# Lunar Theory

## Contents

# Lunar Theory

(glossary entry)

## Description

The termlunar theory(referred to almost always as "thelunar theory") is used to describe attempts to mathematically describe the Moon's motions relative to the stars. Typically the Moon's irregular motions are approximated by a long and complex series of terms, each based on theoretical considerations but fit to observations.

## Additional Information

- A very large number of the persons honored by names on the Moon made contributions either to the development or testing of lunar theories.

- In antiquity, the Moon was apparently first approximated as an object traveling around an inclined circle whose center was offset from the center of the Earth. The name of Hipparchus, very little of whose original work survives, is closely associated with this model.

- Over the centuries, increasingly careful comparison of the Moon's expected trajectory to pre-telescopic observations of the stars and planets led to an increasingly complex computational system of spheres, eccentrics and epicycles, taken by some as the true machinery of the universe, but by most as a somewhat arbitrary mathematical construct that yielded approximately correct results.

- Newton's proposal in the mid 1600's of laws of motion and an inverse-square law force of gravity made it possible to replace the old system of computation with a new one having a simpler and more understandable basis, and indeed calculation of the Moon's motions was a major test-bed and means of validating the new theory. Again many of the best minds in mathematics and astronomy contributed to this effort. But again, when confronted with discrepancies between the predictions and increasingly precise telescopic data the new system eventually became even more complex than the old involving vast numbers of obscure periodic terms obtained by algebraic manipulation of the equations of motion to put them in a form suitable for computation.

- Although work on such "analytic" theories of the motions of the Moon and planets continues to this day, with notable contributions from authors such as J. Chapront, M. Chapront-Touze, P. Bretagnon and G. Francou at the Bureau des Longitudes, Paris, classic lunar theory seems to have reached maturity with the publications of E. W. Brown near the start of the 20th century.

- When electronic computers became available in the mid 20th century they seem first to have been applied to reducing the drudgery, and increasing the precision of the results obtainable with extensions of Brown's series, but they were soon applied to a new and different approach (undoubtedly contemplated, but not practical in pre-computer days) in which solar system positions are computed by applying the laws of motion directly to an observed initial condition subject to the known forces acting between the bodies, bypassing the analytic manipulations almost entirely. The JPL Ephemeris (Newhall
*et al.*, 1983; Standish, 1990), the basis of spacecraft navigation and the lunar and planetary positions printed in modern almanacs, is the most widely recognized outcome of this new approach, although their results have been confirmed by lesser known ones produced independently by an MIT group now at the Harvard-Smithsonian Center for Astrophysics (Ash*et al.*, 1967), and more recently by scientists at the Russian Academy of Sciences (Pitjeva, 2001) and the Paris Observatory (Fienga*et al.*, 2008).

## Ephemerides in the Astronomical Almanac

**Lunar theory**is used to generate the tables of phenomena related to the Moon (primarily position, librations and eclipses) published in national ephemerides, such as the*Astronomical Almanac*in the US and UK. The method of the producing the numbers has changed many times there is generally a brief description of the current method in each volume.- Beginning in 1923, the lunar positions were computed using Brown's 1919
*Tables*, which were themselves a simplification of his theory geared towards practical computation. - By the early 1950's advances in electronic computation had made it possible to generate the positions directly from the equations of Brown's theory, rather than from the
*Tables*derived from them. In 1952 the IAU recommended changing to this new system, which made possible the incorporation of an improved time parameter. At least in the US and UK the suggestion was rapidly implemented as told in a special 422 page*Supplement*to the*Almanac*called the*Improved lunar ephemeris: 1952-1959*(published 1954). Fitting the new computations to observation led to extensive revision of the many constants used by Brown, and the new values are listed in that volume. Positions based on this new method began appearing in the 1960 edition of the*Almanac*itself. - New solar system constants recommended by the IAU were incorporated into the calculations in the 1968 and 1972-3 volumes.
- By the early 1980's scientists working in the field of celestial mechanics had come to recognize the superiority of planetary positions derived by numeric integration over those produced by analytic theories. Beginning with the
*Almanac'*s 1984 volume, computations based on Brown's equations were abandoned in favor of a JPL numeric ephemeris called DE200/LE200, computed in 1980. The 1984 edition of the*Almanac*contains, at its end, a 39-page*Supplement*explaining many changes including the transition to numeric integrations, detailed on pages S26-S29. - Despite the superior accuracy of numerical integrations (Cappallo
*et al.*, 1981; Newhall and Williams, 1996), the "physical librations" of the Moon (the very small amount by which the orientation of the Moon's axes in space differs from that expected for a top spinning at a uniform rate) continue to be based on semi-analytical theories. Since 1985, D. Eckhardt's (1981-2) formulas have been used. - In 2003, the DE200/LE200 ephemeris was replaced with a newer one, DE405/LE405.
- The clearest explanation of the way the tables have been generated in recent years can be found in the article by Standish
*et al.*(Chapter 5) in the 1992 and 2005*Explanatory Supplement*to the*Almanac*. The JPL ephemeris in use at the time of that writing (DE200/LE200) is a combined computation for the Sun, Moon and planets including the forces generated by five asteroids. 175 unknowns (including such things as planet masses and radii, but more often empirical corrections to observational data) were adjusted to obtain a best fit to 50424 observations (mostly transit timings from the US Naval Observatory, but also 2954 lunar laser ranges).

## LPOD Articles

## Bibliography

- Wikipedia article
- Ash, Michael E.; Shapiro, Irwin I.; Smith, William B. 1967. "Astronomical constants and planetary ephemerides deduced from radar and optical observations".
*Astronomical Journal*, Vol. 72, pp. 338-350. - Brown, Ernest W., and Henry Benjamin Hedrick. 1919.
*Tables of the motion of the moon. Transactions*, New Haven: Yale University Press. (Google Books: Vol. 1, Vol. 3; Internet Archive: Vol. 1-2, Vol. 3, Vol. 4). - Cappallo, R. J., King, R. W., Counselman III, C. C., and Shapiro, I. I. 1981, "Numerical Model of the Moon"s Rotation",
*Moon and Planets*24, 281-289. - Eckhardt, D. H. 1981. "Theory of the libration of the moon".
*Moon and the Planets*, vol. 25, Aug. 1981, pp. 3-49. (also pp. 193-198 in O. Calame, ed. 1982.*High-precision Earth rotation and Earth-Moon dynamics*) - Fienga, A.; Manche, H.; Laskar, J.; Gastineau, M. 2008. "INPOP06: a new numerical planetary ephemeris".
*Astronomy and Astrophysics*, Volume 477, pp.315-327. - Gutzwiller, Martin C. 1998. "Moon-Earth-Sun: The oldest three-body problem".
*Reviews of Modern Physics*, Volume 70, Issue 2, April 1998, pp. 589-639. (PDF) - Newhall, X. X.; Standish, E. M.; Williams, J. G. 1983. "DE 102 - A numerically integrated ephemeris of the moon and planets spanning forty-four centuries".
*Astronomy and Astrophysics*, vol. 125, pp. 150-167. - Newhall, X. X., and Williams, J. G. 1996. "Estimation of the Lunar Physical Librations",
*Celestial Mechanics and Dynamical Astronomy*66: 21-30. - Pitjeva, E. V. 2001. "Modern Numerical Ephemerides of Planets and the Importance of Ranging Observations for Their Creation".
*Celestial Mechanics and Dynamical Astronomy*, v. 80, p. 249-271. - Standish, E. M., Jr. 1990. "The observational basis for JPL"s DE 200", the planetary ephemerides of the Astronomical Almanac".
*Astronomy and Astrophysics*, vol. 233, pp. 252-271. - Standish, E. M., Newhall, X. X.; Williams, J. G. and Yeomans, D. K. 2005. "Orbital Ephemerides of the Sun, Moon, and Planets" in Seidelmann, P. Kenneth.
*Explanatory Supplement to the Astronomical Almanac: A Revision to the Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac*. University Science Books (the 2005 paperback is the same as the 1992 hardcover edition -- earlier*Explanatory Supplements*describe the methods used prior to 1984). - United States Naval Observatory. 1954.
*Improved lunar ephemeris*, 1952-1959. Washington: U.S. Govt. Print. Off.