Calculating Density of Black Holes
How Much Must Mass & Energy Be Compressed to Form an Event Horizon
Sep 18, 2008
A black hole forms when a star or other matter collapses to such a small size that the escape velocity equals the speed of light. This point where the escape velocity equals the speed of light is called the event horizon or Schwarzschild radius. To what density must a star be compressed so that its escape velocity equals the speed of light?
Escape Velocity
The escape velocity from a planet or star is the minimum speed at which something must travel to escape its gravitational force. If a rocket is launched from Earth at a speed less than the escape velocity it falls back down to Earth. If it is launched at a speed greater than the escape velocity, it can escape into space.
The formula for the escape velocity from a star or planet with a mass, M, and a radius, R, is:
v(escape) = square root(2GM/R)
Where G=6.67e-11 newtons meters^2 / kilogram^2 is the universal gravitational constant. (e-11 indicates times 10 raised to the 11th power. The symbol ^ indicates an exponent, so ^2 means the quantity is squared.)
Schwarzschild Radius-Event Horizon
To find the Schwarzschild Radius, which is the radius where the escape velocity equals the speed of light, of a black hole set the escape velocity in the above formula equal to the speed of light. Then solve for the radius, which will be the Schwarzschild radius.
Doing so gives
R(Schwarzschild) = 2GM/c^2
The Schwarzschild radius should technically be solved for using the principles of general relativity, but for this particular problem ignoring relativity gives the same answer.
Using the mass of the Sun as 2e30 kilograms, gives 3000 meters for the Schwarzschild radius of a solar mass black hole.
Density and Energy Density of a Black Hole
After a black hole forms, all the mass is compressed into a central point, called the singularity, which is infinitely dense. However to find the density needed to form a black hole divide the mass of the collapsing star by the volume enclosed by the event horizon.
In a black hole space and time are so distorted that ordinary rules of geometry, such as the formula for the volume of a sphere, do not technically apply. However this order of magnitude calculation will ignore that technical detail. The formula for the volume of a sphere of radius, R, is V=(4/3)piR^3. So a sphere of radius 3000 meters will have a volume of about 1e11 meters^3.
Density is the mass divided by the volume. Dividing the mass of the Sun, 2e30 kilograms, by this volume gives the density to which a solar mass star would need to be compressed to form a black hole. This gives the density of a black hole as about 2e19 kilogram/meter^3.
Using Einstein's mass energy equivalence formula, E=mc^2, the mass of 2e30 kilograms is equivalent to an energy of about 2e47 joules. Dividing this energy by the volume gives the energy density needed to form a solar mass black hole, which is 2e36 joules/meter^3.
As the mass of the black hole increases the density decreases, and as the mass decreases the density increases.
Many people have speculated that the Large Hadron Collider (LHC) could form micro black holes that would destroy Earth. However the density or energy density required to form a black hole is many orders of magnitude greater than the energy density produced in the Large Hadron Collider.
Zeilik, M. and Gregory, S. Astronomy & Astrophysics, 4th ed. Saunders, 1998.
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