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

Question #94544
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
1
Expert's answer
2019-09-19T09:49:13-0400

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.



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