Expert: Courtney Seligman - 6/20/2009

Question
Hi, this is a question from my astronomy homework and I can't seem to figure out how to go about it...please help!

You discover a planet orbiting a distant star that has about the same mass as the Sun. Your observations show that the planet orbits the star every  72 days . What is its orbital distance in meters?

Thank you

Answer
This is about Kepler's Third Law of Planetary Motion, which says that the mass of the bodies (essentially the mass of the star, since the planet would be much less massive), times the square of the period, is proportional to the cube of the semi-major axis (which is the "orbital distance" you are supposed to determine).

Since the star is presumed to be the same mass as the Sun, the only difference between the Earth going around the Sun and the planet going around the star is the period (365 1/4 days for us, versus 72 days for your imaginary planet). Squaring the ratio of the periods gives you the value for the cube of the semi-major axis, and you take the cube root of that number to get the orbital size.

Since this is a homework question, I can't give you the actual answer, but as an example, suppose the orbital period was 1/4 of ours (that is, about 91 1/4 days). Then the period squared would be 1/16, and the size of the orbit would be the cube root of that (the inverse of the cube, on a calculator), or 0.397 of the size of our orbit. Multiplying that by the size of our orbit in meters (just under 150 billion) would yield a value of a little less than 60 billion meters for the orbital distance.

For your problem, since the period is smaller, the orbit would also be smaller, but the idea is the same.