Question

One of the satellite of Jupiter has an orbital period of 1.769 days and the radius of the orbit is $4.88\times {10}^{8}m$. Show that the mass of the Jupiter is about one-thousand that of the sun.

Answer 1

The time period is, $T=\frac{2\pi R}{V}$,
$T^2=\frac{4{\pi }^2{R}^2}{V^2}\dots \dots \left(1\right)$
The centripetal force for the satellite's motion is the gravitational force between satellite and Jupiter, $\frac{m{V}^2}{R}=\frac{GMm}{R^2}$, where m is the mass of the satellite and M is the mass of Jupiter.
$V^2=\frac{GM}{R}$
Using this $V^2$ in (1),
$T^2=\frac{4{\pi }^2{R}^3}{\left(GM\right)}$
$M=\frac{4{\pi }^2{R}^3}{\left(G{T}^2\right)}M=\frac{\left(4\right)\left(3.14{)}^2\right(4.22\times {10}^8{)}^3}{(6.67\times {10}^{-11})(1.762\times 24\times 3600{)}^2}=1.90\times {10}^{27}kg=2\times {10}^{27}kg$
Sun's Mass is, $2\times {10}^{30}kg$.
Hence, the Jupiter's mass is 1000th that of the sun.