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 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 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, 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

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, 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 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-major axis — half the distance between the apoapsis and periapsis. This value is positive for elliptical orbits, undefined for parabolic trajectories, and negative for hyperbolic trajectories, which can hinder its usability when working with different types of trajectories.
  • 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 for parabolic and hyperbolic trajectories, as they continue moving away from the central body forever. This value is sometimes given the symbol.
  • Periapsis — the closest point in the orbit from the central body. Unlike with apoapsis, this quantity is defined for all orbit types. This value is sometimes given the symbol.
For perfectly circular orbits, there is no distinct apoapsis or periapsis, as all points of the orbit have the same distance from the central body. Additionally, it is common to see the affix for "apoapsis" and "periapsis" changed depending on the central body.
Other parameters can also be used to describe the size and shape of an orbit, such as the linear eccentricity, flattening, and focal parameter, but the use of these is limited.

Relations between elements

This section contains the common relations between these orbital elements, but more relations can be derived through manipulations of one or more of these equations. The variable names used here are consistent with the ones described above.
Eccentricity can be found using the semi-minor and semi-major axes as
Eccentricity can also be found using the apoapsis and periapsis through the following relation:
The semi-major axis can be found using the fact that the lines that connects the apoapsis to the center of the conic and from the center to the periapsis both combined span the length of the conic, and thus the major axis. This is then divided by 2 to get the semi-major axis:
The semi-minor axis can be found using the semi-major axis and eccentricity through the following relations:
Two formula are needed to avoid taking the square root of a negative number. Alternatively, use
The semi-parameter can be found using the semi-major axis and eccentricity:
Apoapsis can be found using the following equation using the semi-major axis and eccentricity:
Periapsis can be found with the semi-major axis and eccentricity using the following equation:

Element Animations

Rotation-describing elements

Three parameters are required to describe the orientation of the plane of the orbit and the orientation of the orbit within that plane.
  • Inclination — vertical tilt of the orbital plane with respect to the reference plane, typically the equator of the central body, measured at the ascending node. Inclinations near zero indicate equatorial orbits, and inclinations near 90° indicate polar orbits. Inclinations from 90 to 180° are typically used to denote retrograde orbits.
  • Longitude of the ascending node — describes the angle from the ascending node of the orbit to the reference frame's reference direction. This is measured in the reference plane, and is shown as the green angle in the diagram. This quantity is undefined for perfectly equatorial orbits, but is often set to zero instead by convention. This quantity is also sometimes referred to as the right ascension of the ascending node.
  • Argument of periapsis — defines the orientation in the orbital plane, as an angle measured from the ascending node to the periapsis, the purple angle in the diagram. This quantity is undefined for circular orbits, but is often set to zero instead by convention.
These three elements together can be described as Euler angles defining the orientation of the orbit relative to the reference coordinate system. Although these three are the most common, other elements do exist and are useful to describe other properties of the orbit.
  • Longitude of periapsis — describes the angle between the vernal point and the periapsis, measured in the reference plane. This can be described as the sum of the longitude of the ascending node and the argument of periapsis:. Unlike the longitude of the ascending node, this value is defined for orbits where the inclination is zero.

    Elements describing motion over time

One parameter is required to describe the speed of motion of the orbiting object around the central body. However, this can be omitted if only a description of the shape of the orbit is required. Various quantities that do not directly describe a speed can be used to satisfy this condition, and it is possible to convert from one to any other.
  • Mean motion — quantity that describes the average angular speed of the orbiting body, measured as an angle per unit time. For non-circular orbits, the actual angular speed is not constant, so the mean motion will not describe a physical angle. Instead this corresponds to a change in the mean anomaly, which indeed increases linearly with time.
  • Orbital period — the time it takes for the orbiting body to complete one full revolution around the central body. This quantity is undefined for parabolic and hyperbolic trajectories, as they are non-periodic.
  • Mass of the 2-body system — the total mass of the central body and orbiting body. If the mass of the orbiting object is insignificant compared to the mass of the primary, then the mass of the central body only can be used.
  • Standard gravitational parameter — quantity equal to the mass of the 2-body system times the gravitational constant. This quantity is often used instead of mass, as it can be easier to measure with precision than either mass or. This is also often not included as part of orbital element lists, as in some cases it can assumed to be known based on the central body.

    Relations between elements

This section contains the common relations between the set of orbital elements described above, but more relations can be derived through manipulations of one or more of these equations. The variable names used here are consistent with the ones described above.
Mean motion can be calculated using the standard gravitational parameter and the semi-major axis of the orbit.
This equation returns the mean motion in radians and will need to be converted if  is desired to be in a different unit.
Because the semi-major axis is related to the mean motion and standard gravitational parameter, it can be calculated without being specified. This is especially useful if is assumed to be known, as then can be used to calculate, and likewise for specifying. This can allow one less element to specified.
Orbital period can be found from given the fact that the mean motion can be described as a frequency, which is the reciprocal of period:
The standard gravitational parameter can be found given the mean motion and the semi-major axis through the following relation :
The mass of the 2-body system can be found given the standard gravitational parameter using a rearrangement of its definition as the product of the mass and the gravitational constant,