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A circular orbit is depicted in the top-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy remains constant throughout the constant speed circular orbit.
Orbital mechanics is a core discipline within space-mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets.
An orbiting body's mean longitude is calculated L = Ω + ω + M, where Ω is the longitude of the ascending node, ω is the argument of the pericenter and M is the mean anomaly, the body's angular distance from the pericenter as if it moved with constant speed rather than with the variable speed of an elliptical orbit.
The mean anomaly at epoch, M 0, is defined as the instantaneous mean anomaly at a given epoch, t 0. This value is sometimes provided with other orbital elements to enable calculations of the object's past and future positions along the orbit. The epoch for which M 0 is defined is often determined by convention in a given field or discipline.
The orbital equation can be derived from the Hamilton–Jacobi equation. [15] The advantage of this approach is that it equates the motion of the particle with the propagation of a wave, and leads neatly into the derivation of the deflection of light by gravity in general relativity , through Fermat's principle .
The mean anomaly changes linearly with time, scaled by the mean motion, [2] =. where μ is the standard gravitational parameter. Hence if at any instant t 0 the orbital parameters are (e 0, a 0, i 0, Ω 0, ω 0, M 0), then the elements at time t = t 0 + δt is given by (e 0, a 0, i 0, Ω 0, ω 0, M 0 + n δt).
In the case of circular orbits it is often assumed that the periapsis is placed at the ascending node and therefore ω = 0. However, in the professional exoplanet community, ω = 90° is more often assumed for circular orbits, which has the advantage that the time of a planet's inferior conjunction (which would be the time the planet would ...
The diagram shows a Hohmann transfer orbit to bring a spacecraft from a lower circular orbit into a higher one. It is an elliptic orbit that is tangential both to the lower circular orbit the spacecraft is to leave (cyan, labeled 1 on diagram) and the higher circular orbit that it is to reach (red, labeled 3 on diagram).