In part 1 we found that the intensity of the Hawking radiation of a very small black hole will overpower its own ability to pull in matter by gravity. Essentially this is the problem of the
Eddington Limit, where if an object is too luminous, then it will blow its surroundings away. For astrophysical black holes, this places a limit on how quickly they can grow by accretion.
So our next step is to make the black hole large enough such that the intensity of its Hawking radiation,
plus the radiation emitted by the accretion rate necessary to balance its mass loss from Hawking radiation, is less than or equal to the Eddington Limit for that mass of black hole:
The intensity of the Hawking radiation is:
What mass accretion rate does it need to balance that? From E=Mc
^{2} we can say M = E/c
^{2} and by differentiation, dM/dt = 1/c
^{2} dE/dt = P/c
^{2}, where P is power. That is, the rate of mass loss by Hawking radiation is the power radiated, divided by c
^{2}. Therefore,
Mass loss rate by evaporation =
(where a dot is shorthand for rate of change with time).
So the accretion rate must be equal to that in order for the black hole to sustain itself. But some fraction (call it
ε, "epsilon") of the accreted mass will be converted to energy and radiated away, adding to the strength of the Hawking radiation. The intensity of radiation given off from the required accretion rate is
which is simply
εL_{Hawk}The Eddington Limit for a particular mass M is
where
m_{p} is the mass of a proton and
σ_{T} is the Thompson scattering cross section of an electron. I should say that the definition for the Eddington Limit depends
a lot on assumptions about the system -- how the radiation is emitted, and what the surroundings are. Black hole accretion rates can actually greatly exceed what you would expect from the Eddington limit. But anyway, we're just going to go with this because it's probably the best we can do without simulating what a microscopic black hole falling through Earth would do using complex models.
So now let's set the sum of these luminosities to be less than or equal to the Eddington Limit, and solve for the mass, thereby obtaining
Epsilon is probably on the order of 0.1 (~10% of the accreted mass is converted to energy for most black holes), and this value this doesn't change the result much.
This formula tells us that in order for the black hole to be big enough that its radiation doesn't cut off the accretion, it must have a mass of at least 40 million metric tons. What would a 40 million ton black hole be like? It would have an event horizon about 1/10th the size of a proton, and a gravitational pull of 1g at a distance of half a meter. This grows to 2700g at a centimeter, and the surface gravity of a neutron star at a distance of a micrometer. So it is only when you get
really close that the hole's gravity becomes strong. It would also be emitting radiation at about 200 gigawatts, or 2.5 milligrams of mass energy per second. That's about 50 tons of TNT going off every second... from a region smaller than a proton! (If you think antimatter is a high energy-density substance, it's a pittance to the power of a subatomic black hole!) And this little devil lasts a long time. Without eating anything, it would last about 170 million years!
Now because we balanced the inward pull of gravity with the outward pressure of the radiation, this 40 million ton black hole would grow not by pulling things into it, but instead just by sweeping up the material that was in its way as it fell through the Earth, blasting a channel of hot vaporized rock as it goes. Since the hole itself is smaller than a proton, it's really not reasonable that it would sweep up enough to sustain itself. Rather than eating Earth, it would be more of a nuisance -- a freefalling, continuously exploding 40-ton bomb that briefly pops up at the surface every ~42 minutes.
It does raise the interesting question of what the effects of this would be on the planet and on civilization, which I'll leave to you to think about.
For now let's make the black hole a little bigger and see what that does.