Question

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 1

Orbital period of Io, TIo = 1.769 days = 1..769 x 24 x 60 x60 s
Orbital radius of Io , RIo = 4.22 x 108 m
Satellite Io is revolving around the Jupiter
Mass of the latter is given by the relation:
$M_j =\frac{ 4π^2 R_{Io }^3}{ GT_{Io}^2}$...(i)

Where,
Mj = Mass of Jupiter
G = Universal gravitational constant

Orbital radius of the Earth,
Te = 365.25 days = 365.25 x 24x 60 x60 s

Orbital radius of the Earth,
Re = 1 AU = 1.496 x 1011 m
Mass of Sun is given as:
$Ms = \frac{4π^2 \,R_2^3}{ GT_e^2}$ ...(ii)

∴ $\frac{ M_s}{ M_j} = \frac{4π^2 R_e^3}{ GT_e^2 }× \frac{GT_{Io}^2}{ 4π^2 R_{Io}^ 3} = \frac{R_e^3}{ R_{Io}^3 }× \frac{T_{Io}^2}{ T_e^2}$

$= (\frac{1.769 × 24 × 60×60 }{365.25 × 24× 60 ×60 })^2 × (\frac{1.496 ×10^{11} }{4.22 ×10^8})^3$
= 1045.04

∴ $\frac{M_s}{ M_j} ∼ 1000$
Ms∼ 1000 x Mj

Hence, it can be inferred that the mass of Jupiter is about one-thousandth that of the Sun.