Note on tidal evolution of the Earth-Moon system

The need is pointed out of a re-discussion of the past tidal evolution of the Earth-Moon system as a boundary-value problem on the time-scale indicated by radiometric dating of lunar soils returned by successive space missions from different localities on the Moon's surface.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Tidal evolution in close binary systems

The aim of the present paper will be to give a mathematical outline of the theory of tidal evolution in close binary systems of secularly constant total momentum — an evolution activated by viscous friction of dynamical tides raised by the two components on each other. The first section contains a general outline of the problem; and in Section 2 we shall establish the basic expressions for the energy and momenta of close binaries consisting of components of arbitrary internal structure. In Section 3 we shall investigate the maximum and minimum values of the energy (kinetic and potential) which such systems can attain for given amount of total momentum; while in Section 4 we shall compare these results with the actual facts encountered in binaries with components whose internal structure (and, therefore, rotational momenta) are known to us from evidence furnished by the observed rates of apsidal advance.The results show that all such systems — be these of detached or semi-detached type — disclose that more than 99% of their total momenta are stored in the orbital momentum. The sum of the rotational momenta of the constituent components amounts to less than a percent of the total — a situation characteristic of a state close to the minimum energy for given total momentum. This appears, moreover, to be true not only of the systems with both components on the Main Sequence, but also of those possessing evolved components in contact with their Roche limits.Under such conditions, a synchronism between rotation and revolution (characteristic of both extreme states of maximum and minimum energy) is not only possible, but appears to have been actually approached — if not attained — in the majority of cases. In other words, it would appear that — in at least a large majority of known cases — the existing close binaries have already attained orbits of maximum distension consistent with their momenta; and tidal evolution alone can no longer increase the present separations of the components to any appreciable extent.The virtual absence, in the sky, of binary systems intermediate between the stages of maximum and minimum energy for given momentum leads us to conjecture that the process of dynamical evolution activated by viscous tides may enroll on a time-scale which is relatively short in comparison with their total age — even for systems like Y Cygni or AG Persei, whose total age can scarcely exceed 10⁷ yr. A secular increase of the semi-major axes of relative orbits is dynamically coupled with a corresponding variation in the velocity of axial rotation of both components through the tidal lag arising from the viscosity of stellar material. The differential equations of so coupled a system are given in Section 5; but their solution still constitutes a task for the future.The Lunar Science Institute Contribution No. 90. The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration.     

Computation of the elements of eclipsing binary systems in the frequency-domain

The aim of the present paper will be to translate the essential parts of the theory of Fourier analysis of the light changes of eclipsing variables into more practical terms; and describe procedures (illustrated by numerical examples) which should enable their users to obtain the desired results with maximum accuracy and minimum loss of information by processes which can be fully automated.In order to unfold in steps how this can be done, the scope of the present paper-the first of two-will be restricted to an exposition of the analysis of light changes caused by eclipses of spherical stars; while between minima due to this cause the light of the system should remain sensibly constant. An extension of our analysis to incorporate photometric effects arising from mutual distortion of the components of close eclipsing systems between minima as well as within eclipses is being postponed for the second communication.In developing this subject we shall single out for the user's attention only those parts of the whole theory which are of direct relevance to practical work. Their justification can be largely found in sources already published; and new developments essential for our work, not yet made public, will be relegated to several Appendices at the end of the text, in order not to render its text too discursive and deflect the reader's attention from the main theme of its narrative.After a brief outline of the subject given in Section 1, Section 2 will introduce the reader to practical aspects of the Fourier analysis of the light curves; and Section 3 will be devoted to its use to determine the numerical values of the momentsA _2m of the light curves which constitute the cornerstones for all subsequent work. Section 4 will describe an algebraization of the process of determination of the elements for the case of total (annular) eclipses; while Section 5 will do the same for partial eclipses. The concluding Section 6 will be devoted to an error analysis of our problem, and to an outline of the way by which the errors of the individual observations will compound to the uncertainty of the final results. Lastly, Appendices 1–5 concluding the paper will contain additional details of some aspects of our work, or proofs of new processes made use of to obtain our results, whose earlier inclusion would have made the main text too discursive.     

Moments of inertia of the lunar globe,and their bearing on chemical differentiation of its outer layers

It is pointed out that the observed moments of inertia of the Moon, disclosed by its librations, are influenced mainly by the distribution of mass in the outer zone in which the lithostatic pressure is less than 10 kb (i.e., in the outer shell not more than 200 km deep); and a conspicuous departure of such moments from those expected in hydrostatic equilibrium disclosed that these layers could never have been fluid. In the same way, the actual shape of the lunar surface cannot represent a solidified surface of a fluid, petrified at any distance from the Earth.The shape of the Moon, and differences of its moments of inertia must reflect the way in which the initial process of cold accretion fell short of producing a globe with strictly spherically-symmetrical stratification of material; and has nothing to do with tides - present or fossil. Such melting or lava flows as may have occurred at the Moon's surface from time to time must have remained localized, and without much effect on the dynamical properties of the Moon. A global ocean of molten magma some 200 km in depth (postulated sometimes to provide a reservoir in which the differentiation of elements exhibited by surface rocks could have taken place) at any time in the past is incompatible with the dynamical evidence on the motion of the Moon about its center of gravity.Bellcomm, Inc., 955 L'Enfant Plaza North, S.W. Washington, D.C. 20024, U.S.A.     

