Classical orbital elements
The classical orbital elements (COE), or Keplerian elements, are six numbers that fully describe an orbit and a body's position on it: semi-major axis, eccentricity, inclination, right ascension of the ascending node, argument of periapsis, and true anomaly. Two set the ellipse's size and shape, three orient it in space, and one locates the body.
The six elements
Think of building an orbit in three stages — shape the ellipse, orient it around the Earth, then place the body on it.
Semi-major axis (a) — size
Half the length of the ellipse's long axis. It sets the orbit's overall size and, with the central body's gravity, its period: a larger a means a slower, higher orbit. In the Orbit Visualizer a is entered in kilometres.
Eccentricity (e) — shape
How far the ellipse departs from a circle. e = 0 is a perfect circle; 0 < e < 1 is an ellipse that gets more elongated as e approaches 1; e = 1 is a parabola (an escape trajectory). Together, a and e fix the periapsis and apoapsis distances.
Inclination (i) — tilt
The tilt of the orbital plane relative to Earth's equator, in degrees. i = 0° is equatorial and prograde, 90° is polar, and i > 90° is retrograde (moving opposite Earth's spin).
Right ascension of the ascending node (Ω, RAAN) — swivel
Where the orbital plane crosses the equator going north (the ascending node), measured as an angle from a fixed reference direction (the vernal equinox), in degrees. It swivels the whole plane around Earth's axis.
Argument of periapsis (ω) — ellipse orientation
The angle within the orbital plane from the ascending node to periapsis (the closest point), in degrees. It rotates the ellipse inside its plane, setting where the low and high points fall.
True anomaly (ν) — where the body is now
The angle from periapsis to the body's current position, measured at the focus (Earth), in degrees. It is the one element that changes continuously as the body moves; the other five stay fixed for an ideal two-body orbit.
True, mean, and eccentric anomaly
Position along the orbit can be expressed three ways. True anomaly (ν) is the real geometric angle to the body. Mean anomaly (M) is a fictitious angle that advances uniformly in time (M = n(t − tp), with n the mean motion), which makes it convenient for time propagation but not a physical direction. The eccentric anomaly (E) bridges the two through Kepler's equation, M = E − e sin E, which is solved iteratively; ν then follows from E and e. For a circle (e = 0) all three are equal. The Orbit Visualizer's form takes true anomaly, and its deep links also accept a mean anomaly (M) that the app converts for you.
In the Orbit Visualizer
The Classical Orbital Elements input maps one-to-one onto the six elements, plus an epoch:
- Semi-major Axis (km) — a
- Eccentricity — e (0 to just under 1)
- Inclination (deg) — i
- Right Ascension of the Ascending Node (deg) — Ω
- Argument of Periapsis (deg) — ω
- True Anomaly (deg) — ν
- Epoch (UTC) — the instant those elements are valid
You can also deep-link an orbit: ?a=, ?e=, ?i=, ?raan=, ?argp=, and ?nu= (or ?M= for mean anomaly). A geostationary example — circular, equatorial, one sidereal day:
▶ Open a GEO orbit (a = 42,164 km, e = 0, i = 0°)
Related
Mini-FAQ
Are RAAN and argument of periapsis defined for every orbit?
Not cleanly. A perfectly equatorial orbit has no ascending node, so Ω is undefined, and a circular orbit has no periapsis, so ω is undefined. In those degenerate cases the missing angle is conventionally set to zero — which is why the GEO example above uses Ω = ω = 0.
Are these the same as the numbers in a TLE?
Related, but not identical. A TLE stores inclination, RAAN, eccentricity, argument of perigee, and mean anomaly (not true anomaly), plus mean motion instead of semi-major axis — and they are mean elements for the SGP4 model. See How to read a TLE.