Equation of the center

In two-body, Keplerian orbital mechanics, the equation of the center is the angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the same period. It is defined as the difference true anomaly, ν, minus mean anomaly, M, and is typically expressed a function of mean anomaly, M, and orbital eccentricity, e. Since antiquity, the problem of predicting the motions of the heavenly bodies has been simplified by reducing it to one of a single body in orbit about another. In calculating the position of the body around its orbit, it is often convenient to begin by assuming circular motion. This first approximation is then simply a constant angular rate multiplied by an amount of time. There are various methods of proceeding to correct the approximate circular position to that produced by elliptical motion, many of them complex, and many involving solution of Kepler's equation. In contrast, the equation of the center is one of the easiest methods to apply. In cases of small eccentricity, the position given by the equation of the center can be nearly as accurate as any other method of solving the problem. Many orbits of interest, such as those of bodies in the Solar System or of artificial Earth satellites, have these nearly-circular orbits. As eccentricity becomes greater, and orbits more elliptical, the equation's accuracy declines, failing completely at the highest values, hence it is not used for such orbits. The equation in its modern form can be truncated at any arbitrary level of accuracy, and when limited to just the most important terms, it can produce an easily calculated approximation of the true position when full accuracy is not important. Such approximations can be used, for instance, as starting values for iterative solutions of Kepler's equation, or in calculating rise or set times, which due to atmospheric effects cannot be predicted with much precision.