Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used (derived from Newton's laws of motion and Newton's law of universal gravitation). 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) changes over time due to gravitational perturbations by other objects and the effects of relativity. A Keplerian orbit is merely a mathematical approximation at a particular time.
This is because the problem contains six degrees of freedom. These correspond to the three spatial dimensions which define position (the x, y, z in a Cartesian coordinate system), plus the velocity in each of these dimensions. These can be described as orbital state vectors, but this is often an inconvenient way to represent an orbit, which is why Keplerian elements (described below) are commonly used instead.
Sometimes the epoch is considered a "seventh" orbital parameter, rather than part of the reference frame.
If the epoch is defined to be at the moment when one of the elements is zero, the number of unspecified elements is reduced to five. (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.)
The main Two elements define the shape and size of the ellipse:
Two define the orientation of the orbital plane in which the ellipse is embedded:
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.
Keplerian elements can be obtained from orbital state vectors (x-y-z coordinates for position and velocity) by manual transformations or with computer software.
Other orbital parameters can be computed from the Keplerian elements such as the period, apoapsis and periapsis. (When orbiting the earth, the last two terms are known as the apogee and perigee.) It is common to specify the period instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameter, GM, is given for the central body.
Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time t must be specified as a "seventh" orbital element. Sometimes it is assumed that mean anomaly is zero at the epoch (by choosing the appropriate definition of the epoch), leaving only the five other orbital elements to be specified.
Euler angle transformations
The angles Ω,i,ω are the Euler angles (α,β,γ with the notations of that article) characterizing the orientation of the coordinate system
with in the orbital plane and with in the direction to the pericenter.
The transformation from the euler angles Ω,i,ω to is:
The transformation from to Euler angles Ω,i,ω is:
Perturbations and elemental variance
Unperturbed, two-body orbits are always conic sections, so the Keplerian elements define an ellipse, parabola, or hyperbola. Real orbits have perturbations, so a given set of Keplerian elements accurately describes an orbit only at the epoch. 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 can often be used to produce useful predictions at times near the epoch. Alternatively, real trajectories can be modeled as a sequence of Keplerian orbits that osculate ("kiss" or touch) the real trajectory. They can also be described by the so-called planetary equations, differential equations which come in different forms developed by Lagrange, Gauss, Delaunay, Poincaré, or Hill.
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 , originally designed for use with 80-column punched cards, but still in use because it is the most common format, and works as well as any other.
Line 1 Column Characters Description ---------------
Example of a two line element:
1 27651U 03004A07083.49636287.0000011900000-030706-4 02692 2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249
Published - July 2009
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