# Is there a limit as to how fast a black hole can grow?

Astronomers find ancient black hole 12 billion times the size of the Sun.

According to the article above, we observe this supermassive black hole as it was 900 million years after the formation of the universe, and scientists find its extreme specifications mysterious because of the relatively young age of the Universe at that time.

Why would the 12 billion Solar Masses mass value be mysterious, unless there was a limit of sorts to the rate of mass consumption by a black hole? (naive point: Why would 900 million years not suffice for this much accumulation, keeping in mind that most supermassive stars which form black holes have life-spans of a few tens of millions of years at most?)

The accretion of matter onto a compact object cannot take place at an unlimited rate. There is a negative feedback caused by radiation pressure.

If a source has a luminosity $L$, then there is a maximum luminosity - the Eddington luminosity - which is where the radiation pressure balances the inward gravitational forces.

The size of the Eddington luminosity depends on the opacity of the material. For pure ionised hydrogen and Thomson scattering $$L_{Edd} = 1.3 \times 10^{31} \frac{M}{M_{\odot}}\ W$$

Suppose that material fell onto a black hole from infinity and was spherically symmetric. If the gravitational potential energy was converted entirely into radiation just before it fell beneath the event horizon, the "accretion luminosity" would be $$L_{acc} = \frac{G M_{BH}}{R}\frac{dM}{dt},$$ where $M_{BH}$ is the black hole mass, $R$ is the radius from which the radiation is emitted (must be greater than the Schwarzschild radius) and $dM/dt$ is the accretion rate.

If we say that $L_{acc} \leq L_{Edd}$ then $$\frac{dM}{dt} \leq 1.3 \times10^{31} \frac{M_{BH}}{M_{\odot}} \frac{R}{GM_{BH}} \simeq 10^{11}\ R\ kg/s \sim 10^{-3} \frac{R}{R_{\odot}}\ M_{\odot}/yr$$

Now, not all the GPE gets radiated, some of it could fall into the black hole. Also, whilst the radiation does not have to come from near the event horizon, the radius used in the equation above cannot be too much larger than the event horizon. However, the fact is that material cannot just accrete directly into a black hole without radiating; because it has angular momentum, an accretion disc will be formed and will radiate away lots of energy - this is why we see quasars and AGN -, thus both of these effects must be small numerical factors and there is some maximum accretion rate.

To get some numerical results we can absorb our uncertainty as to the efficiency of the process and the radius at which the luminosity is emitted into a general ignorance parameter called $\eta$, such that $$L_{acc} = \eta c^2 \frac{dM}{dt}$$ i.e what fraction of the rest mass energy is turned into radiation. Then, equating this to the Eddington luminosity we have $$\frac{dM}{dt} = (1-\eta) \frac{1.3\times10^{31}}{\eta c^2} \frac{M}{M_{\odot}}$$ which gives $$M = M_{0} \exp[t/\tau],$$ where $\tau = 4\times10^{8} \eta/(1-\eta)$ years (often termed the Salpeter (1964) growth timescale). The problem is that $\eta$ needs to be pretty big in order to explain the luminosities of quasars, but this also implies that they cannot grow extremely rapidly. I am not fully aware of the arguments that surround the work you quote, but depending on what you assume for the "seed" of the supermassive black hole, you may only have a few to perhaps 10 e-folding timescales to get you up to $10^{10}$ solar masses. I guess this is where the problem lies. $\eta$ needs to be very low to achieve growth rates from massive stellar black holes to supermassive black holes, but this can only be achieved in slow-spinning black holes, which are not thought to exist!

A nice summary of the problem is given in the introduction of Volonteri, Silk & Dubus (2014). These authors also review some of the solutions that might allow Super-Eddington accretion and shorter growth timescales - there are a number of good ideas, but none has emerged as a front-runner yet.

• Good answer. I would just note that "speculative" means that we aren't sure which details are right, not that we have no good ideas. Overcoming Eddington is easy in principle -- just break spherical symmetry, letting matter flow inward in some places and radiation flow outward elsewhere. It's not like accretion disks are spherically symmetric anyway.
– user10851
Commented Feb 26, 2015 at 20:54
• @ChrisWhite Of course. But most such get-outs are small numerical factors, not the order(s) of magnitude required. But you are correct - no shortage of ideas. Commented Feb 26, 2015 at 21:03
• The Eddington radiation would keep out gas. I don't see how it could stop heavy infalling objects, though--say an area of really massive stars that left behind neutron stars and black holes. Or even galactic mergers. Commented Feb 27, 2015 at 3:05
• @LorenPechtel You are right, though I have not heard that suggested as a solution. I think the problem with the idea is that you need most of the gas to have already turned into stars in the first 900 million years. This sounds like an even bigger problem than growing the black hole. It takes most galaxies much longer to assemble even a fraction of their gas into stars. Commented Feb 27, 2015 at 7:06
• @HritikNarayan Well you still have to grow the smaller black holes. So you have a slightly smaller individual growth timescale, but then you have to factor in some sort of collisional timescale. I don't think there is ever a problem in explaining one particular object in a variety of ways; but there are actually a population of these things. Commented Feb 27, 2015 at 15:06

A 12 billion Solar mass black hole sounds massive, but actually it's not all that big. The radius of the event horizon is given by:

$$r_s = \frac{GM}{c^2}$$

and for a 12 billion Solar mass black hole this works out to be about $1.8 \times 10^{13}$m. This seems big, but it's only about 0.002 light years. For comparison, the radius of the Milky way is 50,000 to 60,000 light years, so the black hole is only 0.00000003% the size of the Milky Way.

Black holes can't just suck in stars. A star orbiting in a galaxy has an orbital angular momentum, and it can't dive into the centre of the galaxy where the black hole is unless it can shed that angular momentum. In fact, given what a small target an 0.001 light year black hole makes, a star would have to shed almost all its angular momentum to hit the event horizon.

But shedding angular momentum is hard because angular momentum is conserved. You can't just make angular momentum disappear, you have to transfer it to something else. Typically a star does this by interacting with other stars. Generally speaking, in an interaction the more massive star emerges with less angular momentum and the lighter star with a higher angular momentum. This process is known as dynamical friction.

And all this takes time. The interactions are random and you need lots of them. Interactions are far more frequent in the central bulge of galaxies than our where we are in the suburbs, but even so the surprise is that there has been enough time for billions of stars to hit the black hole and merge with it.

• Plausible, but not relevant. Commented Feb 26, 2015 at 18:36
• The reason I say it cannot be relevant is that we have known for some considerable length of time that it is possible for AGN/quasars to have huge luminosities that require them to be fed by huge amounts of mass at a rapid rate. So funnelling huge quantities of matter into a small volume does not appear to be a major obstacle. The real difficulty is in growing the black hole because the Eddington rate is smaller for smaller black holes and the seeds for SMBH cannot have been more than of order 1000 solar masses. Radiation pressure is likely what limits the growth of a black hole. Commented Feb 26, 2015 at 19:30
• Interesting answer, although I do agree with @RobJeffries Commented Feb 27, 2015 at 7:23
• A better comparison for the size of the event horizon might be 120 AU (which is about four times Neptune's semi-major axis). Commented Feb 27, 2015 at 12:11