Growth of Super Massive Black Holes

Black holes can be divided into two groups. The first is stellar mass black holes. These were created as a result of a stellar collapse, and therefore have a mass of a few solar masses. The other group is super massive black holes. The masses in this group range between 1e6 and 1e10 solar masses. It is very unlikely that a super massive black hole was created in a single event, but rather they started as stellar black holes, and accreted the remaining mass. In this page we will discuss a few models and limits for black hole growths.

Eddington Limit

One limit on the accretion rate $\dot{M}$ comes from the Eddington luminosity.

$L_E = 4 \pi G M c /\kappa$

where $G$ is the gravitational constant, $M$ is the mass, $c$ is the speed of light and $\kappa$ is the opacity. The idea is that the excess energy of accreted matter has to be evacuated by radiation, but the radiation cannot exceed Eddington's luminosity. If $\eta$ is the efficiency with which in-falling mass is converted to radiation, then

$\frac{\eta}{1-\eta} \dot{M} c^2 = L_E$

because it is assumed that whatever energy was converted to radiation, was not accreted. Solving this equation for $M$ (assuming constant efficiency) yields exponential growth in mass

$M = M_0 \exp \left( \frac{4 \pi G}{c \kappa \eta} \left( 1- \eta \right) t \right)$

For Thomson opacity, and typical efficiency 0.1, the time scale is about fifty million years. This might seem like ample time for massive black holes to grow even from stellar masses. However, there is evidence for super - massive black holes at high redshifts (i.e. when the universe was very young). One example is a 1e9 solar mass black hole discovered at red - shift 6, which corresponds to a time when the universe was just a billion years old. To reach such a mass under the Eddington limit a black hole would have to be created at the big bang with a mass of 100 solar masses, and maybe even 10 times as massive.

However, it was shown that the efficiency decreases due to capture of photons by the black hole, so a more rapid growth might be possible.

Light Cone Limit

An upper limit on the accretion rate can be set by set by condition that a black hole can only accrete matter within its light cone. If the mass density of ambient matter is $\rho_a$, then the mass of the black hole is bound by $M < \frac{4 \pi}{3} \rho_a c^3 t^3$.

Bondi Accretion

It has recently been suggested that super - exponential accretion rates can be obtained with Bondi Accretion. The Bondi capture radius is $r_B = \frac{G M}{v^2}$, where $v$ is the velocity scale (more on that later on). Assuming initial density throughout $\rho_a$, the mass accretion rate is

$\dot{M} = \pi \rho_a \left(\frac{G M}{v^2}\right)^2 v = \pi \rho_a \frac{G^2 M^2}{v^3}$

Solving for the mass yields

$M = \frac{M_0}{1-t/\tau}$

where $\tau = \frac{v^3}{\pi \rho_a M_0 G^2}$. This implies that the mass diverges at a finite time. This, of course, is not physical. In the next subsection we will discuss how the divergence is averted in each type of accretion.

Wind Accretion

In case of a wind accretion, the wind exerts a drag force on the black hole, accelerating it and reducing the velocity difference. Examining the equation above, one might get the impression that reducing the relative velocity might enhance the accretion. However, this is true as long as the bulk velocity $u$ is much larger than the speed of sound $a$. If this is not the case, then the equations have to be amended. The Bondi radius should be $\frac{G M}{a^2}$, and the mass accretion rate would be

$\dot{M} = \pi \rho_a \frac{G^2 M^2}{c^4} v$

From the conservation of momentum

$\frac{d}{d t} \left( M v\right) = 0 \Rightarrow v = v_0 \frac{M_0}{M}$

Plugging this velocity back into the mass accretion rate yield

$\dot{M} = \pi \rho_a \frac{G^2 M M_0}{c^4} v_0$

so the mass will grow exponentially, but would not diverge at a finite time.

Spherical Accretion

As time passes the black hole will accrete matter from ever increasing distances, and hence with higher velocity, which will diminish the accretion rate. Also, it is likely that a shock wave will form around the black hole and brake incoming material.