Orbital elements

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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 an idealized, mathematical approximation at a particular time.

Keplerian elements[edit]

In this diagram, the orbital plane (yellow) intersects a reference plane (gray). For earth-orbiting satellites, the reference plane 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 main two elements that define the shape and size of the ellipse:

  • Eccentricity (e\,\!) - shape of the ellipse, describing how much it is elongated compared to a circle. (not marked in diagram)
  • Semimajor axis (a\,\!) - the sum of the periapsis and apoapsis distances divided by two. For circular orbits, the semimajor axis is the distance between the centers of the bodies, not the distance of the bodies from the center of mass.


  • Inclination - vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node (where the orbit passes upward through the reference plane) (green angle i in diagram).
  • Longitude of the ascending node - horizontally orients the ascending node of the ellipse (where the orbit passes upward through the reference plane) with respect to the reference frame's vernal point (green angle Ω in diagram).

And finally:

  • Argument of periapsis defines the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis (the closest point the second body comes to the first during an orbit). (blue angle \omega\,\! in diagram)
  • Mean anomaly at epoch (M_o\,\!) defines the position of the orbiting body along the ellipse at a specific time (the "epoch").

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 \nu\,\!, 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 \nu\,\! in the diagram, and the mean anomaly is not shown.



Required parameters[edit]


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 are commonly used instead.



Alternative parametrizations[edit]

Keplerian elements can be obtained from orbital state vectors (x-y-z coordinates for position and velocity) by manual transformations or with computer software.[1]


Instead of the mean anomaly at epoch, the mean anomaly M\,\!, mean longitude, true anomaly \nu_o\,\!, or (rarely) the eccentric anomaly might be used.

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.


It is also quite common to see either the Mean Anomaly (M) or the Mean Longitude (L) expressed directly, without either M0 or L0 as intermediary steps, as a polynomial function with respect to time. This method of expression will consolidate the mean motion (n) into the polynomial as one of the coefficients. The appearance will be that L or M are expressed in a more complicated manner, but we will appear to need one fewer orbital element.

Mean motion can also be obscured behind citations of the orbital Period, P.

Sets of Elements for Astronomical Objects
Object Elements used
Major Planet e,a, i, Ω, ϖ, L0
Comet e, q,i,Ω, ω,T0
Asteroid e,a,i,Ω, ω, M0
TLE e,i,Ω,ω, n,M0

Euler angle transformations[edit]

The angles \Omega , i, \omega are the Euler angles (\alpha , \beta, \gamma with the notations of that article) characterizing the orientation of the coordinate system

\hat{x},\hat{y},\hat{z} from the inertial coordinate frame \hat{I},\hat{J},\hat{K}


\hat{I},\hat{J} is in the equatorial plane of the central body. \hat{I} is in the direction of the vernal equinox. \hat{J} is perpendicular to \hat{I} and with \hat{I} defines the reference plane. \hat{K} is perpendicular to the reference plane.

\hat{x},\hat{y} are in the orbital plane and with \hat{x} in the direction to the pericenter (periapsis). \hat{z} is perpendicular to the plane of the orbit. \hat{y} is mutually perpendicular to \hat{x} and \hat{z}.

Then, the transformation from the \hat{I},\hat{J},\hat{K} coordinate frame to the \hat{x},\hat{y},\hat{z} frame with the Euler angles \Omega , i, \omega 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\,


\hat{x}= x_1\hat{I} + x_2\hat{J} + x_3\hat{K}
\hat{y}= y_1\hat{I} + y_2\hat{J} + y_3\hat{K}
\hat{z}= z_1\hat{I} + z_2\hat{J} + z_3\hat{K}

The transformation from \hat{x},\hat{y},\hat{z} to Euler angles \Omega , i, \omega 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) available in many programming languages.

Orbit prediction[edit]

Under ideal conditions of a perfectly spherical central body, and zero perturbations, all orbital elements, with the exception of the Mean anomaly are constants, and Mean anomaly changes linearly with time, scaled by the Mean motion, n=\sqrt{\frac{\mu } {a^3}}.[2] Hence if at any instant t_0 the orbital parameters are [e_0,a_0,i_0,\Omega_0,\omega_0,M_0], then the elements at time t_0+\delta t is given by [e_0,a_0,i_0,\Omega_0,\omega_0,M_0+n\delta t]

Perturbations and elemental variance[edit]



Two-line elements[edit]

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 can be handled easily by all modern data storages as well.


Example of a two line element:[5]

 2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249

See also[edit]


  1. ^ For example, with VEC2TLE
  3. ^ Danby, J. M. A. (1962). Fundamentals of Celestial Mechanics. Willmann-Bell. ISBN 978-0-943396-20-0. 
  5. ^ SORCE - orbit data at Heavens-Above.com

External links[edit]