Toroidal quarter waves in the Earth’s magnetosphere: theoretical studies

An analytic model has been developed for toroidal quarter wave (QW) oscillations in the Earth’s magnetosphere using idealistic and highly asymmetric ionospheric boundary condition. The background magnetic field is dipolar and plasma density distribution is governed by a power law 1/r ^m where r is the geocentric distance of any point along the field line and m is the density index. The solution thus obtained has been compared with the numerical solutions. Earlier workers had developed the analytic model for trivial 1/r ⁶ (m=6) type of plasma density distribution along the field line for which the period of the fundamental is twice that of corresponding half wave (i.e. toroidal oscillation in the symmetric ionospheric boundary). The present analytic model does reproduce this feature. In addition, it is seen that this ratio decreases for lower values of m. Moreover, for a particular value of m, this ratio shows a decreasing trend with increased harmonic number. The spatial characteristics of QW obtained from present analytic model are in excellent agreement with those computed numerically, thereby validating the model. The departure of frequency computed analytically from that obtained numerically is significant only for the fundamental and this departure reduces sharply with the increased harmonic number. It should be noted that there is no such departure for 1/r ⁶ type of plasma density distribution and spatial structures as well as frequency computed from the present analytic model match perfectly with those computed numerically.     

Polarization of ULF waves in the earth’s magnetosphere

Conditions for the realization of ULF waves with different polarization are formulated and verified with the use of observational data from the spacecrafts which performed measurements in the earth’s magnetosphere. It is shown that the conditions formulated are in good agreement with observed wave parameters.     

Asymmetric effects on Earth’s polar motion

Differential equations ruling the Earth’s polar motion are slightly asymmetric with respect to the pole coordinates. This is not only associated with the lack of axial symmetry around the Earth figure axis (triaxiality) but also with the longitude dependency of the pole tide (the main contribution). We propose a consistent handling of both asymmetric contributions, formulating a unique equation in the complex equatorial plane, of which we derive a general solution. Difference with respect to the usual symmetric solution is discussed and found significant in light of the present accuracy of the observed pole coordinates. For the same geophysical excitation, the prograde Chandler wobble is accompanied by a retrograde component up to 2 milliarcseconds (mas), transforming it in a slight elliptic motion. The asymmetric contribution is relatively larger in the geodetic excitation function, for Chandler wobble excitation mixes prograde and retrograde components of comparable level (1 mas).     

Peculiarities of the variations in the coordinates of the Earth’s pole

The appearance of features with cusp points on the diagrams of changes in the coordinates of the Earth’s instantaneous pole (polhodes) is considered as the result of mapping onto the plane of its displacement over the surface during the Earth’s rotational-translational motion. The results of qualitative and quantitative analyses of the data on the coordinates of the Earth’s instantaneous pole are discussed. The basic principles of the theory of Whitney singularities and their application for explaining the bifurcations of the equilibrium positions for the Zeeman catastrophe machine (Arnold 1990) are used in the analyses.     

Statistical Characteristics of Meteoroid Parameters in the Earth’s Atmosphere

Statistical characteristics of meteoroids with kinetic energy from 0.1 to 440 kt TNT are estimated based on NASA satellite observations made in 1994–2016. The distributions of the number of falling meteoroids are constructed and analyzed based on the values of their initial kinetic energy, initial velocity, initial mass, altitude, geographic coordinates of the maximum total radiated energy region, and the year of the fall. Correlation dependences “mass–initial kinetic energy,” “maximum total radiated energy region altitude–initial kinetic energy,” and “maximum total radiated energy region altitude–initial velocity (the square of the initial velocity)” are constructed.     

The effect of the Earth’s oblateness on the Moon’s physical libration in latitude

The Moon’s physical libration in latitude generated by gravitational forces caused by the Earth’s oblateness has been examined by a vector analytical method. Libration oscillations are described by a close set of five linear inhomogeneous differential equations, the dispersion equation has five roots, one of which is zero. A complete solution is obtained. It is revealed that the Earth’s oblateness: a) has little effect on the instantaneous axis of Moon’s rotation, but causes an oscillatory rotation of the body of the Moon with an amplitude of 0.072″ and pulsation period of 16.88 Julian years; b) causes small nutations of poles of the orbit and of the ecliptic along tight spirals, which occupy a disk with a cut in a center and with radius of 0.072″. Perturbations caused by the spherical Earth generate: a) physical librations in latitude with an amplitude of 34.275″; b) nutational motion for centers of small spiral nutations of orbit (ecliptic) pole over ellipses with semi-major axes of 113.850″ (85.158″) and the first pole rotates round the second one along a circle with radius of 28.691″; c) nutation of the Moon’s celestial pole over an ellipse with a semi-major axis of 45.04″ and with an axes ratio of about 0.004 with a period of T = 27.212 days. The principal ellipse’s axis is directed tangentially with respect to the precession circumference, along which the celestial pole moves nonuniformly nearly in one dimension. In contrast to the accepted concept, the latitude does not change while the Moon’s poles of rotation move. The dynamical reason for the inclination of the Moon’s mean equator with respect to the ecliptic is oblateness of the body of the Moon.     

