Kepler’s Third Law Explained (P² = a³)
Kepler’s third law links an orbit’s size to its period: P² = a³ in AU and years. See why planets speed up near the Sun, with diagrams and worked examples.
The idea behind it
In astronomical units, solar masses, and Earth years, Kepler’s third law connects the size of an orbit to its period. Increasing eccentricity stretches the orbit without changing its period when the semi-major axis stays fixed.
The relationship to remember
P = √(1³ / 1) = 1 years. Closest approach: 0.7 AU; farthest: 1.3 AU.
Make your own discovery
Set eccentricity to 0.8, then press Play. Compare the speed near the star with the speed at the far end.
Open the Orbit Simulator Online — Kepler’s Laws ↗Key formulas
| Quantity | Formula | Note or example |
|---|---|---|
| Kepler’s third law | P² = a³ / M★ | P in years, a in AU, M★ in solar masses |
| Closest and farthest distance | r_min = a(1 − e) · r_max = a(1 + e) | Perihelion and aphelion |
| Orbital speed (vis-viva) | v = √(GM (2/r − 1/a)) | Fastest at closest approach |
Worked examples
Each row uses one of this lab’s presets and the same model as the interactive tool.
| Example | Orbital period | Current speed | Star distance |
|---|---|---|---|
| Earth-like | 1 years | 30.3 km/s | 0.98 AU |
| Mars-like | 1.88 years | 26.51 km/s | 1.38 AU |
| Stretched orbit | 5.2 years | 51.59 km/s | 0.6 AU |
What this model leaves out
An isolated star and a negligible-mass planet in a fixed, two-dimensional Keplerian orbit. The diagram rescales to fit; star and planet sizes are exaggerated.
Check your understanding
Why is the star off-center?
The star sits at one focus of an elliptical orbit, not at the ellipse’s center. A circular orbit has coincident foci.
Does an eccentric orbit take longer?
Not if the semi-major axis and stellar mass stay the same. Eccentricity changes the shape and the variation in speed.
What is Kepler’s third law?
The square of a planet’s orbital period is proportional to the cube of its average distance from its star. With periods in years and distances in AU around a Sun-like star, P² = a³: Mars, at 1.52 AU, takes about 1.88 years per orbit.
Continue with the formula reference or test yourself in space practice.