On constructing the general Earth’s rotation theory

In the present paper the equations of the orbital motion of the major planets and the Moon and the equations of the three–axial rigid Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution). Hence, the theory of the Earth’s rotation can be presented by means of the series in powers of the evolutionary variables with quasi-periodic coefficients with respect to the planetary–lunar mean longitudes. This form of the Earth’s rotation problem is compatible with the general planetary theory involving the separation of the short–period and long–period variables and avoiding the appearance of the non–physical secular terms.     

Precession/nutation solution consistent with the general planetary theory

In the present paper the equations of the translatory motion of the major planets and the Moon and the Poisson equations of the Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution). Hence, the theory of the Earth’s rotation is presented by means of the series in powers of the evolutionary variables with quasi-periodic coefficients.     

Relation between solar sidereal and synodic rotation axes

Kinematic picture concerning the solar synodic and sidereal rotation axes has been considered in some detail. Large changes in the synodic angular rotation velocity and the position of the synodic rotation pole have been found for some hypothetical cases of out-of-ecliptic intra-Mercurian orbits. The influence of solar differential rotation and variable planetary velocity along the orbit have been taken into account and a continuous set of co-existing synodic poles oscillating around a mean position has been found. The relevant numerical values for the Earth are given and the possibility of detecting the existence of the two rotation axes has been pointed out.     

Analytical theory of Earth’s rotation

An analytical theory of the rotation of the rigid Earth is developed in a form compatible with the general planetary theory. Numerical estimates of the constants of integration of the Poisson equations, which are a particular case of the equations of the Earth’s rotation, are given.     

On the rotation of a moonlet embedded in planetary rings

We examine the rotation of a small moonlet embedded in planetary rings caused by impacts of ring particles, using analytic calculation and numerical orbital integration for the three-body problem. Taking into account the Rayleigh distribution of particles' orbital eccentricities and inclinations, we evaluate both systematic and random components of rotation, where the former arises from an average of a large number of small impacts and the latter is contribution from large impacts. Calculations for parameter values corresponding to inner parts of Saturn's rings show that a moonlet would spin slowly in the prograde direction if most impactors are small particles whose velocity dispersion is comparable to or smaller than the moonlet's escape velocity. However, we also find that the effect of the random component can be significant, if the velocity dispersion of particles is larger and/or impacts of large particles comparable to the moonlet's size are common: in this case, both prograde and retrograde rotations can be expected. In the case of a small moonlet embedded in planetary rings of equal-sized particles, we find that the systematic component dominates the moonlet rotation when m/M?0.1 (m and M are the mass of a particle and a moonlet, respectively), while the random component is dominant when m/M?0.3. We derive the condition for the random component to dominate moonlet rotation on the basis of our results of three-body orbital integration, and confirm agreement with N-body simulation.     

Interpretation of grazing impacts on Mars by the axial rotation theory

This paper calls attention to the fact that the distribution of grazing impacts on Mars, as described by Schultz and Lutz-Garihan (1982), agrees with the distribution predictable by the theory of planetary axial rotation based on the transmission of mass and angular momentum from a satellite system to its primary (Barricelli, 1972). A few additional implications of the theory are pointed out.     

Canonical planetary perturbation equations for velocity-dependent forces,and the Lense-Thirring precession

A form of planetary perturbation theory based on canonical equations of motion, rather than on the use of osculating orbital elements, is developed and applied to several problems of interest. It is proved that, with appropriately selected initial conditions on the orbital elements, the two forms of perturbation theory give rise to identical predictions for the observable coordinates and velocities, while the orbital elements themselves may be strikingly different. Differences between the canonical form of perturbation theory and the classical Lagrange planetary perturbation equations are discussed. The canonical form of perturbation theory in some cases has advantages when the perturbing forces are velocity-dependent, but the two forms of perturbation theory are equivalent if the perturbing forces are dependent only on position and not on velocity. The canonical form of the planetary perturbation equations are derived and applied to the Lense Thirring precession of a test body in a Keplerian orbit around a rotating mass source.     

Probing the 2D kinematic structure of early-type galaxies out to three effective radii

We detail an innovative new technique for measuring the two-dimensional (2D) velocity moments (rotation velocity, velocity dispersion and Gauss–Hermite coefficients h ₃ and h ₄) of the stellar populations of galaxy haloes using spectra from Keck DEIMOS (Deep Imaging Multi-Object Spectrograph) multi-object spectroscopic observations. The data are used to reconstruct 2D rotation velocity maps.
Here we present data for five nearby early-type galaxies to ∼three effective radii. We provide significant insights into the global kinematic structure of these galaxies, and challenge the accepted morphological classification in several cases. We show that between one and three effective radii the velocity dispersion declines very slowly, if at all, in all five galaxies. For the two galaxies with velocity dispersion profiles available from planetary nebulae data we find very good agreement with our stellar profiles. We find a variety of rotation profiles beyond one effective radius, i.e. rotation speed remaining constant, decreasing and increasing with radius. These results are of particular importance to studies which attempt to classify galaxies by their kinematic structure within one effective radius, such as the recent definition of fast- and slow-rotator classes by the Spectrographic Areal Unit for Research on Optical Nebulae project. Our data suggest that the rotator class may change when larger galactocentric radii are probed. This has important implications for dynamical modelling of early-type galaxies. The data from this study are available on-line.     

