What If?

When I was but a wee lad of five or six, so my mother has said, I liked to ask questions of the following sort: What would happen if a 300 mm cannon shell hit an elephant right smack in the tail? While researching another matter recently, I ran across an interesting parallel question: What would happen if a mini-black hole, say with a marble sized horizon, collided with the Earth at say 100, 000 mph?

An awful lot of people who didn’t have a clue ventured answers. As it happens, a marble sized black hole (MSBH) ( 8 mm to 9.5 mm) is not particularly mini, though it is quite a bit smaller than those likely to be formed by astrophysical processes. Our MSBH would have a mass nearly the same as that of the Earth, so the gravitational forces would be roughly as disastrous for the Earth as an (exploding) 300 mm projectile would have been for my hypothetical elephant.

How about a smaller mini, though, say one with a mass 1/10000 that of the Earth? The Schwarzschild Radius (SR) for Earth is about 9 mm, and scales directly with mass, so our BH junior would have an SR of about 0.9 micrometers, the size of a quite small bacterium. The gravitational acceleration of a BH of that size would be equal to Earth’s surface gravity g = (GM/Re^2) = (GM/10000)/(Re/100)^2 at a distance of 1/100 of the Earth’s radius, or 64 km, and at 6 km would be 100 g. Consequently, we can conclude that gravitational effects alone predict that it would be mightily disruptive, smashing and vaporizing a huge tunnel through the planet.

So would it be slowed enough to be captured by the Earth and nibble it to death from the inside? Not likely. I can’t quite do the necessary math here, but the small size of the BH should make it difficult for it to ingest matter too rapidly. The infalling matter needs to crowd together and gets in the way of other infalling matter.

And what about Hawking radiation ? For our 10^(-4) Earth mass BH (5.97 * 10^20 kg) BH, the Hawking temperature is only 205 K, so it would not be an intrinsic radiation source, though the infalling mass should be heated to very high temperature near the tiny horizon, and that might emit a little radiation. Dial down the mass by another factor of 10^4 (to 5.97 * 10^16 kg.) and things change a bit. The 100 g radius of maximum gravitational destruction is now down to 640 meters, and the horizon has shrunk to 90 picometers (9e-11 m), intermediate between an atom and an atomic nucleus in size. Temperature has cranked up to a couple of million Kelvins – hot but too small to be much of a power source. If we assume Stefan-Boltzmann radiation from a spherical surface of the horizon's diameter, it only produces about a tenth of a Watt.

Yet another factor of 10,000 takes us down to about six billion tonnes mass, and sub nuclear size. At that size radiation is expected to be tens of millions of watts, or perhaps much more, and so gravity would no longer be the only major threat.

The sizes of BH that might be produced in the Large Hadron Collider are still many steps away in this iteration, and they would be hot indeed, but if they follow similar laws, would essentially instantly evaporate, returning their energy to other forms. For almost any other size black hole, though, close encounters with Earth are bad news.