Secular tidal changes in lunar orbit and Earth rotation

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.     

Interpretation of vector magnetograph data including magneto-optic effects

In this paper, the presence of Faraday rotation in measurements of the orientation of a sunspot's transverse magnetic field is investigated. Using observations obtained with the Marshall Space Flight Center's (MSFC) vector magnetograph, the derived vector magnetic field of a simple, symmetric sunspot is used to calculate the degree of Faraday rotation in the azimuth of the transverse field as a function of wavelength from analytical expressions for the Stokes parameters. These results are then compared with the observed rotation of the field's azimuth which is derived from observations at different wavelengths within the Fei 5250 Å spectral line. From these comparisons, we find: the observed rotation of the azimuth is simulated to a reasonable degree by the theoretical formulations if the line-formation parameter η _o is varied over the sunspot; these variations in η _o are substantiated by the line-intensity data; for the MSFC system, Faraday rotation can be neglected for field strengths less than 1800 G and field inclinations greater than 45°; to minimize the effects of Faraday rotation in sunspot umbrae, MSFC magnetograph measurements must be made in the far wings of the Zeeman-sensitive spectral line.     

Relation of a contracting earth to the apparent accelerations of the sun and moon

The tidal theory of the evolution of the lunar orbit has remained inconsistent with the observational values of the apparent secular accelerations of the Sun and Moon since it was first developed by Jeffreys in 1920. Allowance for a changing moment of inertia of the Earth enables the discrepancy to be completely removed if a decrease is occurring at a rate of just about the amount already required by the phase-change theory of the nature of the terrestrial core. The agreement of the resulting theory with the latest determinations of the lunar acceleration increases confidence in the phase-change hypothesis. On the other hand the theory renders it most unlikely that a changing constant of gravitation will prove necessary to account for the observations. On the present theory of itself the Moon would have been extremely close to the Earth only about 10⁹ yr ago which suggests that some additional process may at times have influenced the lunar orbit.     

Variability of the dispersion measure of the pulsar NP 0532

It is hypothesized that NP 0532 is seen from the Earth through a filament of high density of the Crab nebula. The variability of the dispersion measure of the pulsar NP 0532 is explained as due to variation in ionization produced in the filament by varying soft X-ray flux of NP 0532.     

Distant bow shock observations by explorer 33

Locations, orientations and magnetic field changes are given for 135 bow shock crossings at distances downstream from Earth between 84 and 117 Earth radii. The shock locations bracket those calculated for the hypersonic analogue by Dryer and Heckman for a Mach number of 3.8. The shock normal vectors have been calculated using magnetic coplanarity. The average normal vectors have a greater inclination by ～17±5 deg from the symmetry axis than the Dryer and Heckman shock orientations for a 3.8 Mach number. Over a range of downstream distances from 60 to 115 Earth radii, the median magnetic field magnitude jump across the shock changes from 1.90 to 1.70 times.     

Atmospheric angular momentum variations of Earth, Mars and Venus at seasonal time scales

Atmospheric angular momentum variations of a planet are associated with the global atmospheric mass redistribution and the wind variability. The exchange of angular momentum between the fluid layers and the solid planet is the main cause for the variations of the planetary rotation at seasonal time scales. In the present study, we investigate the angular momentum variations of the Earth, Mars and Venus, using geodetic observations, output of state-of-the-art global circulation models as well as assimilated data. We discuss the similarities and differences in angular momentum variations, planetary rotation and angular momentum exchange for the three terrestrial planets. We show that the atmospheric angular momentum variations for Mars and Earth are mainly annual and semi-annual whereas they are expected to be “diurnal” on Venus. The wind terms have the largest contributions to the LOD changes of the Earth and Venus whereas the matter term is dominant on Mars due to the CO₂ sublimation/condensation. The corresponding LOD variations (ΔLOD) have similar amplitudes on Mars and Earth but are much larger on Venus, though more difficult to observe.     

