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
where is the total orbital energy, is the angular momentum, 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 is the specific orbital energy (total energy divided by the reduced mass), the standard gravitational parameter based on the total mass, and 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 , 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.
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 yields 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 degrees. 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 ekkentros "out of the center", from ek-, "out of" + 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:
where: is the eccentricity vector ("Hamilton's vector").
For elliptical orbits it can also be calculated from the periapsis and apoapsis since and where is the length of the semi-major axis,
where: is the radius at apoapsis (also "apofocus", "aphelion", "apogee"), i.e., the farthest distance of the orbit to the center of mass of the system, which is a focus of the ellipse. is the radius at periapsis (or "perifocus" etc.), the closest distance.
The eccentricity of an elliptical orbit can also be used to obtain the ratio of the apoapsis radius to the periapsis radius:
For Earth, orbital eccentricity , apoapsis is aphelion and periapsis is perihelion, relative to the Sun.
For Earth's annual orbit path, the ratio of longest radius () / shortest radius () is
Examples
The eccentricity of Earth's orbit is currently about ; its orbit is nearly circular. Venus and Neptune have even lower eccentricities. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly to almost 0.058 as a result of gravitational attractions among the planets.