The planat Mars has two moons, phobos and delmos. (i) Phobos has a period 7 hours. 39 minutes and an orbital radius of `9.4xx10^(3)km`. Calculate the mass of mars. (ii) Assume that earth and mars move in circular orbits around the sun. with the martian orbit being 1.52 times the orbital radius of the earth. what is the length of the martian year in days?

Explanation

(i) We know that `T^(2) = (4pi^(2))/(GM_(m)) R^(3)`
`M_(m) = (4pi^(2))/(G)(R^(3))/(T^(2)) = (4 xx (3.14)^(2) xx (9.4)^(3) xx 10^(8))/(6.67 xx 10^(-11) xx (459 xx 60)^(2))`
`M_(m) = (4 xx (3.14)^(2)(9.4)^(3) xx 10^(18))/(6.67 xx (4.59 xx 6)^(2) xx 10^(-5))`
`= 6.48 xx 10^(23) kg`.
(ii) According to Kepler.s third law `(T_(M)^(2))/(T_(E)^(2)) = (R_(MS)^(3))/(R_(ES)^(3))`
where `R_(MS)` is the mars-sun distance and `R_(ES)` is the earth-sun distance.
`therefore T_(M) = (1.52)^(3//2) xx 365 = 684` days