Suppose a planet is in an elliptical orbit around the sun with semimajor axis *a* and semiminor axis *b*. Then the average distance of the planet to the sun over time equals

*a*(1 + *e*²/2)

where the eccentricity *e* satisfies

*e*² = 1 − *b*²/*a*².

You can find a proof of this statement in [1].

This post will look at the set of all orbits with a fixed average distance *r* to the sun. Without loss of generality we can choose our units so that *r* = 1.

Clearly one possibility is to set *a* = *b* = 1 so the orbit is a circle. The distance is constantly 1, so the average is 1.

We can also maintain a distance of 1 by reducing *a* but increasing the eccentricity *e*. The possible orbits of average distance 1 satisfy

*a*(1 + *e*²/2) = 1

with 0 < *b* ≤ *a* ≤ 1. A little algebra shows that

*b* = √(3*a*² – 2*a*),

and that 2/3 < *a* ≤ 1. As *a* approaches 2/3, *b* approaches 0.

Let’s put the center of our coordinate system at the sun and assume the other focus of the elliptical orbits is somewhere along the positive *x*-axis. When *e* is 0 we have a unit circle orbit. As *e* approaches 1, the orbits approach a horizontal line with the sun on one end.

## Related posts

[1] Sherman K. Stein. “Mean Distance” in Kepler’s Third Law. Mathematics Magazine, Vol. 50, No. 3 (May, 1977), pp. 160–162

The post Set of orbits with the same average distance to sun first appeared on John D. Cook.