Activity 2.1.1 ended with an uncomfortable fact: Copernicus's model wasn't measurably more accurate than Ptolemy's, only simpler. What finally broke the tie wasn't a better argument — it was better data. Tycho Brahe, working without a telescope (it hadn't been invented yet), spent decades recording planetary positions with a precision no one had matched before, accurate to a fraction of a degree using nothing but careful naked-eye instruments.
After Tycho's death, his assistant Johannes Kepler inherited that dataset — and spent years trying to force Mars's observed positions into a perfect circle. They wouldn't fit. The gap was small, but it was real, and Kepler trusted the data over the assumption. That decision is what led him to ellipses.
Every planet orbits the Sun in an ellipse, not a circle, with the Sun sitting at one focus — one of two special points inside the ellipse, never at the center. How stretched the ellipse is gets measured by its eccentricity (e), running from 0 (a perfect circle) up toward 1 (a very elongated shape). Most planets in our solar system have fairly small eccentricities — nearly circular, but not quite.
Drag the eccentricity from a perfect circle (0) toward a stretched ellipse. The Sun stays fixed at one focus — never the center — no matter how the shape changes.
An imaginary line connecting the Sun to a planet sweeps out equal areas in equal times, no matter where in the orbit the planet is. Near perihelion, where the planet is close to the Sun, that line is short — so to sweep the same area in the same time, the planet has to move through a wide angle quickly. Near aphelion, the line is long, so a much smaller angle sweeps the same area — the planet crawls.
Drag through one full orbit and watch the planet move — fast near the Sun, slow far away. The two shaded wedges each represent the same slice of time (8% of the orbit); notice how different their angular width is, even though their areas match.
Published years after the first two, the Third Law connects every planet's orbital period (P, in Earth years) to its semi-major axis (a, in astronomical units — Earth's own average distance from the Sun):
Farther planets don't just have longer orbits to travel — they also move more slowly, so the relationship isn't linear. This single equation, checked against real data, is remarkably exact across the entire solar system.
Set a semi-major axis in AU and see the period Kepler's Third Law predicts — then check it against the real solar system below.
| Planet | a (AU) | Actual P (yr) | P²=a³ predicts |
|---|---|---|---|
| Mercury | 0.39 | 0.24 | 0.24 |
| Venus | 0.72 | 0.62 | 0.61 |
| Earth | 1 | 1 | 1.00 |
| Mars | 1.52 | 1.88 | 1.87 |
| Jupiter | 5.2 | 11.86 | 11.86 |
| Saturn | 9.58 | 29.4 | 29.65 |
| Uranus | 19.2 | 84 | 84.13 |
| Neptune | 30.05 | 164.8 | 164.73 |
An asteroid orbits the Sun with a semi-major axis of 4 AU. What's its orbital period?
A newly discovered dwarf planet takes 125 years to orbit the Sun. What's its semi-major axis?
Kepler's three laws are remarkably accurate, and they were a genuine triumph — but they're descriptions, not explanations. Kepler could tell you exactly how a planet moves; he had no idea why it should move that way at all. That answer — gravity, and the force pulling any two masses toward each other — wouldn't arrive for decades, from Isaac Newton.