Two planets orbit the Sun in circular orbits, with their radius of orb

Two planets orbit the Sun in circular orbits, with their radius of orbit as R₁ = R and R₂ = 4R. Ratio of their periods (T₁/T₂) around the Sun will be

1/16

1/8

1/4

1/2

Answer is Right!

Answer is Wrong!

This question was previously asked in

UPSC NDA-2 – 2020

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This problem can be solved using Kepler’s Third Law of planetary motion, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its orbit: T² ∝ R³. For circular orbits, the semi-major axis is simply the radius (R). Thus, (T₁/T₂)² = (R₁/R₂)³. Given R₁ = R and R₂ = 4R, we have (T₁/T₂)² = (R / 4R)³ = (1/4)³ = 1/64. Taking the square root of both sides, T₁/T₂ = √(1/64) = 1/8.

Kepler’s Third Law relates the orbital period and orbital radius of planets orbiting the same central body: T² ∝ R³.

Kepler’s Laws are empirical laws describing the motion of planets around the Sun. Newton’s Law of Universal Gravitation provides the theoretical basis for Kepler’s Laws. For circular orbits, the speed v is constant, and the period T = 2πR/v. The gravitational force provides the centripetal force: GMm/R² = mv²/R. Substituting v = 2πR/T gives GMm/R² = m(2πR/T)²/R, which simplifies to T² = (4π²/GM) R³, confirming T² ∝ R³.

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