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Black hole - Calcualtion of Schwartzschild radius

(a) Calculate the Schwarzschild radius of a supermassive black hole of mass 3.7 x 10^6 M., the estimated mass of the black hole at the galactic center. Give your answer in both kilometers and astronomical units.

(b) What is the angular diameter of such a black hole as seen at a distance of 8 kpc, the distance from the Earth to the galactic center? Give your answer in arcseconds.

c) What is the angular diameter of such a black hole as seen from a distance of 45 AU, the closest that the star SO-16 comes to Sagittarius A*? Again, give your answer in arcseconds. Would it be discernible to the naked eye at that distance? (A normal human eye can see details as small as about 60 arcseconds.)

I believe this question makes use of Kepler's Third law. P^2=((4*pi^2/G(m1=m2)0a^3 and Newton's law of universal gravitaion, F=Gm1m2Ir2. In addition, the Schwartzschild radius (Rsch) is the distance from the center of a non-rotating black hole to its event horizon. The Schwarzstschild radius is related to the mass M of the black hole by Rsch=2GM/c^2.

I'm unsure of the correct formulas and I'm getting confused if gravitational redshift is also taking place. Any guideance you can lend would be appreciated.

Solution Preview

The Schwarzschild radius is proportional to the mass M with a proportionality constant involving the gravitational constant and the speed of light:


R_sch is the Schwarzschild radius;
G is the gravitational constant;
M is the mass of the gravitating object;
c is the speed of light in vacuum.
The proportionality constant, 2G /c^2, is approximately 1.48x10^-27 m/kg

So ...

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