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The planet Mars has two moons, phobos an...

The planet Mars has two moons, phobos and delmos. (i) phobos has a period 7 hours, 39 minutes and an orbital radius of 9.4 `xx`103 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 ?

Text Solution

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(1) We employ Eq. (8.38) with the sun’s mass replaced by the martian mass `M_(m)`
`T^(2) = (4 pi^(2))/(GM_(m)) R^(3)`
`M_(m) = (4 pi^(2))/(G) (R^(3))/(T^(2))`
= `(4 xx (3.14)^(2) xx (9.4)^(3) xx 10^(18))/(6.67 xx 10^(-11) xx (459 xx 60)^(2))`
`M_(m) = (4 xx (3.14)^(2) xx (9.4)^(3) xx 10^(18))/(6.67 xx (4.59 xx 6)^(2) xx 10^(-5))`
(ii) Once again Kepler.s third law comes to our. atd.
`(T^(2)_(m))/(T^(2)_(E)) = (R^(3)_(MS))/(R^(3)_(RS))`
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`
= 648 days
we note that the ornits of all planets except Mercury. Mars and pluto are very close to being ctruclar. For example. the ratio of the semi-minor to semi-major axis for our Earth is . b/a = 0.99986.
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