Empiric models of the Earth’s free core nutation

Free core nutation (FCN) is the main factor that limits the accuracy of the modeling of the motion of Earth’s rotational axis in the celestial coordinate system. Several FCN models have been proposed. A comparative analysis is made of the known models including the model proposed by the author. The use of the FCN model is shown to substantially increase the accuracy of the modeling of Earth’s rotation. Furthermore, the FCN component extracted from the observed motion of Earth’s rotational axis is an important source for the study of the shape and rotation of the Earth’s core. A comparison of different FCN models has shown that the proposed model is better than other models if used to extract the geophysical signal (the amplitude and phase of FCN) from observational data.     

New problems in nutation calculation

The effects of Oppolzer terms on the determination of latitude variation, polar motion and declination are discussed. We suggest that the celestial pole may be defined as the intersection of the rotation axis, corrected for the lunisolar polar motion with the celestial sphere. Or approximately, it may be defined as the intersection of the diurnal mean direction of rotation axis with the celestial sphere.     

Tides and nutation of the Earth: II. Realistic models of the liquid core

In the first part of this study (Molodensky, 2004; hereinafter, paper I), a survey of the theory of tides and nutation of the Earth was given with the inclusion of the dynamical effects of the elastic mantle and liquid core in the framework of a very simple model of a homogeneous, incompressible core. Here, effects of the inhomogeneity, compressibility, and viscosity of the liquid core are considered, along with those of electromagnetic coupling of the liquid core with the mantle and solid inner core. Errors in the present-day measurements of the amplitudes of forced nutation (of the order of 20 arc s) are well below not only the dynamical effects of the Earths liquid core but also the effects of the inelasticity of the mantle, the dynamical effects of the solid inner core, and the possible effects of electromagnetic coupling between the liquid core, solid inner core, and mantle. This opens up new avenues for astrometric studies of the mechanical properties and electrical conductivity of the mantle and solid inner core at very low frequencies. The modern data on the amplitudes and phases of the Earths forced nutation cannot be interpreted entirely unambiguously, since the following factors remain unknown: (i) the role of the core-mantle electromagnetic coupling compared to the coupling due to core viscosity near the core-mantle boundary and (ii) the frequency dependence of the creep function of the mantle at low frequencies. In large measure, the effects of electromagnetic and viscous coupling can be separated if high-precision data on the tidal variations in the gravitational force at periods of about a day are invoked and allowances are made for the differences between the effects of viscous and electromagnetic coupling on the amplitudes and phases of forced nutation and on the tidal variations in the gravitational force. Here, ranges of possible values of the creep function are constructed for periods from one hour to one day; for these calculations, values consistent with the entire set of data on the forced nutation of the Earth are assumed for the effective dynamical flattening of the liquid core-mantle and liquid core-solid inner core boundaries (these values describe the ellipticity of the boundaries of the inner core, outer core, and mantle, as well as the electromagnetic coupling between the liquid core, mantle, and solid inner core).Translated from Astronomicheskii Vestnik, Vol. 39, No. 1, 2005, pp. 61–80.Original Russian Text Copyright © 2005 by Molodensky.     

The precession and nutation of deformable bodies

The aim of the present paper will be to derive from the fundamental equations of hydrodynamics the explicit form of the Eulerian equations which govern the motion about the centre of gravity of self-gravitating bodies, consisting of compressible fluid of arbitrary viscosity, in an arbitrary external field of force. If the problem is particularized so that the external field of force represents the attaction of the sun and the moon, this motion would represent the luni-solar precession and nutation of a fluid viscous earth; if, on the other hand, the external field of force were governed by the earth (and the sun), the motion would define the physical librations of the moon regarded as a deformable body. The same equations are, moreover, equally applicable to the phenomena of precession and nutation of rotating fluid components in close binary systems, distorted by mutual tidal action; and the present paper contains the first formulation of the effects of viscosity on such phenomena.Investigation supported in part by the U.S. National Aeronautics and Space Administration under Contract No. NASW-1470.     

