Friday, November 13, 2009

Black Hole Basics Part 2

First a quick review of part 1.

A black hole is an object with a density sufficient to cause a gravitational acceleration greater than the speed of light.

The point to which all mass is drawn at the center of the black hole is called the singularity.

The surface beyond which light cannot escape the black hole is called the event horizon.

The event horizon is a sphere whose radius is called the Schwarzschild radius which is determined for non rotating black holes by the equation R = 2GM/c2 here G is the gravitational constant 6.77 x 10-11 m3/(Kg*s2) M is the mass of the black hole and c = 299792458 m/s is the speed of light.

For part two we will begin with a more thorough analysis of the Schwarzschild radius. If you ever need to remember the equation for the Swarzchild radius is just remember that you combine the speed of light the gravitational constant and the mass of the black hole in such a way as to give you units of meters and you have the equation modulo a factor of 2.

The derivation of the Swarzchild radius is actually somewhat complicated since it involves general relativity theory. But as often happens a simple calculation using just Newtonian gravity gives the right answer. A Newtonian gravitational well of a spherical object has a potential of -G*M/r where r is the distance from the center of the sphere. This means that it would require at least m*G*M/r joules of energy to completely remove an object of mass m from the sphere of mass M if that object was originally a distance r away. This and the formula 1/2m*V2 give us all we need to calculate the Schwarzchild radius or rather the newtonian estimate of it.

We find that at a radius r we require a certain minimum escape velocity in order to not be trapped by the gravitational potential. Specifically we have

m*G*M/RSchwarzchild = 1/2m*V2escape

therefore RScwarzchild = 2*G*M/V2escape

But the condition we are interested in is the condition that the escape velocity is the velocity of light whereupon we recover our previous formula for the Swarzchild radius. This calculation is just a classical approximation but conveniently gives us the correct answer.

Black holes really are perfectly black. That is to say the event horizon of a black hole is a perfect absorber of light. This of course is not surprising since there is nothing at the event horizon for the light to reflect off of. In physics a body with this property of being a perfect absorber of light is also expected to be something called a blackbody emitter. A blackbody emits light according to a certain characteristic spectra which was discovered by Max Planck. Originally it was assumed that a black hole would not have a temperature and therefore would not emit radiation (meaning light). But careful thought about what might happen at the event horizon gave rise to the idea that the black hole could allow virtual particles to become real. Meaning that black holes really do emit radiation and therefore have a non zero temperature. This line of reasoning was followed by Stephen Hawking who calculated the temperature that a black hole would have to have to correspond to this emission. This leads us to the equation for the temperature of a black hole

T = K/M

where K = 1.227 x 1023 kilograms kelvin. (note this is not the boltzmann constant it is just an accumulation of a bunch of terms I didn't feel like writing out)

K may seem to be an extremely large constant temperatures but when you consider the masses involved it actually predicts ridiculously small temperatures. A one solar mass black hole would have a temperature of about 0.00000006 kelvin. Any natural black hole would have a larger mass than this and therefore have an even smaller temperature. So one can safely ignore the temperature of large black holes. Such small temperatures are virtually undetectable. Even for much smaller black holes say one the size of Jupiter the temperature is about 64 microkelvin.

But for very very small black holes hawking radiation causes them to rapidly evaporate though explode might be a more apt term. A black hole of a mass on the order of a kilogram or less would have a temperature of around 1023 and would essentially evaporate instantly. I bring up such a tiny mass because people frequently worry about cern or some other powerful particle accelerator generating a black hole which eats the earth. While it would be great if it were possible for cern to generate black holes because of some as yet unknown phenomenon if it did those black holes would have energies of at most say 1010 J which is being rather generous. Such an energy corresponds to a mass around a thousandth of a gram. So there could be no danger from such a black hole as it would evaporate as soon as it formed.

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