In 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 the mass of Jupiter is about one-thousandth that of the sun.

Explanation

`implies` Orbital period of Jupiter
T = 1.769 days
= `= 1.769 xx 24 xx 3600 s`
`= 1.528 xx 10^5 s`
Radius of satellite,
` r = 4.22 xx 10^(8) m`
` G = 6.67 xx 10^(-11) Nm^(2)//kg^(2)`
Mass of sun `= 2 xx 10^(30)` kg
Suppose mass of Jupiter = `M_J`
Centripetal = Gravitational force
`:. (mv^2)/r = (GM_J.m)/r^2`
`:. v^2 =(GM_J)/r`
but `v = r omega`
`:. r^(2)omega^2=(GM_J)/r`
`:. omega^2=(GM_J)/r^3 " "[ :. omega=(2pi)/T]`
`:. (4pi^2)/T^2=(GM_J)/r^3`
`:. M_(J) = (4pi^2r^3)/(T^2G)`
`:. M_J = (4 xx (3.14 )^2 xx(4.22xx10^8)^3)/((1.528xx10^5)^2xx6.67 xx10^(-11))`
`~~ 1.903 xx 10^(27)kg `
`~~ 2 xx 10^(27) kg `
`:. M_J = M_s/1000`