The planet Mars has two moons. Phobos and Delmos (i) phobos has period `7` hours, `39` minutes and an orbital radius of `9.4 xx 10^(3) km`. Calculate the mass of Mars. (ii) Assume that Earth and mars move in a circular orbit 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? `(G = 6.67 xx 10^(-11) Nm^(2) kg^(-2))`

Explanation

(i) Orbital radius, `r = 9.4 xx 10^(6) m`
Time period = 7h 39 min = 459 min `= (459 xx 60)s`
Mass of Mars is M `= (4pi^(2)r^(3))/(G T^(2))`
`= (4 xx 9.87 xx (9.4)^(3) xx 10^(18))/(6.67 xx 10^(-11) xx (459 xx 60)^(2))`
`= 6.48 xx 10^(23) kg`
(ii) Now, `R_(M) = 1.52 R_(E)`
Also, `T_(E) = 365` days
Using Kepler.s third law
`(T_(M)^(2))/(T_(E)^(2)) = (R_(M)^(3))/(R_(E)^(3))`
`:. T_(M) = ((R_(M))/(R_(E)))^((3)/(2))T_(E) = (1.52)^(3//2) xx 365`
= 684 days