Say I have some planet in a circular orbit around a star, and I give it a small radial push (directly toward or directly away from the star). How would I describe the new orbit? I.e. how would I determine the equations of motion? h
Kepler orbital radius: r(φ)=c/(1+εcos(φ)) where c=(L^2)/(γμ).
Also possibly relevant, orbital energy: E=(γ^2)μ((ε^2)-1)/(2L^2)
The Attempt at a Solution
My intuitive hypothesis is that the new orbit should be elliptical. A circular orbit has constant radius r(φ)=c, and we have already proved in this class that the radius after a small push will oscillate with a period equal to the orbital period, so it seems like if we just add a sinusoidal term, r(φ)=c+Asin(φ) we should be able to get the equation in the elliptical form r(φ)=c/(1+εcos(φ)) but that doesn't seem to work. I have a hunch that I'm supposed to do some sort of a Taylor expansion, but I don't know which function I would expand. It also seems like, since a push would change the angular momentum L, we should be able to replace L with L+δ and again get something of the form r(φ)=c/(1+εcos(φ)). Is any of this heading in the right direction? Have spent many hours on this problem and could use a hint.