This problem is still being typed. 
Mechanics}Stability of Orbits
The gravitational force suspect to a bit of perturbation is given as .
One can narrow down most choices by recalling some basic facts from central force theory:
(A) No mention is made of frictional effects, and thus energy should be conserved.
(B) Angular momentum is always conserved since the net torque is 0 (to wit: the force and moment arm are parallel).
(C) This is just Kepler's Third Law applied to this force. (Recall the following bromide: The square of the period is equal to the cube of the radiusfor the inverse square law force. For a perturbed force, the bromide becomes: The square of the period is equal to the power of the radius.)
(D) Recall Bertrand's Theorem from Goldstein. Stable noncircular orbits can only occur for the simple harmonic potential and the inversesquare law force. This is of neither form, and thus this choice is FALSE.
(E) Circular orbits exist for basically all potentials. A stationary orbit exists if and only if the following conditions are satisfied: . Recall that the potential is related to the force by . Use , and recalling the extra term added to the effective potential to be , one chunks out the derivatives to get the condition that , as a potential exponent, (, as a force exponent) for stable orbit. One can remember this result or rederive it whenever necessary. For , (the power exponent of the force equation), a stable circular orbit exists. Since is presumably less than 1, the planet does, indeed, move in a stationary circular orbit about the sun.
