Question

You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its diameter to be 17000 km and its rotation period to be 22.8 hours. You have previously determined that the planet orbits 240000000 km from its star with a period of 500 Earth days. Once on the surface you find that the free-fall acceleration is 11.8 m/s22. What is the mass of the planet? What is the mass of the star?

Answer 1

To determine the mass of the planet we will use the acceleration in free fall and gravitational force.

Since the magnitude of the force acting on an object of mass m in free fall near the surface of the earth is known by

where is the mass of the planet and is its radius.

You can match this force with the force of gravity , where m is the mass of the object and we get

eliminating m and we obtain:

where G is a constant called universal gravitational constant. Its value in units
of the SI is

Rt is determined by dividing the diameter of the planet by 2

substituting all the values we get

to solve the second part of the problem we will use Kepler's third law, where the mass of the sun is related to the orbital size and the period of a planet

where T is the period and r is the distance of the planet-star orbit, where

substituting we have: