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.
Definition 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
e = 1 + 2 E L 2 m rdc α 2 {\displaystyle e={\sqrt {1+{\frac {\ 2\ E\ L^{2}\ }{\ m_{\text{rdc}}\ \alpha ^{2}\ }}}}}
where E is the total orbital energy, L is the angular momentum, mrdc is the reduced mass, and α {\displaystyle \alpha } the coefficient of the inverse-square law central force such as in the theory of gravity or electrostatics in classical physics: F = α r 2 {\displaystyle F={\frac {\alpha }{r^{2}}}} ( α {\displaystyle \alpha } is negative for an attractive force, positive for a repulsive one; related to the Kepler problem) or in the case of a gravitational force: e = 1 + 2 ε h 2 μ 2 {\displaystyle e={\sqrt {1+{\frac {2\varepsilon h^{2}}{\mu ^{2}}}}}}
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 the hyperbolic case, (when 1 < e < ∞ {\displaystyle 1<e<\infty } ) the hyperbola branch makes 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). Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits have zero angular momentum and hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic, parabolic, and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1 (or in the parabolic case, remains 1). For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable. For elliptical orbits, a simple proof shows that arcsin ( e ) {\displaystyle \arcsin(e)} gives the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury (e = 0.2056), one must simply calculate the inverse sine to find the projection angle of 11.86°. Then, tilting any circular object by that angle, the apparent ellipse of that object projected to the viewer's eye will be of the same eccentricity.
Etymology The word "eccentricity" comes from Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros "out of the center", from ἐκ- ek-, "out of" + κέντρον kentron "center". "Eccentric" first appeared in English in 1551, with the definition "...a circle in which the earth, sun. etc. deviates from its center". In 1556, five years later, an adjectival form of the word had developed.
Calculation The eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector: e = | e | {\displaystyle e=\left|\mathbf {e} \right|} where:
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