Orbital Elements - Keplerian Elements

The traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of planetary motion.
When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass. When viewed from the non-inertial frame of one body only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body is called the primary, the other body is called the secondary. The primary is not necessarily more massive than the secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary.

The main two elements that define the shape and size of the ellipse:

• Eccentricity - shape of the ellipse, describing how much it elongated compared to a circle. (not marked in diagram)
• Semimajor axis - the sum of the periapsis and apoapsis distances divided by two. For circular orbits the semimajor axis is the distance between the center of the bodies, not the distance of the bodies to the center of mass.

Two elements define the orientation of the orbital plane in which the ellipse is embedded:

• Inclination - vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node (where the orbit passes upward through the reference plane) (green angle i in diagram).
• Longitude of the ascending node - horizontally orients the ascending node of the ellipse (where the orbit passes upward through the reference plane) with respect to the reference frame's vernal point (green angle Ω in diagram).

And finally:

• Argument of periapsis defines the orientation of the ellipse (by the direction of the minor axis) in the orbital plane, as an angle measured from the ascending node to the semiminor axis. (violet angle in diagram)
• Mean anomaly at epoch defines the position of the orbiting body along the ellipse at a specific time (the "epoch").

The mean anomaly is a mathematically convenient "angle" which varies linearly with time, but which does not correspond to a real geometric angle. It can be converted into the true anomaly, which does represent the real geometric angle in the plane of the ellipse, between periapsis (closest approach to the central body) and the position of the orbiting object at any given time. Thus, the true anomaly is shown as the red angle in the diagram, and the mean anomaly is not shown.

The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system.

Note that non-elliptic trajectories also exist but are not closed and are thus nor orbits. If the eccentricity is greater than one, the trajectory is a hyperbola. If the eccentricity is equal to one and the angular momentum is zero, the trajectory is radial. If the eccentricity is one and there is angular momentum, the trajectory is a parabola.