Question

A planet of mass m revolves around the sun of mass M in an elliptical orbit. The maximum and minimum distances of the planet from the sun are r₁ and r₂, respectively. The time period of the planet in terms of r₁ and r₂ is

(a) T ∝ (r₁ + r₂)²

(b) T ∝ (r₁ + r₂)3

(c) T ∝ (r₁ + r₂)1/2

(d) T ∝ (r₁ + r₂)3/2

Answer 1

Answer is : (d) \(T\propto (r_1+r_2)^{\frac{3}{2}}\)

The semi-major axis of the elliptical orbit around the sun is given by

\(r=\frac{r_1+r_2}{2}\)

According to Kepler’s third law,

\(T^2\propto r^3\)

\(\Rightarrow T^2\propto (\frac{r_1+r_2}{2})^3\)

\(or\: T\propto (\frac{r_1+r_2}{2})^\frac{3}{2}\)

\(or\:T\propto (r_1+r_2)^{\frac{3}{2}}\)