Orbital elements

In celestial mechanics these elements are considered in two-body systems using a Kepler orbit.

A real orbit and its elements change over time due to gravitational perturbations by other objects and the effects of general relativity.

When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories.

When viewed from a non-inertial frame centered on one of the bodies, 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.

Two elements define the shape and size of the ellipse: Two elements define the orientation of the orbital plane in which the ellipse is embedded: The remaining two elements are as follows: The mean anomaly M is a mathematically convenient fictitious "angle" which does not correspond to a real geometric angle, but rather varies linearly with time, one whole orbital period being represented by an "angle" of 2π radians.

It can be converted into the true anomaly ν, 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 body at any given time.

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.

Regardless of eccentricity, the orbit degenerates to a radial trajectory if the angular momentum equals zero.

Sometimes the epoch is considered a "seventh" orbital parameter, rather than part of the reference frame.

(The sixth parameter is still necessary to define the orbit; it is merely numerically set to zero by convention or "moved" into the definition of the epoch with respect to real-world clock time.)

[1] Other orbital parameters can be computed from the Keplerian elements such as the period, apoapsis, and periapsis.

It is common to specify the period instead of the semi-major axis a in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameter, GM, is given for the central body.

The eccentricity, e, and either the semi-major axis, a, or the distance of periapsis, q, are used to specify the shape and size of an orbit.

The longitude of the ascending node, Ω, the inclination, i, and the argument of periapsis, ω, or the longitude of periapsis, ϖ, specify the orientation of the orbit in its plane.

The choices made depend whether the vernal equinox or the node are used as the primary reference.

The semi-major axis is known if the mean motion and the gravitational mass are known.

This method of expression will consolidate the mean motion (n) into the polynomial as one of the coefficients.

Mean motion can also be obscured behind citations of the orbital period P.[clarification needed] The angles Ω, i, ω are the Euler angles (corresponding to α, β, γ in the notation used in that article) characterizing the orientation of the coordinate system where: Then, the transformation from the Î, Ĵ, K̂ coordinate frame to the x̂, ŷ, ẑ frame with the Euler angles Ω, i, ω is:

According to the rules of matrix algebra, the inverse matrix of the product of the 3 rotation matrices is obtained by inverting the order of the three matrices and switching the signs of the three Euler angles.

where arg(x,y) signifies the polar argument that can be computed with the standard function atan2(y,x) available in many programming languages.

Under ideal conditions of a perfectly spherical central body, zero perturbations and negligible relativistic effects, all orbital elements except the mean anomaly are constants.

Unperturbed, two-body, Newtonian orbits are always conic sections, so the Keplerian elements define an ellipse, parabola, or hyperbola.

Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, the nonsphericity of the primary, atmospheric drag, relativistic effects, radiation pressure, electromagnetic forces, and so on.

Keplerian elements parameters can be encoded as text in a number of formats.

The most common of them is the NASA / NORAD "two-line elements" (TLE) format,[4] originally designed for use with 80 column punched cards, but still in use because it is the most common format, and 80-character ASCII records can be handled efficiently by modern databases.

Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable.

Orbital positions can be calculated from TLEs through simplified perturbation models (SGP4 / SDP4 / SGP8 / SDP8).

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 (