Central Force Orbit

a) Find the central force which results in the following orbit for a particle:

b) A particle of mass is acted on by an attractive force whose potential is given by  Find the total cross section for capture of the particle coming from infinity with an initial velocity v_∞.

SOLUTION

a) We see that the orbit describes a cardioid as shown in Figure 1.1. Invoking the orbit equation yet again, we may find the

Figure 1.1

force:

(1)

where u = 1/r Calculating the derivatives of u :

(2)

and substituting into (1), we obtain

b) The initial impulse to solve for the scattering angle as a function of the impact parameter leads one astray into the realm of elliptic integrals. Instead, realize that the operative word is “capture” and construct the effective potential of the particle, where

and A is a constant of proportionality, and l is the angular momentum. Those particles whose kinetic energy exceeds U_eff (r₀) will be captured. At r₀, dU(r₀)/dr = 0, so we obtain

and

The condition for capture becomes

Where l = mv_∞b, and is the impact parameter. Rearranging, we find that  The cross section is given by πb², so