Evolution; Astrophysics; Black Hole Neighborhood:Star Distribution; Black Holes:Stellar Systems; Stellar Systems: Star Distribution full text sources ADS |