# Orbital elements

The elements of an orbit are the parameters needed to specify that orbit uniquely, given a model of two point masses obeying the Newtonian laws of motion and the inverse-square law of gravitational attraction. Because there are multiple ways of parameterising a motion, depending on which set of variable you choose to measure, there are several different ways of defining sets of orbital elements, sets each of which will specify the same orbit.

This problem contains three degrees of freedom (the three Cartesian coordinates of the orbiting body). Therefore, each particular Keplerian ( = unperturbed) orbit is fully defined by six quantities - the initial values of the Cartesian components of the body's position and velocity. For this reason, all sets of orbital elements contain exactly six parameters. For a mathematically accurate explanation of this fact see the Discussion and references therein. (See also: orbital state vectors).

## Keplerian elements

The traditionally used set of orbital elements is called the set of Keplerian elements, after Johannes Kepler and his Kepler's laws. The Keplerian elements are six:

We see that the first three orbital elements are simply the Eulerian angles defining the orientation of the orbit relative to some fiducial coordinate system.

The above six elements parameterise a conic orbit emerging in an unperturbed two-body problem - an ellipse, a parabola, or a hyperbola. A realistic perturbed trajectory is represented as a sequence of such instantaneous conics that share one of their foci. In case the orbital elements are postulated to parameterise a sequence of conics that are always tangent to the trajectory, these orbital elements are called osculating.

Instead of the mean anomaly at epoch they often employ the mean anomaly. Sometimes the mean longitude, or the true anomaly or, rarely, the eccentric anomaly are used instead of the mean anomaly at epoch. Sometimes the epoch itself is used as the sixth orbital element, instead of the mean anomaly at epoch.

Keplerian elements can be obtained from orbital state vectors using VEC2TLE software or by some direct computations.

Other orbital parameters, such as the period, can then be calculated from the Keplerian elements. In some cases, the period is used as an orbital element instead of semi-major axis. The elements can be seen as defining the orbit by degrees:

• The semi-major axis (or the period, interchangeably) fixes the size of the orbit.
• The eccentricity fixes its shape.
• The inclination (orange in Fig. 1) and longitude of the ascending node (green) fix its plane.
• The argument of perihelion (blue) orients the orbit within its plane.
• The epoch (or mean anomaly, interchangeably) (red) fixes the object in time on its orbit.

Because the simple Newtonian model of orbital motion of idealised points in free space is not exact, the orbital elements of real objects tend to change over time. Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, due to the nonsphericity of the primary, due to the atmospheric drag, relativistic effects, radiation pressure, electromagnetic forces, etc... This evolution is described by the so-called planetary equations, which come in the form of Lagrange, or in the form of Gauss, or in the form of Delaunay, or in the form of Poincare, or in the form of Hill. (The latter is a very exotic option, emerging in the case when the true anomaly enters the set of six orbital elements. Hill considered this kind of orbit parameterisation back in 1913.)

## Two line elements

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[1] , 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.
TLEs older than 30 days become considerably inaccurate. Orbital positions and heights can be calculated from TLEs through the SGP/SGP4/SDP4/SGP8/SDP8 algorithms.

Line 1
Column Characters Description
-----  ---------- -----------
1        1       Line No. Identification
3        5       Catalog No.
8        1       Security Classification
10        8       International Identification
19       14       YRDOY.FODddddd
34        1       Sign of first time derivative
35        9       1st Time Derative
45        1       Sign of 2nd Time Derivative
46        5       2nd Time Derivative
51        1       Sign of 2nd Time Derivative Exponent
52        1       Exponent of 2nd Time Derivative
54        1       Sign of Bstar/Drag Term
55        5       Bstar/Drag Term
60        1       Sign of Exponent of Bstar/Drag Term
61        1       Exponent of Bstar/Drag Term
63        1       Ephemeris Type
65        4       Element Number
69        1       Check Sum, Modulo 10

Line 2
Column Characters Description
-----  ---------- -----------
1       1        Line No. Identification
3       5        Catalog No.
9       8        Inclination
18       8        Right Ascension of Ascending Node
27       7        Eccentricity with assumed leading decimal
35       8        Argument of the Perigee
44       8        Mean Anomaly
53      11        Revolutions per Day (Mean Motion)
64       5        Revolution Number at Epoch
69       1        Check Sum Modulo 10


Reference:

• Explanatory Supplement to the Astronomical Almanac. 1992. K. P. Seidelmann, Ed., University Science Books, Mill Valley, California.

## Orbiter Elements

Example:
ELEMENTS 6731158 0.0004046 51.6412 163.8597 185.4817 215.0447 54745.343391

where

6731158 - Semi-major axis in meters.
0.0004046 - Eccentricity.
51.6412 - Inclination.
163.8597 - Longitude of Ascending Node.
185.4817 - Argument of Periapsis.
215.0447 - Mean Anomaly.
54745.343391 - Epoch in Modified Julian Date.

Note that the lack of a semi-major axis element or MJD epoch element means that conversion between TLE format and Orbiter's format requires a small amount of math.

$a = \sqrt[3]{(T\div2\pi)^2 \mu}$

(The equation for $a$ is based on T because T can be easily derived from the
number of seconds in a day divided by a TLE's 'Revolutions per Day' element.)

$a$ is the semi-major axis in meters.
$\mu$ is the standard gravitational parameter. (398,600,441,800,000 $m^3s^{-2}$ on Earth)
T is the orbital period in seconds.