# Lower limits for steady-state black holes

Stellar mass (and larger) black holes emit Hawking radiation below the temperature of the cosmic microwave background; thus, they should always absorb more energy from space than they emit, and always grow (at a very slow rate).

A black hole somewhere around the mass of the moon, on the other hand, were such an object to exist, should radiate at a high-enough temperature to remain in a steady state, emitting as much as it absorbs.

Going further, a black hole of $\sim4\times10^{20}$ kg (a little more than 1/200th the mass of the moon) should radiate at around a toasty 30 degrees Celsius. On its own, such a black hole would slowly evaporate, but, if we were to find one, it would not be terribly difficult to maintain it in a steady state by shining bright lights at it, or dropping in a slow stream of hydrogen atoms. At that point it is effectively acting as a very low-power but high efficiency matter-to-energy converter, turning hydrogen into thermal photons.

As a black hole becomes smaller and smaller, it seems that there are two basic factors which would make it increasingly difficult to arrange a steady-state situation. The first is increasing radiation pressure, blowing potential feed material away from the event horizon. The second is that the steadily decreasing size of the event horizon will make it more difficult to feed matter back in to the black hole.

The first issue is contemplated in the question Could micro black holes obey the Eddington limit?. Some back-of-the-envelope calculating suggests that the second issue may, however, present a more significant bottleneck. For reference, a black hole massing $4\times10%{15}$ kg should radiate about 22 watts of power, which is really not all that bright, but is only 2.97 picometers in radius - about 1/18th the size of a hydrogen atom. It seems to me that the black hole would have to be immersed in a fairly dense medium in order to make the rate of particle capture high enough to offset that 22 watts. How dense, I am uncertain; would air suffice? Or water? Or solid iron? Or electron degenerate matter, or neutronium? The issue of just how quickly you can force material into a sub-atomic sized blackhole is partially addressed by the paper Fermion absorption cross section of a Schwarzschild black hole, but I'm not quite sure how to get from that to calculating the necessary material density.

So, the question is: just how small can a black hole get, such that it can be kept in a steady state by some means?

(A necessary corollary to that is "what does the steady-state environment look like?"- i.e., does it have to be embedded in neutronium, or whatever. I realize it is preferable to ask one question at a time, but I'm not really sure how these can be separated cleanly; I welcome suggestions for improvement.)

• Note that another potential difficulty is the control issue. A steady-state black hole in an environment of uniform density is at an unstable equilibrium, in that if it gets a bit too big it will start to grow, and if it gets a bit too small it will shrink. This can be countered by feedback control, but the smaller the black hole gets the more sensitive this is, and if it's really small it will almost instantly evaporate the moment there's a small negative fluctuation in the inflow rate. The fact that inserting matter gets harder as the size decreases only makes this worse. – Nathaniel Apr 3 '15 at 5:32
• True. "How do you effectively control a black hole powered mass/energy converter" is, however, both an additional question and one which seems more suited to engineering than pure physics, the complication of which I hoped to avoid here. I'll be satisfied with an instantaneous solution. – Logan R. Kearsley Apr 3 '15 at 5:48
• Have you considered a more massive central black hole orbited by one or more appropriately small ones? The small one(s) should pick up all the matter, just like Jupiter sweeps up more material than the sun does. – pyramids Apr 3 '15 at 7:46
• Uh, no. Why should I? That seems like a completely unrelated problem. Or do you mean to suggest that a large black hole could attract and compress material to enhance the feeding of a smaller one? – Logan R. Kearsley Apr 3 '15 at 8:00

Take the example of a black hole of mass $M$, with a Hawking radiation output. Let's immerse it in a gas of hydrogen at temperature $T$ and density $\rho$ with no bulk motion or turbulence.

Ignoring radiation pressure, the mass-energy accreted at the Bondi-Hoyle accretion rate could be equated with the Hawking luminosity. $$\frac{4\pi\rho (GM)^2 c^2}{c_s^3} \simeq \frac{\hbar c^6}{15360 \pi (GM)^2}$$

Assuming the sound speed $c_s = (5kT/3m_u)^{1/2}$ $$\rho T^{-3/2} = \left(\frac{5k}{3m_u}\right)^{3/2} \left( \frac{\hbar c^4}{61440 \pi^2 (GM)^4}\right)$$ $$\rho = 1.1\times 10^7 \left(\frac{T}{10^4\ K}\right)^{3/2} \left(\frac{M}{10^{10}\ kg}\right)^{-4}\ kg\ m^{-3}$$

It therefore seems (and a big caveat is someone should check my numerical calculation!) accumulating enough mass inside the Bondi-Hoyle radius is really not a problem for a black hole of mass $\sim 10^{15}--10^{20}$ kg.

At $M=4\times 10^{15}$ kg, and $L=22$ Watts, you only need to accrete $1.5 \times 10^{11}$ atoms per second, from a Bondi-Hoyle radius of $R = R_{Sch} (c/c_s)^2$. For a gas at $10^{4}$ K this means you only need to keep the gas flowing into a sphere of radius $\sim 10^9$ times that of the Schwarzschild radius. This is roughly the radius at which the gravitational potential energy becomes larger than the thermal energy of the gas particles. Note though that I'm happier with this answer at $10^{20}$ kg, than $10^{15}$ kg, since I don't have a quantum theory of GR for when the Schwarzschild radius is smaller than an atom!

A much bigger issue is how would you then dump enough angular momentum to allow the material to get within a few Schwarzschild radii (after which, accretion is inevitable)? I don't know enough about the scalability of the Physics to say much about the formation of mini accretion discs.

• Would this be complicated by having a Schwarzschild radius of only an Angstrom? i.e. the fluid approximation might start to breakdown, but would quantum effects be important for the particles being accreted? – DilithiumMatrix Dec 22 '15 at 0:54
• Hmm, I'm getting the prefactor to be $1.1\times10^7\ \mathrm{kg/m^3}$, rather than $130$. Though I'm no more certain of my numerics than you are. – user10851 Dec 22 '15 at 1:45
• Scaling of accretion disks is interesting and could be its own full question. Normal astrophysical disks are large enough to have effectively infinite Reynolds numbers, so traditional viscosity can't transport angular momentum. Instead the magnetorotational instability comes in, which has its own complicated scaling depending on what magnetic fields do. – user10851 Dec 22 '15 at 1:53
• It's unfortunate that I've been doing astro long enough to be perturbed by the lack of $\textrm{cm}$ and $M_\odot$ in these equations... – DilithiumMatrix Dec 22 '15 at 4:14
• @LoganR.Kearsley Seems from this paper there is no problem using a classical approach until your black hole drops to around $10^{12}$ kg. – Rob Jeffries Dec 22 '15 at 21:02