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 I_o, T_Io = 1.769 days = 1..769 x 24 x 60 x60 s
Orbital radius of I_o , R_Io = 4.22 x 10⁸ m
Satellite I_o 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,
M_j = Mass of Jupiter
G = Universal gravitational constant 

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

Orbital radius of the Earth, 
R_e = 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 

∴ \(\frac{M_s}{ M_j} ∼ 1000\)
M_s∼ 1000 x Mj

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