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10, one of the stallites of Jupiter, has...

10, one of the stallites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is `4.22xx10(8)m`. Show that the mass of jupiter is about one-thousandth that of the sun.

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For a satellite of Jupiter, orbital period, `T_1 = 1.769 `days = `1.769 xx 24 xx 60 xx 60 s`
Radius of the orbit of satellite, `r_1= 4.22 xx 10^8 m`, Mass of the Jupiter. `M_1` is given by
`M_1 = (4pi^2 r_1^3)/(GT_1^2) = (4pi^2 xx (4.22 xx 10^8)^3)/(G xx (1.769 xx 24 xx 60 xx 60)^2)` ....(1)
We know that the orbital period of Earth around the Sun,
T= 1 year = 365.25 x 24 x 60 x 60s, Orbital radius, r=1A.U. ` = 1.496xx10^11m ` .
Mass of the Sun is given by `M = (4pi^2 r^2)/(GT^2) = (4pi^2 xx (1.496 xx 10^11 )^3)/(G xx (365.25 xx 24 xx 60 xx 60)^2)` ......(2)
Dividing (2) by (1) , we get
`(M)/(M_1) = (4pi^2 xx (1.496 xx 10^11)^3)/(G xx (365.25 xx 24 xx 60 xx 60)^2) xx (G xx (1.769 xx 24 xx 60 xx 60)^2)/(4pi^2 xx (4.22 xx 10^8)^3) = 1046`
or `(M)/(M_1) = 1046 ~~ 1000 " or " M ~~ 1000M_1 `, which was to be proved
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