Keplerian Motion

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 = 1.092 years 1/P = 1/1 – 1/1.092 = 0.08425 Or P = 1/0.08425 = 11.87 years • So Jupiter takes 11.87 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