Eccentric anomaly

In orbital mechanics, 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.

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 to the right. The eccentric anomaly, E, is observed by drawing a right triangle with one vertex at the center of the ellipse, 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 crosses 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:

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 being the similar triangle with adjacent side through P and the minor auxiliary circle, hypotenuse b, and whose opposite side is y.

The eccentricity e is defined as:

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