Geochemistry and mineralogy of Platinum-group elements in the Ransko gabbro–peridotite massif,Bohemian Massif (Czech Republic)

The Ransko gabbro-peridotite massif in Eastern Bohemia is a strongly differentiated intrusive complex of Lower Cambrian age. The complex hosts low grade Ni-Cu ores mainly developed close to the contact of olivine-rich rocks with gabbros, in troctolites and, to a much lesser extent, in both pyroxene and olivine gabbros and plagioclase-rich peridotites. The ore zone is characterized by strong serpentinization and uralitization. The total Ni + Cu locally reaches up to 4 wt%. Anomalous concentrations of platinum-group elements (PGE's) (maximum 532 ppb Pd, 182 ppb Pt, 53 ppb Rh, 15 ppb Ru, 41 ppb Ir) were detected in samples of Cu-Ni and Ni-Cu ores (maximum 2.63 wt% Ni and 2.31 wt% Cu) from the Jezírka orebody. The main ore paragenesis includes pyrrhotite, pentlandite, chalcopyrite, cubanite, pyrite, magnetite, mackinawite, valleriite, ilmenite and sphalerite. During this work, michenerite, froodite, sperrylite, gold, native bismuth, altaite, tsumoite, hessite, an unnamed Bi-Ni telluride, cobaltite-gersdorffite and galena were newly identified. The host rocks originated through partial melting of a slightly depleted mantle source with noble metals scavenged from this primitive magma prior to the development of these rocks.     

Seismic Activity Around and Under Krakatau Volcano,Sunda Arc: Constraints to the Source Region of Island Arc Volcanics

There is general agreement that calc-alkaline volcanic rocks at convergent plate margins are genetically related to the process of subduction (Ringwood, 1974; Maaloe and Petersen, 1981; Hawkesworth et al., 1997). However, opinions on the mode and site of generation of primary magma for island arc volcanism differ substantially. The site of generation of calc-alkaline magma is thought to be either in the mantle wedge (Plank and Langmuir, 1988; McCulloch and Gamble, 1991) or in the subducting slab (White and Dupré, 1986; Defant and Drummond, 1990; Edwards et al., 1993; Ryan and Langmuir, 1993). We present seismological evidence in favour of the latter concept. A distinctive seismicity pattern around and under the Krakatau volcano was identified during systematic studies of the SE Asian convergent plate margins by means of global seismological data. A column-like cluster of events, probably associated with the dynamics of the volcano, is clearly separated from the events in the Wadati-Benioff zone. The accuracy of hypocentral determinations of the events of the cluster does not differ from the accuracy of the events belonging to the subducting slab. The depths of the cluster events vary from very shallow to about 100 km without any apparent discontinuity. On the other hand, there is a pronounced aseismic gap in the Wadati-Benioff zone directly beneath the volcano at depths between 100-150 km. The Krakatau cluster connects this aseismic gap to the volcano at the surface. The pervasive occurrence of earthquakes in the continental wedge between the subducting slab and the Earth surface bears witness to the brittle character of the continental lithosphere and casts doubt on the existence of large-scale melting of mantle material. The aseismic gap (Hanu and Vank, 1985), interpreted by us as a partially melted domain occurring in subducted slabs in practically all active subduction zones that reach depths greater than 100 km, is here used as evidence for the location of the primary source region of island arc volcanics in the subducting plate.     

The Role of the Magnetic Field in the Ascension of the 22nd and 23rd Cycles of Activity

The first small active regions of the last two new cycles seem to be formed, in its very first, preparatory stage, close to the equator with uncertain magnetic orientation, and as secondary products at the periphery of equatorial `magnetic bubbles' of the new magnetic flux still belonging to the old cycle. At the real beginning of the new cycle, at the periphery of its higher-latitude `magnetic bubbles', the opposite applies to the secondary regions of the ending cycle. It seems that two modes of magnetic flux supply the visible layers of the photosphere during the cycle transition phase: the emergence of the magnetic flux from the depth of the convective zone, and the mutual interaction of the earlier emerging fluxes, or the action of the `local dynamo', giving rise to the new local magnetic field concentrations.     

Topex/Poseidon Altimetry and Dynamics of the Ocean-Atmosphere System

The T/P altimeter data 1993 – 1997 (cycles 11 – 194) has been analyzed with emphases on seasonal variations in sea surface topography (SST). The amplitude of the annual variations amounted to (5.9±0.3) mm when inverted barometer (IB) corrections were applied and (2.0±0.4) mm without any IB corrections. The amplitude of the semi-annual variations in SST was small with IB corrections applied: (0.6±0.3) mm. However, when no IB corrections were applied, it was (1.8±0.4) mm, i.e. the semiannual variations are at the same level as the annual variations with no IB corrections. The phase angle offset of the annual term has shifted by about 180° when IB correction was applied. The dynamics of the ocean-atmosphere system is discussed and it is concluded that it could, at least partly, be responsible for the above observed effects.