Planetary waves on the Sun?

Some recent observations of the Sun suggest a class of wave-like motions moving both eastward and westward at a uniform velocity with respect to the mean solar angular velocity. It is suggested that these may be hydromagnetic planetary waves. An estimate of the mean toroidal magnetic field is made, based on a theoretical treatment of such waves already in the literature, and a slight correction to the mean rate of rotation of the Sun is inferred.     

A Simulation for a Planetary Nebular Plasma

A plasma simulation code is applied to interpret the instabilities in an expanding planetary nebula. The temperature of the central star of a planetary nebula is assumed as above 50,000 K. Most of the atoms are ionized at this temperature. Since ionization cannot be neglected for such a hot plasma, the electrostatic instability should be taken into account. In the one dimensional electrostatic simulation, Maxwell and Vlasov equations are used and the fast Fourier transform is applied. The calculated drift velocity in the simulation is found comparable with the expansion velocity of a planetary nebula. The linear and non-linear behaviors of the simulated nebular plasma have been investigated in phase space; the simulation results agree with the theory. This revised version was published online in September 2006 with corrections to the Cover Date.     

Rotation of asteroids and planetary axial rotation theory

A theory (Barricelli, 1972) developed for the interpretation of planetary axial rotations is here applied to an interpretation of the axial rotations of major asteroids. The interpretation is based on the assumption that also asteroids can have satellite-systems, which may influence the axial rotation of the respective primaries. The reason why smaller asteroids tend to have slower axial rotation than the major ones as an average is discussed. Predictions of the theory can be tested by space-craft exploration of asteroids.     

The seven ages of a planet

An overlooked systematic error exists in the apparent radial velocities of solar lines reflected from regions of Venus near the terminator, owing to a combination of the finite angular size of the Sun and its large (2 km/sec) equatorial velocity of rotation. This error produces an apparent, but fictitious, retrograde component of planetary rotation, typically on the order of 40 m/sec. Spectroscopic, photometric, and radiometric evidence against a 4-day atmospheric rotation is also reviewed. The bulk of the somewhat contradictory evidence seems to favor slow motions, on the order of 5 m/sec, in the atmosphere of Venus; the 4-day “rotation” may be due to a travelling wavelike disturbance, not bulk motions, driven by the uv albedo differences.     

Non-linear equations for the rotation of a viscoelastic planet taking into account the influence of a liquid core

Non-linear equations governing the temporal evolution of the vector of instantaneous rotation are developed for an Earth with a homogeneous mantle having a viscoelastic Maxwell rheology and with a homogeneous inviscid fluid core.This general theory is investigated using the angular momentum theorem applied to the coupled core-mantle system. It allows to study the influence upon the planetary rotation of a quasi-rigid rotational motion in the liquid core. It also enables to investigate the consequences of excitation sources (e.g. pressure), located at the core-mantle interface. Especially, the influence of viscoelastic variations in the inertia tensors resulting from the rotation itself or from various excitation sources are detailed with the help of a Love number formalism. The equations of the linear theory for an elastic Earth with a liquid core, and the non-linear theory for a viscous planet with a quasi-fluid behavior are shown to be particular cases of our generalized system of equations. Some planetological applications may be derived from the quasi-fluid approximation.     

Evidence for the phi-dependent rotation-oscillation of the Sun (and for the driving mechanism of the asymmetric dynamo)

Longitude-dependent oscillations of the solar rotation are derived from the 27-day averages of the photospheric velocity data. Two pairs of prominent periods are obtained. Their harmonic means correspond to a semiannual variation and to the first harmonic of the latter. To explain the origin of the oscillation the corona and the interplanetary material are supposed to rotate parallel to the planetary plane with an inclination to the solar equator. The non-uniform shearing around the equator is assumed to result in oscillation with a period of half of a year.     

The effects of rotation rate on deep convection in giant planets with small solid cores

We study how the pattern of thermal convection and differential rotation in the interior of a giant gaseous planet is affected by the presence of a small solid core as a function of the planetary rotation rate. We show, using 2D anelastic, hydrodynamic simulations, that the presence of a small solid core results in significantly different flow structure relative to that of a fully convective interior only if there is little or no planetary rotation.     

