The moment the black hole is created, it will freefall with virtually no resistance towards the center of the Earth.  The time it takes to fall the one meter to the ground is just under half a second.  Let's figure out what size of black hole would (without eating any mass) last long enough (with Hawking radiation) to fall the one meter to the ground.  Then we'll see how much mass we might expect a black hole of that size to eat along the way, and see if that's enough to sustain it.  

The lifetime for a black hole with initial mass M₀ is



which we can solve for mass to get



This tells us that the black hole must have a mass of 180 metric tons just to survive the half second it takes to reach the ground.  

A black hole of this mass has an event horizon radius of about 2.7x10^-22 meters, which is really small -- more than a million times smaller than a proton.  How much would it eat, though?  Aren't black holes supposed to be powerful vacuum cleaners sucking in everything nearby?

Hah!  Not in this case!  First of all, its gravitational pull drops to 1g at a distance of just 1 millimeter.  There's no way it can pull in the 180 tons necessary to maintain its mass by the time it reaches the ground.  

Secondly, its gravity is completely useless anyway.  Because of its Hawking radiation, it acts less like a vacuum, and more like a tremendous thermonuclear explosion!  It is radiating away its 180 tons of mass energy in half a second!  By Einstein's famous equation E=mc², that's like setting off a few thousand gigatons of TNT!  It would destroy itself and everything on Earth (though the Earth itself will still be intact) in a blinding flash of destruction.

But we want the black hole to actually eat Earth, not explode it.  So clearly we need the black hole to be bigger.  How much bigger?

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=Mc² we can say M = E/c² and by differentiation, dM/dt = 1/c² dE/dt = P/c², where P is power. That is, the rate of mass loss by Hawking radiation is the power radiated, divided by c².  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.

In part 2 we found that the black hole must be at least 40 million metric tons to be able to accrete matter despite its powerful hawking radiation.  But even if we make it bigger than that, it's unclear if this subatomic black hole actually has the reach to be able to pull in enough matter to sustain itself. To help figure that out, let's apply some geometry.

The black hole has a Schwarzschild radius r_s, and as it falls it sweeps up everything in its path, carving out a cylindrical hole with some radius which is presumably larger than the black hole itself.  There are several ways we could try to determine that radius.  As some number of event horizon radii... as a distance for which the freefall time into the hole is reasonably short... as a distance where the gravitational acceleration is some strength?

But before just randomly picking a route, we must note one very important fact.  The black hole is very small relative to the region for which its gravitational pull is strong.  Just like how it is hard for a large body of water to flow straight into a drain, it will be hard for the matter around the hole to fall straight in.  The majority of it will miss and form some kind of accretion flow, just like what happens with real black holes in astronomical settings.  Therefore, although the black hole's radiation might be intense enough to vaporize a significant area around it, the rate at which material is eaten may be much smaller than we expect, and it might be the equivalent of eating up a cylinder only a small number of event horizon radii across.

So I think it is probably best that we define the hole's "eating range" as a number of horizon radii, rather than going by some other definition.

Let the black hole's eating range be some number n of radii from the hole.  Then if the black hole falls for a short time Δt with a velocity v, then it drops a distance vΔt, and sweeps up everything in a cylinder with a volume of π(nr_s)²vΔt.  The mass it eats is that volume times the density ρ of the material.  And we can use the equation for the black hole's radius (r_s = 2GM/c²) to replace r_s with M.  Then we get:

Mass accretion rate  =

Finally, setting the mass eating rate equal to the evaporation rate and solving for M, we obtain our formula:



Ok, let's see if the formula makes sense.  It is dimensionally consistent (kg^4 on the left, and kg^4 on the right), which is always a good check, and the proportionality tells us that if the black hole is falling faster, or the density of the surroundings is greater, or if its "eating range" is bigger, then the black hole does not need to be as large in order to survive.  This makes sense!  So now it's just a matter of putting numbers in.  

The density of rock when its falling through the ground might be 2 to 3 g/cm³, and rises to about almost 13 at the core.  Its speed will start out slow at just ~5m/s when it reaches ground, but increases as it falls deeper.  At its fastest, (at the core), it will be moving at almost 8 km/s!  (Essentially the black hole acts like a mass on a spring, undergoing simple harmonic motion through the Earth.  It is fun to show that the period of this oscillation is exactly the same as the orbital period of a satellite in an orbit just grazing Earth's surface.)

Since we know that the limiting case of a 40 million ton black hole will last for over a hundred million years (which is way longer than the time it takes to pass through the Earth), we only need to consider the average speed of the hole and the average density of material it goes through to see if it survives.  That's about 5 kilometers per second, with 8 grams per cubic centimeter.  Let's apply those numbers to the formula, for different numbers of eating radii.

In the extreme limiting case where n=1 (the black hole only sweeps up what is actually in the path of its event horizon), then its mass must be about 2 trillion tons.  That's about a fifth of the mass of Phobos, packed into an event horizon about a tenth the size of an atom.  It would have 1g of gravitational pull at a distance of about 120 meters, and it would exceed the surface gravity of a neutron star at 0.1 millimeters.  Even though the horizon itself is smaller than an atom, it would have no problem pulling in -- and tearing apart -- atomic nuclei and gaining mass from the nucleons themselves.

The intensity of this black hole's hawking radiation is also quite reasonable.  Less than 100 watts!  Its gravitation will totally dominate the radiation pressure.

Now a more realistic eating range is probably much more than 1 black hole radius, but conveniently, the required mass of the black hole does not depend very strongly on the eating range.  If we increase the eating range by a factor of 100, then the mass requirement is dropped by a factor of 10.  If 10,000 horizon radii, then the mass requirement dropped by a factor of 100, and so forth.  Every factor of 100 more in its range decreases its mass requirement by a factor of 10, up to the point that we approach the Eddington Limit. 


So I believe this gives us our best answer.  The black hole must be at least 180 tons just to make it to the ground (exploding violently in the process), at least 40 million tons (though give or take a few orders of magnitude because of the assumptions with the Eddington Limit) so that its Hawking radiation is not overpowering its gravity, and a most reasonable guess of about a trillion tons to really be able to consume and grow.


The most natural question to ask next is "how long will it take for this atom-sized, trillion ton black hole to actually eat the Earth, and what would it be like?"  I'll leave that for another time.