Orbital elements

The six classical orbital elements are: semi-major axis (a), eccentricity (e), inclination (i), longitude of the ascending node (Ω), argument of periapsis (ω), and mean anomaly at epoch (M₀). Together they define an ellipse in three-dimensional space and the position of the body on that ellipse at a specific time. The semi-major axis and eccentricity determine the size and shape of the orbit, while the three angular elements orient it relative to a reference plane, typically the ecliptic for solar-system bodies or the equatorial plane for Earth satellites. The mean anomaly gives the position along the orbit, often converted to true anomaly via Kepler's equation.¹

For certain applications, alternative element sets are used. Equinoctial elements avoid singularities for circular and equatorial orbits, making them useful for low-thrust trajectory optimization. Delaunay variables are action-angle coordinates used in Hamiltonian mechanics. For hyperbolic and parabolic orbits, the semi-major axis is replaced by the periapsis distance and eccentricity, and the anomaly is replaced by hyperbolic or parabolic anomalies. These sets are essential for cometary orbits and interplanetary trajectories.²

Real orbits are not perfect ellipses due to perturbations from other bodies, solar radiation pressure, and non-spherical mass distributions. To account for these, osculating elements are defined as the Keplerian elements that the body would have if perturbations were suddenly removed at a given instant. They change continuously over time. Perturbation theory, often using Lagrange's planetary equations, describes these variations. For example, the Moon's orbit is significantly perturbed by the Sun, leading to secular variations in its elements.³

Orbital elements have a rich history: Kepler's original formulation used the line of apsides and the node, but the modern set was codified by Euler and Lagrange. The concept of mean anomaly was introduced by Kepler to solve the time-position problem. In asteroid studies, proper elements are used to identify families by filtering out secular perturbations. For artificial satellites, two-line element sets (TLEs) are a compact format used for tracking, though they are less accurate than full state vectors. The choice of reference frame is critical: the J2000 frame is standard, but for interplanetary missions, the ecliptic and equinox of date are used.⁴

Semi-major axis

Half the longest diameter of an elliptical orbit; a measure of orbit size.

Eccentricity

A number between 0 and 1 describing how much an orbit deviates from a circle.

Inclination

The angle between the orbital plane and a reference plane.

Longitude of the ascending node

The angle from a reference direction to the point where the orbit crosses the reference plane from south to north.

Argument of periapsis

The angle from the ascending node to the periapsis point, measured in the orbital plane.

Mean anomaly

A parameter that increases uniformly with time, representing the fraction of an orbit completed.

Osculating elements

The Keplerian elements that describe the instantaneous orbit of a body at a given time, assuming no perturbations.