Equation of the center

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.

then Kepler's equation can be solved by numerical methods, but there are also series solutions involving sine of

In cases of small eccentricity, the position given by a truncated series solution may be quite accurate.

As eccentricity becomes greater, and orbits more elliptical, the accuracy of a given truncation of the series declines.

The series in its modern form can be truncated at any point, and even 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,[1] or in calculating rise or set times, which due to atmospheric effects cannot be predicted with much precision.

The ancient Greeks, in particular Hipparchus, knew the equation of the center as prosthaphaeresis, although their understanding of the geometry of the planets' motion was not the same.

[2] The word equation (Latin, aequatio, -onis) in the present sense comes from astronomy.

[3] The equation of the center in modern form was developed as part of perturbation analysis, that is, the study of the effects of a third body on two-body motion.

[4][5] In Keplerian motion, the coordinates of the body retrace the same values with each orbit, which is the definition of a periodic function.

[7] The series for ν, the true anomaly can be expressed most conveniently in terms of M, e and Bessel functions of the first kind,[8] where The result is in radians.

The Bessel functions can be expanded in powers of x by,[10] and βm by,[11] Substituting and reducing, the equation for ν becomes (truncated at order e7),[8] and by the definition, moving M to the left-hand side,

This formula is sometimes presented in terms of powers of e with coefficients in functions of sin M (here truncated at order e6), which is similar to the above form.

[12][13] This presentation, when not truncated, contains the same infinite set of terms, but implies a different order of adding them up.

Because of this, for small e, the series converges rapidly but if e exceeds the "Laplace limit" of 0.6627... then it diverges for all values of M (other than multiples of π), a fact discovered by Francesco Carlini and Pierre-Simon Laplace.

Simulated view of an object in an elliptic orbit , as seen from the focus of the orbit . The view rotates with the mean anomaly , so the object appears to oscillate back and forth across this mean position with the equation of the center . The object also appears to become smaller and larger as it moves farther away and nearer because of the eccentricity of the orbit. A marker (red) shows the position of the periapsis .
Maximum error of the series expansion of the equation of the center, in radians , as a function of orbital eccentricity (bottom axis) and the power of e at which the series is truncated (right axis). Note that at low eccentricity (left-hand side of the graph), the series does not need to be carried to high order to produce accurate results.
Series-expanded equation of the center as a function of mean anomaly for various eccentricities , with the equation of the center truncated at e 7 for all curves. Note that the truncated equation fails at high eccentricity and produces an oscillating curve. But this is because the coefficients of the Fourier series are inaccurate due to truncation in their calculation.