Transcript Keplerian Motion - George Mason University

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