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 along with time and magnitude of acceleration) 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.

In general, eight parameters are necessary to unambiguously define an arbitrary and unperturbed orbit. This is because the problem contains eight degrees of freedom. These correspond to the three spatial dimensions which define position (x, y, z in a Cartesian coordinate system), the velocity in each of these dimensions, the magnitude of acceleration (only magnitude is needed, as the direction is always opposite the position vector), and the current time (epoch). The mass or standard gravitational parameter of the central body can specified instead of the acceleration, as one can be used to find the other given the position vector through the relation ${\displaystyle a=\mu /r^{2}}$. These parameters can be described as orbital state vectors, but this is often an inconvenient and opaque way to represent an orbit, which is why orbital elements are commonly used instead.

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, one is needed to describe the speed of motion, and two elements are needed to describe the position of the body around its orbit along with the epoch time at which this occurs. However, if the epoch time is chosen to be the time at which the position-describing element of choice (e.g. mean anomaly) is equal to some constant (usually zero), then said element can be omitted, meaning that only seven elements are required in total.

Commonly only 6 variables are specified for a given orbit, as the motion-describing variable can be the mass or standard gravitational parameter of the central body, which is often already known and does not need specifying, and the epoch time can be considered part of the reference frame and not as a distinct element. However, in any case, 8 values will need to be known, regardless of how they are categorized.