Eccentric anomaly
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In celestial mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.
For the point p=(x,y) on an ellipse with the equation
the eccentric anomaly is the angle E such that
The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit; the other two being the true anomaly and the mean anomaly.
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[edit] Formulas
[edit] From the true anomaly
The eccentric anomaly can be computed from the true anomaly by the formulas
hence
where 
is the angular coordinate of point (X,Y) in polar coordinates.
[edit] From the mean anomaly
The eccentric anomaly E is related to the mean anomaly M by the formula
This equation does not have a closed-form solution for E given M. It is usually solved by numerical methods, e.g. Newton-Raphson method.
[edit] Radius and eccentric anomaly
The radius (distance from the focus of attraction to the orbiting body) is related to the eccentric anomaly by the formula
[edit] 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.)
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Eccentric anomaly
