Understanding Semi-Major Axis
From mean motion to orbital size
Because a Two-Line Element set reports an orbit's mean motion (n, in revolutions per day) rather than its size, the semi-major axis has to be derived. Converting n to radians per second and inverting Kepler's relation n = √(μ/a³) gives a = (μ/n²)^⅓, with μ = 398,600 km³/s² the standard gravitational parameter of Earth. This is exactly the step orbit propagators perform before they can project a satellite forward in time. Because the period depends only on a, two orbits of very different shape but equal semi-major axis complete a revolution in identical time:
| Orbit (example) | Altitude | Semi-major axis (a) | Period |
|---|---|---|---|
| LEO — ISS | ~420 km | ~6,790 km | ~93 min |
| MEO — GPS | ~20,200 km | 26,560 km | ~11 h 58 min |
| GEO | 35,786 km | 42,164 km | ~23 h 56 min |
| GTO (transfer) | 250 × 35,786 km | ~24,400 km | ~10.5 h |
One number for the whole orbit's energy
Beyond the period, the semi-major axis fixes an orbit's total energy: the specific orbital energy is ε = −μ/(2a), so a larger a means a higher-energy, more loosely bound orbit. The speed at any distance r then follows from the vis-viva equation, v² = μ(2/r − 1/a). Raising a satellite means increasing a — precisely what a Hohmann transfer does, using two burns and an intermediate ellipse to lift the semi-major axis from one circular orbit to a higher one. The reverse happens passively in low orbits, where atmospheric drag bleeds energy away, steadily shrinking a and driving orbital decay until the satellite re-enters.
Altitude, apogee and perigee: what a really measures
Altitude and semi-major axis are easy to confuse. Altitude is height above Earth's surface, whereas a is measured from Earth's centre, so for a circular orbit the two differ by one Earth radius. An elliptical orbit has no single altitude at all: the craft sweeps from a low perigee to a high apogee, and the semi-major axis is the average of those two radii. A geostationary satellite, for example, has a = 42,164 km yet hovers 35,786 km above the equator — the gap is Earth's equatorial radius (6,378 km). Reading an orbit's size this way is what separates the standard orbit classes, from crowded low Earth orbit up to the geostationary belt.