Orbital elements

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

A real orbit and its elements change over time due to gravitational perturbations by other objects and the effects of general relativity. A Kepler orbit is an idealized, mathematical approximation of the orbit at a particular time.

( $e$ ) — shape of the ellipse, describing how much it is elongated compared to a circle (not marked in diagram).

Eccentricity

( $a$ ) — half the distance between the apoapsis and periapsis. The portion of the semi-major axis extending from the primary at one focus to the periapsis is shown as a purple line in the diagram; the rest (from the primary/focus to the center of the orbit ellipse) is below the reference plane and not shown.

Semi-major axis

Orbit prediction[edit]

Under ideal conditions of a perfectly spherical central body, zero perturbations and negligible relativistic effects, all orbital elements except the mean anomaly are constants. The mean anomaly changes linearly with time, scaled by the mean motion,^[2] $n={\sqrt {\frac {\mu }{a^{3}}}}.$ where $μ$ is the standard gravitational parameter. Hence if at any instant $t 0$ the orbital parameters are $(e 0, a 0, i 0, Ω 0, ω 0, M 0)$ , then the elements at time $t = t 0 + δt$ is given by $(e 0, a 0, i 0, Ω 0, ω 0, M 0 + n δt)$ .

the : $\ell =M+\omega +\Omega$ ,

mean longitude

the : $g=\omega +\Omega$ , and

longitude of periapsis

the : $h=\Omega$

longitude of the ascending node

The Delaunay orbital elements were introduced by Charles-Eugène Delaunay during his study of the motion of the Moon.^[7] Commonly called Delaunay variables, they are a set of canonical variables, which are action-angle coordinates. The angles are simple sums of some of the Keplerian angles:

along with their respective conjugate momenta, $L$ , $G$ , and $H$ .^[8] The momenta $L$ , $G$ , and $H$ are the action variables and are more elaborate combinations of the Keplerian elements $a$ , $e$ , and $i$ .

Delaunay variables are used to simplify perturbative calculations in celestial mechanics, for example while investigating the Kozai–Lidov oscillations in hierarchical triple systems.^[8] The advantage of the Delaunay variables is that they remain well defined and non-singular (except for $h$ , which can be tolerated) when $e$ and / or $i$ are very small: When the test particle's orbit is very nearly circular ( $e\approx 0$ ), or very nearly "flat" ( $i\approx 0$ ).

Apparent longitude

asteroids that share similar proper orbital elements

Asteroid family

Beta angle

Ephemeris

Geopotential model

Orbital inclination

Orbital state vectors

Proper orbital elements

Osculating orbit

Gurfil, Pini (2005). "Euler parameters as nonsingular orbital elements in Near-Equatorial Orbits". Journal of Guidance, Control, and Dynamics. 28 (5): 1079–1084. :2005JGCD...28.1079G. doi:10.2514/1.14760.