Answer: M = 42.553·
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 =
, where:
v is velocity;
G is the gravitational constant( = 6.67·![10^{-11}](https://tex.z-dn.net/?f=10%5E%7B-11%7D)
)
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}](https://tex.z-dn.net/?f=v%5E%7B2%7D%20%3D%20%5Cfrac%7BGM%7D%7Br%7D)
![M = \frac{v^{2}.r }{G}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7Bv%5E%7B2%7D.r%20%7D%7BG%7D)
Transforming light-years in metres and dividing by 2 to find the radius:
r = (15.9.461 x 10¹⁵)·
= 70.9575
M = ![\frac{(2.10^{5}) ^{2} .70.9575 }{6.67.10^{-11} }](https://tex.z-dn.net/?f=%5Cfrac%7B%282.10%5E%7B5%7D%29%20%5E%7B2%7D%20.70.9575%20%7D%7B6.67.10%5E%7B-11%7D%20%7D)
M = 42.553.
kg
The mass of the massive object at the center of the ring is 42.553.
kg.