Mean anomaly

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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.

Formulas

The mean anomaly M can be computed from the eccentric anomaly E and the eccentricity 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 equation (1) and then the corresponding eccentric anomaly is found by solving equation (2) numerically, e.g. with the Newton-Raphson algorithm.

See also

References

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