The physical librations of the moon,including higher harmonic effects

The equations of the physical libration of the Moon are developed using a representation of the Earth-Moon orbit as a Kepler ellipse referred to the lunar equator and expanding the lunar potential in terms of these Kepler elements. TheImproved Lunar Ephemeris is used to calculate solar perturbations, and a linear integration of all effects arising from lunar gravitational harmonics through the fourth degree performed. Aside from unobservable constant offsets of the principal axes, the main effects of the higher harmonics on longitude are: 10 six-yearly (argument), 1.2 three-yearly, 0.5 annual, and 0.1 monthly; on pole direction they are on the order of 0.5 six-yearly and 1.0 monthly. The higher harmonics must hence be taken into account in analyzing ranging data of 10 cm accuracy.Paper dedicated to Prof. Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

New orbital elements for Moon and planets

The results of a simultaneous solution for the orbital elements of Moon and planets are given and their derivation is discussed. A modern Cowell integrator is used for orbit computations, and least-squares fits are made to some 40000 optical observations taken since 1913. The model includes relativistic terms, the leading zonal harmonics of Earth and Moon, the precession of the lunar equator, and the tidal couple between Earth and Moon. The tidal term in the Moon's mean longitude is found to be –19±4 per century squared. The solution also yields an extrapolation of the atomic time scale back to 1912.5. At that time, the difference between atomic and ephemeris time is about 6±2 s. Lunar declinations observed by the Washington transit circles, after receiving limb corrections and thus with respect to the center of Watts' reference sphere, are smaller than computed values by 0.33±0.01. It is found that solar oblateness cannot quite be determined with optical data covering about 50 yr, butJ ₂ is unlikely to be much larger than 10^–5. The advance of Mercury's perihelion is verified to within our resolution of 2 per century to match that predicted by Einstein.The solution presented here is believed to be the only simultaneous improvement of the orbits of Moon and planets. This simultaneity is found to be an essential feature in separating the Moon's mean motion, the lunar tidal deceleration, and the corrections to the Earth rotation rate. It is now possible to refer all astronomical events of the past 60 yr to a time with uniform rate, namely the atomic clock system. Considering the long baseline, this model should facilitate the prediction of fast variables, such as the lunar longitude, with considerably increased confidence. The planetary orbital elements compete with efforts of similar scope and accuracy at the Massachusetts Institute of Technology and the Jet Propulsion Laboratory.     

Some effects of elasticity on lunar rotation

A general Hamiltonian for a rotating Moon in the field of the Earth is expanded in terms of parameters orienting the spin angular momentum relative to the pricipal axes of the Moon and relative to coordinate axes fixed in the orbital plane. The effects of elastic distortion are included as modifications of the moment of inertia tensor, where the magnitude of the distortion is parameterized by the Love numberk ₂. The principal periodic terms in the longitude of a point on the Moon due to variations of the tide caused by the Earth are shown to have amplitudes between 3.9 × 10^–3 and 1.6 × 10^–2 with a period of an anomalistic month, 3.0 × 10^–4 and 1.2 × 10^–3 with a period of one-half an anomalistic month and 2.4 × 10^–4 and 9.6 × 10^–4 with a period of one-half of a nodical month. The extremes in the amplitudes correspond to rigidities of 8 × 10¹¹ cgs and 2 × 10¹¹ cgs, respectively, the former rigidity being comparable to that of the Earth. Only the largest amplitude given above is comparable to that detectable by the projected precision of the laser ranging to the lunar retrorereflectors, and this amplitude corresponds to an improbably low rigidity for the Moon. A detailed derivation of the free wobble of the lunar spin axis about the axis of maximum moment of inertia is given, where it is shown that elasticity can alter the period of the free wobble of 75.3 yr by only 3 × 10^–4 to 10^–3 of this period. Also, the effect of elasticity on the period of free libration is completely negligible by many orders of magnitude. If the Moon's rigidity is close to that of the Earth there is no effect of elasticity on the rotation which can be measured with the laser ranging and, therefore, no elastic properties of the Moon can be determined from variations in the rotation.Currently on leave from the Dept. of Physics, University of California, Santa, Barbara, California.Communication presented at the conference on Lunar Dynamics and Observational Coordinate Systems held January 15–17, 1973 at the Lunar Science Institute, Houston, Tex., U.S.A.     

