1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 16
G. A. Krafft
Jefferson Lab

2. Elliptical Motion
First determine the constant Laplace-Runge-Lenz vector

3. Orbit as function of θ Determine the vector magnitude
Equation for radius r(θ)

4. Parametric Equation for Ellipse
Sum of the lengths to the two foci constant 2L
Eccentricity is ε = f/L =
aphelion
perihelion

5. More on Ellipse
In Homework show using Cartesian coordinates centered on dot get the ellipse equation (Kepler’s First Law)
Ellipse orientation in plane handled by θ0

6. Method Two Most common analysis, integrate the energy function
Check: 2D harmonic oscillator

7. Linear Restoring Force
Equation for ellipse centered at origin
semi-major axis semi-minor axis

8. Period Use area result. Total area of orbit ellipse
Time derivative of area swept out by radius vector
Period
Works!

9. Kepler Problem
Case of gravitational attraction

10. Method Three (Taylor) Write a differential equation for u = 1/r
For Newton gravity
Not so “attractive” because need to write a differential equation for 1/r2 for linear restoring force

11. Relations Among Parameters
For gravitational attraction and bound orbit

12. Kepler’s Third Law Revolution period T is
Only power laws for the potential with closed (elliptical) orbits are linear restoring force (kr2) and gravitation/Coulomb force (k/r)

13. Unbound Motion
E = 0 means the motion is a parabola with focus at sun. Minimum radius is rm/2
For E > 0, motion is a hyperbola, ε > 1. Orbit asymptotes are
Minimum radius

14. Impact Parameter
Distance of closest approach for particle without interacting is called the impact parameter bim
Equation for hyperbola
Equation for asymptotes
Distance of closest approach and
eccentricity