I was doing my Gravitation problems and I found this one that I'm unable to solve.

Yukawa's theory for nuclear forces states that the potential energy corresponding to the attraction force produced by a proton and a neutron is: $$U(r) = \frac{k}{r}e^{-\alpha r},\ k<0,\ \alpha > 0$$ From the expression of it's effective potential, find the module of it's angular momentum and it's energy, for which it's possible a circular movement with a radius $r_0​$.

I've tried several things, none of them leading to something meaningful. In fact, I know that expression for effective potential is: $$U_{ef}(r)=U(r)+\frac{L}{2r^2}$$

So I imagine I would need to find $L$ fist in order to get the expression for $U_ef$, but I'm not able to remember nor find any kind of formula linking $U$ and $L$. Would you please help me out?

PS: Once I know how to find $L$ I know how to end it, since: $$\frac{dU_{ef}}{dr} = 0 \Leftrightarrow r = r_0$$ is the expression of the energy of a circular movement with a radius $r_0$


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Since this is a homework problem, I won't provide a full solution, but here's a nudge in the right direction. Take a look at these two plots of the effective potential:

k = -1, $\alpha$ = 1, L = 0.25

k = -1, $\alpha$ = 1, L = 1

What's different about these two effective potentials? We only changed $L$ between the two graphs; what does that imply about the allowed values of $L$ if we want a circular orbit? (If you can't answer this question immediately, try plotting the effective potential for a few more values of $L$. If you have access to Mathematica and know how to use it, the Manipulate function is your friend.)

  • $\begingroup$ I think I get it. The value of L should be $r_0$. Meaning that L = d(Uef)/dr at $r_0$ $\endgroup$ – Ignasi Sánchez May 20 '15 at 14:45
  • $\begingroup$ Not quite. It's more that for certain values of $L$, no $r_0$ will exist. This is different than the usual Newtonian potential, for which the effective potential has a minimum for every value of $L$. $\endgroup$ – Michael Seifert May 20 '15 at 17:34
  • $\begingroup$ Okay, so, how should I get the angular momentum with the potential energy? I really don't know how to solve it. $\endgroup$ – Ignasi Sánchez May 20 '15 at 17:54
  • $\begingroup$ Given the above information, you should be able to find a relationship between $L$ and $r_0$. Remember, the question you're being asked is "for a given $r_0$, what should $L$ be to have a circular orbit at that radius?" (At least, I think that's what the question is asking. It's no model of clarity.) $\endgroup$ – Michael Seifert May 20 '15 at 18:29
  • $\begingroup$ Hmm I don't think you're getting it right. In theory, $$L=(-mkr_0(1+\alpha r_0)e^{-\alpha r_0})^{1/2}$$ $\endgroup$ – Ignasi Sánchez May 20 '15 at 21:38

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