I'm working on a homework question in which I am supposed to discuss the types of motion that orbits may have under a certain potential. What I've done so far is draw the energy diagram for such a situation, and looked at the different types of motion depending on the initial energy.
What I'm wondering though is if it's possible to figure out if the elliptical orbits that the potential allows are precessing or not. Furthermore, I want to know if I can tell whether the elliptical orbits are closed (i.e., $r(t) = r(t+\Delta t)$ for some $t$) from my energy diagram and/or the potential given?
I'm trying to answer this question without being too quantitative. I'm also just looking for pointers in the right direction, not necessarily a full answer.
I've also graphed the corresponding energy diagram, and I get something that I'm not quite sure how to interpret. Essentially, there seems to be a certain tuning of the parameters where there can be a bound elliptical orbit with $E \gt 0$. I was under the impression that a bound orbit requires negative energy (in the sense that we define zero energy to be at infinity). Am I missing something here?
Update: I think my question about closing orbits is mostly answered. I was just wondering if something about my energy diagram could point to the fact that a certain elliptical orbit could precess. In particular, I've added two points on my graph that shows an initial given energy, which is positive. If I understand the diagram correctly, the particle should oscillate between the radii of these two points on the graph. However, unlike with regular $1/r$ potentials, these orbits seem to be bound, yet still have positive energy. Is this allowed? I was under the impression that the energy needs to be negative in order to stay bound, but it seems like it's possible to have a positive energy and still have a bound elliptical orbit.