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मान लीजिए एक ऐसा ग्रह है जो सूर्य के परितः पृथ्वी की तुलना में दो गुनी चाल से गति करता है, तब पृथ्वी की कक्षा की तुलना में इसका कक्षीय आमाप क्या है ?
more by a factor of $2.67$
more by a factor of $1.24$
Lesser by a factor of $0.63$
Lesser by a factor of $0.33$
Solution
Time taken by the Earth to complete one revolution around the Sun,
$T_{ e }=1$ year
Orbital radius of the Earth in its orbit, $R_{e}=1$ $AU$
Time taken by the planet to complete one revolution around the Sun, $T_{p}=\frac{1}{2} T_{e}=\frac{1}{2}$ year
Orbital radius of the planet $=R_{ p }$
From Kepler's third law of planetary motion, we can write:
$\left(\frac{R_{p}}{R_{e}}\right)^{3}=\left(\frac{T_{r}}{T_{e}}\right)^{2}$
$\frac{R_{p}}{R_{e}}=\left(\frac{T_{p}}{T_{e}}\right)^{\frac{2}{3}}$
$=\left(\frac{\frac{1}{2}}{1}\right)^{\frac{2}{3}}=(0.5)^{\frac{2}{3}}=0.63$
