Question

See problem 8.12 in Classical Mechanics by Taylor

3. Radial Nudges and Oscillations: In your homework, you saw that for the Kepler Problem, circular orbits are stable. If you give the planet a small radial nudge, it oscillates about its original orbit with simple harmonic motion with a period equal to the orbital period. However, after deriving the conic solutions to the Kepler Problem, your professor said that the only possible trajectories are circles, ellipses, parabolas and hyperbolas. a) How can this be compatible with a planet oscillating about a circular orbit? Is the conic section orbit with a small perturbation a possible solution to the Kepler problem in addition to the circles, ellipses etc. or is a circle plus a "specific" sinusoidal perturbation ellipse. Prove one or the other of these propositions. b) Draw two pictures: The energy diagram with positions before and after the nudge, and the actual orbits. Points (10, 10 20) Hint: In either case you need to use the solution of the Kepler Problem in polar (r, 0) coordinates and circular orbit: 0= @t, @ = constant rotational velocity.

Answer 1

The general solution can be put in the form

1/r A1e cos(-)]

for the Kepler problem. In perturbative analysis, the deviation to circular orbit gives the following solution

r=ro+ a cos(-'

and a is very small. There will be more terms for better and better accuracy.

The second statement is correct. Circle + perturbation = ellipse.