Two of the most basic parameters for characterizing a closed ORBIT are the PERIOD of the orbit and the SIZE. The PERIOD is the time it takes the orbiting body to make one complete trip around the orbit. For an elliptical orbit, the SIZE is usually measured by giving the semimajor axis of the ellipse. For the special case of a circular orbit, this is just the radius of the circle.

Kepler's Third Law of Planetary Motion asserts that, for all bodies orbiting around the same massive center, the ratio of the SQUARE of the PERIOD to the CUBE of the SEMIMAJOR AXIS is a constant. That is,

(SQUARE OF ORBITAL PERIOD) / (CUBE OF SEMIMAJOR AXIS) = CONSTANT

The constant takes one value when used to compare the orbits of all planets orbiting the sun, another value when comparing the orbits of the 16 or so moons of Jupiter, and so on. (Actually, the value of the constant depends on the MASS of the thing in the center.)

The value of the constant depends, of course, on the UNITS used to measure the period and the semimajor axis. For comparing the orbits of planets orbiting the sun, it is particularly convenient to measure the PERIOD of the planet in YEARS, and the SEMIMAJOR AXIS in ASTRONOMICAL UNITS, where 1 Astronomical Unit is defined as the semimajor axis of the Earth's orbit around the sun. With this choice of units, the value of the constant is ONE.

1. Explain in a sentence or two why the last statement is true.

2. The orbital period of the planet MARS is 687 days. What is this in years? Use Kepler's Third Law to calculate the semimajor axis of the orbit of Mars in Astronomical Units.

3. Suppose that you wish to travel from Earth to Mars. How long would it take? You should travel along an elliptical orbit where the end closest to the sun (the perihelion) is tangent to the orbit of Earth, and the end furthest from the sun (the aphelion) is tangent to the orbit of Mars. Find the semimajor axis of this orbit, then use Kepler's Third Law to find the period (and remember that a one-way trip requires just one-half of the period!). Convert your answer to days.

How does your result compare with newspaper accounts of the travel time to Mars of space probes? Go to the Jet Propulsion Laboratory web site (there is a link from our class site) and check out the Mars Exploration Rovers, launched last year.

(Of course, you are making some approximations here. Since both Earth and (especially) Mars have elliptical orbits, the "trip" orbit will be different for departures at different times of the year. Also, you have to leave Earth at just the right time so that when you get to the orbit of Mars, Mars meets you and is not somewhere else in its orbit. And if you have a huge rocket and can blast through space as you wish, you can travel faster- this method is the one that takes the least fuel, called a "Hohmann Transfer".)