Question

In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun ($M_{sun}$ = 1.99 × 10³⁰ kg) at a distance of 1.50 × 10¹¹ m, called 1 astronomical unit (1 au). Others have extreme orbits that are much different from anything in our solar system. The following problem relates to one of these planets that follows circular orbit around its star. Assume the orbital period of earth is 365 days. Part A HD 10180g orbits with a period of 600 days at a distance of 1.4 au from its star. What is the ratio of the star's mass to our sun's mass?

Answer 1

Answer:

1.02

Explanation:

By the third Kepler's Law for an elliptic orbital, such as the Earth and the Sun:

T² = R³/k

Where T is the period of the rotation (in years), R is the distance between the planet to the star (in au), and k is the constant which depends on the mass of the star. For Earth and Sun, T = 1 year, and R = 1 au, so k = 1.

For HD 10180g, T = 600/365 = 1.64 years, R = 1.4

(1.64)² = (1.4)³/k

k = 2.744/2.6896

k = 1.02

Because k only depends on the mass, we can conclude that the ratio of the star's mass to our sun's mass is the ratio of the k values, so it will be:

1.02/1

1.02