# 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.

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#### 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=2GM/c^2

where:

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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