PhD comics and mini black holes

Contributed by
Nov 20, 2007

The second part of the PhD comic strip I mentioned yesterday is out, and lots of astrobloggers are linking to it. But no one seems to have noticed the glaring error in it:

Black holes are at the center of a lot of misconceptions. Basically, they are objects where the escape velocity is greater than the speed of light. If you throw a rock straight up off the surface of the Earth, you must give it a velocity of 11 km/sec for it to be able to get away from the Earth and not fall back down*. That's because the Earth's escape velocity is 11kps. It depends on the size of the object and its mass, or, if you prefer just its density.

Black holes are so small and so massive (or just plain dense) that their escape velocity is faster than light. You literally cannot escape them once inside their ravenous maw. The size of a black hole is actually rather simple to calculate if you know its mass:

radius = 2 x G x mass / c2

G is Newton's Gravitational constant of the Universe, and is just a number. c is the speed of light. It turns out that for a star like the Sun, it would have to be crushed to a diameter of about 6 kilometers to becomes a black hole. But look at the equation! If I double the mass, the size of the black hole doubles. So it's really easy to scale this equation to different mass black holes.

The one in the center of the Milky Way Galaxy has a mass of roughly 4 million times the Sun's mass, so it must be 4 million times bigger, or 24 million kilometers across, far less than the distance between the Sun and Mercury (for comparison, the Sun is about 1.4 million kilometers across right now).

Black holes are small.

However, they can get even smaller. There is a hypothesis that just after the Big Bang, fluctuations in the density of matter may have compressed small amounts of material so much they collapsed into black holes. These are called mini black holes. If you plug in the mass of, say, a typical mountain or asteroid into the equation above, you'll see that a mini black hole is actually far smaller than an atom!

I won't even go into Hawking radiation, which says that a black hole that small would have a surface temperature of 10 billion Kelvins. That would make them a bit obvious if they were pelting us; they'd be pretty bright.

I will add that from a centimeter away, the gravity of a mini black hole can be hundreds of times that of the Earth, but from a few meters away you'd hardly notice it.

So the author of PhD comics got tripped up a bit by the scale of things (a black hole the size of a pea would have a mass comparable to that of the Earth). But that's not too surprising; obviously, black holes are weird, and difficult to comprehend. Now, if only someone were writing a book that had a whole chapter (plus part of another chapter) dealing with the potential dangers of black holes including an easy-to-understand description of how they form, what escape velocity is, and what happens when you fall into one, and what would happen if a mini black hole hit the Earth ...

Mwuhahahahaha. But you'll have to wait until next year for that book.

* Actually, it's a bit more complicated than this. Escape velocity is for an impulse, a forcer applied all at once. If you apply a smaller force, but do it over a long period of time, you can escape the Earth never having gone 11 kps.

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