Circular Orbits/Effective Potential

[Brian-san]

Homework Statement

Given potential V(r)=−ke^−ar/r; k,a>0. Using the method of the equivalent one-dimensional potential discuss the nature of the motion, stating the ranges of l and E appropriate to each type of motion. When are circular orbits possible? Find the period of small radial oscillations about the circular motion.

Homework Equations

E=K+V
L=K-V
p_q=dL/dq'
Taylor series expansion

The Attempt at a Solution

After the usual Lagrangian/momentum stuff I get E=0.5mr'²+(l²/2mr²)-(ke^-ar/r), giving the effective potential of V_eff=(l²/2mr²)-(ke^-ar/r)

I only have a rough idea of what the graph of V_eff looks like, infinity at zero, zero at infinity, and I assume it has only one extrema (a minimum). The only thing that seems obvious is that for E>=0, the motion is unbounded, and for a circular orbit, r'=0, so E=V_eff. The problem is the exponential function makes it difficult to solve for zeroes of dV/dr to locate and determine this minimum value. Also, I'm unsure of what the limits of l will be. All the examples I've seen of this process focused on the energy only.

Also, if you expand the potential around the minimum at point r₀, since V'=0, you get V(r₀)+0.5V''(r₀)(r-r₀)²

Other than that, I'm pretty much stuck at where to go next with this.

 

[turin]

If you are clever, you can plot V_eff in terms of a single parameter, where the parameter will determine the characteristic shape.

Hint:
Convert to a dimensionless coordinate variable and an overall energy scale. The exponent should be dimensionless (hint). The parameter should depend on angular momentum. Converting to dimensionless quantities makes comparisons meaningful, and it gives you a guideline for deciding what values (of the coordinate, angular momentum, etc.) are extremely large and small.

Regarding the condition for circular orbits, I am stumped. It is a transcendental equation (as you seem to have realized) that I am unfamiliar with.

 

[chrisk]

As suggeted, find V effective. Another approach is to plot V effective vs. r. The plots will show the turning points and indicate closed or open orbits depending on E. For a cirular orbit, the first derivative of V effective with respect to r has to equal zero. How is the period of oscillation related to V effective? (Hint: examine the second partial derivative of V effective with respect to r).

 

[turin]

chrisk,

Are there closed orbits (orbits on which the particle returns to its original position after one orbit) other than the circular one? By "closed" and "open", do you mean "bound" and "unbound"?

 

[chrisk]

turin,

Yes, I meant bounded and unbounded orbits. Thanks.