1. Keplerian Motion
Lab 3

2. Kepler’s Laws
Kepler’s laws are kinematic, or descriptive, as they describe planetary motion
Dynamic laws are prescriptive, describe a cause and effect
Kepler’s laws apply to any 2 celestial objects locked in mutual orbit with each other

3. Definitions of an Ellipse
Ellipse – a squashed circle
Major axis of an ellipse – line which divides it into 2 parts
Minor axis of an ellipse – short axis or line ┴ to major axis which also ÷ ellipse into 2 equal but different parts
Center of ellipse – where major and minor axes cross

4. Points on an Ellipse
Foci – 2 points along the major axis, one of which is empty, the other occupied by the Sun
Perihelion – when planet is closest to Sun while orbiting elliptically around it
Aphelion – point farthest from Sun while planet is orbiting around it
Radius Vector – line joining planet and Sun

5. animation http://www.geocities.com/literka/planet.htm
Shows Kepler’s Laws, radius vector, elliptical orbits

6. Angles in an Ellipse
True Anomaly – angle between radius vector to perihelion point AND radius vector to the planet
When the true anomaly is equal to 0˚, then the Earth is closest to the Sun (perihelion)
When the true anomaly is equal to 180˚, then the Earth is furthest from the Sun (aphelion)
Mean Anomaly - mean anomaly is what the true anomaly would be if the object orbited in a perfect circle at constant speed

7. True anomaly is the angle between the direction z-s and the current position p of an object on its orbit, measured at the focus s of the ellipse, or the angle ZSP

8. Semi-major axis Semi-major axis – analogous to radius of a circle
= average distance of planet from the Sun divided by 2
a = [(perihelion+aphelion)/2]

9. Eccentricity
Eccentricity of an ellipse (e) – how squashed the circle is
If both foci are in center of ellipse, it is a circle with e = 0
If both foci are max distance away from each other, the ellipse would no longer be an ellipse but a straight line

10. Periods
Synodic Period – time it takes for 2 identical successive celestial configurations as seen from Earth
Sidereal Period – true orbital period of a planet, or the time it takes to complete one orbit around the Sun

11. Relationship between synodic and sidereal periods
For inferior planets –
1/P = 1/E + 1/S
P = sidereal period of inferior planet
E = Earth;s sidereal period (1 yr)
S = inferior planet’s synodic period
For superior planets –
1/P = 1/E – 1/S

12. Example of synodic-sidereal relationship
Jupiter’s synodic period = years
1/P = 1/1 – 1/1.092 =
Or P = 1/ = years
So Jupiter takes years to orbit the Sun

13. Kepler’s First Law The orbit of a planet around the Sun is an ellipse,
The Sun is at one focus of the ellipse

14. Kepler’s Second Law Speed of planet varies along its orbit
Planet moves faster at perihelion, slower at aphelion
But in same amount of time, same amount of area is covered
This is the law of equal areas
For a circular orbit, the planet would have to move at constant speed at all times

15. Kepler’s Third Law
Relationship between semi-major axis a (size of orbit) and sidereal period P
P2 = a3
P is in years, a is in AU units