Testing the Distance Scale of the Gaia TGAS Catalogue by the Kinematic Method

We have studied the simultaneous and separate solutions of the basic kinematic equations obtained using the stellar velocities calculated on the basis of data from the Gaia TGAS and RAVE5 catalogues. By comparing the values of Ω'₀ found by separately analyzing only the line-of-sight velocities of stars and only their proper motions, we have determined the distance scale correction factor p to be close to unity, 0.97 ± 0.04. Based on the proper motions of stars from the Gaia TGAS catalogue with relative trigonometric parallax errors less than 10% (they are at a mean distance of 226 pc), we have found the components of the group velocity vector for the sample stars relative to the Sun (U, V,W)_⊙ = (9.28, 20.35, 7.36) ± (0.05, 0.07, 0.05) km s^?1, the angular velocity of Galactic rotation Ω⁰ = 27.24 ± 0.30 km s?1 kpc?1, and its first derivative Ω'₀ = ?3.77 ± 0.06 km s^?1 kpc^?2; here, the circular rotation velocity of the Sun around the Galactic center is V₀ = 218 ± 6 km s^?1 kpc (for the adopted distance R₀ = 8.0 ± 0.2 kpc), while the Oort constants are A = 15.07 ± 0.25 km s^?1 kpc^?1 and B = ?12.17 ± 0.39 km s^?1 kpc^?1, p = 0.98 ± 0.08. The kinematics of Gaia TGAS stars with parallax errors more than 10% has been studied by invoking the distances from a paper by Astraatmadja and Bailer-Jones that were corrected for the Lutz–Kelker bias. We show that the second derivative of the angular velocity of Galactic rotation Ω'₀ = 0.864 ± 0.021 km s^?1 kpc^?3 is well determined from stars at a mean distance of 537 pc. On the whole, we have found that the distances of stars from the Gaia TGAS catalogue calculated using their trigonometric parallaxes do not require any additional correction factor.     

Origin and evolution of the atmospheres of early Venus,Earth and Mars

We review the origin and evolution of the atmospheres of Earth, Venus and Mars from the time when their accreting bodies were released from the protoplanetary disk a few million years after the origin of the Sun. If the accreting planetary cores reached masses \(\ge 0.5 M_\mathrm{Earth}\) before the gas in the disk disappeared, primordial atmospheres consisting mainly of H\(_2\) form around the young planetary body, contrary to late-stage planet formation, where terrestrial planets accrete material after the nebula phase of the disk. The differences between these two scenarios are explored by investigating non-radiogenic atmospheric noble gas isotope anomalies observed on the three terrestrial planets. The role of the young Sun’s more efficient EUV radiation and of the plasma environment into the escape of early atmospheres is also addressed. We discuss the catastrophic outgassing of volatiles and the formation and cooling of steam atmospheres after the solidification of magma oceans and we describe the geochemical evidence for additional delivery of volatile-rich chondritic materials during the main stages of terrestrial planet formation. The evolution scenario of early Earth is then compared with the atmospheric evolution of planets where no active plate tectonics emerged like on Venus and Mars. We look at the diversity between early Earth, Venus and Mars, which is found to be related to their differing geochemical, geodynamical and geophysical conditions, including plate tectonics, crust and mantle oxidation processes and their involvement in degassing processes of secondary \(\hbox {N}_2\) atmospheres. The buildup of atmospheric \(\hbox {N}_2\), \(\hbox {O}_2\), and the role of greenhouse gases such as \(\hbox {CO}_2\) and \(\hbox {CH}_4\) to counter the Faint Young Sun Paradox (FYSP), when the earliest life forms on Earth originated until the Great Oxidation Event \(\approx \) 2.3 Gyr ago, are addressed. This review concludes with a discussion on the implications of understanding Earth’s geophysical and related atmospheric evolution in relation to the discovery of potential habitable terrestrial exoplanets.     

