iBook The Future of the Universe - 11. Appendix B

Appendix B

Appendix B: An Accreting Black Hole Model The accretion-driven luminosity of a black hole is derived from Einstein’s well-known mass-energy equivalence formula E = M c2, where E is the energy, M is the mass and c is the speed of light. If a quantity of mass M falls into a black hole in timet, liberating energy E, then the accretion luminosity Lacc = E / t will be Lacc = f M t c2 (B.1) where f is an efficiency factor accounting for the conversion of accreted material into energy, and where M / t is the amount of material accreted into the black hole per unit time (i.e., the accretion rate). Now, the maximum accretion rate is typically taken to be that which produces a luminosity Lacc = LEdd, where LEdd is the so-called Eddington luminosity. The Eddington luminosity for an accreting black hole corresponds to the situation in which the radiation pressure on the infalling gas exactly balances the gravitational attraction of the black hole. For a black hole of mass Mbh, accreting at the Eddington rate the luminosity will be Lacc = LEdd = 4Gc 0 02 1+X Mbh (B.2) where it is assumed that the opacity is that due to electron scattering = 0.02 (1 + X) m2/kg. The actual accretion rate of material into the black hole now becomes, Mbh t = 1−f M t (B.3) Combining Equations (B.1), (B.2), with (B.3) and integrating reveals that the mass of the black hole increases exponentially with time. 221 222 Rejuvenating the Sun and Avoiding Other Global Catastrophes Indeed, the black hole mass, a time t after emplacement within a star, will be Mbh t = Mbh 0 exp t/ acc (B.4) where Mbh(0) is the initial black hole mass, and the e-folding time, during which the black hole mass increases by a factor of e = 2.718…, is acc = 0 02 1+X c 4 G f 1−f ≈ 3 4x108 f 1−f yrs (B.5) where we have taken the hydrogen mass fraction in the core to be X = 0.5. For a typically assumed mass conversion efficiency of 10 percent (f = 0.1) the e-folding time is about 4 x 107 years. If the efficiency is as high as, say 25 percent then the e-folding time increases by a modest amount to about 108 years. In comparison to the main-sequence lifetime TMS of the host star [given by Equation (2.1)], the time Tconsume for the black hole to fully consume the host star, is Tconsume TMS = 0 03 f 1−f M3 star ln Mstar Mbh (B.6) where Mstar and Mbh are the initial masses of the star and the black hole.