One of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 xx 10^8 m. Show that mass of Sun is about one thousand times that of the Jupiter. (Take 1 year = 365.25 mean solar day).

One of the satellites of Jupiter, has an orbital period of 1.769 days

1.	One of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 xx 10^8 m. Show that mass of Sun is about one thousand times that of the Jupiter. (Take 1 year = 365.25 mean solar day).
Answer» Solution :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