Adiabatic pulsations and convective instability of rotating gaseous masses

Third order virial equations have been used to investigate the oscillations and the stability of the sequence of differentially rotating, compressible Maclaurin spheroids in the presence of toroidal magnetic fields. It is shown that the neutral point occurring at eccentricitye=0.731 13, which is the analogue of the first point of bifurcation along the Dedekind sequence, remains unaffected by the presence of differential rotation or a toroidal magnetic field. The point of onset of dynamical instability corresponding to the third harmonic deformations does, however, depend upon the magnetic field. It is shifted to values higher thane=0.966 96, the value that obtains in the case of uniform rotation; and a sufficiently large magnetic field can suppress this point. Complete frequency spectra (‘Kelvin’ modes belonging to the harmonicsl=3 and compressible modes belonging tol=1) are obtained in two cases of interest: when the equilibrium state is one of equipartition, and when toroidal magnetic and velocity fields (vanishing at the surface) are present in a configuration rotating with a constant angular velocity.     

Fine Structure of the Core of the Blazar OJ 287. III. Polarized Emission

This conclusive paper summarizes the results of our studies of the fine and superfine structure of the blazar OJ 287 at wavelengths of 7 mm and 2 cm in polarized emission with angular resolution is 20 μas. The orientation of the polarization of its fragments is almost orthogonal to the motion of the flows, suggesting that the magnetic field of the structures is oriented along the direction of the flow velocity. This is determined by the rotation of the flows—the excitation of ring currents and the generation of a solenoidal magnetic field, which applies both to the arms along which the surrounding matter is transferred to the center, the northern (m = 16%) and southern (m = 5%) ones, and to the ejected flows carrying away an excess angular momentum. The polarization level of the jet and counterjet flows reachesm = 15–20%and rises as one recedes fromthe nozzle due to a decrease in the optical depth of the fragments. The polarization level of the counterjet at the nozzle exit reaches 10%, while that of the jet is considerably lower. This is related to the location of the jet nozzle in the opposite direction relative to the observer, the influence of the screen. The special position refers to the nozzles. The polarization level is m - 2%. In the case of outbursts, the polarization increases with brightness, λ = 7 mm. At λ = 2 cm there is an inverse dependence. The spectral index of outbursts lies within the range α = 0–0.8.     

Seasonal changes on Jupiter: 1. Factor of activity of the hemispheres

To identify temporal variations of the characteristics of Jupiter’s cloud layer, we take into account the geometric modulation caused by the rotation of the planet and planetary orbital motion. Inclination of the rotation axis to the orbital plane of Jupiter is 3.13°, and the angle between the magnetic axis and the rotation axis is β ≈ 10°. Therefore, over a Jovian year, the jovicentric magnetic declination of the Earth φ _m varies from–13.13° to +13.13°, and the subsolar point on Jupiter’s magnetosphere is shifted by 26.26° per orbital period. In this connection, variations of the Earth’s jovimagnetic latitude on Jupiter will have a prevailing influence in the solar-driven changes of reflective properties of the cloud cover and overcloud haze on Jupiter. Because of the orbit eccentricity (e = 0.048450), the northern hemisphere receives 21% greater solar energy inflow to the atmosphere, because Jupiter is at perihelion near the time of the summer solstice. The results of our studies have shown that the brightness ratio A _j of northern to southern tropical and temperate regions is an evident factor of photometric activity of Jupiter’s atmospheric processes. The analysis of observational data for the period from 1962 to 2015 reveals the existence of cyclic variations of the activity factor A _j of the planetary hemispheres with a period of 11.86 years, which allows us to talk about the seasonal rearrangement of Jupiter’s atmosphere.     