太阳多方模型对行星轨道要素变化的长期摄动影响

本文利用天体力学中的摄动理论和天体物理学中的气体星多方模型理论研究了太阳多方模型对行星轨道要素变化的长期摄动影响。文中给出了太阳日多方指数ｎ＝３的模型由于自转、扁度和内部密度分布等因素对行星轨道要素变化的长期摄动效应的理论结果。研究结果表明：行星轨道要素除长轴、偏心率和轨道倾角不受长期摄动外升交点经度、近日点经度以及平近点角均受长期摄动的影响。最后利用理论结果对行星轨道要素的长期投动效应做了数值计算，数值结果在表１中给出。     

Planetary nebula velocities in the disc and bulge of M31

We present radial velocities for a sample of 723 planetary nebulae in the disc and bulge of M31, measured using the WYFFOS fibre spectrograph on the William Herschel Telescope. Velocities are determined using the [O  iii ] λ5007 emission line. Rotation and velocity dispersion are measured to a radius of 50 arcmin (11.5 kpc), the first stellar rotation curve and velocity dispersion profile for M31 to such a radius. Our kinematics are consistent with rotational support at radii well beyond the bulge effective radius of 1.4 kpc, although our data beyond a radius of 5 kpc are limited. We present tentative evidence for kinematic substructure in the bulge of M31 to be studied fully in a later work. This paper is part of an ongoing project to constrain the total mass, mass distribution and velocity anisotropy of the disc, bulge and halo of M31.     

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).     

Rotation rate and velocity dispersion of planetary ring particles with size distribution II. Numerical simulation for gravitating particles

We examine rotation rates of gravitating particles in low optical depth rings, on the basis of the evolution equation of particle rotational energy derived by Ohtsuki [Ohtsuki, K., 2006. Rotation rate and velocity dispersion of planetary ring particles with size distribution. I. Formulation and analytic calculation. Icarus 183, 373-383]. We obtain the rates of evolution of particle rotation rate and velocity dispersion, using three-body orbital integration that takes into account distribution of random velocities and rotation rates. The obtained stirring and friction rates are used to calculate the evolution of velocity dispersion and rotation rate for particles in one- and two-size component rings as well as those with a narrow size distribution, and agreement with N-body simulation is confirmed. Then, we perform calculations to examine equilibrium rotation rates and velocity dispersion of gravitating ring particles with a broad size distribution, from 1 cm up to 10 m. We find that small particles spin rapidly with 〈ω²〉^1/2/Ω?¹⁰²-¹⁰³, where ω and Ω are the particle rotation rate and its orbital angular frequency, respectively, while the largest particles spin slowly, with 〈ω²〉^1/2/Ω?1. The vertical scale height of rapidly rotating small particles is much larger than that of slowly rotating large particles. Thus, rotational states of ring particles have vertical heterogeneity, which should be taken into account in modeling thermal infrared emission from Saturn's rings.     

Venus' superrotation,mixing length theory and eddy diffusion: A parametric study

The Hadley mechanism is adopted to describe the axisymmetric four day superrotation in the Venus atmosphere, with solar driven meridional winds redistributing energy and momentum, and eddy diffusion describing the actions of three dimensional transient eddies. We address the question how the eddy diffusion coefficients are related to the properties of the circulation. For the atmosphere of a slowly rotating planet such as Venus, we show that a form of the non-linear closure is suggested by the mixing length hypothesis, which constrains the magnitude of the eddy diffusion coefficients. Combining this constraint with the concept of the Rossby radius of deformation yields zonal velocities on the order of 100 m sec^–1. A steady state, non-linear, one-layer spectral model is used for a parametric study to find a relationship between heat source, meridional circulation and eddy diffusion coefficients, which yields the large zonal velocities observed. This analysis leads to the following conclusions: (1) Proportional changes in the heat source and eddy diffusion coefficients do not significantly change the zonal velocities. (2) The meridional velocity is virtually constant for large eddy diffusion coefficients. (3) Below a threshold in the diffusion rate, the meridional velocity decreases, commensurate with the mixing length hypothesis. Eddy heat conduction becomes important and shares with the Hadley cell advection in balancing the solar heating. The zonal velocities then reach large values near 100 m sec^–1. (4) For large eddy diffusion and small heating rates, the zonal velocities decrease with decreasing planetary rotation rates. However, under condition (3), the zonal velocities are independent of the planetary rotation rate. Ramifications are discussed for related parameterizations in GCMs, emphasizing that eddy diffusion coefficients are governed by solar forcing and cannot be chosen independently.