The perihelion and aphelion are the points on the orbit closest to and farthest from the sun,

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The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted vper. Similarly, we write rap and vap for the distance and speed at the aphelion. The semimajor axis is denoted a.

Aphelion Vap F2 r y F Vper a Semimajor axis X Perihelion


Prove thatVper GM 1 + e 1-e a as follows:

(a) Use Conservation of Energy (Exercise 17) to show that

v per -1 v = 2GM(re-rap) ap

Data From Exercise 17

The total mechanical energy (kinetic energy plus potential energy) of a planet of mass m orbiting a sun of mass M with position r and speed v = ΙΙr'ΙΙ is

(a) Prove the equations d 1 di zmv. 1 GMm E = mv_ ||r|| -mv = v. (ma), 2 d GMm dt ||rl| = V GMm ||r|| dE (b)


(b) Show thatper 2e -1 ap a(1-e) =using Exercise 13. 

Data From Exercise 13

Use the polar equation of an ellipse P 1 + ecos 0 to show that per = a(1-e) and rap = a(1+e). Hint: Use the



(c) Show that12 per e (1 + e)2 12 4- = ap -12 per using Exercise 15. Then solve for vper using (a) and (b).

Data From Exercise 15

Use the fact that J = rx r' is constant to prove Vper (1 e) Vap(1 + e) r is perpendicular to r' at the


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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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