Answer to Question #94544 in Astronomy | Astrophysics for chris H

Question

Prove that in a properly elliptic orbit, the angle between r ⃗ and v ⃗ is π/2 only at the apogee and perigee.

Clue: Do so by showing v ⃗⊥r ⃗ if and only if ε≠0, then τ is maximized (apogee) or minimized (perigee)when θ=0 or π

Answer 1

As we know, we assume that the position vector is always directed from the central body (which is in one focus of an elliptic orbit) to a smaller body orbiting a planet or a star. The velocity vector, however, is directed along the tangent to the trajectory of motion. For a purely circular orbit the position and velocity vectors are always perpendicular. Draw what happens to an elliptic orbit:

So we see that at the apogee and perigee the two vectors are perpendicular to each other.