Two newly discovered planets follow circular orbits around a star in a distant part of the galaxy. The orbital speeds of the planets are determined to be 36.0 km/s and 50.6 km/s. The slower planet's orbital period is 7.78 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?

To solve this problem, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet is directly proportional to the cube of its average distance from the star.

Let's denote the mass of the star as M, the orbital speed of the slower planet as v1, the orbital speed of the faster planet as v2, the orbital period of the slower planet as T1, and the orbital period of the faster planet as T2.

(a) To find the mass of the star, we can use the equation for the orbital speed (v) of a planet:

v = √(GM/r)

Where G is the universal gravitational constant, M is the mass of the star, and r is the average distance between the star and the planet.

For the slower planet:

v1 = √(GM/r1) ----(1)

For the faster planet:

v2 = √(GM/r2) ----(2)

Dividing equation (2) by equation (1), we get:

(v2^2) / (v1^2) = (r1/r2)

Substituting the given values, we get:

(50.6^2) / (36.0^2) = (r1/r2)

Solving for r1/r2, we find:

r1/r2 = (50.6^2) / (36.0^2) ≈ 1.984

Now, we can use Kepler's Third Law to find the mass of the star. According to the law:

(T1^2) / (T2^2) = (r1^3) / (r2^3)

Substituting the given values and the ratio of r1/r2, we have:

(7.78^2) / (T2^2) = (1.984^3)

Solving for T2^2, we get:

T2^2 = (7.78^2) / (1.984^3)

Taking the square root of both sides, we find:

T2 ≈ √[(7.78^2) / (1.984^3)]

Now, we can calculate the values.

(b) Let's substitute the values into the equation to find the mass of the star:

(50.6^2) / (36.0^2) = (r1/r2)
r1/r2 ≈ 1.984

(7.78^2) / (1.984^3) ≈ 1.9992

T2 ≈ √(1.9992) ≈ 1.413 years (rounded to three decimal places)

Therefore, the mass of the star is approximately 1.9992 times the mass of the slower planet, and the orbital period of the faster planet is approximately 1.413 years.