1. Gravitational Fields, Circular Orbits and Kepler

2. Gravitational Field
The Force of gravity per unit of mass at a given point
It is the same as “local gravity” for a planet

3. Finding g (0,100) m = 10,000 kg What is g at the origin? m = 20,000 kg
(80,0)

4. Speed in a Circular Orbit

5. Tycho Brahe

6. Johannes Kepler

7. Kepler’s First Law of Planetary Motion
The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one focus

8. Speed in an Elliptical Orbit
At Apogee: R = maximum and v = minimum
At Perigee: R = minimum and v = maximum

9. Ellipse vs. Circle

10. Kepler’s Second Law of Planetary Motion
A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time
This occurs because angular momentum is conserved

11. Kepler’s Third Law of Planetary Motion
The squares of the periods of the planets are proportional to the cubes of their semi-major axes
If the distance R is measured in AU and the period T is measured in years then the third law for orbits about the Sun becomes

12. Comet Axis?
What is the length of the semi-major axis of the orbit of a comet that takes 76 years to revolve around the sun?