__Question 8.4__:
Io, one of the satellites of Jupiter, has an orbital period of
1.769 days and the radius of the orbit is 4.22 × 10

^{8}m. Show that the mass of Jupiter is about one-thousandth that of the sun.

__Solution__:Orbital period of I

_{0}, T

_{I0}= 1.769 days = 1.769 X 24 X 60 X 60 s

Orbital radius of
I

_{0}, R_{I}_{0}= 4.22 X 10^{8}m
Satellite I

_{0}is revolving around the Jupiter
Mass of the latter is given by the relation:

M

_{J}= 4π^{2}R_{I0}^{3}/ GT_{I0}^{2}.....(i)_{}
Where,

M

_{J}= Mass of Jupiter
G = Universal gravitational constant

Orbital period of the earth,

T

_{e }= 365.25 days = 365.25 X 24 X 60 X 60 s
Orbital radius of the Earth,

R

_{e }= 1 AU = 1.496 X 10^{11 }m
Mass of sun is given as:

M

_{s}= 4π^{2}R_{e}^{3}/ GT_{e}^{2 }......(ii)
∴ M

_{s}/ M_{J}= (4π^{2}R_{e}^{3}/ GT_{e}^{2}) X (GT_{I0}^{2}/ 4π^{2}R_{I0}^{3}) = (R_{e}^{3}X T_{I0}^{2}) / (R_{I0}^{3}X T_{e}^{2})_{}Substituting the values, we get:

= (1.769 X 24 X 60 X 60 / 365.25 X 24 X 60 X 60)

^{2}X (1.496 X 10^{11}/ 4.22 X 10^{8})^{3}
= 1045.04

∴ M

_{s}/ M_{J}~ 1000
M

_{s }~ 1000 X M_{J}
Hence, it can be inferred that the mass of Jupiter is about
one-thousandth that of the Sun.

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