LearnMechanics & orbitsSome maths
Kepler's laws
Orbits are ellipses with the Sun at one focus; a planet sweeps equal areas in equal times; and the square of the period is proportional to the cube of the semi-major axis.
Why it matters
These three statements position every planet in this site's planetarium. They were extracted from Tycho Brahe's naked-eye observations decades before anyone knew why they held.
Try it
The maths, in layers
1 · Intuition
The second law is conservation of angular momentum wearing a disguise: close to the Sun a planet must move faster, or it would sweep out less area in the same time.
2 · The equation
T² = 4π²·a³ / (G·M)
- T
- orbital period, seconds
- a
- semi-major axis, metres
- M
- mass of the central body, kilograms
3 · Where it comes from
Set the gravitational force equal to the centripetal force for a circular orbit: GMm/r² = mv²/r. Substitute v = 2πr/T and rearrange. The result holds for ellipses too, with a in place of r — which is not obvious, and took Newton to prove.
What people usually get wrong
Earth is closest to the Sun in summer.
Perihelion falls in early January, during the northern winter. Seasons come from axial tilt, not distance — the 3.4% variation in distance is swamped by the 23.4° tilt.
The Sun sits at the centre of the ellipse.
It sits at one focus, which is offset from the centre by a·e.
Rests on
Leads to
- Orbital resonanceTwo orbiting bodies exerting regular, repeating gravitational tugs on one another because their periods form a ratio of small whole numbers.
- Stellar parallaxThe tiny annual shift in a nearby star's apparent position, caused by the Earth viewing it from opposite sides of its orbit.
- Synodic periodThe time between successive identical alignments of two bodies as seen from a third — for a planet, the gap between one opposition and the next.
- The discovery of NeptuneA planet found in 1846 by mathematics rather than by searching — predicted from discrepancies in Uranus's orbit, then located within one degree on the first night of looking.