Libration Research Articles

A new approach to the problem of the lunar libration

Modern techniques permit a new approach to the solution of the problem of the lunar physical libration. The equations which express the phenomenon can be reconstructed and solved with the use of computers, while higher accuracy in its detection has been achieved with the development of a new observational method.

A nonlinear analysis of the moon's physical libration in longitude

The Euler equations for the forced physical librations of the Moon have already been solved by using a digital computer to perform the semi-literal mathematical manipulations. Very near resonance, the computer solution for the physical libration in longitude is complemented by the solution of the appropriate Duffing equation with a dissipation term. Because of its apparent proximity to a resonant frequency, the term whose argument is 2ω - twice the mean angular distance of the Moon's perigee from the ascending node of its orbit - is especially important. Its phase, which soon should be measurable, is related to the Moon's anelasticity. The term's frequency, in units of the sidereal month, increases as the semi-major axis of the Moon's orbit about the Earth increases. Using the Moon's mechanical ellipticity of Koziel and the rate of increase of the semi-major axis of MacDonald, it is estimated that the 2ω term will cross the resonant frequency in 130 million years and, if the rate of energy dissipation is sufficiently low, a transient libration will be induced.

Lunar libration tables

The selenographic direction cosines of the Earth and of the pole of the ecliptic are developed in trigonometric series; the time dependence of each term enters through the Delaunay arguments. These and other series are tabulated for different libration parameters to provide means for calculating at any epoch the lunar librations and their partial derivatives with respect to the parameters.

A lunar libration point experiment

Photographs of the lunar libration point L4 were made during the lunar eclipse of 6 October 1968, using the 48-inch Schmidt telescope at Palomar Mountain, California. An area about 6.6° square was covered. Neither discrete objects nor clouds, associated with the L4 point, were found. The detection limit for discrete objects was at least 15th magnitude and probably 17th magnitude; for clouds, the limit was determined by short-term variations in the local pattern of night sky-glow.

Detectability of Lunar Libration Clouds at Small Phase Angles

THE existence of “trapped” interplanetary particles in the libration centres of the Earth–Moon system was reported by Kordilewski1 in 1961. He observed two very faint clouds close to the Lagrangian point L5 and a large one at L4.

VISUAL OBSERVATIONS OF THE LUNAR LIBRATION CLOUDS AND AN OBSERVING EPHEMERIS

Lunar libration clouds visual observations and observing ephemeris, discussing sightings from aircraft and photographic attempts

The constants of the moon's physical libration derived on the basis of four series of heliometric observations from the years 1877 to 1915

A new method for the adjustment of heliometric libration observations of the Moon is given; it is simple from the mathematical point of view, satisfies the principles of the least-squares method, and can easily be applied to programming of calculations of electronic computers. On the basis of 3282 observations of the Moon carried out in four heliometric series from the years 1877 to 1915, the author has determined the forced libration constants, simultaneously with those of free libration in longitude, by using his new method of adjustment of heliometric observations. A joint adjustment has been performed by eliminating the Moon's mean radius corresponding to each series, as it might be affected by irradiation. At the same time an exact proof is given for the uniqueness of the solution for the mechanical ellipticity of the Moon f, which lies below the critical value 0.662. From the libration constants obtained new values for the differences of the Moon's principal moments of inertia are derived. Moreover, the coordinates of crater Mösting A, which constitutes a first order point of triangulation on the Moon, are obtained with high accuracy. A conclusion is drawn that, in the problem discussed, a simultaneous analysis—based on a correct adjustment method—of several heliometric series with a sufficient number of observations covering a sufficiently long period, is necessary to obtain reliable results, fully satisfying the requirements of scientific exactitude.

Satellite resonances and librations associated with Tesseral harmonics of the geopotential

The motion of a satellite under the influence of the longitude-dependent terms of the geopotential is examined in a frame of reference rotating with the mean motion of the satellite. The orbits are assumed to be near-circular but unrestricted with respect to inclination. In general the effect of the tesseral harmonics is to induce short-period perturbations of small amplitude. Under special conditions, however—when the satellite's mean motion is commensurable with the earth's rotation—two distinct types of resonance may be set up. The physical behavior, as well as the mathematical description, of the motion is inherently different in the separate resonances. Specifically, dynamical resonances occur when the impressed frequency due to some Jnm term is equal to the natural frequency of the satellite's orbital motion. In this case the amplitude of the induced oscillation builds up with time, the eccentricity increases, and the orbit is unstable in the sense that it will be driven out of the resonance state. For other commensurabilities librational resonances occur when the satellite is in a localized potential well in the moving frame. In this situation the periodic motion of the satellite in the potential trough is analogous to pendulum motion. In turn, this manifests itself as a long-period libration in longitude of the satellite's ground track. For near-equatorial motion, only the 24-hour orbit is librational; the orbits in dynamical resonance are confined to the range from 12 to 48 hours, clustering around but not including the 24-hour orbit.

Theory of Lunar Librations Re-Examined.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Theory of Lunar Librations Re-Examined. Michelson, Irving Abstract The principal features of lunar rotation were concisely summarized with a high degree of precision by Cassini in 1693 in the set of three empirical laws that are known by his name. Laplace one century later furnished a dynamical theory for the actual small departures of the motion from the idealized form enunciated by Cassini, and suggested measurements from which lunar mass distribution characteristics might be inferred. In modern times Commission 17 of the International Astronomical Union, recognizing the inconclusive character of all such subsequent efforts, has urged the application of a rigorous method for the entire process of data reduction and interpretation. A generalized formulation utilizing contemporary methods of analysis has been carried out which now permits discussion in fundamental dynamical terms of physical librations of the moon and also of artificial satellites in arbitrary elliptic orbits. Among the features of the lunar motion which were inaccessible on the earlier theory is the dependence of the Cassini motion upon given orbital characteristics (eccentricity and inclination) and inclination of the lunar equator to the ecliptic; this limitation is now removed. Restrictions on small-amplitude librations are also relaxed, both in relation to libration mode coupling and the role of arbitrary principal moment of inertia differences. Finally the complete dynamical basis of Cassini motion is demonstrated with due regard for significant but earlier ignored effects of relative motion of lunar centroid. Publication: The Astronomical Journal Pub Date: 1965 DOI: 10.1086/109584 Bibcode: 1965AJ.....70R.144M full text sources ADS |

On the libration orbits of a particle near the triangular point in the semirestricted three-body problem

On the libration orbits of a particle near the triangular point in the semirestricted three-body problem

High-Contrast Electronic Image-Orthicon Techniques Applied to the Lunar Surface.

The current interest in the structure and mapping of the lunar surface has led to the increased use of various methods of investigating the lunar surface, e.g., photographic stereoscopy over the entire range of lunar libration and the sequential photometry of lunar shadows, as presently employed by the Army and Air Force, respectively. The advent of high-contrast image-orthicon techniques appears to offer a valuable and versatile variation of the conventional point-by-point photometry of the lunar surface. By a combination of the subtraction of a constant bac~ground and the subsequent variable expansion of the residual range of brightness, it is possible to emphasize relatively minor differences in contrast, without the loss of gray-scaling, nearly simultaneously over the entire lunar surface. The effects of the inevitable increase in noise attendant in this technique can be effectively minimized by the use of long-persistence phosphors on the viewing screen and the use of relatively long exposures with the recording camera. The application of these techniques is at present a part of the image conversion program at the Dearborn Ubservatory. They are particularly adaptable to the detection of minor differences in lunar albedo and are capable of delineating fine differences particularly in the lunar maria, especially at full lunar phase, when lac~ of shadows limits the effectiveness of conventional photography. The method, therefore, offers more promise in the detection of differences in texture, and by inference, composition, of the lunar surface rather than in contour determination. Movies illustrating a run across the full moon at constant contrast, the effects of variation of contrast discrimination near the subsolar point, and preliminary studies of relative photometry of line elements using the video signal itself, have been made.

The libration of a satellite: V. V. Beletskii: (pp. 13–31)

The libration of a satellite: V. V. Beletskii: (pp. 13–31)

Roughness of the Moon as a Radar Reflector

THE early work on radar reflexions from the Moon was carried out at metre wave-lengths with long pulses or with continuous wave giving a returned signal which was the total scattering from the lunar surface. Measurements showed that apart from a long-period fading produced by Faraday rotation of the linearly polarized waves in passing through the ionosphere, the reflected signal also had a short-period amplitude modulation produced by lunar libration. If a frequency analysis is carried out on these more rapid fluctuations, the different frequency components can be related to the Doppler frequency from the various regions on the Moon1, and in this way Evans2 indicated that most of the radiation was scattered from a central part of the optical disk of a radius only one-third of the lunar radius. More recently, with radars of higher peak power and µsec. pulse-lengths, it has been possible to confirm this result by direct measurement over the wave-length range of at least 10 cm. to 1.5 m. (refs. 3–5).

