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) We employ Eq. (8.38) with the sun's mass replaced by the martian mass `M_(m)`
`T^(2)=(4pi^(2))/(GM)R^(3)`
`M_(m)=(4pi^(2))/(G)(*R^(3))/(T^(2))`
`=(4xx(3.14)^(2)xx(9.4)^(3)xx10^(18))/(6.67xx10^(-11)xx(459xx60)^(2))`
`M_(m)=(4xx(3.14)^(2)xx(9.4)^(3)xx10^(18))/(6.67xx(4.59xx6)^(2)xx10^(-5))` ltbr. (ii) once again kepler's third law comes to our aid.
`(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 R_(M)=(1.52)^(3//2)xx365` ltbr. =684 days
We note that the orbits of all planets except Mercuty. Mars and Pluto are very close to being circular. for example, the ratio of the semi-minor to wsemi-major axis for our Earth is b/a=0.99986.