ABSTRACT The Equatorial Ionospheric Anomaly (EIA) occurs within the ionosphere on both sides of the Earth's magnetic equator, typically at magnetic latitudes of approximately ± 10° to ± 20°. In this region, the spatiotemporal evolution of ionospheric total electron content (TEC) is highly complex, leading to anomalous enhancements in electron density. These ionospheric disturbances become particularly pronounced during geomagnetic storms, posing significant challenges for high‐precision prediction. To address this prediction challenge, we introduce a deep learning model termed an Encoder‐Decoder with Self‐Attention Convolutional Gated Recurrent Unit (ED‐SA‐ConvGRU), where multiple physical indices were incorporated, including solar activity indices (F10.7, Solar Radio Flux at 10.7 cm; SSN, Sunspot Number) and geomagnetic indices (Kp, K‐index; Dst, Disturbance Storm Time Index), with three distinct input combinations constructed. This study utilised GNSS data from 70 stations of the Australian Regional GNSS Network (ARGN) from 2023 to 2025. Test results indicate that, compared to the GRU, ConvGRU, and ED‐ConvGRU models, the proposed ED‐SA‐ConvGRU model reduced the Root Mean Square Error (RMSE) by 16.8%, 2.4%, and 6.8%, respectively. Moreover, the prediction accuracy of the ED‐SA‐ConvGRU model was further enhanced through the incorporation of physical indices. Specifically, Input Combination III (historical TEC + geomagnetic indices Kp and Dst + solar activity indices F10.7 and SSN) yielded the lowest RMSE. These findings indicate that the ED‐SA‐ConvGRU model has competitive performance in ionospheric TEC prediction over the low‐to mid‐latitude EIA region.

Asteroids are celestial bodies widely present within the solar system. Light curves, which are plots of brightness variation over time obtained from continuous observations of an asteroid, contain information related to its orbital, rotational, shape, and thermophysical parameters. Through light curves, shape inversion of asteroids can be performed to investigate the relationship between the characteristics of light curves and the asteroid's rotational state. Firstly, by comprehensively considering the shape, rotational state, and phase angle of the asteroid, a photometric model is established to simulate and generate asteroid light curves. Based on the triaxial ellipsoid model, the influence of various parameters on light curves is analyzed, and the variations in light curves under principal-axis rotation and non-principal-axis rotation are studied, thereby obtaining the correspondence between light curve features and rotational and shape parameters. Subsequently, parameter fitting of asteroid light curves is performed using both the triaxial ellipsoid model and the convex polyhedron model to invert and obtain the rotational and shape parameters.

Variations in polar motion (PM) observed through space geodetic techniques imply comprehensive information on geophysical excitation. Traditional geodetic excitation series derived from PM observations typically excluded the calculation of the excitation derivative term dψ/dt in the classical Liouville equation (LE) and relied on the low-accuracy numerical integration method. In this paper, we derive the complete LE (CLE) geodetic excitation solution, in which the integral term involves a highly oscillatory integral with a daily-frequency oscillatory kernel eiΩτ. To address the limitations of traditional numerical integration methods when applied with daily PM observations as input, we apply two Filon-type methods—one based on Lagrange polynomials and the other on cubic spline interpolation to derive the CLE geodetic excitation. Using both simulated and observed datasets, we perform a recovery process for CLE solutions (input PM → excitation → recovered PM) to assess numerical calculation errors under the accuracy standard in the PM domain. The observed dataset demonstrates that the recovered PM errors, with standard deviations (STD) greater than 30 microarcseconds (μas) for the X and Y components, primarily vary within tens of μas, comparable to the observed PM errors during the same period. This also proves that the derived CLE geodetic excitation, with the geophysical completeness, satisfies the analysis research under the current PM observational accuracy, providing a valuable reference indicator for various geophysical processes. The differences between CLE and traditional geodetic excitations, with STDs of about 5–6 milliarcseconds (mas), arise from high-frequency biases associated with numerical calculations. Regarding future PM investigations in daily and sub-daily frequency bands, simulation tests indicate that a time sampling interval of less than 4 h for the input PM can achieve accuracies below the μas level, which demonstrates the capability of CLE numerical calculations to support future high-resolution PM applications.

