2.4. Black hole growth by star accretion

Let us compute the time required to reach the critical mass M_c where R_T = R_g, above which stars are swallowed by the black hole without any gas radiation (M_c = 3 10⁸ M). When a tidal breakup of a star (of mass m, radius R) occurs, the energy required is taken from the orbital energy of the star

E_b = 3/4 G m²/R

then the gas coming from the disruption will have an orbit of typical semi-major axis

For our own Galaxy, with a 2 10⁶ M black hole, this means a typical radius of the gas disk of 0.03 pc.

The black hole cannot swallow the gas too fast, the maximum rate occurs when it radiates at Eddington luminosity (above which the radiation pressure prevents the material to fall in). This maximum luminosity is: L_E = 3.2 10⁴ (M / M) L. For a mass M_c, the maximum is 10¹³ L (close to the peak luminosity of QSOs). Then the corresponding accretion rate, assuming an efficiency of = 10-20% is dM / dt_E = 1.1 10^-8 (M / M) M/yr. This implies an exponential growth of the black hole; it takes only 1.6 10⁹ yr to grow from a stellar black hole of 10 M to M_c:

t_E= 9.3 10⁷ ln(M_c/M) yr

Note that this very simple scheme would lead to a maximum at z = 2.8 of the number of quasars. This maximum rate, however, is not realistic, since the black hole quickly gets short of fuel, as the neighbouring stars (in particular at low angular momentum) are depleted. Then it is necessary to consider a growth limited by stellar density _s :

DM/dt = _s V

where is the accretion cross-section, and V the typical stellar velocity. The corresponding time-scale to grow from M to M_c is

t_D = 1.7 10¹⁵ yr (_s / Mpc^-3)^-1 M / M^-1/3 (1 - M / M_c^1/3) <V² > ^1/2 (km/s)

Typically in galaxy nuclei, _s = 10⁷ M/pc³, <V² > ^1/2 = 225 km/s. A black hole could grow up to M_c in a Hubble time, and the luminosity at the end could be of the order of 10⁴⁶ erg/s (see figure 2). More detailed considerations (Frank & Rees 1976, Lightman & Shapiro 1977) introduce the loss-cone effect: the angular momentum can diffuse faster than the energy (faster than a stellar relaxation time t_R). Stars with low angular momentum, or very excentric orbits, will be swallowed first. Since the low angular momentum stars are replenished faster, the loss-cone effect increases the accretion rate by: t_l = t_R (1-e²), with e the excentricity of the orbits. This is significant inside a critical radius r_crit, where the loss-cone angle becomes larger than the diffusion angle _D ~ (t_dyn / t_R)^1/2. This critical radius is also plotted in figure 1.

Figure 2. Growth of a supermassive black hole in two simple models: accretion at Eddington luminosity (time-scale tE and corresponding luminosity LE, as a function of black hole mass MBH), and when accretion is limited by diffusion (tD and LD) (from Hills, 1975).