13.18 13.6: SUMMARY

• Kepler’s three laws of planetary motion:

  • Law of Ellipses

  • Law of Equal Area in Equal Time

  • Law of the Period , where T is the period (time to complete one full revolution) and a is the semimajor axis (Figure 7).

    

    Planetary orbit.

• According to Newton’s Universal Law of Gravitation and Second Law of Motion, the position vector r(t) of a planet satisfies the differential equation

  

  770

• Properties of J = r(t) × r′(t):

  • J is a constant of planetary motion.

  • Let J = ∥J∥. Then J = r(t)²θ′(t).

  • The planet sweeps out area at the rate .

• A planetary orbit has polar equation , where e is the eccentricity of the orbit.