New estimates of the number of flare stars

New estimates are proposed for the number of flare stars. They are compared with Ambartsumian’s well-known estimate. Using the new estimates, it is found that flare stars of the same luminosity in the Orion association have the same average flare frequency. Translated from Astrofizika, Vol. 41, No. 1, pp. 73–80, January-March. 1998.     

An observational determination of the principal nutation constants

The longitude and obliquity components of the principal nutation terms were derived from observations made in 1955–1977 of 43 star-pairs common to the ILS Programmes VI and VII. The observations were reduced to a uniform system appropriate to these 43 star-pairs. The results are 6'.'850 for the longitude component and 9'.'212 for the obliquity component.     

Comparison of new nutation series with numerical integration

To check the results given by the new tables of the nutation for the rigid Earth model in (Kinoshita and Souchay, 1990), we perform a calculation of this nutation by numerical integration, using the basic equations of Kinoshita (1977). Results show very small long period residuals, with a discrepancy of 0.06 mas sin in longitude, -0.04 mas cos in obliquity, without out-of-phase components. Moreover, the total residuals of short period do not exceed 0.1 mas in longitude, 0.06 mas in obliquity. All these results are considerably better than those found by previous numerical integrations, thus confirming the validity of the new tables.     

Tides and nutation of the earth: I. Models of an earth with an inelastic mantle and homogeneous,inviscid, liquid core

The use of very-long-baseline radio interferometers during the past 10–15 years has increased the accuracy of amplitude measurements of the Earths forced nutation by more than two orders of magnitude (from 3–5 arc ms to 20 arc s). At the same time, cryogenic gravimeters (which depend for their action on the repulsion of two superconductive rings in a gravitational field) have made it possible to also improve the accuracy of measurements of tidal variations in the gravitational force by two orders of magnitude. This opens up new avenues for the investigation of the mechanical properties of the Earths interior at ultralow frequencies. A brief review is given here of the basic results of interpreting astrometric data and measurements of tidal variations in the gravitational force. Problems suggested by the new observational techniques are formulated.Translated from Astronomicheskii Vestnik, Vol. 38, No. 6, 2004, pp. 542–558.Original Russian Text Copyright © 2004 by Molodensky.     

The precession and nutation of deformable bodies,II

In a previous paper of this series (Kopal, 1968a) the Eulerian equations have been set up which govern the precession and nutation of selfgravitating bodies of viscous fluid in inertial coordinates which are at rest in space. In order to facilitate their solution, in the present investigation we shall transform these equations to the rotating body-axes; and shall explicitly evaluate all their coefficients arising as a result of second-harmonic dynamical tides.Following the introductory Section 1 which contains a mathematical statement of the problem, the requisite transformation of coordinates will be outlined in Section 2, and applied to the equations of motion in Section 5. The corresponding moments and products of inertia appropriate for selfgravitating configurations of arbitrary internal structure will be formulated in Section 4; while the deformation terms arising from second-harmonic dynamical tides raised on centrally-condensed configurations will be evaluated in Sections 3 and 6. The concluding Section 7 will then contain a specification of the components of the disturbing force.The next stage of our investigation — namely, a construction of the actual solutions of the equations governing precession and nutation of fluid bodies in different cases of astrophysical interest — has been postponed for a separate paper.     

