Equation of the center

For further closely related mathematical developments see also Two-body problem, also Gravitational two-body problem, also Kepler orbit, and Kepler problem

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.

The "equation" in the present sense comes from astronomy. It was specified and used by Kepler, as that variable quantity determined by calculation which must be added or subtracted from the mean motion to obtain the true motion. It is based on aequatio, -onis, f. in Latin. In the expression "equation of time" used in astronomy, the term "equation" has a similar meaning.

Analytical expansions

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

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

Related expansions may be used to express the true distance $r$ of the orbiting body from the central body as a fraction of the semi-major axis $a$ of the ellipse,

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

or the inverse of this distance $a/r$ has sometimes been used (e.g. it is proportional to the horizontal parallax of the orbiting body as seen from the central body):

$\frac{a}{r} = 1 + (e -e^3/8) \cos M + e^2 \cos 2 M + \frac{9}{8} e^3 \cos 3 M + ...$ .

Series such as these can be used as part of the preparation of approximate tables of motion of astronomical objects, such as that of the moon around the earth, or the earth or other planets around the sun, when perturbations of the motion are included as well.

Moon's equation of the center

In the case of the moon, its orbit around the earth has an eccentricity of approximately 0.0549. The term in $\sin(M)$, known as the principal term of the equation of the center, has a coefficient of 22639.55", approximately 0.1098 radians, or 6.289° (degrees).

The earliest known estimates of a parameter corresponding to the Moon's equation of the center are Hipparchus' estimates, based on a theory in which the Moon's orbit followed an epicycle or eccenter carried around a circular deferent. (The parameter in the Hipparchan theory corresponding to the equation of the center was the radius of the epicycle as a proportion of the radius of the main orbital circle.) Hipparchus' estimates, based on his data as corrected by Ptolemy yield a figure close to 5° (degrees).

Most of the discrepancy between the Hipparchan estimates and the modern value of the equation of the center arises because Hipparchus' data were taken from positions of the Moon at times of eclipses. He did not recognize the perturbation now called the evection. At new and full moons the evection opposes the equation of the center, to the extent of the coefficient of the evection, 4586.45". The Hipparchus parameter for the relative size of the Moon's epicycle corresponds quite closely to the difference between the two modern coefficients, of the equation of the center, and of the evection (difference 18053.1", about 5.01°).

References

Bibliography

• 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.
• O Neugebauer, A History of Ancient Mathematical Astronomy (Springer, 1975), vol.1, pp. 315–319.

Notes

•  This article incorporates text from a publication now in the public domain: