Lunar theory

After centuries of being problematic, lunar motion can now be modeled to a very high degree of accuracy (see section Modern developments).

The history can be considered to fall into three parts: from ancient times to Newton; the period of classical (Newtonian) physics; and modern developments.

[3] Surviving ancient writings of Pliny had made bare mention of three astronomical schools in Mesopotamia – at Babylon, Uruk, and 'Hipparenum' (possibly 'Sippar').

[5] 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 (no trace has yet been found of anything from the third school mentioned by Pliny).

This system involved calculating daily stepwise changes of lunar speed, up or down, with a minimum and a maximum approximately each month.

[6] 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.

[8] This helped them build a numerical theory of the main irregularities in the Moon's motion, reaching remarkably good estimates for the (different) periods of the three most prominent features of the Moon's motion: The Babylonian estimate for the synodic month was adopted for the greater part of two millennia by Hipparchus, Ptolemy, and medieval writers (and it is still in use as part of the basis for the calculated Hebrew (Jewish) calendar).

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.

[14] 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 (i.e. opposite to the direction of annual and monthly apparent movements of the Sun and Moon relative to the fixed stars) once in 182⁄3 years.

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 (1st term) 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 (which subtracts under those conditions from the equation of the center) was at that time unknown and overlooked.

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.

But this theory, applied to its logical conclusion, would make the distance (and apparent diameter) of the Moon appear to vary by a factor of about 2, which is clearly not seen in reality.

[20][21] Tycho Brahe and Johannes Kepler refined the Ptolemaic lunar theory, but did not overcome its central defect of giving a poor account of the (mainly monthly) variations in the Moon's distance, apparent diameter and parallax.

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.

According to Brewster, Edmund Halley also told John Conduitt that when pressed to complete his analysis Newton "always replied that it made his head ache, and kept him awake so often, that he would think of it no more" [Emphasis in original].

In summary, line LS in Newton's diagram as shown below represents the size and direction of the perturbing acceleration acting on the Moon in the Moon's current position P (line LS does not pass through point P, but the text shows that this is not intended to be significant, it is a result of the scale factors and the way the diagram has been built up).

The resulting difference, after subtracting SQ from LQ, is therefore the vector sum of LM and MS: these add up to a perturbing acceleration LS.

Shown here is an often-presented form of the diagram that summarises sizes and directions of the perturbation vectors for many different positions of the Moon in its orbit.

The main aim of Newton's successors, from Leonhard Euler, Alexis Clairaut and Jean d'Alembert in the mid-eighteenth century, down to Ernest William Brown in the late nineteenth and early twentieth century, was to account completely and much more precisely for the moon's motions on the basis of Newton's laws, i.e. the laws of motion and of universal gravitation by attractions inversely proportional to the squares of the distances between the attracting bodies.

They also wished to put the inverse-square law of gravitation to the test, and for a time in the 1740s it was seriously doubted, on account of what was then thought to be a large discrepancy between the Newton-theoretical and the observed rates in the motion of the lunar apogee.

However Clairaut showed shortly afterwards (1749–50) that at least the major cause of the discrepancy lay not in the lunar theory based on Newton's laws, but in excessive approximations that he and others had relied on to evaluate it.

Most of the improvements in theory after Newton were made in algebraic form: they involved voluminous and highly laborious amounts of infinitesimal calculus and trigonometry.

From the time of the earliest gravitational analysts among Newton's successors, Euler, Clairaut and d'Alembert, it was recognized that nearly all of the main lunar perturbations could be expressed in terms of just a few angular arguments and coefficients.

However, work in the nineteenth and twentieth century led to very different formulations of the theory so these terms are no longer current.

In the meantime, Brown's theory was improved with better constants and the introduction of Ephemeris Time and the removal of some empirical corrections associated with this.