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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» Answer:Orbital period of I0 , TI0 = 1.769 days = 1.769 × 24 × 60 × 60 s Orbital radius of I0 , RI0 = 4.22 × 108 m Satellite I0 is REVOLVING around the Jupiter Mass of the latter is GIVEN by the relation: MJ = 4π2RI03 / GTI02 .....(i) Where, MJ = Mass of Jupiter G = Universal gravitational constant Orbital period of the earth, Te = 365.25 days = 365.25 × 24 × 60 × 60 s Orbital radius of the Earth, Re = 1 AU = 1.496 × 1011 m Mass of sun is given as: Ms = 4π2Re3 / GTe2 ......(ii) ∴ Ms / MJ = (4π2Re3 / GTe2) × (GTI02 / 4π2RI03) = (Re3 × TI02) / (RI03 × TE2) Substituting the values, we GET: = (1.769 × 24 × 60 × 60 / 365.25 × 24 × 60 × 60)2 × (1.496 × 1011 / 4.22 × 108)3 = 1045.04 ∴ Ms / MJ ~ 1000 Ms ~ 1000 × MJ Hence, it can be inferred that the mass of Jupiter is about one-thousandth that of the Sun. |
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