Kepler's equation occurs in the context of the Newtonian two-body problem. The relative orbit of one body with respect to the other is easily characterized with the true anomaly as the independent variable. The true anomaly is just the angle: pericenter – focus — body, where focus is the ellipse focus around which the body moves. This is adequate for determining the orbit in space – its shape, size, and orientation. However, if one wishes to determine the orbit in time, things get more complicated, and one must solve Kepler's equation:

where is the eccentric anomaly , e is the orbital eccentricity, is the mean anomaly , is the time of pericenter passage, and is the mean motion. The true and eccentric anomalies are related by

and

Hence, to determine we must first solve Kepler's equation for .

Since Kepler's equation is transcendental, we require iterative or series expansion approaches. Numerically, one can employ various fast algorithms. Here, we are not concerned with numerical methods but rather the analytical properties of the solution, so we will consider algebraic approximations.

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