Definition
The major axis of an ellipse is its longest diameter. For an orbit it joins the two extreme points: perihelion, where the body is closest to the Sun, and aphelion, where it is farthest. The semi-major axis, written a, is half of that length. It is the first of the six orbital elements and the one that gives the size of the orbit.
The Sun does not sit at the centre of the ellipse but at one focus, offset from the centre by a × e, where e is the eccentricity. The semi-major axis is therefore not a distance from the Sun. It is the average of the perihelion and aphelion distances, which is why it is often loosely called the mean distance.
From a and e to the shape of the orbit
With the eccentricity e, the semi-major axis gives every other dimension of the ellipse:
q = a × (1 − e) perihelion distanceQ = a × (1 + e) aphelion distanceb = a × √(1 − e²) semi-minor axisa = (q + Q) / 2Example: Mars has a = 1.5237 AU and e = 0.0934. Its perihelion distance is 1.381 AU (206.6 million km) and its aphelion distance 1.666 AU (249.2 million km). The semi-minor axis is 1.517 AU, only 0.4 % shorter than a: even a noticeably eccentric planetary orbit looks almost circular, and what shows to the eye is the offset of the Sun from the centre.
Averaged over time, the distance to the Sun is slightly larger than a, at a × (1 + e²/2), because a planet spends more time near aphelion where it moves slowly. For Mars the difference is about 0.4 %.
Kepler's third law: a sets the period
The orbital period depends on the semi-major axis alone, not on the eccentricity. Two orbits with the same a take the same time to go round, whether one is a circle and the other a long thin ellipse. For a body orbiting the Sun, with the period P in years and a in astronomical units, Kepler's third law is simply P² = a³. Newton's general form works around any central mass M:
P[years]² = a[AU]³P = 2π × √(a³ / (G × (M + m)))Doubling the semi-major axis makes the period 2√2 ≈ 2.83 times longer. Halley's Comet, with a ≈ 17.8 AU, returns every 17.8^1.5 ≈ 75 years, even though its eccentricity of 0.967 takes it from inside the orbit of Venus to beyond Neptune.
The vis-viva equation: a sets the energy
The semi-major axis also fixes the orbital energy. The vis-viva equation gives the speed v of a body at distance r from the Sun:
v² = GM × (2/r − 1/a)GM(Sun) = 1.32712 × 10²⁰ m³/s²At the same point of space, a faster body is on an orbit with a larger semi-major axis. When r = a the formula reduces to the circular speed √(GM/a), about 29.8 km/s at 1 AU. When v² reaches 2GM/r, the term 1/a becomes zero: a is infinite, the orbit is a parabola and the body escapes. Hyperbolic orbits are given a negative semi-major axis by convention.
Worked example: the cheapest coast from Earth to Mars is half an ellipse touching Earth's orbit at perihelion (1.000 AU) and Mars's orbit at aphelion (1.524 AU). Its semi-major axis is (1.000 + 1.524) / 2 = 1.262 AU, so the full ellipse would take 1.262^1.5 = 1.418 years. The ship flies half of it: about 259 days, which is the transfer time drawn on the Earth to Mars page.
Semi-major axes of the planets
| Planet | a (AU) | Eccentricity | Perihelion q (AU) | Aphelion Q (AU) |
|---|---|---|---|---|
| Mercury | 0.3871 | 0.2056 | 0.3075 | 0.4667 |
| Venus | 0.7233 | 0.0068 | 0.7184 | 0.7282 |
| Earth | 1.0000 | 0.0167 | 0.9833 | 1.0167 |
| Mars | 1.5237 | 0.0934 | 1.3814 | 1.6660 |
| Jupiter | 5.2026 | 0.0485 | 4.9503 | 5.4549 |
| Saturn | 9.5549 | 0.0555 | 9.0245 | 10.0853 |
| Uranus | 19.2184 | 0.0463 | 18.3286 | 20.1082 |
| Neptune | 30.1104 | 0.0090 | 29.8397 | 30.3811 |
Why a barely changes
The planets constantly perturb one another, and most orbital elements drift: the perihelia advance and the nodes regress. The semi-major axes are a notable exception. Laplace and Lagrange showed in the eighteenth century that, to first order, planetary perturbations produce no secular change in a, only small periodic oscillations. This is one reason the solar system has stayed orderly for billions of years, and why tables of mean orbital elements such as those in Meeus' Astronomical Algorithms give each planet an almost constant semi-major axis.
Since 2012 the astronomical unit is a fixed number of metres rather than a quantity derived from the Earth's orbit, so the Earth's semi-major axis is no longer exactly 1 AU by definition. It is about 1.000001 AU, a difference of about 150 km.
See it in 3D
The view above draws each planetary orbit with its major axis. Open the heliocentric page and switch on the major axis, perihelion and aphelion in the Orbits menu to see how far the Sun sits from the centre of each ellipse: clearly for Mercury and Mars, almost not at all for Venus and Neptune.
Explore it on AstroAlgos
Frequently asked questions
Is the semi-major axis the average distance from the Sun?
It is the average of the perihelion and aphelion distances, so it is often called the mean distance. Averaged over time the distance is slightly larger, a × (1 + e²/2), because a planet moves slowly and lingers near aphelion. For nearly circular orbits the difference is negligible.
How do you find the semi-major axis from the period?
Invert Kepler's third law. For a body orbiting the Sun, a in astronomical units is the period in years raised to the power 2/3. A period of 8 years gives a = 4 AU; Jupiter's 11.86 years gives 5.20 AU.
Does a more eccentric orbit take longer?
No. The period depends only on the semi-major axis. A circular orbit and a very elongated one with the same a take exactly the same time, although the elongated orbit has a much wider range of speeds.
