Q. 44.0( 28 Votes )

Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Answer :

If a body orbits around other heavier body due to gravitational force of attraction then we have a relation between mass of heavier body, time period of revolution and, radius of the orbit as


Where M is the mass of heavier body, R is the radius of orbit, G is universal gravitational Constant and T is time period of revolution


Here we will consider two cases, motion of Satellite Io around Jupiter and motion of earth around sun


So for motion of Io around Jupiter we have



Where, Mj is the Mass of Jupiter


G is Universal gravitational constant


RIo is radius of Io’s orbit, we are given


RIo = 4.22 × 108 m


TIo is the time period of revolution of Io around Jupiter


TIo = 1.769 days


So for motion of Earth around Sun we have



Where, Ms is the Mass of sun


G is Universal gravitational constant


Re is radius of Earth’s orbit


Re = 1 AU = 1.496 × 1011 m


Te is the time period of revolution of Earth around sun


Te = 365.25 days


Diving equation of mass of sun with equation of mass of Jupiter to compare



So putting values of Re, RIo, TIo, Te in above equation we get



NOTE: Time periods are not converted to SI units second, but are in years, but would not make any difference to result because we are taking ration and units and converting factors are going to be cancelled out ultimately


Solving above equation we get



Or we can say


Ms ≈ 1000Mj


i.e. mass of Sun is nearly 1000 times mass of Jupiter


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