Answer 5. Using Kepler's Laws


Mercury's orbit nicely illustrates all three of Kepler's laws.

Law I. Planets move in elliptical orbits with the Sun at one focus of the ellipse.

Mercury's orbit is a good example of Law I. As you can see, the orbit is not a circle, and the Sun, indicated by the black dot, is offset to one side. The other focus of the ellipse, indicated by the `+' sign, is empty.

Law II. A line between the Sun and a planet sweeps out equal areas in equal times.

The shaded triangles shown at right represent the areas swept out in a time equal to 4% of Mercury's orbital period (about 3.5 days). These two triangles must have equal areas; as a result, Mercury moves about 50% faster at perihelion (closest to the Sun) than at aphelion (furthest from the Sun).

Law III. The relationship between a planet's orbital period P and the semimajor axis of its orbit a is


P2 = a3

where P is measured in years and a is measured in AU.

Given that Mercury's orbit has a semimajor axis a = 0.3871 AU, its orbital period P can be calculated:


P = (0.38713) = 0.05801 = 0.2408 year.


Joshua E. Barnes (barnes@ifa.hawaii.edu)
Last modified: September 30, 1999
http://www.ifa.hawaii.edu/~barnes/ast110_99/homework/ans5.html