Keplerian Motion

1
Keplerian Motion

• Lab 3

2
Keplers Laws

• Keplers laws are kinematic, or descriptive, as
  they describe planetary motion
• Dynamic laws are prescriptive, describe a cause
  and effect
• Keplers 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 Keplers 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 (perihelionaphelion)/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 Earths sidereal period (1 yr)
• S inferior planets synodic period
• For superior planets
• 1/P 1/E 1/S

12
Example of synodic-sidereal relationship

• Jupiters 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
Keplers First Law

• The orbit of a planet around the Sun is an
  ellipse,
• The Sun is at one focus of the ellipse

14
Keplers 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
Keplers 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