Size of the hard X-ray source in the Crab Nebula

The size of the hard X-ray source in the Crab Nebula was observed with scintillation counters on board two balloons at a lunar occulatation on 24 January, 1975. The Gaussian width of the source is 34 (+17, –14) and the center thereof is offset from the pulsar by 6±4 at position angle 102°. The observed time profile can also be fitted to an alternative model of two line sources whose intensities are 48% at 11 and 25.5% at 7 on both sides of the pulsar.Paper presented at the COSPAR Symposium on Fast Transients in X- and Gamma-Rays, held at Varna, Bulgaria, 29–31 May, 1975.     

Lunar physical librations due to rotation of the ecliptic

Irregularities of the Moon's rotation due to rotation of the ecliptical plane were derived for all three libration components. Optimum values of coefficients in the linearized equations of motion, which strongly affect the magnitude of the resulting terms, were determined by comparing the rotation relative to the moving and to the fixed reference ecliptical system. Ecliptic rotation causes monthly oscillations of the Moon's polar axis of 6.1 and deviations of 0.3 from uniform rotation along that axis with a period of 18.6 yr.     

Note on the Earth-figure perturbations in the lunar theory

Thej=2 lunar ephemeris is based on a flattening of the Earth which is slightly different from the 1964 IAU recommended value. The total effects of Earth-oblateness on the analytical lunar theory are determined, and the corrections, which are about 0.02 in lunar longitude and latitude, are derived.     

Free librations of the moon determined by an analysis of laser range measurements

Among the lunar laser range measurements obtained during the past six years at McDonald Observatory, those available cover the period August 1969-November 1974, being 1377 normal observations made on the three Apollo reflectors and that of Lunokhod II. The fit of these data led to a rms residual of 55 cm. In this study, a large number of parameters have been resolved, including the geocentric coordinates of the telescope, the selenocentric coordinates of each of the reflectors, as well as orbital elements of the Moon. In addition, the interest has been directed more specially towards the study of the rotational motion of the Moon and particularly the problem of its free librations. The performed resolutions give the evidence of the three modes of free oscillations. The determined amplitudes arise to 1.7 in longitude and 0.5 and 8.7 in latitude, with the respective periods of 2.9 years, 27.3 days and 75 years. In connection with these parameters, the fittings determined also the most of part of elements of lunar gravitational field: the moment of inertia parameters and, and a number of the third degree harmonics. These new results should now permit a research on the implications of these oscillations effects, concerning the impact history of the Moon and the properties of its internal structure. On the basis of the amplitudes determined here, one can already estimate an order of the magnitude for theQ dissipation coefficient comparable with that determined from seismic studies of the Moon.     

Perturbations by the oblateness of the Earth and by the planets in the motion of the Moon

Perturbations in the motion of the Moon are computed for the effect by the oblateness of the Earth and for the indirect effect of planets. Based on Delaunay's analytical solution of the main problem, the computations are performed by a method of Fourier series operation. The effect of the oblateness of the Earth is obtained to the second order, partly adopting an analytical evaluation. Both in longitude and latitude are found a few terms whose coefficient differs from the current lunar ephemeris based on Brown's theory by about 0.01. While, concerning the indirect effect of planets, several periodic terms in the current ephemeris seem to have errors reaching 0.05.As for the secular variations of and due to the figure of the Earth and the indirect effect of planets, the newly-computed values agree within 1/cy with Brown's results reduced to the same values of the parameters. Further, the accelerations in the mean longitude, and caused by the secular changes in the eccentricity of the Earth's orbite and in the obliquity of the ecliptic are obtained. The comparison with Brown shows an agreement within 0.3/cy² for the former cause and 0.02/cy² for the latter. An error is found in the argument of the principal term for the perturbations due to the ecliptic motion in the current ephemeris.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

CASPIR: A cryogenic array spectrometer imager for the MSSSO 2.3 m telescope

CASPIR is a near-infrared spectrometer/imager being built for the Mount Stromlo and Siding Spring Observatories' 2.3 m telescope. The instrument is based on a SBRC 256×256 InSb detector array and uses AR-coated Sapphire, MgO, CaF₂, and BaF₂ optics to produce two imaging focal plane scales with 0.5/pixel and 0.25/pixel. Spectral resolving powers of 500 will be achieved through a 1×128 slit with three grisms designed for the J, H, and K bands. IJ, JH, and HK cross-dispersed échelle grisms will achieve resolving powers of 1100 through a 1×15 slit. Coronograph and imaging polarimetry modes will also be available. The various observing configurations are selected via five remotely controlled wheels. The instrument design and system architecture are discussed, and preliminary detector performance figures reported.     

