A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is

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Solution

Q.
A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius orbit of planet is

(Given = Radius of geo-stationary orbit for earth is $4.2 \times 10^{4}$ km) [2024]

1 $1.05 \times 10^{4}$ km

2 $1.4 \times 10^{4}$ km

3 $1.68 \times 10^{5}$ km

4 $8.4 \times 10^{4}$ km

Ans.
(1)

By Kepler's law, $T^{2} = \frac{4 π^{2} R^{3}}{G M} \Rightarrow T^{2} \propto \frac{R^{3}}{M}$

${(\frac{T_{1}}{T_{2}})}^{2} = {(\frac{R_{1}}{R_{2}})}^{3} (\frac{M_{2}}{M_{1}})$

${(\frac{24}{6})}^{2} = {(\frac{4.2 \times 10^{4}}{R_{2}})}^{3} \times (\frac{\frac{M_{E}}{4}}{M_{E}})$

$4^{2} \times 4 = {(\frac{4.2 \times 10^{4}}{R_{2}})}^{3} \Rightarrow R_{2} = 1.05 \times 10^{4}$ km

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