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

## Graphical representation

Consider the ellipse with equation given by:

where *a* is the *semi-major* axis and *b* is the *semi-minor* axis.

For a point on the ellipse, *P* = *P*(*x*, *y*), representing the position of an orbiting body in an elliptical orbit, the eccentric anomaly is the angle *E* in the figure. The eccentric anomaly *E* is one of the angles of a right triangle with one vertex at the center of the ellipse, its adjacent side lying on the *major* axis, having hypotenuse *a* (equal to the *semi-major* axis of the ellipse), and opposite side (perpendicular to the *major* axis and touching the point *P′* on the auxiliary circle of radius *a*) that passes through the point *P*. The eccentric anomaly is measured in the same direction as the true anomaly, shown in the figure as *f*. The eccentric anomaly *E* in terms of these coordinates is given by:^{[1]}

and

The second equation is established using the relationship

- ,

which implies that sin *E* = ±*y*/*b*. The equation sin *E* = −*y*/*b* is immediately able to be ruled out since it traverses the ellipse in the wrong direction. It can also be noted that the second equation can be viewed as coming from a similar triangle with its opposite side having the same length *y* as the distance from *P* to the *major* axis, and its hypotenuse *b* equal to the *semi-minor* axis of the ellipse.

## Formulas

### Radius and eccentric anomaly

The eccentricity *e* is defined as:

From Pythagoras' theorem applied to the triangle with *r* (a distance *FP*) as hypotenuse:

Thus, the radius (distance from the focus to point *P*) is related to the eccentric anomaly by the formula

With this result the eccentric anomaly can be determined from the true anomaly as shown next.

### From the true anomaly

The *true anomaly* is the angle labeled *f* in the figure, located at the focus of the ellipse. In the calculations below, it is referred as *θ*. The true anomaly and the eccentric anomaly are related as follows.^{[2]}

Using the formula for *r* above, the sine and cosine of *E* are found in terms of *θ*:

Hence,

Angle *E* is therefore the adjacent angle of a right triangle with hypotenuse 1 + *e* cos *θ*, adjacent side *e* + cos *θ*, and opposite side √1 − *e*^{2} sin *θ*.

Also,

Substituting cos *E* as found above into the expression for *r*, the radial distance from the focal point to the point *P*, can be found in terms of the true anomaly as well:^{[2]}

### From the mean anomaly

The eccentric anomaly *E* is related to the mean anomaly *M* by Kepler's equation:^{[3]}

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

## See also

## Notes and references

**^**George Albert Wentworth (1914). "The ellipse §126".*Elements of analytic geometry*(2nd ed.). Ginn & Co. p. 141.- ^
^{a}^{b}James Bao-yen Tsui (2000).*Fundamentals of global positioning system receivers: a software approach*(3rd ed.). John Wiley & Sons. p. 48. ISBN 0-471-38154-3. **^**Michel Capderou (2005). "Definition of the mean anomaly, Eq. 1.68".*Satellites: orbits and missions*. Springer. p. 21. ISBN 2-287-21317-1.

## Sources

- Murray, Carl D.; & Dermott, Stanley F. (1999);
*Solar System Dynamics*, Cambridge University Press, Cambridge, GB - Plummer, Henry C. K. (1960);
*An Introductory Treatise on Dynamical Astronomy*, Dover Publications, New York, NY (Reprint of the 1918 Cambridge University Press edition)