A planet moves around sun in elliptical orbit with eccentricity π/4 as shown. Find the ratio of time taken by planet to travel ABC to that of CDA.
Physics · Mixed Chapter
JEE Mains

Question

A planet moves around sun in elliptical orbit with eccentricity π/4 as shown. Find the ratio of time taken by planet to travel ABC to that of CDA.

G1 Physics · Accepted Answer

Step 1: By Kepler's second law, orbital time is proportional to the area swept by the radius vector from the Sun (at one focus). Step 2: Let the ellipse have semiaxes a,b and eccentricity e=π/4. The swept area for the right-hand arc ABC is S₁=πab/2+abe, while for the complementary left-hand arc CDA it is S₂=πab/2−abe. Step 3: The time ratio is tₐᵦ꜀/t꜀ᵈₐ=S₁/S₂=(π/2+e)/(π/2−e). Step 4: Substituting e=π/4 gives (3π/4)/(π/4)=3.