Orbital mechanics

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Introduction to Orbital mechanics
Orbital Mechanics is a fairly broad category involving the study of orbiting bodies. Most commonly, orbital mechanics refers to astronomical bodies orbiting one another such as planets, asteroids, space debris, artificial satellites and spacecraft, and stars. However, orbital mechanics can, in general, also be taken to mean the interactions on a cosmic scale such as those of black holes, quasars, galaxies, and galactic structures as well as interactions on a quantum scale involving atoms and other elementary particles. However, since entities in these two extreme realms do not typically follow classical orbital dynamics (those driven primarily by gravity) this topic will focus primarily on working with bodies on the first scale mentioned.

Orbital mechanics has its foundation in areas such as physics, dynamics, and advanced mathematics involving differential equations. That being said, it would behoove students to properly educate themselves in at least these three areas before trying to understand much of what is covered in orbital mechanics.

Types of Orbits
Generally speaking, there are four types of orbits that a single body can follow when acted upon by a central body. These four types or orbits each have unique characteristics and applications. They are named after the four conic sections of mathematics. Each one is described in more detail as follows:

Circular Orbits
Circular orbits are the only other kind of orbit in which the orbiting body cannot naturally escape the central body. Circular orbits have an eccentricity of 0. Generally speaking, circular orbits do not exist naturally and are very impractical artificially. Rather, near circular orbits (orbits with eccentricities so close to 0 that they can be assumed circular) are more commonly occurring.

Elliptical Orbits
Elliptical orbits have an eccentricity less than one. These are the most commonly referred to orbits as these orbits describe the motion of a satellite body in a captured orbit around a central body. That is, elliptically orbiting bodies do not naturally escape from their orbit without some external force.

The central body of an elliptical orbit resides at one of the focal points of the ellipse that the orbiting body traces out.

Parabolic Orbits
Parabolic orbital trajectories have an eccentricity exactly equal to 1. Like circular orbits, parabolic orbits are not realistically common. However, they are very useful in that they represent the first escape trajectory of an orbiting body. That is, for an orbiting body to escape its orbit around a central body, it must at least achieve a parabolic orbit. As such, parabolic orbit calculations are very common in problems involving transfer orbits.

Hyperbolic Orbits
Hyperbolic orbits, like parabolic orbits, represent an escape orbit. That is, after orbiting a central body once (at the focal point of the hyperbola) the orbiting body will "slingshot" around the central body and move away from the central body continuously to never return again (unless influenced by some external force). Hyperbolic orbits have an eccentricity greater than one and are the most common type of orbit employed to transfer an orbiting body from one central body to another. For the purposes of orbital trajectory analysis between two central bodies, numerous hyperbolic orbits are often studied and compared to derive an optimal path for the orbiting body (optimization may take into factors such as, but not limited to, fuel consumption, solar heating, debris avoidance, and simplicity of control methods).

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