Equation of the center

The equation of the center, in astronomy and elliptical motion, is equal to the true anomaly minus the mean anomaly, i.e. the difference between the actual angular position in the elliptical orbit and the position the orbiting body would have if its angular motion was uniform. It arises from the ellipticity of the orbit, is zero at pericenter and apocenter, and reaches its greatest amount nearly midway between these points.

Analytical Expansions

For small values of orbital eccentricity, $e$, the true anomaly, $u$, may be expressed as a sine series of the mean anomaly, $M$. The following shows the series expanded to terms of the order of $e^3$:

:$u\; =\; M\; +\; (2\; e\; -\; frac\{1\}\{4\}\; e^3)\; sin\; M\; +\; frac\{5\}\{4\}\; e^2\; sin\; 2\; M\; +\; frac\{13\}\{12\}\; e^3\; sin\; 3\; M\; +\; ...$

A related expansion may be used to express the true distance, $r$, of the orbiting body from the central body, as a ratio of the semi-major axis, $a$, of the ellipse:

:$frac\{r\}\{a\}\; =\; (1\; +\; e^2)\; -\; (e\; -\; frac\{3\}\{8\}e^3)\; cos\; M\; -\; frac\{1\}\{2\}\; e^2\; cos\; 2\; M\; -\; frac\{3\}\{8\}\; e^3\; cos\; 3\; M\; -\; ...$

Series such as these can be used to prepare tables of motion of astronomical objects, such as that of the moon around the earth, or the earth or other planets around the sun. In the case of the moon, its orbit around the earth has an ecentricity of approximately 0.055. The term in $sin\; M$, known as the principal term of the equation of the center, therefore has a value of approximately 0.11 radians, or 6.3 degrees.

References

*Brown, E.W. "An Introductory Treatise on the Lunar Theory." Cambridge University Press, 1896 (republished by Dover, 1960).
*Brown, E.W. "Tables of the Motion of the Moon." Yale University Press, New Haven CT, 1919.
*1911