Abstract
Let x, y, z represent a set of selenocentric rectangular coordinates, the axes of which coincide with the principal axes of inertia of the Moon, and their origin with the center of gravity of its mass. Let, moreover, these axes share the mean rotation of our satellite (i.e., be fixed in its globe and share its motion — the body axes); while
represent the moments of inertia about the x, y, z-axes (the products of inertia being by definition, zero). If so, the well-known Eulerian equations of motion of a solid body about its center of gravity assume the forms
where ω x, y, z stand for the angular velocities of rotation about the respective axes; and F x, y, z , for the components of the forces acting on the Moon from outside.
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Bibliographical Notes
The possibility that, apart from optical librations, the Moon may librate physically around its center of gravity was first mentioned by Newton in the third volume of his Principia which appeared in 1687; but such a motion remained undetectable from the observations for almost 150 years. Throughout the 18th century the observations by Mayer and Lalande failed to detect any significant displacements of the apparent positions on the Moon which could be due to this cause; and the same was true of subsequent efforts by Bouvard and Arago or Nicolet, undertaken in Paris (at the encouragement of Laplace) in the first two decades of the 19th century, or somewhat later by Kreil in Milan.
The success in this age-long quest for the detection of physical librations of the Moon was finally scored by Bessel (1839), the founder of modern astrometry, who devised for this purpose a new type of telescope (i.e., the helimeter) which remained in use for visual observations of this kind for almost a century. Bessel (1839, 1941) worked out also in full detail the basic procedures for the reductions of the observations which were continued after his death by his pupil Schlüter. Death claimed Schlüter only two years after the passing of his master; and their work fell into the hands of Wichmann, another pupil of Bessel, who in 1846–1847 published the combined results of the three (Wichmann, 1846, 1847). In subsequent decades, heliometric studies of the lunar librations in Germany became associated with the lifelong efforts of Ernst Hartwig, the founder and first director of the Bamberg Observatory. Hartwig’s interest in lunar librations has apparently been evoked by Winnecke, under whose encouragement Hartwig completed at Strasbourg between 1877–1879 his first series of 42 heliometric observations of the positions of Mösting A (known as the Hartwig Strasbourg Series), eventually published in his doctoral thesis (Hartwig, 1880). In 1884 Hartwig moved from Strasbourg to the Dorpat (now Tartu) Observatory in Russia, where between 1884–1885 he carried out his second (Dorpat) series of 36 heliometric observations of the Moon. In 1885 Hartwig returned to Germany to become director of the newly founded observatory at Bamberg, where between 1885 and 1922 he carried out his last and longest (Bamberg) series of such observations. On the whole, over an interval of 42 years between March 1880 and May 1922 Hartwig carried out 266 sets of heliometric observations of the Moon (and, between 1841 through 1922, three German observers — Schlüter, Wichmann and Hartwig — performed a total of 552 such observations) representing an important contribution to our science.
Since Hartwig’s death in 1923 heliometric observations of the Moon in Germany have come to an end; for no successor was ready to follow in his footsteps. However, the earlier series of observations made in the 19th century were re-discussed by subsequent investigators several times. Thus Hartwig’s Strasbourg series was later re-discussed by Franz (1887) who re-discussed later also the observations by Schlüter (Franz, 1899). Schlüter’s measurements were re-discussed once more by Stratton (1909); while Hartwig’s Bamberg series was re-reduced later by Naumann (1939); and, still more recently, Hartwig’s Dorpat series was thoroughly re-discussed by Koziel (1948, 1949). Of other work in this field carried out at Leipzig in preceding years cf. Hayn (1902–23); but is has since come to the end owing to a complete destruction of the Leipzig Observatory during the second world war.
Apart from the German contributions to the study of the physical librations of the Moon which over a century were on a massive scale, the second most important school of this subject has arisen in Russia and has been centered around the Engelhardt Observatory in Kazan. Moreover, unlike the German school (going back to Bessel) which has now become extinct, the Russian school has remained active to this date and continues to produce important results in this field. Following the Hartwig Dorpat series of 1884–1885, heliometric observations of the relative positions of Mösting A were commenced at Kazan by A. V. Krasnov between 1895 and 1898, consisting of 112 sets of observations; while between 1900 and 1905 A. A. Mikhailovsky completed the second Kazan series consisting of 54 sets of observations. This latter series of observations was subsequently reduced by Völkel (1908) and Belkovich (1936). A third Kazan series of 130 such observations was carried out between 1910–1915 by Th. Banachiewicz, whose observations were reduced by Yakovkin (1928). Yakovkin’s contributions to the study of the figure and motion of the Moon over the past forty years have been on a massive scale. Between 1916–1931 he performed 251 sets of heliometric measurements of Mösting A which he reduced himself in two separate publications (Yakovkin, 1939, 1945). Since 1931 the Kazan heliometric observations were continued by I. V. Belkovich, who between 1931–1948 secured further 247 sets of positional measurements of Mösting A, as well as reduced jointly all Kazan and Hartwig series (Belkovich, 1949). In addition, since 1938 heliometric observations of the Moon have been carried out at Kazan by A. A. Nefediev and Sh. T. Habibullin. Nefediev (1955) reduced also the old lunar heliometric observations by Krasnov; while Habibullin (1958) is the author of the only comprehensive monograph on physical librations of the Moon which has appeared so far in any language.
Photographic determinations of the physical librations have been attempted early in the history of lunar photography by Pritchard and Warren de la Rue (cf. Pritchard, 1879, 1882); and, more recently, by Puiseux (1916, 1925), Chandon (1941) and Habibullin (1958). With the exception of the work reported in this last reference, the results of photographic work compared less than favourably with visual heliometric observations (cf. Weimer, 1954). This is not, however, due to any basic unsuitability of the photographic technique to this type of work, but rather to the fact that (at least until the work of Habibullin) photographic techniques in selenodesy have not yet been properly developed — in contrast to the care with which the reductions of visual observations have been worked out.
Of other more theoretical, investigations of the rotation of the Moon we may refer to Jönsson (1917), Hayn (1923), Yakovkin (1923, 1934, 1939, 1945, 1950, 1952), Belkovich (1948, 1949), Koziel (1948, 1949, 1957, 1962), Nefediev (1951, 1954, 1957), Fridland (1959, 1961), Gorynia (1960), Jeffreys (1957, 1961), Drofa (1962) or Eckhardt (1965). This latter paper develops a new method for treatment of the perturbing function, due to the terrestrial attraction, which is well amenable to solution by automatic computing machines; and such represents a significant advance in further development of our subject.
Another significant advance was initiated by Makover (1962) and Mietelski and Maslowski (1963) in treating Equation (4–54) for forced librations in longitude as one of Mathieu type, in place of a nonhomogeneous Equation (4–97) with constant coefficients.
For a more expository treatment of the problem of physical librations of the lunar globe, cf., e.g., Tisserand (1891), Chapter 28; Plummer (1918), Chapter 23; or Danby (1962), Chapter 14.
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© 1966 Springer Science+Business Media Dordrecht
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Kopal, Z. (1966). Motion of the Moon about its Center of Gravity; Physical Librations. In: An Introduction to the Study of the Moon. Astrophysics and Space Science Library. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-6320-2_4
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DOI: https://doi.org/10.1007/978-94-017-6320-2_4
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