Energy Flux and Particle Flux in Steady-state Solutions of Nuclear Star Clusters

We examine the effects of two-body interactions in a nuclear star cluster surrounding a supermassive black hole. We evaluate the energy flux, analogously to the particle flux calculation of Bahcall and Wolf. We show that there are two types of power-law steady-state solutions: one with zero energy flux and constant particle flux and the other with constant energy flux and zero particle flux. We therefore prove that a zero particle flux solution, which corresponds to the case of an accreting supermassive black hole, can be obtained by requiring a constant energy flux. Consequently, this solution can be derived by simple dimensional analysis, bypassing the need for detailed calculation.