Find mass of black hole in center of galaxy given eccentricity+

[oddjobmj]

Homework Statement

Astronomers believe that there is a massive black hole at the center of the Milky Way galaxy. What evidence is there for that?
A group of astronomers have observed a star "S2" in a 15.2-year orbit around the center of the galaxy. They measured the period of revolution T = 15.2 years, the eccentricity of the elliptical orbit ε = 0.87, and the semimajor axis of the orbit a = 4.62 mpc. [The unit: mpc = milli-parsec = 0.001 parsec.]

(A) Calculate the mass of the compact object about which S2 is revolving.

Homework Equations

sm=semi-minor axis
sM=semi-major axis

Eccentricity=$\sqrt{1-(sm/sM)^2}$

Mass=$\frac{4π^2r^3}{GT^2}$

The Attempt at a Solution

If I plug in my values to the eccentricity equation and solve for the semi-minor axis I get 7.031*10¹⁰ km. With that I can find r which is the average of the semi-major and semi-minor axis:

r=1.06455*10¹¹ km

Plugging in my knows to the mass equation I get 3.11*10^36 kilograms which is incorrect. What am I doing wrong?

Thank you!

 

[oddjobmj]

Apparently the way I figured out r was incorrect. It should be 1.426 *10^11 km. My answer is now correct!

 

[Basic_Physics]

Why does the correct mass imply its a black hole? Density from known size?