Orbital elements

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same orbit, but certain schemes are commonly used in astronomy and orbital mechanics.

A real orbit and its elements change over time due to gravitational perturbations by other objects and the effects of general relativity. A Kepler orbit is an idealized, mathematical approximation of the orbit at a particular time.

When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass. When viewed from a non-inertial frame centered on one of the bodies, only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body (usually the most massive) is called the primary, the other body is called the secondary. The primary does not necessarily possess more mass than the secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary.

Orbital elements can be obtained from orbital state vectors (position and velocity vectors of the orbiting object) by manual transformations or with computer software through a process known as orbit determination.

Non-closed orbits exist, although these are typically referred to as trajectories and not orbits, as they are not periodic. The same elements used to describe closed orbits can also typically be used to represent open trajectories.

Required parameters

A set of six orbital elements are needed to unambiguously define a Keplerian orbit. This is because the problem contains six degrees of freedom. These correspond to the six parameters defined in a set of orbital state vectors: three spatial dimensions which define position (', ', ' in a Cartesian coordinate system), and the velocity in each of these dimensions. The orbiting object's trajectory is completely defined by the orbital state vectors, but this is often an inconvenient and opaque way to represent the orbit, which is why orbital elements are commonly used instead.

Such a set of 6 elements, however, only describes the starting position of the orbiting object and the shape of its trajectory. If one wants to use a set of orbital elements to solve Kepler's problem, two additional parameters must be included. This is to say, in order to solve for the position and velocity of the orbiting object at an arbitrary future time, an extended set of eight orbital elements will be required.

When describing an orbit with orbital elements, typically two are needed to describe the size and shape of the trajectory, three are needed describe the rotation of the orbit, and one is needed to describe the starting position along the orbit. These can then be extended to include an element describing the speed of motion, and an element describing the time that the starting position occurs if position as a function of time needs to be solved.

Common orbital elements by type

Size- and shape-describing parameters

alt=Diagram of an orbit with an eccentricity of 0.5|thumb|360x360px|Diagram of an orbit with an eccentricity of 0.5. The variable names used are consistent with the names described, except for the apoapsis and periapsis, which are labeled as Ap. and Pe. respectively. The circular dot is the center of the ellipse, and the black diamond is the primary focus.

Two parameters are required to describe the size and the shape of an orbit. Generally any two of these values can be used to calculate any other (as described below), so the choice of which to use is one of preference and the particular use case.

Eccentricity — shape of the ellipse, describing how much it deviates from a perfect a circle. An eccentricity of 0 (zero) describes a perfect circle, values less than 1 describe an ellipse; a value of describes a parabola; values greater than 1 describe a hyperbola.
Semi-minor axis — half the short axis through the geometric center of the ellipse. This value shares the same limitations as with the semi-major axis: it is undefined for parabolic trajectories and negative for hyperbolic trajectories.
Semi-parameter — half the width of the orbit perpendicular to the periapsis direction, crossing the primary focus (the orbital radius for a true anomaly of or This value is useful for its use in the general orbit equation, which can return the distance from the central body given and the true anomaly for any type of orbit or trajectory. This value is also commonly referred to as the semi-latus rectum and given the alternate symbol Additionally, this value will always be defined and positive unlike the semi-major and semi-minor axes.

r_\mathsf{a} = a \left( 1 + e \right) ~.

</math>

Periapsis can be found with the semi-major axis and eccentricity using the following equation: This iteration can be repeated until a desired level of tolerance is reached.

True anomaly can be found from the eccentric anomaly using the following relations:

\begin{align}

\sin\nu\ &=\ \frac{\ \sqrt{1 - e^2\ } \sin E\ }{ 1 - e \cos E }\ , \\

\cos\nu\ &=\ \frac{ \cos E - e }{\ 1 - e \cos E\ } ~.

\end{align}

</math>

The quadrant of the solution can be resolved using an atan2(y,  x) function.

It is common to specify the period () or mean motion () instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameter (<math>\mu</math>) is known for the central body though the relations above.

For the epoch, the epoch time () along with the mean anomaly (), mean longitude (), true anomaly (<math>\nu_0</math>) or (more rarely) the eccentric anomaly () are often used. The time of periapsis passage () is also sometimes used for this purpose, since using it eliminates the need to specify position at the epoch.

{| class="wikitable"

|+Sets of orbital elements

!Object

!Elements used

|Major planet

|Comet

|Asteroid

=== Two-line elements ===

Orbital elements 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 80-character ASCII records can be handled efficiently by modern databases.

The two-line element format lists the eccentricity (), inclination (), longitude of the ascending node (), argument of periapsis (), mean motion (), epoch (), and mean anomaly at epoch ().

Example of a two-line element for the SORCE satellite:

<pre>

1 27651U 03004A 07269.09107561 .00000015 00000-0 17636-4 0 4191

2 27651 039.9956 188.8112 0026975 282.9289 076.8483 14.81973121252789

</pre>

Delaunay variables

The Delaunay orbital elements were introduced by Charles-Eugène Delaunay during his study of the motion of the Moon. Commonly called Delaunay variables, they are a set of canonical variables, which are action-angle coordinates. The angles are simple sums of some of the Keplerian angles and are often referred to with different symbols than in other applications:

the mean anomaly: <math>\ell = M</math>,
the argument of periapsis: <math>g = \omega </math>,
the longitude of the ascending node: <math>h = \Omega</math>,

along with their respective conjugate momenta , , and . The momenta , , and are the action variables and are more elaborate combinations of the Keplerian elements , , and .

Delaunay variables are used to simplify perturbative calculations in celestial mechanics, for example, while investigating the Kozai–Lidov oscillations in hierarchical triple systems.

Orbital elements

Required parameters

Common orbital elements by type

Size- and shape-describing parameters

Delaunay variables

See also

References