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 r_per and the speed at this point is denoted v_per. Similarly, we write r_ap and v_ap for the distance and speed at the aphelion. The semimajor axis is denoted a.

Show that the total energy [Eq. (8)] of a planet in a circular orbit of radius Use Exercise 9.

Data From Exercise 9

Prove that if a planetary orbit is circular of radius R, then vT = 2πR, where v is the planet’s speed (constant by Exercise 7) and T is the period. Then use Kepler’s Third Law to prove that

Data From Exercise 7

Show that a planet in a circular orbit travels at constant speed. Use the facts that J is constant and that r(t) is orthogonal to'(t) for a circular orbit.

Transcribed Image Text:

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