The precession and nutation of deformable bodies,III

In preceding papers of this series (Kopal, 1968; 1969) the Eulerian equations have been set up which govern the precession and nutation of self-gravitating fluid globes of arbitrary structures in inertial coordinates (space-axes) as well as with respect to the rotating body axes; with due account being taken of the effects arising from equilibrium as well as dynamical tides.In Section 1 of the present paper, the explicit form of these equations is recapitulated for subsequent solations. Section 2 contains then a detailed discussion of the coplanar case (in which the equation of the rotating configuration and the plane of its orbit coincide with the invariable plane of the system); and small fluctuations in the angular velocity of axial rotation arising from the tidal breathing in eccentric binary systems are investigated.In Section 3, we consider the angular velocity of rotation about theZ-axis to be constant, but allow for finite inclination of the equator to the orbital plane. The differential equations governing such a problem are set up exactly in terms of the time-dependent Eulerian angles and , and their coefficients averaged over a cycle. In Section 4, these equations are linearized by the assumption that the inclinations of the equator and the orbit to the invariable plane of the system are small enough for their squares to be negligible; and the equations of motion reduced to their canonical form.The solution of these equations — giving the periods of precession and nutation of rotating components of close binary systems, as well as the rate of nodal regression which is synchronised with precession — are expressed in terms of the physical properties of the respective system and of its constituent components; while the concluding Section 6 contains a discussion of the results, in which the differences between the precession and nutation of rigid and fluid bodies are pointed out.     

Numerical derivation of forced nutation terms for a rigid earth

We compare the results of a numerical integration of the Euler equations for a rigid Earth model covering a time span of 250 years with Kinoshita's theory for the forced nutations and with a new nutation series by Kinoshita and Souchay. We also present numerical corrections to some of the analytically derived nutation terms.     

New age estimates of M31 globular clusters from multicolour photometry

The large majority of extragalactic star cluster studies performed to date essentially use multicolour photometry, combined with theoretical stellar synthesis models, to derive ages, masses, extinction estimates and metallicities. M31 offers a unique laboratory for studies of globular cluster (GC) systems. In this paper, we obtain new age estimates for 91 M31 GCs, based on improved photometric data, updated theoretical stellar synthesis models and sophisticated new fitting methods. In particular, we used photometric measurements from the Two Micron All Sky Survey (2MASS), which, in combination with optical photometry, can partially break the well-known age–metallicity degeneracy operating at ages in excess of a few Gyr. We show robustly that previous age determinations based on photometric data were affected significantly by this age–metallicity degeneracy. Except for one cluster, the ages of our other sample GCs are all older than 1 Gyr. Their age distribution shows populations of young- and intermediate-age GCs, peaking at ∼3 and 8 Gyr, respectively, as well as the 'usual' complement of well-known old GCs, i.e. GCs of similar age as the majority of the Galactic GCs. Our results also show that although there is significant scatter in metallicity at any age, there is a notable lack of young metal-poor and old metal-rich GCs, which might be indicative of an underlying age–metallicity relationship among the M31 GC population.     

Analytical developments of rigid Mars nutation and tide generating potential series

In this paper, series of rigid Mars nutations for the angular momentum axis, the rotation axis and the figure axis, as well as series of rigid Mars tide generating potential (TGP) are computed. The method used is based on the calculation of the forces produced by the external bodies on the rigid Mars. We have included the direct effect of the sun, Phobos and Deimos. We have also included the indirect effects associated with these bodies and planets of the Solar System which are given at the level of the ephemerides. For the nutation series, with a truncation level of 0.1mas (milliarcsecond), related to the present-day precision of the Martian precession constant, 24 terms in longitudes (*****) and 10 terms in obliquity (*****) are computed. The value of the dynamical flattening used is H _D=(C–A)/C=0.00536, derived from the value p _a=–7576±35mas/yr for the precession rate. Our results show a perfect agreement with those of Bouquillon and Souchay (1996) to the truncation level of Bouquillon and Souchary (1mas). For the TGP series, we have set the truncation level to 10^–6m²/s² which corresponds to an effect on the vertical acceleration on Mars surface of about 10^–12m/s²=0.1nanogal (1nanogal=10^–11m/s². With this truncation level, 134 terms are computed.     

