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

CONCEPT:

Kepler’s laws of planetary motion:

• The law of Orbits: Every planet moves around the sun in an elliptical orbit with the sun at one of the foci.
• The law of Area: The line joining the sun to the planet sweeps out equal areas in equal interval of time. i.e. areal velocity is constant.
• According to this law, the planet will move slowly when it is farthest from the sun and more rapidly when it is nearest to the sun. It is similar to the law of conservation of angular momentum.
• The law of periods: The square of the period of revolution (T) of any planet around the sun is directly proportional to the cube of the semi-major axis of the orbit i.e. T2 ∝ r3

CALCULATION:

Given - 1st orbital radius (R1) = R, and 2nd orbital radius (R2) = 4R 

• According to the law of periods

⇒ T2 ∝ r3

\(⇒ (\frac{T_1}{T_2})^2=(\frac{R_1}{R_2})^3\)

\(⇒ (\frac{T_1}{T_2})^2=(\frac{R}{4R})^3=\frac{1}{64}\)

\(\Rightarrow \frac{T_1}{T_2}=(\frac{1}{64})^\frac{1}{2}=\frac{1}{8}\)