Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 3 / v / Solution

The assumed central black hole mass is $M_{\bullet }\sim 10^8{M}_{\odot }\simeq 2\times 10^{38}\mathrm{kg}$. A mass alone does not fix radiated energy: specify a formation mechanism and its radiative efficiency of black-hole accretion. For radiatively efficient accretion, if a fraction $ϵ$ of the supplied rest-mass energy escapes and the remainder increases the black hole mass, then $M_{\bullet }=(1-ϵ)M_{\mathrm{supplied}}$ andTaking the customary order-of-magnitude efficiency $ϵ\sim 0.1$ givesThe coeval accretion phase would be a quasar-scale source, hundreds of times brighter than a typical present $L_{\ast }$ galaxy. Compared directly with the preceding gravitational-collapse estimate,Despite its small mass fraction, black-hole accretion extracts a fraction of rest-mass energy, while galactic collapse releases only an energy scale $M{\sigma }^2\ll M{c}^2$. The numerical luminosity is conditional: a radiatively inefficient accretion flow, direct collapse or assembly by mergers need not radiate this amount over the galactic clock, and the question supplies no unique efficiency.