Question

Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s.
Determine the mass M of the massive object at the center of the Milky Way galaxy. Take the distance of one light-year to be 9.461 x 10¹⁵m.

Answer 1

Answer: M = 42.553·$10^{36}$ kg

Explanation: A orbit where a body remains traveling around a gravitating mass at constant radius is called Circular Orbit. Although in reality the orbit is more like an ellipse, the circular orbit is a good approximation to the real one.

In that system, it is possible to determine the velocity needed to maintain the orbit. The formula is:  v = $\sqrt{\frac{GM}{r} }$ , where:

v is velocity;

G is the gravitational constant( = 6.67·$10^{-11}$$\frac{N.m^{2} }{kg^{2} }$)

M is the mass of the gravitating mass;

r is the distance between the center of the massive object and the orbiting object;

But, this question is asking for the mass M, so, rearraging:

$v^{2} = \frac{GM}{r}$

$M = \frac{v^{2}.r }{G}$

Transforming light-years in metres and dividing by 2 to find the radius:

r = (15.9.461 x 10¹⁵)·$\frac{1}{2}$ = 70.9575

M = $\frac{(2.10^{5}) ^{2} .70.9575 }{6.67.10^{-11} }$

M = 42.553.$10^{36}$kg

The mass of the massive object at the center of the ring is 42.553.$10^{36}$kg.