We have examined the effect of slow growth of a central black hole on spherical galaxies that obey Sérsic or R ^ { 1 / m } surface-brightness profiles .
During such growth the actions of each stellar orbit are conserved , which allows us to compute the final distribution function if we assume that the initial distribution function is isotropic .
We find that black-hole growth leads to a central cusp or “ excess light ” , in which the surface brightness varies with radius as R ^ { -1.3 } ( with a weak dependence on Sérsic index m ) , the line-of-sight velocity dispersion varies as R ^ { -1 / 2 } , and the velocity anisotropy is \beta \simeq - 0.24 to -0.28 depending on m .
The excess stellar mass in the cusp scales approximately linearly with the black-hole mass , and is typically 0.5–0.85 times the black-hole mass .
This process may strongly influence the structure of nuclear star clusters if they contain black holes .