New estimates of scale heights and spiral structures for non-edge-on spiral galaxies

On the basis of Poisson＇s equation for the logarithmic perturbation of matter density, we provide improved estimates of scale heights and spiral structures for non-edge-on spiral galaxies by subtracting the surface brightness distributions from observed images. As examples, the non-edge-on spiral galaxies PGC 24996, which is face-on, and M31, which is inclined, are studied. The scale height, pitch angle and inclination angle of M31, our nearest neighbor, that are presented in this work, agree well with previous research.     

New estimates for Io eruption temperatures: Implications for the interior

The initial interpretation of Galileo data from Jupiter's moon, Io, suggested eruption temperatures . Tidal heating models have difficulties explaining Io's prodigious heat flow if the mantle is , although we suggest that temperatures up to may be possible. In general, Io eruption temperatures have been overestimated because the incorrect thermal model has been applied. Much of the thermal emission from high-temperature hot spots comes from lava fountains but lava flow models were utilized. We apply a new lava fountain model to the highest reported eruption temperature, the SSI observation of the 1997 eruption at Pillan. This resets the lower temperature limit for the eruption from 1600 to . Additionally, viscous heating of the magma may have increased eruption temperature by as a result of the strong compressive stresses in the ionian lithosphere. While further work is needed, it appears that the discrepancy between observations and interior models is largely resolved.     

New estimates for the sublimation rate for ice on the Moon

The strong hydrogen signal that the Lunar Prospector saw at the Moon's poles suggests that water ice may be present near the surface of the lunar regolith. A robotic mission to obtain in situ samples and to quantify the amount of this valuable resource must be designed carefully to avoid dissipating too much heat in the regolith during coring or drilling and, thus, causing the ice to sublimate before it is processed. Here I use new results for the saturation vapor pressure of water ice to extend previous estimates of its sublimation rate down to a temperature of 40 K, typical of the permanently shaded craters near the lunar poles where the water ice is presumed to be trapped. I find that, for temperatures below 70 K, the sublimation rate of an exposed ice surface is much less than one molecule of water vapor lost per square centimeter of surface per hour. But even if a small ice sample (∼4 ng) were heated to 150 K, it could exist for over two hours without sublimating a significant fraction of its mass. Hence, carefully designed sampling and sample handling should be able to preserve water ice obtained near the lunar poles for an accurate measurement of its in situ concentration.     

The effects of viscous friction on the precession and nutation of celestial bodies

The aim of the present study has been to set the system of differential equations which govern the precession and nutation of self-gravitating globes of compressible viscous fluid, due to the attraction exerted on the rotating configuration by its companion; and to construct their approximate solution which are correct to terms of the second order in small dependent variables of the problem. Section 2 contains an explicit formulation of the effects of viscosity arising in this connection, given exactly as far as the viscosity remains a function of radial distancer only; but irrespective of its magnitude. In Section 3 the equations of motion will be linearized for the case of near-circular orbits and small inclinations andi of the equator of the rotating configuration, and of its orbital plane, to the invariable plane of the system; while in Section 4 further simplifications will be introduced which are legitimate for studies of secular (or long-periodic) motions of the nodes and inclinations. The actual solutions of so simplified a system of equations are constructed in Section 5; and these represent a generalization of the results obtained in our previous investigation (Kopal, 1969) of the inviscid case.The physical significance of the new results will be discussed in the concluding Section 6. It is demonstrated that the axes of rotation of deformable components in close binary systems are initially inclined to the orbital plane, viscous dissipation produced by dynamical tides will tend secularly to rectify their positions until perpendicularity to the orbital plane has been established, and the equators as well as orbit made to coincide with the invariable plane of the system-in a similar manner as other effects of tidal friction are bound eventually to synchronize the velocity of axial rotation with that of orbital revolution in the course of time.An application of the results of the present study to the dynamics of the Earth-Moon system discloses that the observed inclination of 1°.5 of the lunar equator to the ecliptic cannot be regarded as being secularly constant, but representing the present deviations from perpendicularity of oscillatory motion of very long period.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. This paper constitutes the Lunar Science Institute Contribution No. 85.