Morphological properties and origin of the photospheric facular granules

From time series of high resolution photographs, morphological properties of the photospheric facular granules were derived. The facular granules are cells of the common granular pattern, brighter than the normal granules when seen between cos = 0.6 and the limb. Their apparent diameter, which decreases towards the limb, is smaller than that of the normal granules: 0.65 and 1.25 respectively at cos = 0.55; their lifetime is 25 min but their bright stage lifetime is only 15 min; they are visible closer to the limb than the normal granules: 1.2 compared to 2–5; the brightening of the facular granules occurs at a faster rate than their fading. From the great similitude of both morphological properties and temperature models of facular and normal granules, it appears possible that the photospheric facular granules are convective cells modified by the presence of a magnetic field of some hundreds Gauss.     

The solar minimum temperature determined by the CO (A 1Π-X 1Σ)

The synthetic Voigt profile of the following transitions (v=0,v=0), (v=0,v=1), (v=1,v=1), (v=1,v=0) have been computed for different concentrations and temperatures of CO and compaed to the measured intensities of the UV sunspot spectrum by a high resolution spectrograph. From this comparison the solar minimum temperature has been determined.     

Automatic solar image motion measurements

The solar seeing image motion has been monitored electronically and absolutely with a 25 cm telescope at three sites along the ridge at the southern end of the Magdalena Mountains west of Socorro, New Mexico. The uncorrelated component of the variations of the optical flux from two points at opposite limbs of the solar disk was continually monitored in 3 frequencies centred at 0.3, 3 and 30 Hz. The frequency band of maximum signal centred at 3 Hz showed the average absolute value of image motion to be somewhat less than 2 although wide variations from 20 to an extraordinarily quiet day of less than the measurement limit of 1/2 were observed. The observer estimates of combined blurring and image motion were well correlated with electronically measured image motion, but the observer estimates gave a larger value × 2 presumable because the electronic measurement gave only the uncorrelated motion of opposite limbs. Approximately 30% of the total solar time would allow spatial position measurements of solar features to a precision 2 and, from the visual estimates, blurring limited measurements to a precision 4.     

The secular acceleration of the Moon determined from lunar laser ranging data

Lunar Laser Ranging data covering the interval from August 1969 to December 1987 were used to determine the seculer acceleration in the mean longitude of the Moon (\.n). In our analysis, the DE200/LE200 planets and lunar ephemerides were adopted for calculating the theoretical distance between the observing station and reflector. The method of stepwise regression was used in the processing of the data and the value of –25.4 ± 0. 1/cy² was obtained by a weighted least squares fit.Our result is in good agreement with that derived by other authors using various methods. The uncertainty (\.n) estimated from LLR data would be decreasing rapidly with increasing the data span. The high precision obtained in this paper is mainly due to the longer span and higher measuring accuracy of data.     

On the possible amplitudes of the moon's free libration

The relations between probable amplitude of the Moon's free libration and its exciting action are obtained using a theory of the stochastic process. Among the three modes of the free libration, the one in longitude is most prominent if the exciting effect is isotropic. If the excitations are caused by crater-producing meteoritic impacts, it is very probable that the free librations in longitude exist whose semi-amplitude exceeds 1.     

The solar O i λ 7773 triplet

The reduction of observations of the O i 7773 triplet obtained at high spatial resolution (0.5) at two disk positions is described. Two sets of triplet profile data are presented at each disk position. One set represents data taken from 0.5 regions centered on the brightest granules, while the other set represents data taken from 0.5 regions centered on the cooler infalling intergranular material.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

Theory of the rotation of the rigid earth

An analytical theory is developed for planes normal to the angular-momentum axis, to the figure axis, and to the rotational axis of the triaxial rigid Earth. One of the purposes of this paper is to determine the effect on nutation and precession of Eckertet al.'s improvement to Brown's tables of the Moon and to check Woolard's theory from a different point of view. The present theory is characterized by the use of Andoyer variables, a moving reference plane, and Hori's averaging perturbation method. A comparison with Woolard's results shows that (1) the maximum difference in nutation for the plane normal to the angular-momentum axis, calculated from the same constants as Woolard adopted, reaches 0.0017, (2) the discrepancy in Oppolzer terms is large compared with the discrepancy in nutation for the plane normal to the angular-momentum axis, and (3) the present theory does not include some of the secular terms that are incorporated in Woolard's theory and that have an effect on the establishment of a reference system. The nutation coefficients 0.0001 for the three above-mentioned planes are calculated by using the numerical values recommended at the working meeting of the International Astronomical Union held in Washington in September 1974. The effects on precession and nutation due to the higher geopotential (n3) are also investigated. Any future revision of the lunar theory will not alter the values of the coefficients of the nutational terms derived here.     

