Orbital elements

Orbital elements are a set of mathematical parameters used to identify a specific orbit. In the field of celestial mechanics, scientists study how two bodies interact in a two-body system. They often use a Kepler orbit to model this movement. A Kepler orbit is an idealized mathematical approximation of a path at a specific moment. While a real orbit changes over time, these elements provide a clear way to describe a trajectory. They are much more useful than using position and velocity vectors alone. Using these elements allows astronomers to define the shape, orientation, and timing of an object's path through space.

To understand these elements, we must first define the bodies involved. In an orbital system, the larger or more central body is called the primary. The smaller object moving around it is the secondary. Even if the two bodies have equal mass, the orbital elements will change depending on which one is chosen as the primary. When we view the system from a non-inertial frame centered on the primary, we only see the trajectory of the secondary. This is what Keplerian elements describe. These elements can be calculated through a process called orbit determination. This process uses computer software or manual transformations to turn state vectors into orbital elements.

To uniquely define a Keplerian orbit, a set of six orbital elements is required. This is because the problem involves six degrees of freedom. These six degrees correspond to three spatial dimensions for position and three dimensions for velocity.

These six elements describe the shape and the starting position of the object. However, they do not tell us where the object will be at a future time. To solve Kepler's problem for a future moment, we must use an extended set of eight elements. This extended set includes parameters for the speed of motion and the specific time the starting position occurs.

Six parameters are grouped into categories based on what they describe. The first category involves the size and shape of the trajectory. Eccentricity is a key parameter here. It describes how much an orbit deviates from a perfect circle. An eccentricity of zero represents a circle. Values less than one describe an ellipse. A value of one describes a parabola, and a value greater than one describes a hyperbola.

Another shape parameter is the semi-major axis. This is half the distance between the apoapsis and the periapsis. The apoapsis is the farthest point from the central body. The periapsis is the closest point. For circular orbits, these two points are not distinct.

Another way to describe shape is through the semi-minor axis or the semi-parameter. The semi-parameter, also called the semi-latus rectum, is half the width of the orbit perpendicular to the periapsis direction. Unlike the semi-major axis, the semi-parameter is always positive and defined. The third category of elements describes the rotation or orientation of the orbit. This requires three parameters. The first is inclination, which is the vertical tilt of the orbital plane. This is measured against a reference plane, such as the Earth's equator.

The second is the longitude of the ascending node. This is the angle from a reference direction to the point where the orbit crosses the reference plane.

The third rotation element is the argument of periapsis. This defines the orientation of the orbit within its own plane. It is the angle measured from the ascending node to the periapsis. Together, these three rotation elements act as Euler angles to set the orbit's direction in space.

The final category of elements describes motion over time. One parameter is the mean motion, which describes the average angular speed. Another is the orbital period, which is the time required for one full revolution. For non-periodic paths like parabolas or hyperbolas, the orbital period is undefined.

It is important to remember that orbits are not permanent. Real orbits change due to gravitational perturbations. These are the gravitational pulls exerted by other objects in space. Changes can also occur because of the effects of general relativity. Because of these forces, a Kepler orbit is only an approximation of a real path at a specific time. Some paths are not closed loops at all. These are called trajectories rather than orbits because they are not periodic. Even so, the same six elements can be used to represent these open trajectories. Understanding these elements allows us to track everything from satellites to distant planets.