Eccentric anomaly

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).

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:

For small values of () this equation can be solved iteratively, starting from and using the relation . The first few terms of the expansion in powers of are:

.
For references on details of this derivation, as well as other more efficient methods of solution, see Murray and Dermott (1999, p.35).

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

or equivalently

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

and

Reference

Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge.

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