The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipse's circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. In the diagram below, it is E (the angle zcx).

Variables used in this article

Calculation

In astrodynamics eccentric anomaly E can be calculated as follows:

where:

• is the orbiting body's position vector (segment sp),
• is the orbit's semi-major axis (segment cz), and
• is the orbit's eccentricity.

The relation between E and M, the mean anomaly, is:

This equation can be solved iteratively, starting from E₀ = M and using the relation .

The equation can also be expanded in powers of e, as long as e < 0.6627434. The first few terms of the expansion are:

• 
• 
• .

For references on details of this derivation, as well as other more efficient methods of solution, see Murray and Dermott (1999, p.35). For a derivation of the limiting value of e see Plummer (1960, section 46).

The relation between E and ν, the true anomaly, is:

or equivalently

The relations between the radius (position vector magnitude) and the anomalies are:

and

See also

• Kepler's laws of planetary motion
• Mean anomaly
• True anomaly

References

• Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge.
• Plummer, H.C., 1960, An Introductory treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)

Orbits   Types General Box · Capture · Circular · Elliptical / Highly elliptical · Escape · Graveyard · Hyperbolic trajectory · Inclined / Non-inclined · Osculating · Parabolic trajectory · Parking · Synchronous (semi · sub) Geocentric Geosynchronous · Geostationary · Sun-synchronous · Low Earth · Medium Earth · Molniya · Near-equatorial · Orbit of the Moon · Polar · Tundra Other Areosynchronous · Areostationary · Halo · Lissajous · Lunar · Heliocentric · Heliosynchronous   Parameters Classical  Inclination ·  Longitude of the ascending node ·  Eccentricity ·  Argument of periapsis ·  Semi-major axis ·  Mean anomaly at epoch Other  True anomaly ·  Semi-minor axis ·  Linear eccentricity ·  Eccentric anomaly ·  Mean longitude ·  True longitude ·  Orbital period   Maneuvers Bi-elliptic transfer · Geostationary transfer · Gravity assist · Hohmann transfer · Inclination change · Phasing · Rendezvous · Transposition, docking, and extraction   Other orbital mechanics topics Apsis · Celestial coordinate system · Delta-v budget · Epoch · Ephemeris · Equatorial coordinate system · Gravity turn · Ground track · Interplanetary Transport Network · Kepler's laws of planetary motion · Lagrangian point · Low energy transfers · n-body problem · Oberth effect · Orbit equation · Orbital state vectors · Perturbation · Retrograde and direct motion · Specific orbital energy · Specific relative angular momentum List of orbits

Source: adapted by the editor from Wikipedia, the free encyclopedia; from the article "Eccentric anomaly". Image Credit.

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