In this section we explain the inverse scaling relation between the mass of a black hole and related frequencies. This relation was used, amongst other things, in the measurement of the mass of a medium size black hole.

If an object circle around a black hole of mass $ M $ at radius radius $ r $, its frequency would be

$ \omega = \sqrt{\frac{GM}{r^3}} $

where $ G $ is the universal constant of gravitation. The radius at which accretion occurs is of the same order of magnitude as the Schwartzschild radius $ R_s = \frac{GM}{c^2} $ where $ c $ is the speed of light in vacuum. Substituting the Schwartzschild radius in the expression for the frequency yields

$ \omega = \frac{c^3}{GM} $

Hence the frequency scales as the reciprocal of the mass.

In the measurement mentioned above, it turns out that some black holes exhibit two simultaneous oscillations in their x ray. The frequencies of said oscillations change with the mass of the black hole, but the ratio between them remains constant. Hence, by measuring those oscillations it is possible to infer the mass of the black hole.