Physics 101- Introduction to Mechanics
February 22- 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). 
 

Kepler's 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