Eccentric anomaly

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 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

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.

The eccentricity e is defined as: From Pythagoras's 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.

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] where is called "the semi-latus rectum" in classical geometry.

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.