The structure of the lunar crust preserves key records of the Moon's origin and long-term evolution. Using the latest high-resolution gravity field from the Gravity Recovery and Interior Laboratory (GRAIL) mission together with Lunar Orbiter Laser Altimeter (LOLA) topography, we investigate the shallow crustal structure of the lunar farside. Applying the effective density spectrum technique with a two-layer formulation, we constrain the densities of the upper and lower layers and the depth of their interface. Our results reveal a thin (average 4.7 km), low-density (average 2,111 kg m(-3)) surficial layer, likely representing a highly porous megaregolith formed by basin ejecta and impact fragmentation, overlying a higher density unit (average 2,739 kg m(-3)) consistent with a less porous, anorthositic crust. From the inferred upper-layer density, we estimate crustal porosity and find that the porosity decreases systematically with increasing interface depth, while exhibiting substantial lateral variability at a given depth. This variability is closely linked to differences in cumulative impact modification. Residual porosity below similar to 2 km shows a statistically significant negative correlation with N(20), indicating that repeated small impacts have progressively compacted the shallow crust and reduced its pore space.

The conventional Liouville equation (LE), commonly used to elucidate Earth's polar motion (PM), has subsequently developed into the observable LE (OLE); however, the OLE is not applicable for full-band PM investigations. By integrating the OLE with relationship between the theory and observations of PM, we obtained a comprehensive resolution to the LE, which is referred to as the complete LE (CLE) solution. In the CLE solution, the effects of the incalculable excitation derivative term ${\rm{d}}\psi /{\rm{d}}t$ have been incorporated and can now be easily estimated. Afterwards, we did an extensive review of the geophysical implications of the CLE solution in order to figure out how this excitation derivative term impacts the full-band PM. According to this CLE solution, we have established a solid relationship between the observed PM and the celestial intermediate pole. Furthermore, we recreated the PM by employing both the simulated and real geophysical fluid excitation series. The simulation example demonstrates the application of the CLE solution for full-band PM examinations with a numerical accuracy of less than 1 ${\rm{\mu as}}$. Real data reconstructions have shown that the CLE solution outperforms the OLE solution in both low- and high-frequency PM bands, with amplitude enhancements of around hundreds ${\rm{\ \mu as}}$ and several ${\rm{\ \mu as}}$, respectively. These findings enable us to precisely evaluate the comprehensive PM investigations.

Precise Point Positioning (PPP) provides static positioning at the millimeter level and kinematic positioning ranging from millimeters to decimeters globally. Unlike the traditional network solution, PPP does not require data from other reference stations. This flexibility enhances the convenience of densifying the reference frame while maintaining the accuracy of solutions. In this study, Precise Point Positioning with Ambiguity Resolution (PPP-AR) was employed instead of a network solution, utilizing the combined orbit, clock, and bias products from IGS Repro3 to resolve the long-term station coordinates and derive their velocities, thereby contributing to the maintenance and densification of the terrestrial reference frame. We selected 46 globally distributed stations and performed PPP-AR over a 5-year period, from 2015.0 to 2020.0. The results show that differences in station coordinates between PPP-AR and IGS Repro3 are almost within 2 mm in the horizontal direction and within 5 mm in the vertical direction after Helmert transformation, which is roughly equivalent to the formal error of IGS solutions. The velocity uncertainty of PPP-AR solutions and the difference between PPP-AR and IGS Repro3 are nearly equal to the formal error of the ITRF horizontal velocity field and slightly exceed that of the IGS horizontal velocity field. The seasonal amplitudes of the remaining stations demonstrate strong consistency. Compared to PPP solutions, PPP-AR solutions provide improved coordinate and velocity precision, particularly in the east component. The consistency between the IGS Repro3 orbit/clock combination and IGS Repro3 position solutions is relatively high. These findings indicate that the PPP-AR technique can derive high-precision station coordinates with a similar level of accuracy to network solutions for supporting the maintenance and densification of the terrestrial reference frame.

Combining the Liouville equations for polar motion (PM) with forecasted geophysical effective angular momentum (EAM) functions can significantly improve the accuracy of Earth's PM predictions. These predictions rely on deconvolution and convolution methods. Deconvolution derives the geodetic EAM function from the PM observations, while convolution uses both the geodetic and geophysical EAM functions to reproduce and predict the PM values. However, there are limitations in existing numerical realisations of deconvolution and convolution that must be addressed. These limitations include low-frequency biases, high-frequency errors, and edge errors, which can negatively impact the accuracy of PM prediction. To overcome these concerns, we develop the Convolution Least Squares (Conv-LS) scheme through a multi-perspective analysis in the frequency domain, PM domain, and EAM domain. By comparing representative approaches for reproducing three different PM series (IERS C01, IERS C04, and IGS) with varying sampling intervals (18.25 days, daily, and 6 h), we demonstrate that the Conv-LS scheme can effectively eliminate the usually present spurious signals and also integrate high-accuracy deconvolution algorithms to reduce reproduced errors further. Compared to the traditional approacsh (using a low-accuracy discrete PM equation for deconvolution and numerical integration for convolution), our new approach (utilising a high-accuracy deconvolution algorithm along with the Conv-LS scheme for convolution) reduces the standard deviations of the residuals' x-component by 31.0

