Keplerian Motion Lab 3

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

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

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

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

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

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

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]

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

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

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

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

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

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

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