Topocentric Librations for an Eclipse Station

Topocentric Librations for an Eclipse Station

On Critical Terms associated with the Libration

On Critical Terms associated with the Libration

(1) An Introductory Treatise on Dynamical Astronomy (2) The Destinies of the Stars

(1) THERE has long been a need for a general book on celestial mechanics on a smaller scale and at a more accessible price than the standard work of Tisserand, and Prof. Plummer's recent publication is a very successful effort to satisfy that need. It is so concisely written that a most remarkable amount of material is made available, within a limited space, with only occasional loss of clearness. Halphen's theorem, that if the acceleration of a particle is a function of its position alone, and all the trajectories are plane curves, then the acceleration is always directed towards a fixed point, forms the commencement of the problem of two bodies. This constitutes a most welcome innovation, as Kepler's second law now becomes a consequence of the first, and its truth a confirmation of it. The methods of determining from observation the orbits of planets, comets, and visual and spectroscopic binaries are treated in detail, and then the author passes on to the treatment of perturbations, which is dealt with much as usual; but it is pleasant to see that chap, xvi., on secular perturbations, includes a table of numerical results. Very useful features are the account of recent work on methods of numerical interpolation and integration, and the description of Cowell and Crommelin's method of computing special perturbations. The lunar theory and the theories of precession, nutation, and lunar libration are also discussed in some detail. Misprints are few and the index is good. The present writer would like to suggest, however, that in a future edition tables of the best available values of the elements of the solar system, in astronomical and metrical units, should be included; for these are the fundamental data of dynamical astronomy, and are not usually presented in any convenient form. (1) An Introductory Treatise on Dynamical Astronomy. By Prof. H. C. Plummer. Pp. xx + 344. (Cambridge: At the University Press, 1918.) Price 18s. net. (2) The Destinies of the Stars. By Prof. Svante Arrhenius. Authorised translation from the Swedish by J. E. Fries. Illustrated. Pp. xviii + 256. (New York and London: G. P. Putnam's Sons, The Knickerbocker Press, 1918.) Price 7s. 6d. net.

Libration and the asteroids

Libration and the asteroids

On planetary librations

On planetary librations

Libration of Venus and Mercury

Libration of Venus and Mercury

Our Astronomical Column

A CAUSE OF LUNAR LIBRATION.—A paper by Mr. S. E. Peal, “On a Possible Cause for Lunar Libration other than an Ellipsoidal Figure, and on Lunar Snow Mountains,” has recently been published by Messrs. Dulau and Co. It is shown that evidence of several kinds points to the existence of a vast shoal, or submerged continent, some 1500 miles long by 400 across along the prime meridian. This is presumed to be of greater specific gravity than the refrigerated maria east and west of it, and to have been at one time situated in the southern hemisphere. The difference of attraction upon the shoal and the surrounding maria is shown to be sufficient both to cause and maintain libration. Since libration began, the shoal has placed itself geocentrically, in which case the south pole must have been drawn forward about 30°. The possibilities of the case seem to be as follows. The moon formerly had a physical constitution the same as that of the earth at the present time. The lunar ocean beds were then steadily subsiding, the lines of upheaval and weakness being on the continents, and causing a series of quasi-volcanic orifices. Whilst tidal friction was reducing the velocity of rotation, polar snow-caps were formed, and the atmosphere became rarer. The extension of the snow-cap to the equator was for ages prevented by the incidence of solar heat. This struggle between steadily-increasing refrigeration and solar heat should therefore be evidenced by the existence of an irregular belt about the (then) equator. Such a belt is found in the circular maria Smythii, Crisium, Serenitatis Imbrium, and part of Oceanus Procellarum. If the axis of rotation be shifted about 3O°, so that the south pole occurs near Nach or Maginus, all these irregular maria form a chain of seas along the equator, which may represent the belt of solar influence referred to. Eventually these maria were refrigerated, and the meridional shoal, acting as a fixed tide during libration, caused the change in the direction of the axis of rotation, which shifted the belt of seas from their equatorial position to that at present occupied by them.