We finished our analysis of inverse square law orbits with a discussion of Kepler's laws for planetary motion, and of Rutherford scattering and the repulsive Coulomb orbits.
Given our geometric understanding of gravitational orbits, we listed
Kepler's laws. Since the sun is much more massive than an orbiting
planet, the gravitational interaction between the two has negligible
effect on the sun, which to a good approximation remains
stationary. Thus the reduced mass 
is essentially the planet mass
and the separation coordinate 
is essentially the planet
position. Using this interpretation of as the planet position,
and considering only bound orbits, Kepler's laws state: 1.) Planets
move in elliptical orbits with the sun at one focus; 2.) Orbits sweep
out area at a constant rate (angular momentum conservation); and 3)
the square of the period of the orbit is proportional to the cube of
the semimajor axis. This last one we derived with some algebra and the
geometric parameters of the orbit.
Then considered orbits for repulsive Coulomb potential, where k of our
previous problem becomes negative. Showed that we only get scattering
orbits, by considering the effective potential. Our solutions have
the same form as for the Kepler problem, 
, but C
is negative and so 
must be negative to give a physical
r. We found that this solution had a closest approach at 
,
with x negative, and the same asymptotic angles 
, 
as the Kepler case. We thus get hyperbolae that are deflected without
crossing the y-axis.
Discussed Rutherford scattering of alpha particles off of heavy nuclei. We measure the orbit's deflection indirectly, by using an incident beam of many alpha particles and measuring the angular distribution of the outgoing scattered particles. The rate at which deflected particles are seen at a given solid angle is proportional to the differential cross section, which is calculable using geometry and our understanding of the Coulomb orbit for each alpha particle. Rutherford first calculated this cross section, and showed that experimental scattering of alpha particles from atoms was consistent with having all atomic charge focused in a point nucleus, and inconsistent with a homogeneous charge distribution inside the atom.
--
KB