Question

It takes a star, with 24 times the mass of the sun, 3 years to orbit a black hole. If the average distance between the star and the center of the black hole is 9 AU, then the mass contained inside the black hole is _____ Msun.

Answer 1

The mass of the black hole can be estimated using Kepler's third law by relating the orbital period of the star to the average distance between it and the black hole, ultimately allowing us to solve for the mass.

Step-by-step explanation:

To estimate the mass of the black hole, we can apply Kepler's third law, which relates the orbital period of a body around a larger mass to the distance between them. The law states that the square of the orbital period (T) is directly proportional to the cube of the semi-major axis of its orbit (a), and the constant of proportionality depends on the mass of the larger body. In this case, given the star's orbital period (T = 3 years) and the average distance (a = 9 AU), we can calculate the mass of the black hole (M).

Kepler's third law in terms of solar mass and astronomical units is: T² = a³ / M. Plugging in the values we have, (3²) = (9³) / M, we can rearrange to find M = (9³) / (3²). Solving this gives us the mass contained inside the black hole which would allow the star to orbit in the observed manner.