MadSci Network: Astronomy

Re: What is the relationship between black hole mass and 'surface gravity'?

Date: Tue Dec 16 18:57:02 2003
Posted By: Bryan Mendez, Education and Public Outreach Scientist
Area of science: Astronomy
ID: 1062992942.As

Hello Robert,

To answer your questions I will use the formulas derived from General Relativity for a non-spinning black hole, called a Schwarzchild black hole after the German physicist who first worked out the formulas.

For a Schwarzchild black hole there is a radius beneath which light cannot escape it, called the Schwarzchild Radius, RS:

RS = 2GM/c2
Where G is the gravitational constant of the Universe (G = 6.67x10- 11 m3 kg-1 s-2), M is the mass of the black hole, and c is the speed of light (c = 3x108 m/s). This formula tells you that the radius of the black hole is directly proportional to the mass of a black hole, i.e. the more massive the black hole the larger the radius. Since nothing can travel faster than light, anything falling beneath this radius is forever lost. The surface at the Schwarzchild Radius is called the Event Horizon. It is called this because beyond this surface all events must be forever unknown.

Now, the circumference, C, of a circle is simply 2*Pi*radius. So the circumference of a black hole is simply

CS = 4(Pi)GM/c2.

The "surface gravity" or gravitational acceleration, g, of a black hole at its surface is given by General Relativity as

g = GM/Rs2= c4/4GM.
"One G" is the acceleration due to gravity at Earth's surface (g = 9.8 m/s2). Therefore, if we use this value in the equation above and solve for the mass of the black hole we get
M = c4/4Gg = (3x108 m/s)4 / 4* (6.67x10-11 m3 kg-1 s-2)*(9.8 m/s2)

= 3.1x1042 kg = 1.5x1012 Suns

So it would be a black hole with the mass of 1.5 trillion Suns that would have the same surface gravity as Earth. That's the mass of a dozen Milky Way Galaxies. This black hole would have a diameter of 9.2x1015 m, which is about 1 light-year!

I hope that was useful to you.

Best regards,
Bryan Méndez, Space Science Laboratory, UC Berkeley

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