On the regime of the Earth’s polar motion based on data for the period 1962–2007

The view of the Earth’s polar motion as a completely deterministic process has been called into question in the past decades, because no long-term prediction can be made. At the same time, no fundamental restrictions currently exist in the problem of a long-term prediction of the Earth’s rotation. Determining the boundaries of predictability is related to identifying the regime of the Earth’s polar motion. IERS data for the period 1962–2007 have been used to study the regime of the Earth’s polar motion. Analysis of the plots of polhodes reveals peculiarities in the variations of the pole’s coordinates X and Y in certain intervals along the time axis. The data in the interval from 2003 to 2006 have been analyzed in greatest detail: a model for the Chandler and annual oscillations has been constructed and relations between the parameters of these oscillations have been determined; the shift of the instantaneous pole on the phase plane and the Poincare plane has been investigated. As a result, we have found features inherent in chaotic motion (intermittency) and calculated the period (32 years) of the possible repetitions of such anomalies, as confirmed by our analysis of the plots of polhodes. The intervals where the peculiarities in the motion of the Earth’s instantaneous pole manifest themselves are compared with the intervals of the inflections on the plots of variations in the length of the day (LOD).     

Study of the interannual variations of the Earth’s rotation

In this work we investigate the variations of the Earth’s rotation in the interval of periods from 2 to 8 years using the longest available observational series obtained both by means of astrometry and space geodesy. We found an abrupt change of the variation pattern in the middle of the 1980s, when classical ground-based astrometric facilities for studying the Earth Rotation Parameters (ERP) were replaced with space geodesy methods. Variations with a 6-year periodicity and ∼0.2-ms amplitude practically disappeared (space geodesy instruments did not detect these variations right from the start), but the 2- to 4-year periodicities increased in amplitude and began to dominate in this frequency range under consideration. In this study, we analyze some possible excitation sources and possible causes of the change in the variability pattern.     

Gamma-ray bursts and the production of cosmogenic radionuclides in the Earth’s atmosphere

An explanation is offered for the impulsive increase in the concentration of cosmogenic radiocarbon in annual tree rings (Δ¹⁴C ～ 12‰) from AD ?775. A possible cause of such an increase could be the high-energy emission from a Galactic gamma-ray burst. It is shown that such an event should not lead to an increase in the total production of ¹⁰Be in the atmosphere, as distinct from the effect of cosmic-ray fluxes on the atmosphere. At the same time, the production of an appreciable amount of ³⁶Cl, which can be detected in Greenland and Antarctica ice samples of the corresponding age, should be expected. This allows the effects caused by a gamma-ray burst and anomalously powerful proton events to be distinguished.     

On the origin of energetic particles in the foreshock region of the Earth’s bow shock

We consider the processes related to the formation of the so-called foreshock region upstream of the Earth’s bow shock. We suggest a model based on the surfing of pick-up ions in the bow shock front in terms of which the ion acceleration mechanism in the front can be explained. We ascertain the physical conditions under which the accelerated ions lie upstream of the shock front and determine the direction of motion of the energetic ions. We conclude that it is this population of energetic ions (longitudinal beams) that plays a major role in forming the ion foreshock boundary.     

Role of the solar wind parameters in changing orbital motion of the Earth’s satellites

The investigation of the solar wind and geomagnetic activity parameters' effect on variations of the orbital motion periods of artificial satellites has been continued. The periods of orbital motion of uncontrolled satellites from the database of the Ukrainian network of optical stations (UNOS) for 2012–2014 was used. The data have been compared with the values of geomagnetic planetary index K and the energy spectra of protons and electrons obtained by the GEOS satellites in events during which the orbital periods have changed. It is shown that, in the energy spectra of the proton and electron fluxes, there is no effect of softening the spectrum with time at the time of the flare appearance. This indicates the possibility of particle accumulation above the active region (AR), which entails further continuous energy emission of the solar flare from AR. Dependences have been obtained between the geomagnetic activity and the solar wind speed at a given interplanetary magnetic field strength during the periods under study for the changes in the orbital motion periods of satellites. The corresponding correlation coefficients are 0.93–0.96.     

A Simulation Study of the Plasma Wave Enhancements in the Earth’s Equatorial Plasmasphere

In the equatorial plasmasphere, plasma waves are frequently observed. To improve our understanding of the mechanism generating plasma waves from instabilities, a comparison of observations, linear growth-rate calculations, and simulation results is presented. To start the numerical experiments from realistic initial plasma conditions, we use the initial parameters inferred from observational data obtained around the plasma-wave generation region by the Akebono satellite. The linear growth rates of waves of different modes are calculated under resonance conditions, and compared with simulation results and observations. By employing numerical experiments by a particle code, we first show that upper hybrid-, Z-, and whistler-mode waves are excited through instabilities driven by a ring-type velocity distribution. The simulation results suggest a possibility that energetic electrons with energies of some tens of keV confined around the geomagnetic equator are responsible for the observed enhancements of Z- and whistler-mode waves. While the comparison between linear growth-rate calculations and observations shows the different tendency of wave amplitude of Z-mode and whistler-mode waves, the wave amplitude of these wave modes in the simulation results is consistent with the observation.     

