Copernicus kept one old idea from the Earth-centered model: orbits had to be perfect circles. His predictions were better than Ptolemy's, but Mars still drifted away from them. Johannes Kepler spent years fitting Tycho Brahe's careful measurements of Mars and in 1609 found the answer: the orbit is not a circle but an ellipse.
- An ellipse is a stretched circle. It has two special points inside it, the foci. For every point on the ellipse, the distances to the two foci add up to the same total, $2a$.
- The semi-major axis $a$ is half the long axis; it is the planet's average distance from the Sun.
- The eccentricity $e = c/a$, where $c$ is the distance from the center to a focus. $e = 0$ is a circle; $e$ close to $1$ is long and thin.
Why it works
Kepler's first law: each planet moves on an ellipse with the Sun at one focus; the other focus is empty. Since the Sun sits a distance $c = ae$ from the center, the closest point (perihelion) is $a - ae = a(1 - e)$ from the Sun and the farthest point (aphelion) is $a + ae = a(1 + e)$. Adding them gives $2a$; subtracting gives $2ae$. Newton later showed that an inverse-square force like gravity produces exactly these ellipses.
Key idea
Orbits are ellipses with the Sun at one focus. Perihelion $= a(1 - e)$, aphelion $= a(1 + e)$, and $a$ is their average. For the planets, $e$ is small: their ellipses are nearly circles.
| Body | $a$ (AU) | $e$ |
|---|---|---|
| Venus | 0.723 | 0.007 |
| Earth | 1.000 | 0.017 |
| Mars | 1.524 | 0.093 |
| Mercury | 0.387 | 0.206 |
| Pluto | 39.5 | 0.249 |
| Halley's Comet | 17.8 | 0.967 |
Worked example
Worked example: Mercury's range
Mercury has $a = 0.387$ AU and $e = 0.206$. How close and how far from the Sun does it get?
- Perihelion: $0.387 \times (1 - 0.206) = 0.387 \times 0.794 = 0.307$ AU.
- Aphelion: $0.387 \times (1 + 0.206) = 0.387 \times 1.206 = 0.467$ AU.
- Check: the average of $0.307$ and $0.467$ is $0.387$ AU, which is $a$.
Answer: From $0.307$ AU to $0.467$ AU.
Common mistake
Textbook drawings stretch planetary orbits so the ellipse is easy to see, and many people come away thinking orbits are long, thin ovals. Real planetary orbits are nearly circles: Earth's $e = 0.017$ means the Sun is only $1.7\%$ of the way from the center to the edge. Long, thin orbits belong to comets. Check: Earth's distance from the Sun changes by only about $3\%$ over the year.