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Black Holes

We tend to think that the escape velocity of a body is a fixed value. However this is not the case it all depends where you start from. It increases the closer to the body that you go.

For example for the Earth the escape velocity at the surface of the planet is 11.3 km per second while 10 000 km above the surface this will have fallen to just under 7 km per second.

For a body like the Earth or the Sun the escape velocity is greatest at its surface and has a finite value. However things are a little different when you consider a Black Hole. A Black Hole is an unimaginable concentration of matter, and as you approach it the escape velocity increases and increases. Eventually a point will be reached where the escape velocity is the speed of light this point is known as the event horizon for the Black Hole.

The radius of the sphere of the event horizon around the Black Hole is known as the Schwarzschild radius. For example although the Sun is too light to form a Black Hole the theoretical "radius" of its event horizon would be just less than 3 km! For a red giant star such as Betelgeuse to become a black hole it would first have to contract to a radius of about 30 km.

For a black hole the Schwarzschild radius = RS = 2GM/c2

where c the speed of light, G the gravitational constant and M the mass of the black hole.

At the event horizon, a distance of RS from the centre of the Black Hole its escape velocity will be equal to that of light closer you fall in and can never escape, further away from the centre than that distance you could theoretically escape if you have a space ship that could travel fast enough.

Black Holes swallow up matter- increasing in mass themselves and so their Shwarzschild radius increases as the event horizon expands. However if anti matter falls into a black hole then its mass decreases and eventually the black hole fills up and disappears in a burst of radiation.

The diagram (Figure 1) represents the gravitational fields outside a large star and then a black hole. The depth of the gravitational vortex is much greater for the black hole than for a heavy star the escape velocity being the speed of light.

Only stars much heavier than the Sun will form black holes. The Sun will become a red giant and then shrink away to a white dwarf and finally a black dwarf.

Example problem
Calculate the Schwarzschild radius of a black hole formed from a star 50 times the mass as the Sun collapsing to form a black hole:

Mass = 100x1030 kg c = [2GM/R]1/2 and so  
R = 2GM/c2 = [2x6.7x10-11x1032]/9x1016 = 1.5x105 m
R = 150 km

© Keith Gibbs 2013