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.
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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