Theorie des planetes inferieures

Resumé La construction de théories planétaires a été entreprise au Bureau des Longitudes pour l'ensemble du système solaire. Il s'agit de théories semi-analytiques à variations séculaires ce qui signifie que les termes à longues périodes (périodes des périhélies et des noeuds comprises entre 50 000 ans et 2 000 000 d'années) ont été développés par rapport au temps. Ce sont donc des théories du type de celles de Le Verrier ou de Newcomb.Les théories de Le Verrier et de Newcomb ont une précision interne d'environ 0,1 pour les planètes inférieures. La théorie de Le Verrier-Gaillot a une précision interne de quelques secondes pour les grosses planètes. Mais les constantes d'intégration de, ces théories, la dégradation des éléments moyens due à l'imprécision des termes séculaires calculés font que la précision réelle est comprise entre quelques 0,1 et plusieurs secondes. La précision de la théorie du soleil de Newcomb, par exemple, est de l'ordre de 0,8.Les objectifs que nous nous sommes fixés sont d'atteindre en précision; pour les planètes inférieures 0,001 sur plusieurs siècles; pour les grosses planètes 0,01 sur un siècle, 0,1 sur 1000 ans.Cela implique de déterminer les perturbations au moins jusqu'a l'ordre 3 des masses pour les planètes inférieures et jusqu'à l'ordre 6 pour les grosses planètes.

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Theory of the inner planets

In the contruction of planetary theories for the whole of the solar system undertaken at the Bureau des Longitudes, the aim is to obtain the precision of: for the inner planets 0.001 over several centuries; for the outer planets 0.01 over one century, 0.1 over 1000 years. To get these precisions one must compute the perturbations at least to the 3rd order of the masses for the inner planets and to the 6th order of the masses for the outer planets.We have used an iterative method which has given the perturbations up to the 6th order of the masses for the outer planets and a method working order after order with respect to the masses. Through the latter, we have built the perturbations up to the 3rd order with respect to the masses for all the planets.In the mean longitudes the precision now obtained is of 0.0005 for Mercury, 0.0030 for Venus and the Earth and 0.0047 for Mars.For Mercury, the obtained precision is about 130 meters. One has therefore to introduce besides the advance of the perihelium due to relativity, the periodic relativistic corrections, whose amplitude is over 3000 meters for that planet.We have completed our theory of the Earth-Moon barycenter by the relativistic effects, as well as by the perturbations due to the Moon. As a whole, our solution is about 100 times better than that of Newcomb. Our solution for the variablesq andp of the Earth shows that the equinox is moved by a periodic motion of 0.04 amplitude and with a period of 883 year-a thing not considered generally.the precision of our solution for the mean longitude of Mars is 0.0047, which means a real progress over the theory of Clemence. We have indeed noticed many arguments missing in that theory we last mentioned. For the years to come we intend to replace to theories of Le Verrier by these solutions in the ephemerides published by the Bureau des Longitudes. Beforehand we are going to improve the constants of integration by a comparison to numerical integrations or directly to observation.

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Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

Global astrometry by space techniques

The European Space Agency project of an astrometric satellite-HIPPARCOS-is shortly described. It will measure the angles between stars situated in fields separated by about 70°. The precision of the elementary measurements is expected to be of the order of 0.005. A similar accuracy is found to apply to the basic reduction giving the abcissae of stars referred to great circles on the sky. The final overall reduction should yield accuracies better than 0.002 in position and parallaxes and 0.002 per year in proper motions.The main features of the final catalogue are described and some possible consequences for fundamental astronomy are given.Presented at the Symposium Star Catalogues, Positional Astronomy and Celestial Mechanics, held in honor of Paul Herget at the U.S. Naval Observatory, Washington, November 30, 1978.     

Perturbations due to the shape of the Moon in lunar theory

We compute the perturbations on the motion of the Moon due to its shape. The accuracy is estimated at 1.10^–5 in longitude and latitude and 5 parts in 10¹¹ in distance.