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).

Contents

Calculation

See also

References

Calculation

In astrodynamics eccentric anomaly ''E'' can be calculated as follows:
:
ight | / 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 $e$ () 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:

★ $E\_1\; =\; M\; +\; e,sin\; M$

★

★
+ rac{1}{8} e^3 (3sin 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
:$an\{T\; over\; 2\}\; =\; sqrt\{\{\{1+e\}\; over\; \{1-e\}\}\}\; an\{E\; over\; 2\}.,$
The relations between the radius (position vector magnitude) and the anomalies are:
:$r\; =\; a\; left\; (\; 1\; -\; e\; cdot\; cos\{E\}$
ight ),!
and
:$r\; =\; a\{(1\; -\; e^2)\; over\; (1\; +\; e\; cdot\; cos\{T\})\}.,!$

See also

★ Kepler's laws of planetary motion

★ Mean anomaly

★ True anomaly

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.)