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In celestial mechanics, the mean anomaly is a parameter that defines the position of a body that is moving in a Kepler orbit. It is defined as the time since the last periapsis (closest approach to the central body) times $2π / T$ , where $T$ is the duration of a full orbit.

The mean anomaly increases uniformly from 0 to $2π$ radians during each orbit. However, it does not have any simple interpretation as a geometric angle; it is merely time measured in radians. Due to Kepler's second law, however, the mean anomaly is proportional to the area swept by the focus-to-body line since the last periapsis.

The mean anomaly is usually denoted by the letter $M$ , and is also given by the formula

$M = \sqrt{\frac{\mu } {a^3}} \,t$ (1)

where a is the length of the orbit's semi-major axis and μ is the standard gravitational parameter.

The mean anomaly is one of three angular parameters ("anomalies") that define a position along an orbit; the other two being the eccentric anomaly and the true anomaly.

[edit] Formulas

The mean anomaly $M$ can be computed from the eccentric anomaly $E$ by the formula

$M = E - e \cdot \sin E$ (2)

To find the position of the object in an elliptic Kepler orbit at a given time t, the corresponding mean anomaly is determined with (1) and then the corresponding eccentric anomaly is found by solving (2) numerically, e.g. with the Newton-Raphson algorithm.

[edit] See also

[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.)