Orientation of galaxies and a magnetic ‘urfield’

Brown's results (1964, 1968) concerning the distribution of orientation angles of spiral galaxies in different areas of the sky are discussed and a graphical statistical test is applied. The deviations from randomness are found to be significant. It seems difficult to ascribe them to selection effects. It is shown that the observed distributions can be explained, if the angular momenta in greater aggregations of galaxies are distributed at random on congruent precession cones with parallel axes. This hypothesis may apply if the following cosmological conditions hold:

1. the matter in the universe was reheated after the recombination at an epoch, when most of the angular momentum was already transferred to the protogalaxies by tidal interaction, and the angular momenta of the protogalaxies in greater aggregations were predominantly parallel at the epoch of reheating;
2. the ‘magnetic’ model of the universe is valid and the ‘urfield’ was uniform at least at the epoch of reheating. Under these assumptions, the ‘frozen-in’ magnetic field will give rise to forces, which — apart from slowing down the rotation of the protogalaxies — will cause precession of their angular momenta around the direction of the ‘urfield’.

For a rigid body approximation the equations of motion are derived and solved numerically. Approximate analytic solutions are also given. The precession period is in the range of 10⁴ to 10⁸ yr for plausible values of the parameters of the problem. The observed distributions in the four regions of the sky investigated are — via the precession hypothesis — compatible with a direction of the ‘urfield’ indicated by the work of Sofueet al. (1969) and Reinhardt and Thiel (1970) ofl ^II≈280°,b ^II≈+30° tol ^II≈100°,b ^II≈?30°.     

Dynamics of rotating triple systems

We investigate the dynamical evolution of 100 000 rotating triple systems with equal-mass components. The system rotation is specified by the parameter ω=?c²E, where c and E are the angular momentum and total energy of the triple system, respectively. We consider ω=0.1,1, 2, 4, 6 and study 20 000 triple systems with randomly specified coordinates and velocities of the bodies for each ω. We consider two methods for specifying initial conditions: with and without a hierarchical structure at the beginning of the evolution. The evolution of each system is traced until the escape of one of the bodies or until the critical time equal to 1000 mean system crossing times. For each set of initial conditions, we computed parameters of the final motions: orbital parameters for the final binary and the escaping body. We analyze variations in the statistical characteristics of the distributions of these parameters with ω. The mean disruption time of triple systems and the fraction of the systems that have not been disrupted in 1000 mean crossing times increase with ω. The final binaries become, on average, wider at larger angular momenta. The distribution of their eccentricities does not depend on ω and generally agrees with the theoretical law f(e)=2e. The velocities of the escaping bodies, on average, decrease with increasing angular momentum of the triple system. The fraction of the angles between the escaping-body velocity vector and the triple-system angular momentum close to 90° increases with ω. Escapes in the directions opposite to rotation and prograde motions dominate at small and large angular momenta, respectively. For slowly rotating systems, the angular momentum during their disruption is, on average, evenly divided between the escaping body and the final binary, whereas in rapidly rotating systems, about 80% of the angular momentum is carried away by the escaping component. We compare our numerical simulations with the statistical theory of triple-system disruption.     

Development of planetary ephemerides EPM and their applications

This paper outlines the progress in development of the numerical planet ephemerides EPM—Ephemerides of Planets and the Moon. EPM was first created in the 1970s in support of Russian space flight missions and constantly improved at IAA RAS. Comparison between various available EPM ephemerides (EPM2004, EPM2008, EPM2011) is shown. The first results of the updated EPM2013 version which takes into account the two-dimensional annulus of small asteroids are presented. Currently two main factors drive the progress of planet ephemerides: dynamical models of planet motion and observational data, with the crucial role of spacecraft ranging. EPM ephemerides are the basis for the Russian Astronomical and Nautical Astronomical Yearbooks, are planned to use in the GLONASS and LUNA-RESOURCE programs, and are being used for determination of physical parameters: masses of asteroids, planet rotation parameters and topography, the \(GM_\odot \) and its secular variation, the PPN parameters, and the upper limit on the mass of dark matter in the Solar System. The files containing polynomial approximation for EPM ephemerides (EPM2004, EPM2008, EPM2011) along with TT–TDB and ephemerides of Ceres, Pallas, Vesta, Eris, Haumea, Makemake, and Sedna are available from ftp://quasar.ipa.nw.ru/incoming/EPM/. Files are provided in IAA’s binary and ASCII formats, as well as in the SPK format.     

