Kepler
Inverse Square Force Force can be derived from a potential. < 0 for attractive force Choose constant of integration so V ( ) = 0. m2m2 r1r1 F 2 int r2r2 R m1m1 F 1 int r = r 1 – r 2
Kepler Orbits Right side of the orbit equation is constant. Equation is integrable.Equation is integrable. Integration constants: e, 0Integration constants: e, 0 Equation describes a conic section. init orientation (often 0) init orientation (often 0) e sets the shape: e 1 hyperbola. e sets the shape: e 1 hyperbola. s is the directrix. s is the directrix. focus r s
Kepler Lagrangian The lagrangian can be expressed in polar coordinates. L is independent of time. The total energy is a constant of the motion.The total energy is a constant of the motion. Orbit is symmetrical about an apse.Orbit is symmetrical about an apse. constant
Apsidal Position Elliptical orbits have stable apses. Kepler’s first lawKepler’s first law Minimum and maximum values of r.Minimum and maximum values of r. Other orbits only have a minimum.Other orbits only have a minimum. The energy is related to e: Set r = r 2, no velocitySet r = r 2, no velocity r s r1r1 r2r2
Angular Momentum Change in area between orbit and focus is dA/dt It is constantIt is constant Kepler’s 2 nd law Area for the whole ellipse relates to the period. semimajor axis: a=(r 1 +r 2 )/2. Kepler’s 3 rd law r dr
Effective Potential Treat problem as a one dimension only. Just radial r term.Just radial r term. Minimum in potential implies bounded orbits. For > 0, no minimumFor > 0, no minimum For E > 0, unboundedFor E > 0, unbounded V eff 0 r 0 r unbounded possibly bounded