Growth of Black Hole

Since the exact relationship between and is uncertain, take The condition that the BH is attempting to accrete at a super-Eddington rate is that:

 
but, from eqns. 2 and 6:

 
thus, in the case = 10 = 10 super-Eddington accretion starts almost immediately and ends at t = 2.27 The Left-hand side of equation (23) is the factor by which the instantaneous accretion rate into the central regions exceeds the Eddington value.

In fig 2 we plot the factor by which the instantaneous accretion rate into the central regions can exceed the Eddington value for this particular case. Typically, in the early evolution of the BH, this factor can exceed 100, which implies the existence of a generalised outflow over and above the radiatively driven wind. On the other hand, sub-Eddington accretion disks will be inefficient in maintaining any form of wind. We may therefore distinguish three phases in the accretion:

1. Super-Eddington, disk wind + radiatively-driven wind, = 4

2. Super-Eddington, eqn. (16) satisfied at its lower limit.

Radiatively-driven wind,

3. Sub-Eddington, classical thin disk accretion + relativistic jet.
In this scheme, the transition between the radio-quiet and radio-loud phases would occur sometime in the second phase, when the radiative wind opens up in the polar direction to allow the free escape of the relativistic jet. If the BH grows at its Eddington-limited rate until the end of this phase, and swallows all the accreting gas when the accretion rate falls below the Eddington-limited value, the relationship between the initial BH and the final BH that is grown is as follows:

= 1	0.16x10	0.89x10
= 2	0.54x10	2.89x10
= 5	1.23x10	7.37x10

The temporal evolution of mass of the BH is shown in Fig 3. Under almost all circumstances, a BH with a final mass 10 can be produced from an initial BH similar to those found in spiral galaxies today. During the super-Eddington accretion phase, the luminosity of the BH remains limited at its Eddington value. At later phases, the luminosity declines thanks to the decreasing inflow rate. This would result in an exponentially increasing luminosity during the phases 1 and 2, followed by a (nearly) exponentially decreasing luminosity in phase 3.

The fact that the final BH mass is not very variable with respect to change in the accretion parameters means that the mass of the BH relative to the mass that is accreted into the central regions (which has presumably been used to form the ``bulge'' of the galaxy) also varies weakly with the accretion parameters.

© Copyright Astronomical Society of Australia 1997