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
The FokkerPlanck 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) angularmomentum (J) space, equations whose equivalence to the fundamental FokkerPlanck 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 EJ 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 (FokkerPlanck equation in phase space versus diffusion equation in EJ 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 "losscone" boundary condition governing consumption of lowJ 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 energyconservation argument which previously suggested a = 7/4 suggests a < 7/4 when losscone consumption is accounted for. The argument suggests a 5/4 if the time scales for energy diffusion and "losscone 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;
 FokkerPlanck 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