Calculating Mass of a Black Hole: A Primer

[tntcoder]

Hi,

Im interested in calculating the mass of a black hole, but I am not actually sure how to approach it. I am guessing i want to take an example black hole, someone told me about the CYGNUS X-1 but i don't know if that's sutible. Can anyone point me to an example, or tell me the relevant equations and how to use them.

Cheers
Jack

 

[$id]

I think to do this you need some doppler shift data. No idea where you could get some. Hoping somebody else can help you there.

This will tell you the speeds of a star orbitting the black.
You will also need to know the orbital radius of the orbitting star.

you can then equate its centripetal accelerationing force to Newtons universal gravitation forumla. the mass of the orbitting star will cancel out allowing you to rearrange and calculate the mass of the black hole that it is orbitting.

Once you have mass of the black hole. you can then work out the schwarzschild radius for the star in question by working out the points were the escape velocity is greater than the speed of light.

 

[$id]

update

can somebody check my calculations please

If the mass of the black hole is 1.2X10^31kg

then the schwarzschild radius that i calculated as 17.7km

For an object to escape the gravitiational field

the kinetic energy must be greater than or equal to gravitational potential energy.


so in this special case.

0.5c^2 >Vgrav

Vgrav<4.5X10^16 J/Kg

If we equate Vgrav to the gravitational potential equation

Vgrav = G * Mass of black holes/ Radius(this is where you will find the above Vgrav, hence its the point of no return)

Rearrange to make R subject

Then i got the answer as 17.7km

It sounds reasonable to me.

However, the calculations use what people here would regard as very basic physics forumulae. I have no considered relativity at all ( I can't but that's not the point )

hope somebody else can help as well.

 

[pervect]

update

can somebody check my calculations please

If the mass of the black hole is 1.2X10^31kg

then the schwarzschild radius that i calculated as 17.7km

Yes, 2GM/c^2 = 17.8 km

For an object to escape the gravitiational field
the kinetic energy must be greater than or equal to gravitational potential energy.

The rest of your calculation, as you point out yourself, isn't based on relativity but on Newtonian mechanics. It's interesting that you wind up with the right answer, but the calculation itself isn't "right" according to GR.

 

[$id]

Yes, 2GM/c^2 = 17.8 km



The rest of your calculation, as you point out yourself, isn't based on relativity but on Newtonian mechanics. It's interesting that you wind up with the right answer, but the calculation itself isn't "right" according to GR.

I thought that 2Gm/c^2 is derived from Newtonian mechanics.

I did explicity state i cannot use general relativity. The answer is correct because Newtonian mechanics is a approximation of general relativity under certain assumptions. Maybe it worked because i am dealing with things outside the black hole rather than inside it??

 

[meteor]

to calculate the mass of the central black hole of a galaxy you can use the Magorrian relation (a relation between the mass of the BH and the mass of the galactic bulge)
Alternatively, in this paper
http://xxx.lanl.gov/abs/astro-ph/0006053
Ferrarese and Merritt found a very precisse correlation between the mass of nuclear BHs and the velocity dispersion of their host bulges

 

[$id]

I managed to derive the forumula for this schwarzschild radius by netwonian mechanics alone

Equating 0.5c^2 to GM/r^2

C^2=2GM/r^2

r^2=2GM/c^2

R= SQRT(2Gm/C^2)

See no need for GR ( This is college level stuff)

R is distance at which the energy needed to escape equals gravitational potential.

 

[meteor]

A new technique to measure the mass of a black hole was demonstrated in september of 2004
http://www.universetoday.com/am/publish/printer_astronomers_watch_black_hole_eat.html
"Scientists have pieced together the journey of a bundle of doomed matter as it orbited a black hole four times, an observational first. Their technique provides a new method to measure the mass of a black hole; and this may enable the testing of Einstein's theory of gravity to a degree few thought possible.

A team led by Dr. Kazushi Iwasawa at the Institute of Astronomy (IoA) in Cambridge, England, followed the trail of hot gas over the course of a day as it whipped around the supermassive black hole roughly at the same distance the Earth orbits the Sun. Quickened by the extreme gravity of the black hole, however, the orbit took about a quarter of a day instead of a year.

The scientists could calculate the mass of the black hole by plugging in the measurements for the energy of the light, its distance from the black hole, and the time it took to orbit the black hole -- a marriage of Einstein's general relativity and good old-fashioned Keplerian physics."