We evaluate the Brownian motion of a massive particle ("black hole") at the center of a galaxy using N-body simulations. Our galaxy models have power-law central density cusps like those observed at the centers of elliptical galaxies. The simulations show that the black hole achieves a steady-state kinetic energy that is substantially different than would be predicted based on the properties of the galaxy model in the absence of the black hole. The reason appears to be that the black hole responds to stars whose velocities have themselves been raised by the presence of the black hole. Over a wide range of density slopes and black hole masses, the black hole’s mean kinetic energy is equal to what would be predicted under the assumption that it is in energy equipartition with stars lying within a distance ∼ rh/2 from it, where rh is the black hole’s influence radius. The dependence of the Brownian velocity on black hole mass is approximately hV2i µ M−1/(3−g) BH with g the power-law index of the stellar density profile, r µ r−g. This is less steep than the M−1 BH dependence predicted in a model where the effect of the black hole on the stellar velocities is ignored. The influence of a stellar mass spectrum on the black hole’s Brownian motion is also evaluated and found to be consistent with predictions from Chandrasekhar’s theory. We use these results to derive a probability function for the mass of the Milky Way black hole based on a measurement of its proper motion velocity. Interesting constraints on MBH will require a velocity resolution exceeding 0.5 km s−1 (Refer to PDF file for exact formulas).

Publication Date



This is the pre-print of an article published by the American Astronomical Society. © 2007. The American Astronomical Society. The final, published version is located here: https://doi.org/10.1086/510294

Archived in: arXiv:astro-ph/0408029 v2 22 Dec 2004

DM was supported by grants NSF AST02-0631 and NASA NAG5-9046.

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Document Type


Department, Program, or Center

School of Physics and Astronomy (COS)


RIT – Main Campus