How fast a (relatively) small black hole will consume the Earth?

In the LHC, we are talking about mini black holes of mass around $10^{-24}kg$, so when you talk about $10^{15}-10^{20}kg$ you talk about something in the range from the mass of Deimos (the smallest moon of Mars) up to $1/100$ the mass of the Moon. So we are talking about something really big.

The Schwarzschild radius of such a black hole (using the $10^{20}$ value) would be

$$R_s=\frac{2GM}{c^2}=1.46\times 10^{-7}m=0.146\mu m$$

We can consider that radius to be a measure of the cross section that we can use to calculate the rate that the BH accretes mass. So, the accretion would be a type of Bondi accretion (spherical accretion) that would give an accretion rate

$$\dot{M}=\sigma\rho u=(4\pi R_s^2)\rho_{earth} u,$$

where $u$ is a typical velocity, which in our case would be the speed of sound and $\rho_{earth}$ is the average density of the earth interior. The speed of sound in the interior of the earth can be evaluated to be on average something like

$$c_s^2=\frac{GM_e}{3R_e}.$$

So, the accretion rate is

$$\dot{M}=\frac{4\pi}{\sqrt{3}}\frac{G^2M_{BH}^2}{c^4}\sqrt{\frac{GM_e}{R_e}}.$$

That is an order of magnitude estimation that gives something like $\dot{M}=1.7\times10^{-6}kg/s$. If we take that at face value, it would take something like $10^{23}$ years for the BH to accrete $10^{24}kg$. If we factor in the change in radius of the BH, that time is probably much smaller, but even then it would be something much larger than the age of the universe.

But that is not the whole picture. One should take also in to account the possibility of having a smaller accretion rate due to the Eddington limit. As the matter accretes to the BH it gets hotter since the gravitational potential energy is transformed to thermal energy (virial theorem). The matter then radiates with some characteristic luminosity. The radiation excerpts some back-force on the matter that is accreting lowering the accretion rate. In this case I don't thing that this particular effect plays any part in the evolution of the BH.

This question is addressed in Giddings and Mangano, http://arxiv.org/abs/0806.3381 . See eq. 4.31 and appendix A. For a $10^{20}$ kg black hole, the rate of accretion comes out to be about $10^{13}$ kg/s.

This is greater than the estimate in Vagelford's answer by a factor of $10^{18}$. The reason for this factor is that Giddings essentially uses the Bernoulli equation to model the mass flow, and in this model the mass does not just flow in at the speed of sound $c_s$ all the way until it hits the event horizon. If I'm getting the gist of the calculation right, mass flows in at the speed of sound up until it reaches a certain radius, which is greater than the Schwarzschild radius by a factor of $(c/c_s)^2$. Even without digging in to the details of the calculation, this sort of makes sense. Infalling matter is going to move at relativistic velocities, $\sim c$, as it approaches the horizon, not at $c_s$. They refer to this effective radius as the Bondi radius, and the difference between their estimate and Vagelford's is basically that they use this radius where Vagelford uses the Schwarzschild radius. This causes their accretion rate to be greater than Vagelford's estimate by a factor of $(c/c_s)^4$, or about $10^{18}$.

Using the Giddings result, it takes on the order of years for the black hole to double its mass. I haven't integrated the relevant differential equation, but since the rate goes like the square of the black hole's mass, it looks like it would only be a matter of decades before the black hole consumed a significant fraction of the earth's mass. (It may not consume all of it because of conservation of angular momentum, the ejection of some mass, and other astrophysical processes that occur near the end where the earth's structure is severely disrupted.)

Since I have much better answer from Vagelford -- I'll write my own version.

When matter falls on the black hole it gets fractioned and radiates. As far as I know (correct me if I'm wrong) one can estimate the radiated energy as $\simeq 0.05mc^2$. Where $m$ is the mass of the falling matter.

The Earth's matter is pulled by the black hole gravitation and pushed away by the radiation. Moreover, for the matter flow $J$ we have "negative feedback" system:

• bigger $J$ -> more radiation -> more matter is "pushed away"
• smaller $J$ -> less radiation -> more matter is "pulled in"

The equilibrium between those forces corresponds to already mentioned Eddington luminosity:
$L (J/s) = 1.3\cdot 10^{21} \frac{M}{M_{sun}}$

Equating $L=0.05Jc^2$ and going to $r_{sh}(m) = 3000 \frac{M}{M_{sun}}$, I obtain:

$J (kg/s) = 100 r_{sh}(m)$

It is remarkable, that the "consumption speed" for the $10^{20} kg$ black hole ($r_{sh} = 148.5\mu m$, look here) will give you $1.48\cdot10^{-5}$ kg/s. Which is just order of magnitude larger than the estimate by Vagelford.