Kepler's third law

Copernicus had noted that it takes the outside planets longer to go around the sun.

How much longer?

              P      a     
             (yr)   (AU)

Mercury     0.241  0.387 
Venus       0.615  0.723 
Earth       1.000  1.000 
Mars        1.881  1.524
Jupiter    11.86   5.203
Saturn     29.46   9.540

How is the period, P, of an orbit related to the length of the semimajor axis, a, of the orbit?

P²/a³ = k

where k is the same for all the planets in the solar system.

Try it:

           P      a     P²      a³      p²/a³
          (yr)   (AU)

Mercury  0.241  0.387  0.0581  0.0580  1.002
Venus    0.615  0.723  0.378   0.378   1.001
Earth    1.000  1.000  1.000   1.000   1.000
Mars     1.881  1.524  3.54    3.54    1.000
Jupiter 11.86   5.203  141     141     1.000
Saturn  29.46   9.540  868     868     1.000

Note that if we measure a in AU and P in years, then the constant k in Kepler's Third Law is 1. This means that

This allows you to find P if you know a and vice versa.

If there were a planet with an (approximately circular) orbit with a radius of 3 AU, how many years would it take to go around the sun?

A) 2 years.
B) 3 years.
C) 5 years.
D) 9 years.

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Davison E. Soper, Institute of Theoretical Science, University of Oregon, Eugene OR 97403 USA soper@bovine.uoregon.edu