Orbital elements

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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 variables 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).

1 Keplerian elements
2 Two line elements
3 See also
4 External links

[edit] 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:

Inclination ( $i\,\!$ )
Longitude of the ascending node ( $\Omega\,\!$ )
Argument of periapsis ( $\omega\,\!$ )
Eccentricity ( $e\,\!$ )
Semimajor axis ( $a\,\!$ )
Mean anomaly at epoch ( $M_o\,\!$ )

Keplerian elements can be obtained from orbital state vectors using VEC2TLE software or by some direct computations. 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 next two establish the shape of the orbit, while the last establishes the location of the orbiting body at a particular time. Altogether, the Keplerian elements parameterise a conic orbit emerging in an unperturbed two-body problem — an ellipse, a parabola, or a hyperbola. In a more realistic setting, a 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.

Notice that the last element listed is "Mean anomaly at Epoch". "Epoch" is simply a specified point in time. Since the Mean anomaly of a satellite constantly changes, we must specify both the angle and the point in time at which we measure it. If we choose a different point in time to make the measurement, then we will generally get a different value for the angle. Further, when working with real satellites, there are many forces acting on the satellite which can cause small changes in any of the orbital elements. Since all of the elements can change, Epoch becomes even more significant.

[edit] Alternative expressions

Instead of the mean anomaly at epoch, $M_o\,\!$ , the mean anomaly $M\,\!$ , mean longitude, true anomaly or, rarely, the eccentric anomaly may also be used. (Sometimes the epoch itself is considered an orbital element.) 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 the semi-major axis. It is also possible to describe an orbit using just five elements at an epoch by making some assumption about the sixth element, such as specifying that epoch will only occur when the Mean anomaly is zero. (Actually, of course, all six elements are known since we required one to be zero. This scheme simply allows one to specify an orbit by only writing down the epoch and five elements.)

Fig. 1: Keplerian orbital parameters.

[edit] Visualizing an Orbit

In Fig. 1, the orbital plane (yellow) intersects a reference plane called the plane of the ecliptic (grey). The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. This plane, together with the Vernal Point, (♈) establishes a reference frame. The elements can be seen as defining the orbit in this frame by degrees:

The semi-major axis (violet line in Fig. 1) fixes the size of the orbit. It connects the geometric center of the orbital ellipse with the periapsis, passing through the focal point where the center of mass resides. ^[1]. As noted above, the orbital period also establishes the size of the orbit.
The eccentricity fixes its shape.
The longitude of the ascending node (green angle $\Omega\,\!$ in Fig. 1) orients the ascending node with respect to the vernal point. Imagine the angle being formed by pivoting the orbital plane through an axis of rotation perpendicular to the plane of the ecliptic and passing through the center of mass.
The inclination (green angle $i\,\!$ in Fig. 1) orients the orbital plane with respect to the plane of the ecliptic. Imagine the angle being formed by pivoting the orbital plane through an axis of rotation coinciding with the line of nodes.
The argument of periapsis (perihelion) (violet angle $\omega\,\!$ in Fig. 1) orients the semimajor axis with respect to the ascending node. Imagine the angle being formed by pivoting the orbital plane through an axis of rotation perpendicular to itself and passing through the center of mass.
The true anomaly (red angle $T\,\!$ in Fig. 1) orients the celestial body in space. Imagine this positioning angle being formed by pivoting the body's position vector through an axis of rotation perpendicular to the orbital plane and passing through the center of mass.

[edit] Variance Among Keplerian Elements and Trajectories of Orbiting Bodies

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, and so on. 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 Poincaré, 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.)

For more information, see the Discussion.

[edit] 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.
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 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 Derivative
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: