Lunar theory

Lunar theory attempts to account for the motions of the Moon. There are many small variations in the Moon's motion, and many attempts have been made to account for them. After centuries of being problematic, lunar motion can now be modeled to a very high degree of accuracy.
Lunar theory includes:

the background of general theory; including mathematical techniques used to analyze the Moon's motion and to generate formulae and algorithms for predicting its movements; and also
quantitative formulae, algorithms, and geometrical diagrams that may be used to compute the Moon's position for a given time; often by the help of tables based on the algorithms.

Lunar theory has a history of over 2000 years of investigation. Its more modern developments have been used over the last three centuries for fundamental scientific and technological purposes, and are still being used in that way.

Applications

Applications of lunar theory have included the following:

In the eighteenth century, comparison between lunar theory and observation was used to test Newton's law of universal gravitation by the motion of the lunar apogee.
In the eighteenth and nineteenth centuries, navigational tables based on lunar theory, initially in the Nautical Almanac, were much used for the determination of longitude at sea by the method of lunar distances.
In the very early twentieth century, comparison between lunar theory and observation was used in another test of gravitational theory, to test Simon Newcomb's suggestion that a well-known discrepancy in the motion of the perihelion of Mercury might be explained by a fractional adjustment of the power -2 in Newton's inverse square law of gravitation.
In the mid-twentieth century, before the development of atomic clocks, lunar theory and observation were used in combination to implement an astronomical time scale free of the irregularities of mean solar time.
In the late twentieth and early twenty-first centuries, modern developments of lunar theory are being used in the Jet Propulsion Laboratory Development Ephemeris series of models of the Solar System, in conjunction with high-precision observations, to test the exactness of physical relationships associated with the general theory of relativity, including the strong equivalence principle, relativistic gravitation, geodetic precession, and the constancy of the gravitational constant.
History

The Moon has been observed for millennia. Over these ages, various levels of care and precision have been possible, according to the techniques of observation available at any time. There is a correspondingly long history of lunar theories: it stretches from the times of the Babylonian and Greek astronomers, down to modern lunar laser ranging.
The history can be considered to fall into three parts: from ancient times to Newton; the period of classical physics; and modern developments.

Babylon

Of Babylonian astronomy, practically nothing was known to historians of science before the 1880s. Surviving ancient writings of Pliny had made bare mention of three astronomical schools in Mesopotamia – at Babylon, Uruk, and 'Hipparenum'. But definite modern knowledge of any details only began when Joseph Epping deciphered cuneiform texts on clay tablets from a Babylonian archive: In these texts he identified an ephemeris of positions of the Moon. Since then, knowledge of the subject, still fragmentary, has had to be built up by painstaking analysis of deciphered texts, mainly in numerical form, on tablets from Babylon and Uruk.
To the Babylonian astronomer Kidinnu has been attributed the invention of what is now called "System B" for predicting the position of the moon, taking account that the moon continually changes its speed along its path relative to the background of fixed stars. This system involved calculating daily stepwise changes of lunar speed, up or down, with a minimum and a maximum approximately each month. The basis of these systems appears to have been arithmetical rather than geometrical, but they did approximately account for the main lunar inequality now known as the equation of the center.
The Babylonians kept very accurate records for hundreds of years of new moons and eclipses. Some time between the years 500 BC and 400 BC they identified and began to use the 19 year cyclic relation between lunar months and solar years now known as the Metonic cycle.
This helped them build a numerical theory of the main irregularities in the Moon's motion, reaching remarkably good estimates for the periods of the three most prominent features of the Moon's motion:

The synodic month, i.e. the mean period for the phases of the Moon. Now called "System B", it reckons the synodic month as 29 days and 3,11;0,50 "time degrees", where each time degree is one degree of the apparent motion of the stars, or 4 minutes of time, and the sexagesimal values after the semicolon are fractions of a time degree. This converts to 29.530594 days = 29^d 12^h 44^m 3.33^s, to compare with a modern value of 29.530589 days, or 29^d 12^h 44^m 2.9^s. This same value was used by Hipparchos and Ptolemy, was used throughout the Middle Ages, and still forms the basis of the Hebrew calendar.
The mean lunar velocity relative to the stars they estimated at 13° 10′ 35″ per day, giving a corresponding month of 27.321598 days, to compare with modern values of 13° 10′ 35.0275″ and 27.321582 days.
The anomalistic month, i.e. the mean period for the Moon's approximately monthly accelerations and decelerations in its rate of movement against the stars, had a Babylonian estimate of 27.5545833 days, to compare with a modern value 27.554551 days.
The draconitic month, i.e. the mean period with which the path of the Moon against the stars deviates first north and then south in ecliptic latitude by comparison with the ecliptic path of the Sun, was indicated by a number of different parameters leading to various estimates, e.g. of 27.212204 days, to compare with a modern value of 27.212221, but the Babylonians also had a numerical relationship that 5458 synodic months were equal to 5923 draconitic months, which when compared with their accurate value for the synodic month leads to practically exactly the modern figure for the draconitic month.

The Babylonian estimate for the synodic month was adopted for the greater part of two millennia by Hipparchus, Ptolemy, and medieval writers.

Greece and Hellenistic Egypt

Thereafter, from Hipparchus and Ptolemy in the Bithynian and Ptolemaic epochs down to the time of Newton's work in the seventeenth century, lunar theories were composed mainly with the help of geometrical ideas, inspired more or less directly by long series of positional observations of the moon. Prominent in these geometrical lunar theories were combinations of circular motions – applications of the theory of epicycles.

Hipparchus

Hipparchus, whose works are mostly lost and known mainly from quotations by other authors, assumed that the Moon moved in a circle inclined at 5° to the ecliptic, rotating in a retrograde direction once in 18 years. The circle acted as a deferent, carrying an epicycle along which the Moon was assumed to move in a retrograde direction. The center of the epicycle moved at a rate corresponding to the mean change in Moon's longitude, while the period of the Moon around the epicycle was an anomalistic month. This epicycle approximately provided for what was later recognized as the elliptical inequality, the equation of the center, and its size approximated to an equation of the center of about 5° 1'. This figure is much smaller than the modern value: but it is close to the difference between the modern coefficients of the equation of the center and that of the evection: the difference is accounted for by the fact that the ancient measurements were taken at times of eclipses, and the effect of the [|evection] was at that time unknown and overlooked. ''For further information see also separate article Evection.''

Ptolemy

Ptolemy's work the Almagest had wide and long-lasting acceptance and influence for over a millennium. He gave a geometrical lunar theory that improved on that of Hipparchus by providing for a second inequality of the Moon's motion, using a device that made the apparent apogee oscillate a little – prosneusis of the epicycle. This second inequality or second anomaly accounted rather approximately, not only for the equation of the center, but also for what became known as the evection. But this theory, applied to its logical conclusion, would make the distance of the Moon appear to vary by a factor of about 2, which is clearly not seen in reality. This defect of the Ptolemaic theory led to proposed replacements by Ibn al-Shatir in the 14th century and by Copernicus in the 16th century.

Ibn al-Shatir and Copernicus

Significant advances in lunar theory were made by the Arab astronomer, Ibn al-Shatir . Drawing on the observation that the distance to the Moon did not change as drastically as required by Ptolemy's lunar model, he produced a new lunar model that replaced Ptolemy's crank mechanism with a double epicycle model that reduced the computed range of distances of the Moon from the Earth. A similar lunar theory, developed some 150 years later by the Renaissance astronomer Nicolaus Copernicus, had the same advantage concerning the lunar distances.

Tycho Brahe, Johannes Kepler, and Jeremiah Horrocks

and Johannes Kepler refined the Ptolemaic lunar theory, but did not overcome its central defect of giving a poor account of the variations in the Moon's distance, apparent diameter and parallax. Their work added to the lunar theory three substantial further discoveries.

The nodes and the inclination of the lunar orbital plane both appear to librate, with a monthly or semi-annual period.
The lunar longitude has a twice-monthly Variation, by which the Moon moves faster than expected at new and full moon, and slower than expected at the quarters.
There is also an annual effect, by which the lunar motion slows down a little in January and speeds up a little in July: the annual equation.

The refinements of Brahe and Kepler were recognized by their immediate successors as improvements, but their seventeenth-century successors tried numerous alternative geometrical configurations for the lunar motions to improve matters further. A notable success was achieved by Jeremiah Horrocks, who proposed a scheme involving an approximate 6 monthly libration in the position of the lunar apogee and also in the size of the elliptical eccentricity. This scheme had the great merit of giving a more realistic description of the changes in distance, diameter and parallax of the Moon.