1. Orbital Geometry

2. Johann Kepler
In 1600, Johann Kepler developed his “Laws of Planetary Motion.” He observed that the planets traveled in closed curves (ellipses), rather than circular paths.

3. Ellipses
The center of an ellipse differs from a circle in that there are two fixed points (foci) rather than one.
Kepler's first law: the orbits of the planets around the sun are ellipses, with the Sun at one of the foci.
The eccentricity of an ellipse can be thought of as the degree of non-roundness, or ovalness, of the orbit.

4. Eccentricity = distance between foci/length of major axis
e = d/L
The smaller the eccentricity, the more circular the orbit.

5. The Solar System
Looking at the Solar System Data table, most of the planets have fairly circular orbits (low eccentricities) with the exception of Mercury.

6. The Earth’s Orbit
The earth actually receives 7% less radiation in the summer of the northern hemisphere than the winter!

7. Why is it warmer here in the summer?

8. Other Solar Systems

9. Comet Eccentricities

10. Kepler’s Second Law
An imaginary line joining a planet to the Sun will sweep over equal areas in equal periods of time.
The consequence of this: planets travel faster when they are closest to the sun, and slower when they are farther away.

11. Closest point: perihelion (Jan 3) Farthest point: aphelion (July 4)
At the perihelion, kinetic energy is at a maximum, potential energy a minimum. At the aphelion, the opposite is true.

12. Kepler's Third Law
The square of any planets period of revolution (orbital period) T2 is proportional to the cube of the mean radius (R3)
In other words, farther the planet is from the sun, the larger the orbit and the longer its period.
The mean radius is expressed in astronomical units (AU)
1 AU equals 150,000,000 km, or 1 earth-sun radius
This is a function of Newton’s Law of gravitation.

13. Find the eccentricity of each of the following ellipses:

14. Lab 4 Ellipses