Question

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

Answer 1

Hint:Firstly we will find the value of mass of sun and Jupiter.After substituting the values of radius and time period of the Jupiter and sun. Then take the ratio of both the masses of Jupiter and sun.

Complete step by step solution:
Take Gravitational force equal to centripetal acceleration, we get
\[\dfrac{{GmM}}{{{r^2}}} = \dfrac{{m{v^2}}}{r}\]
\[v = \sqrt {\dfrac{{GM}}{r}} \]
As we know,
\[T = \dfrac{{2\pi }}{\omega }\]
Here T is the time period.
Also, \[v = \omega r\]
So, \[\omega = \dfrac{v}{r}\]
Time period, T=\[\dfrac{{2\pi r}}{v}\]
Squaring on both sides, we get,
\[{T^2} = \dfrac{{4{\pi ^2}{r^2}}}{{{v^2}}}\]
Now substitute the value of v in the resultant equation.
\[{T^2} = \dfrac{{4{\pi ^2}{r^3}}}{{GM}}\]
Here r is the radius, T is the time period, G is the gravitational force constant and M is the mass of the sun.

\[M = \dfrac{{4{\pi ^2}{r^3}}}{{G{T^2}}}\]
We know, radius of sun(1A.U=\[1.496 \times {10^{11}}\]m), r=\[1.496 \times {10^{11}}\]m and Time period is=\[365.25\,days\].
Now substitute all the values, we get-
\[M = \dfrac{{4{\pi ^2}{{\left( {1.496 \times {{10}^{11}}} \right)}^3}}}{{G{{\left( {365.25 \times 24 \times 60 \times 60} \right)}^2}}}\] ------ (1)

Now, the time period of Jupiter=\[1.769\] days=\[1.769 \times 24 \times 60 \times 60\] s.
Radius of Jupiter=\[4.22 \times {10^8}m\]
So, mass of Jupiter will be \[{M_J} = \dfrac{{4{\pi ^2}{{\left( {4.22 \times {{10}^8}} \right)}^3}}}{{G{{\left( {1.769 \times 24 \times 60 \times 60} \right)}^2}}}\] ------ (2)
Now, divide equation (1) by (2),
\[\dfrac{M}{{{M_J}}} = \dfrac{{4{\pi ^2}{{\left( {1.496 \times {{10}^{11}}} \right)}^3}}}{{G{{\left( {365.25 \times 24 \times 60 \times 60} \right)}^2}}} \times \dfrac{{G{{\left( {1.769 \times 24 \times 60 \times 60} \right)}^2}}}{{4{\pi ^2}{{\left( {4.22 \times {{10}^8}} \right)}^3}}}\]
\[\Rightarrow\dfrac{M}{{{M_J}}} = 1046\]
Here $M$ is the mass of the Sun and ${M_J}$ is the mass of Jupiter. We can write this equation as-
\[\dfrac{M}{{1046}} = {M_J}\]
\[\therefore{M_J} \approx \dfrac{1}{{1000}}M\]

Hence proved, the mass of Jupiter is one-thousand times the mass of the sun.

Note:Jupiter is the fifth planet from the Sun and the largest in the solar system. It is a gas giant with a mass one-thousandth that of the sun. These two bodies affect one another proportionally according to distance and mass. Also Jupiter's gravity pulls on the sun is one-thousandth the amount the Sun’s gravity pulls on Jupiter.