The total mechanical energy of a planet of mass m moving around the Sun along an ellipse depends only on its semi-major axis a . This energy as a function of a is given by

Step-by-step solution

- G m ⁡ M ⁡ s ⁡ r ⁡ 1 + 1 2 mv ⁡ 1 2 = - G m ⁡ M ⁡ s ⁡ r ⁡ 2 + 1 2 mv ⁡ 2 2 At point A, Centripetal force mv ⁡ 1 2 r ⁡ 1 = G m ⁡ M ⁡ s ⁡ r ⁡ 1 2 or 1 2 mv 1 2 = G m ⁡ M ⁡ s ⁡ r ⁡ 1 .... (i) ∴ Total energy, E at A ⁡ = 1 2 mv 1 2 - G m ⁡ M ⁡ s r ⁡ 1 ∴ E ⁡ = - G m ⁡ M ⁡ s ⁡ 2 r ⁡ 2 ....(ii) From the property of ellipse, r 1 + r 2 = 2 a From eqn. (i) and (ii), we get E ( r 1 + r 2 ) = – G M s m or E ⁡ × 2 a = - G M ⁡ s ⁡ m ⁡ ∴ E ⁡ = - G M ⁡ s ⁡ m ⁡ 2 a