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Concept# Orbital eccentricity

Summary

In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit (or capture orbit), and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a rosette orbit through the Galaxy.
In a two-body problem with inverse-square-law force, every orbit is a Kepler orbit. The eccentricity of this Kepler orbit is a non-negative number that defines its shape.
The eccentricity may take the following values:
Circular orbit: e = 0
Elliptic orbit: 0 < e < 1
Parabolic trajectory: e = 1
Hyperbolic trajectory: e > 1
The eccentricity e is given by
where E is the total orbital energy, L is the angular momentum, mred is the reduced mass, and the coefficient of the inverse-square law central force such as in the theory of gravity or electrostatics in classical physics:
( is negative for an attractive force, positive for a repulsive one; related to the Kepler problem)
or in the case of a gravitational force:
where ε is the specific orbital energy (total energy divided by the reduced mass), μ the standard gravitational parameter based on the total mass, and h the specific relative angular momentum (angular momentum divided by the reduced mass).
For values of e from 0 to 1 the orbit's shape is an increasingly elongated (or flatter) ellipse; for values of e from 1 to infinity the orbit is a hyperbola branch making a total turn of 2 arccsc(e), decreasing from 180 to 0 degrees. Here, the total turn is analogous to turning number, but for open curves (an angle covered by velocity vector). The limit case between an ellipse and a hyperbola, when e equals 1, is parabola.

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