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.

In one of the satellites of Jupiter has an orbital period of 1.769 day

1.	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.
Answer» Solution :`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`