Hey, I'm a bit stuck on this question. "It has been observed that a certain star (S2) in the Sagittarius A* (meaning in the direction of the constellation Sagittarius) is orbiting a supermassive black hole. If the period of S2 is 16year and the distance between the two is 1.79e13m. What is the mass of this black hole? (be careful with your units)" I think I know which equation to use, but other than that I don't know how to go about solving it.
force gravity=centripetal force
GMm/r^2=m v^2/r= but v=2PIr/period
= m(2PI)^2 r/period
GM=r^3/T^2 * (2PI)^2
M = (1.79e13)^3 * (2PI)^2/period squared.
so convert 16 years to seconds and solve. In my head, I get period to be 31.6e6 * 16 seconds
so calculate.
To solve this question, we will use Kepler's third law, also known as the law of periods. This law states that the square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis distance (r) between the planet and the object it orbits.
The equation for Kepler's third law is:
T^2 = (4 * π^2 * r^3) / (G * M)
Where:
- T is the period of the orbiting object (given as 16 years)
- r is the distance between the two objects (given as 1.79e13 m)
- G is the gravitational constant, approximately 6.674e-11 m^3/kg/s^2
- M is the mass of the central object (the black hole in this case, what we need to find)
Rearranging the equation to solve for M:
M = (4 * π^2 * r^3) / (G * T^2)
Now, let's plug in the values into the equation:
M = (4 * π^2 * (1.79e13)^3) / (6.674e-11 * (16)^2)
Calculating this equation will give us the mass of the black hole in the appropriate units.