Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 4 / viii / Solution

Many lower-luminosity elliptical galaxies have dense central stellar cusps, giving far more photons and kinematic tracers near a central black hole than the shallow depleted cores common in massive elliptical galaxies, as measured in Central galaxy profiles. Thus central surface brightness can make the smaller black hole easier to measure, despite its smaller mass, whereas sparse central tracers and uncertain stellar orbital anisotropy weaken measurements in a large core. This is not a universal mass ordering: the black-hole sphere of influence $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$ must still be sufficiently resolved in angle, so distance, velocity dispersion and angular resolution also matter.