A re-analysis of the eclipsing binary system VV Orionis

The observations of VV Ori inUBV and inuvby obtained by Chambliss in 1975–79 have been re-analyzed with the use of an updated version of the WINK program of Wood. Several solutions were obtained using the normal points (ca. 80 per light curve). These establish fairly reliable values for the linear limb-darkening coefficient of VV Ori A and approximate values for the luminosity of the third component of this system. Attempts to determine non-linear coefficients of limb darkening for the primary component, however, proved to be unsuccessful. Solutions were also obtained using all observations (ca. 620 per light curve). Very close agreement was found between the values of the geometric elements determined from these solutions and those determined by use of the normal points only. The solutions based on all observations produced reliable values foru ₁, the limb darkening coefficient of VV Ori A, (typically, 0.30±-0.04). These results are in good agreement with theoretical limb darkening coefficients derived from model atmospheres calculations. The contribution of VV Ori C to the light of the system was also ascertained, and it was found that this could be best interpreted, if this component has a spectral type of A3V. The other orbital elements of VV Ori were also discussed, and the differences between the various solutions were noted. Since VV Ori A is one of the very few early-type stars for which reliable limb-darkening coefficients can be empirically determined, this system is viewed as being of considerable importance.     

The quasi-stationary coronal magnetic field and electron density as determined from a Faraday rotation experiment

Pioneer VI was launched into a circumsolar orbit on December 16, 1965, and was occulted by the sun in the latter half of November, 1968. During the occultation period, the 2292-MHz S-band telemetry carrier underwent Faraday rotation due to the interaction of this signal with the plasma and magnetic field in the solar corona. The NASA/JPL 210-ft diameter antenna of the Deep Space Network near Barstow, California, was used for the measurement. The antenna feed was modified for automatic polarization tracking for this experiment. The measurement results are interpreted with a theoretical model of the solar corona. This model consists of a modified Allen-Baumbach electron density and a coronal magnetic field calculated both from Mount Wilson magnetograph observations using a source surface model and field extrapolations from the Explorer 33 satellite magnetometer. The observations and the calculated rotation show general agreement with respect to magnitude, sense, and timing, suggesting the source-surface model and field extrapolations from 1 AU are a valid technique to obtain the magnetic field in the corona from 4 to 12 solar radii. Variations present can easily be ascribed to density enhancements known to be present in the corona. Longitudinal variations of the density in the corona cannot be obtained from coronagraph observations, and thus a purely radial variation was assumed. An improved fit to the Faraday rotation data is obtained with an equatorial electron density $$N = 10^8 \left( {\frac{{6000}}{{R^{10} }} + \frac{{0.002}}{{R^2 }}} \right)...{\text{ cm}}^{{\text{ - 3}}} {\text{ (4 < }}R < 12){\text{ }}...$$ where R is in solar radii. The work of W. V. T. Rusch and J. E. Ohlson was supported in part by research sponsored by the Joint Services Electronics Program through the Air Force Office of Scientific Research under Grant AF-AFOSR 69-1622A at the University of Southern California. The work done by K. H. Schatten was in part supported by the National Academy of Science on a National Research Council postdoctoral fellowship. The work of J. M. Wilcox was supported in part by the Office of Naval Research under Contract Nonr 3656(26), by the National Aeronautics and Space Administration under Grant NGR 05-003-230, and by the National Science Foundation under Grant GA-1319 at the University of California at Berkeley.     

Effect of the Earth’s compression on the physical libration of the Moon

The effect of the Earth??s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 10⁶ than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon??s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon??s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth??s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth??s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L _? = 90° or L _? = 270°. The famous Cassini??s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth??s compression on the Moon??s libration in longitude, a harmonic with an amplitude of 0.03?? and period of T ₈ = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth??s compression on the variation in spin angular velocity of the Moon proves to be negligible.