November 5, 2001
Problem of the Month

The Synodic Period Problem
from the Spring 2001 issue of the Virginia Mathematics Teacher

The orbital periods of Earth and Venus are 365.3 and 224.8 Earth days, respectively.   What is the synodic period for Earth and Venus?   In other words, how long does it take Venus to lap Earth?   If Venus and Earth start at points which are collinear with the Sun, how many days will it take Venus to get to a point which is collinear with Earth and the Sun?

Assume the orbits of the two planets are coplanar circles about the Sun at the center.

Extra Credit: If the two planets started in standard position (on the positive x-axis), determine the measure of the angle formed by the positive x-axis and the line passing through the two planets when Venus overtakes Earth.

Solution to the Problem:

The answer is 584 Earth days.

Let t represent the number of Earth days for Venus to lap Earth.   Then set up an equation so that the number of orbits of Venus minus the number of orbits of Earth is equal to one.

Venus would have moved through 935 degrees of its orbit and Earth would have moved 575 degrees.   In other words, Venus would have gone around the Sun 2 and a half times and Earth would have gone around a little more than one and a half times.   To determine this, use the following proportions:

So, the answer to the extra credit is 215 degrees   (935 - 720 or 575 - 360).

Correctly solved by:

1. David and Judy Dixon	Bennettsville, South Carolina
2. Richard K. Johnson *	La Jolla, California
3. David Powell *	Winchester, Virginia
4. Keith Mealy *	Cincinnati, Ohio
5. Walt Arrison *	Philadelphia, Pennsylvania
6. Janine Oliver *	Winchester, Virginia
7. Nick von Keller	Winchester, Virginia
8. Tony Wu *	Fort Collins, Colorado
9. Joe Heintz *	Manchester, Tennessee
10. Charity Calhoun	Columbus, Georgia
11. Diana Xing	Fort Collins, Colorado
12. Chip Crawford	College of William & Mary, Williamsburg, Virginia

* Solved the extra credit as well

Send any comments or questions to: David Pleacher