Physics 101- Introduction to Mechanics
February 20- Applications of Newton's Laws

Kepler's Laws & The Motion of the Planets

Ellipses

Eccentricity describes the shape of the orbit

For a closed orbit 0< e< 1 (e=0 is circular).

Keplerís first law
Planets travel in elliptical orbits with the sun at one of the foci.  The elliptical path is due to the fact that gravity is a central force with an inverse squared power.  (In reality this "law" is more of an assumption- you are fixing the sun and not allowing it to move.  It does move, just not very much.)

 Planet Eccentricity (e) Semimajor axis a (1010m) Period T (y) T2/a3 (10-34 y2/m3) Mercury1 0.206 5.79 0.241 2.99 Venus 0.007 10.8 0.615 3.00 Earth 0.017 15.0 1.00 2.96 Mars 0.093 22.8 1.88 2.98 Jupiter2 0.049 77.8 11.9 3.01 Saturn2 0.056 143 29.5 2.98 Uranus 0.047 287 84.0 2.98 Neptune 0.009 4.50 165 2.99 Pluto 0.254 590 248 2.99

1 Too close to the sun for observations.
2 Moving too slowly for observations during a 16 year period.
Kepler did not know of the outer planets.

BTW: No telescopes were used in collecting Brahe's data!  He collected the data so carefully that Kepler was able to discover that planets travel in ellipses, not circles.  (Keep this in mind the next time you're in lab collecting data.)  :-)

For our purposes, we can approximate the orbits as being circular.  (This makes life much easier and doesn't sacrifice too much accuracy.)

Kepler's Third Law
This is nothing more than Newton's Second Law with gravitational force and centripetal acceleration.

Now substitute in v= 2pR/ T

So,

We'll examine Kepler's Second Law later, when we discuss angular momentum.

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 Jeff Phillips phillips@lmu.edu Loyola Marymount University Spring 2002