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 (x, y, z 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 to 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 (as described below), so the choice of which to use is one of preference and the particular use case.
Eccentricity ( e ) — 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 exactly 1 describes a parabola; values greater than 1 describe a hyperbola. Semi-major axis ( a ) — half the distance between the apoapsis and periapsis (long axis of the ellipse). 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 ( b ) — 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 ( p ) — half the width of the orbit perpendicular to the periapsis direction, crossing the primary focus (the orbital radius r for a true anomaly of ±+π/2 radians, or ±90° ). This value is useful for its use in the general orbit equation, which can return the distance from the central body given p 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 ℓ Additionally, this value will always be defined and positive unlike the semi-major and semi-minor axes. Apoapsis ( ra ) — the farthest point in the orbit from the central body (at a true anomaly of π radians, or 180° ). This quantity is undefined (or infinity) for parabolic and hyperbolic trajectories, as they continue moving away from the central body forever. This value is sometimes given the symbol Q. Periapsis ( rp ) — the closest point in the orbit from the central body (at a true anomaly of 0). Unlike with apoapsis, this quantity is defined for all orbit types. This value is sometimes given the symbol q. 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 (e.g. "apogee" and "perigee" for orbits of the Earth, and "aphelion" and "perihelion" for orbits of the Sun). 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
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