Orbital elements

Orbital elements are the set of numbers needed to pin down a single, specific orbit. In celestial mechanics, these parameters are studied within two-body systems that follow a Keplerian orbit. While many mathematical schemes can describe the same orbit, astronomers and orbital mechanics tend to rely on a few standard sets.

Over time, a real orbit and its elements shift because of gravitational tugs from other bodies and the effects of general relativity. A Kepler orbit is a clean, mathematical approximation of that orbit at one moment in time.

Seen from an inertial frame, two orbiting bodies each trace their own path. Both paths share a focus at the system’s common center of mass. From a non-inertial frame centered on one body, only the other body’s path is visible; Keplerian elements describe these non-inertial paths. An orbit has two sets of Keplerian elements, depending on which body you pick as the reference point. The reference body (usually the more massive one) is called the primary, and the other is the secondary. The primary doesn’t have to be more massive, and even if the two bodies weigh the same, the orbital elements still depend on which one you choose as primary.

You can derive orbital elements from orbital state vectors—the position and velocity vectors of the orbiting object—either by manual calculations or using computer software in a process called orbit determination.

Non-closed orbits exist, but they’re usually called trajectories rather than orbits because they aren’t periodic. The same elements used for closed orbits can typically describe open trajectories as well.

**Required parameters**

A Keplerian orbit needs six orbital elements to be defined without ambiguity. That’s because the problem has six degrees of freedom, matching the six parameters in a set of orbital state vectors: three spatial coordinates for position (x, y, z in a Cartesian system) and three for velocity in each direction. The state vectors fully define the object’s trajectory, but they’re often clunky and opaque to work with, which is why orbital elements are the usual choice.

However, those six elements only describe the starting position and the shape of the trajectory. If you want to solve Kepler’s problem—finding the object’s position and velocity at any future time—you need two more parameters, making eight orbital elements in total.

When describing an orbit with elements, typically two define the size and shape of the path, three describe how the orbit is rotated in space, and one gives the starting position along the orbit. To solve for position as a function of time, you can extend this set with an element for the speed of motion and another for the time at which the starting position occurs.

**Common orbital elements by type**

**Size- and shape-describing parameters**

Two parameters are needed to describe an orbit’s size and shape. Generally, any two of the values below can be used to calculate any other, so the choice comes down to preference and the specific use case.

- **Eccentricity (e)** — the shape of the ellipse, showing how much it deviates from a perfect circle. An eccentricity of 0 means a perfect circle; values less than 1 describe an ellipse; exactly 1 describes a parabola; values greater than 1 describe a hyperbola. - **Semi-major axis (a)** — half the distance between the apoapsis and periapsis (the ellipse’s long axis). This value is positive for elliptical orbits, undefined for parabolic trajectories, and negative for hyperbolic trajectories, which can make it tricky to use across different trajectory types. - **Semi-minor axis (b)** — half the short axis through the ellipse’s geometric center. It shares the same limitations as the semi-major axis: undefined for parabolas and negative for hyperbolas. - **Semi-parameter (p)** — half the width of the orbit perpendicular to the periapsis direction, crossing the primary focus (the orbital radius r at a true anomaly of ±π/2 radians, or ±90°). This value is handy because it appears in the general orbit equation, which can give the distance from the central body using p and the true anomaly for any orbit or trajectory. It’s also called the semi-latus rectum and given the symbol ℓ. Unlike the semi-major and semi-minor axes, this value is always defined and positive. - **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 infinite) for parabolic and hyperbolic trajectories, since they keep moving away forever. It’s sometimes written as Q. - **Periapsis (rp)** — the closest point in the orbit to the central body (at a true anomaly of 0). Unlike apoapsis, this is defined for all orbit types. It’s sometimes written as q.

For perfectly circular orbits, there’s no distinct apoapsis or periapsis because every point is the same distance from the central body. Also, the suffixes for “apoapsis” and “periapsis” often change depending on the central body—for example, “apogee” and “perigee” for Earth orbits, and “aphelion” and “perihelion” for Sun orbits.

Other parameters, like linear eccentricity, flattening, and focal parameter, can also describe an orbit’s size and shape, but they are used less often.