Mercury orbits the Sun in 88 days. Earth takes 365 days. Jupiter takes 12 years. Neptune takes 165 years. Are these numbers random?
No. Kepler discovered that they follow a precise mathematical relationship.
Kepler’s Third Law
The square of a planet’s orbital period (T) is proportional to the cube of the semi-major axis of its orbit (a):
More precisely: if T is in Earth years and a in astronomical units (AU, the Earth–Sun distance), then:
Planet Data
| Planet | a (AU) | T (years) | T² | a³ |
|---|---|---|---|---|
| Mercury | 0.387 | 0.241 | 0.058 | 0.058 |
| Venus | 0.723 | 0.615 | 0.378 | 0.378 |
| Earth | 1.000 | 1.000 | 1.000 | 1.000 |
| Mars | 1.524 | 1.881 | 3.538 | 3.540 |
| Jupiter | 5.203 | 11.86 | 140.7 | 140.8 |
| Saturn | 9.537 | 29.46 | 867.9 | 866.7 |
T² and a³ are essentially identical — a hidden order in the solar system that Kepler discovered in 1619, 66 years before Newton could prove it with his law of gravitation.
Why Does It Matter?
With this law you can calculate a planet’s distance from the Sun without ever visiting it. Just observe its orbital period — then solve for a. This is exactly how Kepler determined planetary distances.
What Did Newton Add?
Newton showed that this law follows directly from his law of gravitation: T² = (4π²/GM) × a³. So the “constant” in T² = a³ depends on the mass of the central star. This means we can calculate a star’s mass just from the orbits of its planets!
References
- Halliday, Resnick & Krane — Physics, Ch. 13
- Feynman Lectures on Physics — Vol. 1, Ch. 7 & 13
- NASA Planetary Fact Sheet
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