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 × 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.

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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 × 10⁸ m. Show that the mass of Jupiter is about one-thousandth that of the sun.

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Vishal Baghel | Contributor-Level 10

4 months ago

The rotation period of the satellite Io, $T_{j u}$ = 1.769 days = 1.769 x 24 x 60 x 60s

Radius of the Orbit is given by the relation, $R_{j u}$ = 4.22 $\times 10^{8}$ m

Mass is given by the relation: $M_{j}$ = $\frac{4 π^{2} R_{j u}^{3}}{G T_{j u}^{2}}$ ……(i)

Where $M_{j}$ = mass of Jupiter and 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 : $M_{s}$ = $\frac{4 π^{2} R_{e}^{3}}{G T_{e}^{2}}$ …(ii)

$\frac{M_{s}}{M_{j}}$ = $\frac{4 π^{2} R_{e}^{3}}{G T_{e}^{2}}$ $\times$ $\frac{G T_{j u}^{2}}{4 π^{2} R_{j u}^{3}}$ = $\frac{R_{e}^{3}}{T_{e}^{2}}$ $\times$ $\frac{T_{j u}^{2}}{R_{j u}^{3}}$ = ( ${\frac{1.496 x 10^{11}}{4.22 \times 10^{8}})}^{3}$ $\times ({\frac{1.769 x 24 x 60 x 60}{365.25 x 24 x 60 x 60})}^{2}$ = 1045.04

Hence it can be inferred that the mass of Jupiter is about one &nda

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The rotation period of the satellite Io, $T_{j u}$ = 1.769 days = 1.769 x 24 x 60 x 60s

Radius of the Orbit is given by the relation, $R_{j u}$ = 4.22 $\times 10^{8}$ m

Mass is given by the relation: $M_{j}$ = $\frac{4 π^{2} R_{j u}^{3}}{G T_{j u}^{2}}$ ……(i)

Where $M_{j}$ = mass of Jupiter and 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 : $M_{s}$ = $\frac{4 π^{2} R_{e}^{3}}{G T_{e}^{2}}$ …(ii)

Hence it can be inferred that the mass of Jupiter is about one – thousandth that of the Sun

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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 × 10⁸ m. Show that the mass of Jupiter is about one-thousandth that of the sun.

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