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.     

Dynamics of the Earth’s core from VLBI observations

The Earth’s rotation is accompanied by free circadian oscillations of its liquid core in the inner cavity of the lower mantle, which perturb the angular momentum of the entire Earth and produce an additional free nutation of the celestial pole called free core nutation (FCN). Since this nutation causes resonances in the diurnal tides and in the expansions of luni—solar nutation, its study, especially an improvement of the FCN period, is of fundamental importance for the theory of the Earth’s rotation. We have determined the FCN parameters from a joint analysis of equidistant series of coordinates of the celestial pole obtained from the combined processing of VLBI observations on global networks of stations for the interval 1984.0–2008.4 by IERS (International Earth Rotation and Reference System Service, Paris, France) and NEOS (National Earth Orientation Service, Washington, USA). Applying a moving least-squares filter (MLSF) to these data has shown that the FCN period averaged over this time interval differs significantly from the theoretical one and its phase varies over a wide range. Using the mean quadratic collocation (MQC) method, we have obtained a new, more accurate stochastic FCN model. Its analysis by the envelope method has revealed long-term linear phase trends, calling into question not only the adopted FCN period but also its stability and, hence, the stability of the resonant effects in the Earth’s luni—solar nutation.     

Photopolarimetric observations of Jupiter’s polar region

Peculiarities in the use of CCD matrices for photopolarimetric observations are considered. Algorithms for processing the CCD photopolarimetric and polarimetric observations of Jupiter and other bright extended objects are described with an emphasis on the specifics of these photodetectors. We present new estimates of the north-south asymmetry parameter of polarization of Jupiter obtained from the observations in 2006 and 2007. The new data are in good agreement with the previous observations.     

Statistical interrelation of 22-yr-long variations in the data on parameters of the Earth’s rotation and magnetic fields of the Earth and the Sun

By means of spectral analysis, oscillations have been detected in many-year time series of deviations in the duration of days from the standard that cannot be explained within the framework of existing gravitational theory. The solution of the problem of the origin and structure of these oscillations is associated with the essence of the phenomena taking place inside the Earth and the mechanisms of energy transfer by external fields and the Sun. The effect of the Sun on the Earth’s rotation also leads to the formation of a unified cyclic background, resulting in correlated oscillations in all shells of the Earth, as well as in its atmosphere and its nucleus. Ground magnetic fields of the Earth and the Sun play the leading role in the abovementioned unified cyclic process. The results of spectral autoregressive and wavelet analyses of experimental data concerning deviations in the duration of days from the standard in the years 1832–2006, ground geomagnetic field intensities in 1832–2006, and the mean magnetic field of the Sun during the period from 1975 to 2005 have been discussed in this paper to reveal and compare correlating oscillations. To analyze a short-period (31-yr-long) series of daily data on the mean magnetic field of the Sun, the results of wavelet transformations (the Morlet wavelet) of the detected amplitude of a burst envelope with a carrier frequency of 13 cycles/yr have been obtained.     

Influence of the hot oxygen corona on the satellite drag in the Earth’s upper atmosphere

Calculation results on the possible influence of the hot oxygen fraction on the satellite drag in the Earth’s upper atmosphere on the basis of the previously developed theoretical model of the hot oxygen geocorona are presented. Calculations have shown that for satellites with orbits above 500 km, the contribution from the corona is extremely important. Even for the energy flux Q ₀ = 1 erg cm⁻² s⁻¹, the contribution of the hot oxygen can reach tens of percent; and considering that real energy fluxes are usually higher, one can suggest that for extreme solar events, the contribution of hot oxygen to the atmospheric drag of the satellite will be dominant. For lower altitudes, the contribution of hot oxygen is, to a considerable degree, defined by the solar activity level. The calculations imply that for the daytime polar atmosphere, the change of the solar activity level from F _10.7 ∼ 200 to F _10.7 ∼ 70 leads to an increase in the ratio of the hot oxygen partial pressure to the thermal oxygen partial pressure by a factor of almost 30, from 0.85 to 25%. The transition from daytime conditions to nighttime conditions almost does not change the contribution from suprathermal particles. The decrease of the characteristic energy of precipitating particles, i.e., for the case of charged particles with a softer energy spectrum, leads to a noticeable increase of the contribution of the suprathermal fraction, by a factor of 1.5–2. It has been ascertained that electrons make the main contribution to the formation of the suprathermal fraction; and with the increase of the energy of precipitating electrons, the contribution of hot oxygen to the satellite drag also increases proportionally. Thus, for a typical burst, the contribution of the suprathermal fraction is 30% even at relatively high solar activity F _10.7 = 135.     

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.