Eccentric anomaly

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The eccentric anomaly of point p is the angle z-c-x

In celestial mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.

For the point p=(x,y) on an ellipse with the equation

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

the eccentric anomaly is the angle E such that

\cos E = \frac{x}{a}\quad \quad \sin E = \frac{y}{b}

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

Contents

Formulas

From the true anomaly

The eccentric anomaly can be computed from the true anomaly by the formulas

\cos E = \frac{x}{a} = \frac{ e + \cos \theta }{1 + e \cos \theta }
\sin E = \frac{y}{b} = \frac{ \sqrt{1 - e^2} \, \sin \theta }{1 + e \cos \theta }

where e is the eccentricity, hence

E = \mathop{\mathrm{arg}}( e + \cos \theta, \; \sqrt{1 - e^2} \, \sin \theta)

where \mathop{\mathrm{arg}}(X,Y) is the angular coordinate of point (X,Y) in polar coordinates.

The following relation also holds:

\tan E = \frac{ \sin \theta \sqrt{1-e^2} }{e + \cos \theta}

and hence

E = \tan^{-1} \left( \frac{ \sin \theta \sqrt{1-e^2} }{e + \cos \theta} \right)

From the mean anomaly

The eccentric anomaly E is related to the mean anomaly M by the formula

M = E - e \cdot \sin E

This equation does not have a closed-form solution for E given M. It is usually solved by numerical methods, e.g. Newton-Raphson method.

Radius and eccentric anomaly

The radius (distance from the focus of attraction to the orbiting body) is related to the eccentric anomaly by the formula

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

References

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