With this definition, one can compute the radius required for a given mass, to
make a black hole, by writing down the escape velocity:
v2 = 2GM/R
where G is Newton's gravitation constant, M is the mass and
R the radius of the mass. If one subsitutes the mass and radius of the Earth, for instance, he finds that the escape velocity from the surface of the Earth is 40000 km/h. One can turn this equation around to find the radius, for a given mass, at which the escpe velocity equals the speed of light, c.
R = 2GM/c2.
This radius is called the Schwarzschild Radius. Any object whose radius
is smaller than this is a black hole. We can compute the Schwarzshild radius
for some familiar objects: RSch(Earth) = 1cm;
RSch(Sun) = 3km; RSch(person) =
10-23cm. A curiosity is that, since mass increases as the cube of
radius, but linearly with the Schwarzschild radius, very large black holes can
have very low average density. In fact, a black hole with a billion solar
masses would have an average density less than that of water---in some sense,
it would float! Still, a billion solar masses of water seems difficult to come
by, and squashing the Earth down to something pea-sized seems implausible, as
does squashing the sun down to the size of a small town, so how does one ever
make a black hole?
Black holes are thought to originate in principally the same way as do neutron stars, in Type II supernova. In detail, very little is known about the details of black hole formation. A few scenarios seem most plausible.
- Direct core collapse: Just like formation of a neutron star, but at the end there is a black hole. It is unclear whether such supernovae have been observed, but there does seem to be a class of particularly energetic Type II's. This extra energy may result from some exotic physical process which destroys neutrons, and allows further collapse from a neutron star into a black
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