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 true anomaly minus the mean anomaly.^[1]

Discussion

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. The equation of the center is one of the easiest 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.

The ancient Greeks, in particular Hipparchus, knew the equation of the center as prostaphaeresis, 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. 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. In astronomy, the term equation of time has a similar meaning.^[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]

Series expansion

In Keplerian motion, the coordinates of the body retrace the same values with each orbit, which is the definition of a periodic function. Such functions can be expressed as periodic series of any continuously increasing angular variable,^[6] and the variable of most interest is the mean anomaly, $M$ . Because it increases uniformly with time, expressing any other variable as a series in mean anomaly is essentially the same as expressing it in terms of time. Because the eccentricity, $e$ , of the orbit is small in value, the series can be developed in terms of powers of $e$ .^[5]

The series for $\nu$ , the true anomaly can be expressed most conveniently in terms of $M$ , $e$ and Bessel functions of the first kind,^[7]

\nu =M+2\sum _{s=1}^{\infty }{\frac {1}{s}}\left\{J_{s}(se)+\sum _{p=1}^{\infty }\beta ^{p}\left[J_{s-p}(se)+J_{s+p}(se)\right]\right\}\sin {sM},

where

J_{n}(se)

are the Bessel functions and

\beta ={\frac {1}{e}}\left(1-{\sqrt {1-e^{2}}}\right).

^[8]

The Bessel functions can be expanded in powers of $e$ by,^[9]

J_{n}(x)={\frac {1}{n!}}\left({\frac {x}{2}}\right)^{n}\sum _{m=0}^{\infty }(-1)^{m}{\frac {\left({\frac {x}{2}}\right)^{2m}}{m!\prod _{k=1}^{m}(n+k)}}

and $\beta ^{m}$ by,^[10]

\beta ^{m}=\left({\frac {e}{2}}\right)^{m}\left[1+m\sum _{n=1}^{\infty }{\frac {(2n+m-1)!}{n!(n+m)!}}\left({\frac {e}{2}}\right)^{2n}\right].

Substituting and reducing, the equation for $\nu$ becomes (expressed to order $e^{7}$ ),^[7]

{\begin{aligned}\nu =M&+\left(2e-{\frac {1}{4}}e^{3}+{\frac {5}{96}}e^{5}+{\frac {107}{4608}}e^{7}\right)\sin M\\&+\left({\frac {5}{4}}e^{2}-{\frac {11}{24}}e^{4}+{\frac {17}{192}}e^{6}\right)\sin 2M\\&+\left({\frac {13}{12}}e^{3}-{\frac {43}{64}}e^{5}+{\frac {95}{512}}e^{7}\right)\sin 3M\\&+\left({\frac {103}{96}}e^{4}-{\frac {451}{480}}e^{6}\right)\sin 4M\\&+\left({\frac {1097}{960}}e^{5}-{\frac {5957}{4608}}e^{7}\right)\sin 5M\\&+{\frac {1223}{960}}e^{6}\sin 6M+{\frac {47273}{32256}}e^{7}\sin 7M+...\end{aligned}}

and by the definition, moving $M$ to the left-hand side,

\nu -M=\left(2e-{\frac {1}{4}}e^{3}+{\frac {5}{96}}e^{5}+{\frac {107}{4608}}e^{7}\right)\sin M+...

gives the equation of the center.

For small $e$ , the series converges rapidly. If $e$ exceeds 0.6627..., it diverges for some values of $M$ , first discovered by Pierre-Simon Laplace.^[11]

Series such as this 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",^[12] 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).^[13]

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.^[13] 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

^ Vallado, David A. (2001). Fundamentals of Astrodynamics and Applications (second ed.). Microcosm Press, El Segundo, CA. p. 82. ISBN 1-881883-12-4.
^ Narrien, John (1833). An Historical Account of the Origin and Progress of Astronomy. Baldwin and Cradock, London. pp. 230–231.
^ Capderou, Michel (2005). Satellites Orbits and Missions. Springer-Verlag. p. 23. ISBN 978-2-287-21317-5.
^ Moulton, Forest Ray (1914). An Introduction to Celestial Mechanics (second revised ed.). Macmillan Co., New York. p. 165., at Google books
^ ^a ^b Smart, W. M. (1953). Celestial Mechanics. Longmans, Green and Co., London. p. 26.
^ Brouwer, Dirk; Clemence, Gerald M. (1961). Methods of Celestial Mechanics. Academic Press, New York and London. p. 60.
^ ^a ^b Brouwer, Dirk; Clemence, Gerald M. (1961). p. 77.
^ Brouwer, Dirk; Clemence, Gerald M. (1961). p. 62.
^ Brouwer, Dirk; Clemence, Gerald M. (1961). p. 68.
^ Smart, W. M. (1953). p. 32.
^ Moulton, Forest Ray (1914). pp. 171-172.
^ (E W Brown, 1919.)
^ ^a ^b (Neugebauer, 1975.)

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

[1] Vallado, David A. (2001). Fundamentals of Astrodynamics and Applications (second ed.). Microcosm Press, El Segundo, CA. p. 82. ISBN 1-881883-12-4.

[2] Narrien, John (1833). An Historical Account of the Origin and Progress of Astronomy. Baldwin and Cradock, London. pp. 230–231.

[3] Capderou, Michel (2005). Satellites Orbits and Missions. Springer-Verlag. p. 23. ISBN 978-2-287-21317-5.

[4] Moulton, Forest Ray (1914). An Introduction to Celestial Mechanics (second revised ed.). Macmillan Co., New York. p. 165., at Google books

[Smart26-5] Smart, W. M. (1953). Celestial Mechanics. Longmans, Green and Co., London. p. 26.

[6] Brouwer, Dirk; Clemence, Gerald M. (1961). Methods of Celestial Mechanics. Academic Press, New York and London. p. 60.

[B&C77-7] Brouwer, Dirk; Clemence, Gerald M. (1961). p. 77.

[8] Brouwer, Dirk; Clemence, Gerald M. (1961). p. 62.

[9] Brouwer, Dirk; Clemence, Gerald M. (1961). p. 68.

[10] Smart, W. M. (1953). p. 32.

[11] Moulton, Forest Ray (1914). pp. 171-172.

[12] (E W Brown, 1919.)

[ON1975-13] (Neugebauer, 1975.)

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Discussion

Series expansion

Moon's equation of the center

See also

References

Bibliography