III.

Orbital Elements

Six elements describe an orbit (see the accompanying diagram). The first two are size and elongation. The size of the orbit is given by the periapsis distance (SP) and the elongation of the orbit is given by the eccentricity ( e). For the ellipse shown, the eccentricity is the ratio CS/CP, where S is the focus and C the centre of the ellipse. For elliptical orbits, e is greater than 0, but less than 1; for circular orbits, e is exactly 0; and for parabolic orbits, e is exactly 1. A body in a hyperbolic orbit—that is, for which e is greater than 1—makes a single passage by a central body and escapes along a so-called open orbit, never to return.

The next three orbital elements are concerned with the orbit's orientation. For this discussion, however, several parameters need to be defined: the reference plane for objects orbiting the Sun is the plane of the Earth's orbit, also known as the plane of the ecliptic; the vernal equinox (g) is the intersection of the ecliptic and the plane of the celestial equator that the Sun reaches when travelling north, at the beginning of the northern spring; and the ascending node (N) is the northbound intersection of the orbit in question and the reference plane.

The three orbital elements that describe an orbit's orientation are the inclination (i), the longitude of the ascending node (Ω), and the argument of the periapsis (ω). The inclination is the angle between the reference plane and the orbit's plane. The longitude of the ascending node is the angle in the reference plane between the equinox and the ascending node. The argument of the periapsis is the angular displacement in the plane of the orbit between the ascending node and the line that passes through the centre of the orbit (C) and the periapsis (P). Finally, the sixth orbital element is the time at which the celestial body in question is at the periapsis.

An orbit can also be described in terms of its semimajor axis (AC, CP, or a). This axis is half the long axis (AP) of the ellipse, that is, half the distance between the periapsis (P) and apoapsis (A). The semimajor axis is longer than the periapsis distance (SP) and shorter than the apoapsis distance (AS), by an amount (CS) that is equal to the product of the semimajor axis and the eccentricity:

CS = e(AC) = e(CP) = ea