Eccentric anomaly

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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

In astrodynamics eccentric anomaly E can be calculated as follows:

$E=\arccos {{1-\left | \mathbf{r} \right | / a} \over e}$

where:

$\mathbf{r}\,\!$ is the orbiting body's position vector (segment sp),
$a\,\!$ is the orbit's semi-major axis (segment cz), and
$e\,\!$ is the orbit's eccentricity.

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

$M = E - e \cdot \sin{E}.\,\!$

For small values of $e$ ( $e < 0.6627434$ ) this equation can be solved iteratively, starting from $E 0 = M$ and using the relation $E_{i+1} = M + e\,\sin E_i$ . The first few terms of the expansion in powers of $e$ are:

$E_1 = M + e\,\sin M$
$E_2 = M + e\,\sin M + \frac{1}{2} e^2 \sin 2M$
$E_3 = M + e\,\sin M + \frac{1}{2} e^2 \sin 2M + \frac{1}{8} e^3 (3\sin 3M - \sin M)$ .

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 T, the true anomaly, is:

$\cos{T} = {{\cos{E} - e} \over {1 - e \cdot \cos{E}}}$

or equivalently

$\tan{T \over 2} = \sqrt{{{1+e} \over {1-e}}} \tan{E \over 2}.\,$

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

$r = a \left ( 1 - e \cdot \cos{E} \right )\,\!$

and

$r = a{(1 - e^2) \over (1 + e \cdot \cos{T})}.\,\!$

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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Orbital elements:	Inclination ( $i\,\!$ ) \| Longitude of the ascending node ( $\Omega\,\!$ ) \| Eccentricity ( $e\,\!$ ) \| Argument of periapsis ( $\omega\,\!$ ) \| Semimajor axis ( $a\,\!$ ) \| Mean anomaly at epoch ( $M_o\,\!$ )
Other orbital parameters:	True anomaly ( $v\,$ ) \| Semi-major axis ( $a\,$ ) \| Semi-minor axis ( $b\,$ ) \| Linear eccentricity ( $\epsilon\,$ ) \| Eccentric anomaly ( $E\,$ ) \| Mean longitude ( $L\,$ ) \| True longitude ( $l\,$ ) \| Orbital period ( $T\,$ )
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