Question

One of the statellites of Jupiter, has an orbital period of $1.769\text{days}$ and the radius of the orbit is $4.22\times {10}^{8}m$. Show that the mass of Jupiter is about $one-thousandth$ that of the sun

Answer 1

For Jupiter's satellites

Orbital period, $T_J=1.769\text{days}=1.769\times 24\times 60\times 60s$

Radius of the orbit, $R_J=4.22\times {10}^{8}m$

Mass of Jupiter, $M_J=\frac{4{\pi }^2{R}_J^3}{G{T}_J^2}$ ...$\left(i\right)$

Now, orbital period of Earth around sun

$T_E=1\text{year}=365.25\times 24\times 60\times 60s$

Orbital radius, $R_S=1.496\times {10}^{11}m$

Mass of sun, $M_S=\frac{4{\pi }^2{R}^3}{G{T}_E^3}$ ..$\left(ii\right)$

Dividing $\left(ii\right)\text{by}\left(i\right)$

$\frac{M_S}{M_J}=\frac{R^3}{R_J^3}\times \frac{T_J^2}{T_E^2}=\frac{(1.496\times {10}^{11}{)}^3}{(4.22\times {10}^8{)}^3}\times \frac{(1.769\times 24\times 60\times 60{)}^2}{(365.25\times 24\times 60\times 60{)}^2}=1046$

or $\frac{M_J}{M_S}=\frac{1}{1046}\approx \frac{1}{1000}$

or $M_J=\frac{1}{1000}{M}_S$