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 1.8×107m and its rotation period to be 22.3 hours . You have previously determined that the planet orbits 2.2×1011m from its star with a period of 382earth days. Once on the surface you find that the free-fall acceleration is 12.2m/s2.

a) What is the mass of the planet?

b)What is the mass of the star?

Answer 1

we can find the mass of the planet from g=GM/R^2 where G is newtonian gravitational constant, M the mass of the planet, R the radius and g the local acceleration due to gravity

solving for M:

M= gR^2/G =12.2*(9x10^6)^2/6.67x10^(-11)

M= 1.48x10^25 kg (we agree!)

find the mass of the star from Kepler's 3rd law (or the procedure that derives the law)

equate gravitational force to centripetal force

GMm/r^2 = mv^2/r

M= mass of star, m= mass planet, r= size of orbit, v= orbital speed

this becomes M= v^2r/G

v= circumference of orbit/time = 2 pi r/time =2 pi x 2.2x10^11/382 days x 86,400sec/day = 41881.81 m/s

M= (4.188x10^4)^2*2.2x10^11/6.67x10^(-11)

M= 5.786 x10^30 kg which is a perfectly reasonable mass for a star.

Please rate my answer, good luck...