Kepler's Laws

Johann Kepler determined three laws characterizing orbital motion, using Tycho Brahe's planetary observation data.  These laws can be proven mathematically using Newton's law of gravitation. Kepler's laws are paraphrased below along with the corresponding physical implications. These laws apply directly to satellite orbital motion, thus the laws are from the point of view of an Earth-orbiting satellite.

Kepler's First Law:

Satellite orbits are elliptical Paths with the Earth at one focus of the ellipse.

Kepler's Second Law:

A line between the center of the Earth and the satellite sweeps out equal areas in equal intervals of time.

Kepler's Third Law:

The square of the orbital period is proportional to the cube of the the orbit's semi major axis.

Kepler's first law simply states that orbits are shaped like ellipses (elongated circles). This can be proven mathematically, once it's understood that the gravitational force between the Earth and the satellite decreases in proportion to the square of distance between the [centers of] the two.  This law does not preclude a satellite from orbiting in a circular path since a circle is an ellipse with no elongation (or eccentricity). 

Two things are needed for a satellite to orbit in a circular path:

  • It's velocity must be directed perfectly horizontally and
  • It's speed (or velocity magnitude) must be perfectly balanced against the Earth's gravitational acceleration (at the orbital altitude). To get this balance, the speed must have the value that would cause the gravity to be what Physicists call a centripetal acceleration.

Any differences in the velocity (speed or direction) from these two conditions will cause either an elliptical path or an escape trajectory (with a parabolic or hyperbolic shape). A significant lack of speed or a poorly-chosen direction can cause part of the elliptical path it intersect the Earth, in which case the satellite will re-enter the atmosphere and either burn up or impact the surface.

Kepler's Second Law is a consequence of the physical law of conservation of angular momentum.  This is the same principle that figure skaters use by pulling in their arms to speed up their spin rate and extend them out to slow their spin rate down.  This effect is amplified by gravity's inverse square law dictating lower speeds at higher altitudes and vice versa.

This gives elliptical orbits a very distinct characteristic: the satellite moves fastest at its lowest altitude (perigee) and it moves slowest at its highest altitude (apogee). This speed change is dramatic in highly elliptical orbits in which the satellite spends the majority of its time moving rather slowly near apogee.

Kepler's Third Law states that you can compute the time it takes the satellite to make one complete orbit (the period) from half the longest dimension of the orbital ellipse (the semi major axis). This is also known as the harmonic law. 

The mathematical relationship between period and semi major axis in the harmonic law results from two things:

  • larger ellipses have a longer orbital path, so (if everything else was equal) you would expect it to take more time to go around the longer path and
  • gravity's inverse square law dictates lower speeds at higher altitudes and higher speeds at lower altitudes.

Kepler's laws describe orbital motion

Orbital motion described  by Kepler's laws has it roots in basic Physics

Kepler's first law -- orbits are ellipses -- this is a balance of velocity with gravity

Kepler's second law -- equal areas swept out in equal time describes  conservation of angular momentum

Kepler's third law -- the size of the orbit's elipse tells you how much time it takes to complete one orbit