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.
Showing posts with label Black Hole Basics. Show all posts
Showing posts with label Black Hole Basics. Show all posts
Friday, November 13, 2009
Thursday, October 11, 2007
Black hole basics part 1
To fill in those of you who may not know a black hole is an object with such a great density of mass that the gravitational acceleration near enough to the center of mass is so great that light is unable to escape. So more or less a black hole is an object whose escape velocity is the speed of light c. Since a massive body would have to have infinite energy to achieve a speed of c there exists a region around a black hole such that once an object has entered that region it cannot again escape it. The surface of this region of no return is called the event horizon of the black hole. When one talks of the "radius" of a black hole one means the radius of the event horizon.
The event horizon is actually always perfectly spherical and so the radius of the event horizon is actually a perfect description of its geometry. The reason the surface of the region of no return is called the event horizon is that events that happen on the outside of it may have an effect on other events that happen farther away but events that happen inside of the event horizon cannot affect anything outside of it. This is because information can only travel at the speed of light and so information can't get transmitted from inside to outside a black hole.
The event horizon of course has no physical substance and is not the surface of a black hole in the sense that a star has a surface. In fact we do not know if there is a material surface of a black hole or not. It is generally considered that because the gravitational acceleration inside the event horizon exceeds the speed of light no other force could be sufficient to overcome it and therefore all matter inside a black hole must collapse to a single point. This point is known as the singularity. However it is possible that because of quantum effects or effects due to string theory there may be a point beyond which it is not possible to compress matter in which case there would be a tiny nugget of unimaginably dense material at the core of each black hole. As a point of reference it is interesting to note that the density of a neutron star is somewhere in the neighborhood of 1017 Kg/m3 and the primary agent against further collapse of neutron stars is in fact the pauli exclusion principle. So quantum effects are already needed to keep neutron stars from collapsing to a singularity.
The model of a black hole that I have been describing up to this point is what is called the non-rotating or Schwarzschild black hole. It is a solution to Einstein's equations for the case where there is no rotation and no charge. both rotating and charged black holes are fascinating critters but for the moment I am just going to keep on ignoring them maybe I will include a post about them later.
You may have heard the phrase that "black holes have no hair" this simply means that black holes have only three independent properties, mass, angular momentum, and charge. They have no hair in the sense that unlike any other macroscopic object they are totally indistinguishable but for those three properties. Since we are looking at non-rotating and non-charged black holes the only item of interest is the radius of the event horizon and its relation to mass. Interestingly enough the radius that classical mechanics suggests for such an object turns out to be the correct radius. This radius of the black hole for the non-rotating case is called the Schwarzschild radius.
If you compress a mass down to close to its Schwarzschild radius it will collapse into a singularity under its own gravity. The equation for the Schwarzschild radius is R = 2GM/c2 , where G is the gravitational constant and c is of course the speed of light. Now for the grand finale of the post we apply this equation to find the approximate Schwarzschild radius for Jupiter as being 2*6*10-11*1027/(9*1018). Which is about 1.3 meters. Now compressing the entirety of jupiter's mass into a volume of only 1.3 meters may seem just a tad on the side of implausible and I would tend to agree. However keep in mind that this is the radius that one would need to put matter in in order to make it collapse under its own gravitational field alone. At any rate though we have our first rough calculations of what it would take to make jupiter a black hole.
The event horizon is actually always perfectly spherical and so the radius of the event horizon is actually a perfect description of its geometry. The reason the surface of the region of no return is called the event horizon is that events that happen on the outside of it may have an effect on other events that happen farther away but events that happen inside of the event horizon cannot affect anything outside of it. This is because information can only travel at the speed of light and so information can't get transmitted from inside to outside a black hole.
The event horizon of course has no physical substance and is not the surface of a black hole in the sense that a star has a surface. In fact we do not know if there is a material surface of a black hole or not. It is generally considered that because the gravitational acceleration inside the event horizon exceeds the speed of light no other force could be sufficient to overcome it and therefore all matter inside a black hole must collapse to a single point. This point is known as the singularity. However it is possible that because of quantum effects or effects due to string theory there may be a point beyond which it is not possible to compress matter in which case there would be a tiny nugget of unimaginably dense material at the core of each black hole. As a point of reference it is interesting to note that the density of a neutron star is somewhere in the neighborhood of 1017 Kg/m3 and the primary agent against further collapse of neutron stars is in fact the pauli exclusion principle. So quantum effects are already needed to keep neutron stars from collapsing to a singularity.
The model of a black hole that I have been describing up to this point is what is called the non-rotating or Schwarzschild black hole. It is a solution to Einstein's equations for the case where there is no rotation and no charge. both rotating and charged black holes are fascinating critters but for the moment I am just going to keep on ignoring them maybe I will include a post about them later.
You may have heard the phrase that "black holes have no hair" this simply means that black holes have only three independent properties, mass, angular momentum, and charge. They have no hair in the sense that unlike any other macroscopic object they are totally indistinguishable but for those three properties. Since we are looking at non-rotating and non-charged black holes the only item of interest is the radius of the event horizon and its relation to mass. Interestingly enough the radius that classical mechanics suggests for such an object turns out to be the correct radius. This radius of the black hole for the non-rotating case is called the Schwarzschild radius.
If you compress a mass down to close to its Schwarzschild radius it will collapse into a singularity under its own gravity. The equation for the Schwarzschild radius is R = 2GM/c2 , where G is the gravitational constant and c is of course the speed of light. Now for the grand finale of the post we apply this equation to find the approximate Schwarzschild radius for Jupiter as being 2*6*10-11*1027/(9*1018). Which is about 1.3 meters. Now compressing the entirety of jupiter's mass into a volume of only 1.3 meters may seem just a tad on the side of implausible and I would tend to agree. However keep in mind that this is the radius that one would need to put matter in in order to make it collapse under its own gravitational field alone. At any rate though we have our first rough calculations of what it would take to make jupiter a black hole.
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