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Orbital Elements [WIP]

Orbital elements are closely related to the concept of state from Control Theory. A properly defined state representation of a system provides enough information about the system to determine its future behaviour in the absence of any unknown or unmodeled forces affecting the system. Orbital elements are one means of representing the state of a system for orbital trajectories that can provide unique insight into that trajectory's properties.

Osculating Orbital Elements

The properties of a satellite orbit most commonly described in terms of its osculating orbital elements. The osculating orbital elements, frequently just referred to as the orbital elements1, describe the state of an object in space at an instantaneous moment in time with respect to trajectory that it would have around its central body if no perturbations were present. That is, the osculating elements are the orbital elements that fit a Keplerian orbit trajectory of the object with respect to its central body for a given moment in time.

Since Keplerian orbits assume no other forces are present, they do not occur in reality. However, because in many cases the conservative (higher-order gravity, third-body gravity, etc.) and non-conservative (drag, solar radiation pressure, propulsion, etc.) perturbations present are an order-of-magnitude smaller than the point-mass gravitational force or more, present as higher-order perturbations. This makes the orbital element of a trajectory a useful tool in understanding the general motion of an object.

Keplerian Orbital Elements

The most common orbital elements used are the Keplerian Orbital Elements

\[ \vec{x}_{oe} = (a, e, i, \Omega, \omega, M) \]

  1. In addition to osculating orbital elements there are also mean orbital elements. Mean elements average the effect of higher-order perturbations to capture the secular (constant) and long-periodic trends affecting orbital trajectories. ↩

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