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, each of which will specify the same orbit.

This problem contains three degrees of freedom (the three Cartesian coordinates of the orbiting body). Therefore, any given Kepler orbit (unperturbed) is fully defined by six quantities - the initial values of the Cartesian components of the body's position and velocity - and an epoch, a time at which the elements are valid. For this reason, all sets of orbital elements contain exactly six parameters.

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.

[edit] Keplerian elements

The traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of planetary motion.

Two elements define the shape and size of the ellipse:

Eccentricity ( $e\,\!$ )
Semimajor axis ( $a\,\!$ )

Two define the orientation of the orbital plane:

Inclination ( $i\,\!$ )
Longitude of the ascending node ( $\Omega\,\!$ )

And finally:

Argument of periapsis ( $\omega\,\!$ ) defines the orientation of the ellipse in the orbital plane.
Mean anomaly at epoch ( $M_o\,\!$ ) defines the position of the orbiting body along the ellipse.

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 defined inertial coordinate system. The next two establish the size and shape of the orbit, and the last establishes the location of the body within its orbit at the given time (epoch). Unperturbed, two-body orbits are always conic sections, so the Keplerian elements define an ellipse, a parabola, or a hyperbola. Real orbits have perturbations, so a given set of Keplerian elements is valid only at the epoch though the predictions are often adequate at times near the epoch. A real trajectory can be modeled as a sequence of osculating Keplerian element sets defining orbits that osculate ("kiss" or touch) the real trajectory at their respective epoch times.

The last element is "Mean anomaly at Epoch". The mean anomaly steadily increases by 360 degrees per orbit, so we must specify the time (epoch) at which it is measured. As mentioned above, real orbits are generally perturbed by small forces that can cause some or all of the Keplerian elements to change slowly with time, so the other elements are also strictly valid only at the epoch time.

[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 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. An orbit can also be described with just five elements if the epoch always represents the moment at which the mean anomaly is zero. (Actually, all six elements are known, we just constrain one to be zero.)

Fig. 1: Keplerian orbital parameters.

[edit] Visualizing an orbit

In Fig. 1, the orbital plane (yellow) intersects a reference plane. For earth-orbiting satellites this is usually the earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. 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 $\nu\,\!$ in Fig. 1) orients the celestial body in space. Imagine this positioning angle being formed by pivoting the body's position vector, starting at periapsis, 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 planetary orbit 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 forms of Lagrange, Gauss, Delaunay, Poincaré, or Hill. (The last 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.)

[edit] Transformations

The angles $Ω, i,ω$ are the euler angles ( $α,β,γ$ with the

notations of that article) characterising the orientation of the coordinate system

$\hat{x},\hat{y},\hat{z}$

with $\hat{x},\hat{y}$ in the orbital plane and with $\hat{x}$ in the direction to the pericentre.

The transformation from the euler angles $Ω, i,ω$ to $\hat{x},\hat{y},\hat{z}$ is:

$x_1= \cos \Omega \cdot \cos \omega - \ sin \Omega \cdot \cos i \cdot \sin \omega$

$x_2= \sin \Omega \cdot \cos \omega + \cos \Omega \cdot \cos i \cdot \sin \omega$

$x_3= \sin i \cdot \sin \omega$

$y_1=-\cos \Omega \cdot \sin \omega - \sin \Omega \cdot \cos i \cdot \cos \omega$

$y_2=-\sin \Omega \cdot \sin \omega + \cos \Omega \cdot \cos i \cdot \cos \omega$

$y_3= \sin i \cdot \cos \omega$

$z_1= \sin i \cdot \sin \Omega$

$z_2=-\sin i \cdot \cos \Omega$

z 3 = cos i

The transformation from $\hat{x},\hat{y},\hat{z}$ to euler angles $Ω, i,ω$ is:

$\Omega= \operatorname{arg}(\ -z_2\ ,\ z_1\ )$

$i = \operatorname{arg}(\ z_3\ ,\ \sqrt{{z_1}^2 + {z_2}^2}\ )$

$\omega= \operatorname{arg}(\ y_3\ ,\ x_3\ )$

where $\operatorname{arg}(x\ ,\ y)$ signifies the polar argument that can be computed with the standard function ATAN2(y,x) (or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN.

[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.^[2]

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

Example of a two line element:^[3]

1 27651U 03004A   07083.49636287  .00000119  00000-0  30706-4 0  2692
2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249

[edit] Relationship to orbital state vectors

Orbital state vectors are equivalent to Keplerian orbital elements and each can be computed with each other (and used to derive other parameters of the orbit).

[edit] See also

[edit] References

^ The semi-major axis is not completely visible in Fig. 1; the segment between the focal point and the geometric center of the orbital ellipse is occluded by the plane of ecliptic
^ Explanatory Supplement to the Astronomical Almanac. 1992. K. P. Seidelmann, Ed., University Science Books, Mill Valley, California.
^ SORCE - Orbit Data