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

According to Kepler's Third Law of planetary motion, the square of the orbital period (T) of a planet is directly proportional to the cube of its orbital radius (R), expressed as T² ∝ R³ [1]. For two planets orbiting the same star, the ratio of their periods is given by (T1/T2)² = (R1/R2)³. Given the radii R1 = R and R2 = 4R, the ratio of the radii is R1/R2 = 1/4. Substituting this into the formula yields (T1/T2)² = (1/4)³, which simplifies to (T1/T2)² = 1/64. Taking the square root of both sides, we find the ratio of their periods T1/T2 = √(1/64) = 1/8. This law applies to all bodies in circular or elliptical orbits under the influence of gravity [2].

Sources

1. [1] Physical Geography by PMF IAS, Manjunath Thamminidi, PMF IAS (1st ed.) > Chapter 2: The Solar System > Kepler's Laws of Planetary Motion > p. 21
2. [2] https://pwg.gsfc.nasa.gov/stargaze/Kep3laws.htm