Common mode error (CME) arises from various sources, including unknown regional errors, potential geophysical signals, and other factors present in global navigation satellite system (GNSS) coordinate solutions, undeniably affecting the GNSS precision. This research concentrates on the effects of CME correction in global IGS-based reference frame refinement. We first estimated the regional CME with principal component analysis to obtain CME-corrected GNSS coordinate solutions. Subsequently, effects on the global reference frame with the regional CME correction were analyzed in three aspects: accuracy improvement of the coordinate solutions, variation in the velocity field, and accuracy improvement of the Helmert parameters in the reference frame transformation. The results show that after applying CME correction, the GNSS coordinate accuracy was improved by 28.9%, 22.1%, and 29.5% for the east, north, and vertical components, respectively. Regarding the site velocities, the maximum difference in velocity reached 0.48 mm/yr. In addition, the standard deviation of the Helmert transformation parameters between the International Terrestrial Reference Frame (ITRF) and the IGS-based reference frame—exclusively derived from GNSS technology—was reduced by over 30%, indicating CME correction enhanced the accuracy of the transformation parameters and refined the IGS-based reference frame.

We reconstructed the Chandler Wobble (CW) from 1962 to 2022 by combining the Eigen-oscillator excited by geophysical fluids of atmospheric and oceanic angular momentums (AAM and OAM). The mass and motion terms for the AAM are further divided with respect to the land and ocean domains. Particular attention is placed on the time span from 2012 to 2022 in relation to the observable reduction in the amplitude of the CW. Our research indicates that the main contributor to the CW induced by AAM is the mass term (i.e., the pressure variations over land). Moreover, the phase of the AAM-induced CW remains relatively stable during the interval of 1962–2022. In contrast, the phase of the OAM-induced CW exhibits a periodic variation with a cycle of approximately 20 years. This cyclic variation would modulate the overall amplitude of the CW. The noticeable amplitude deduction over the past ten years can be attributed to the evolution of the CW driven by AAM and OAM, toward a state of cancellation. These findings further reveal that the variation in the phase difference between the CW forced by AAM and OAM, may be indicative of changes in the interaction between the solid Earth, atmosphere, and ocean.

ABSTRACT A globally valid analytically averaged Hamiltonian model for the coorbital motion is hard to construct because the analytical expansions of the disturbing function usually diverge in the quasi-satellite domain that is close to collision singularity. In this paper, an analytically averaged model for the coorbital motion is proposed in case of the circular restricted three-body problem, which can describe properly the transitions that occur at small eccentricities and inclinations, such as the transition between the horseshoe orbit and the quasi-satellite orbit. With the help of the numerical averaging method, numerical experiments are carried out to show the validity and accuracy of the analytically averaged model. The averaged model proposed here can be easily extended to more complicated cases such as the elliptic three-body problem or the planetary three-body problem.

In the restricted three-body problem (RTBP), if a small body and a planet stably orbit around a central star with almost exactly the same semimajor axis, and thus almost the same mean motion, this phenomenon is called the co-orbital motion, or equivalently, the 1:1 mean motion resonance. The classical expansion of the disturbing function is divergent when the semimajor axis ratio of the small body to the planet is close to unity. Thus, most of the previous studies on the co-orbital dynamics were carried out through numerical integrations or semi-analytical approaches. In this work, we construct an analytical averaged model for the co-orbital motion in the framework of the circular RTBP. This model is valid in the entire coorbital region except in the vicinity of the collision singularity. The results of the analytical averaged model are in good agreement with the numerical averaged model even for moderate eccentricities and inclinations. The analytical model can reproduce the tadpole, horseshoe and quasi-satellite orbits common in the planar problem. Furthermore, the asymmetry of 1:1 resonance and the compound orbits (Icarus 137:293–314) in the general spatial problem can also be obtained from the analytical model.

The 1:1 mean motion resonance may be referred to as the lowest order mean motion resonance in restricted or planetary three-body problems. The five well-known libration points of the circular restricted three-body problem are five equilibriums of the 1:1 resonance. Coorbital motion may take different shapes of trajectory. In case of small orbital eccentricities and inclinations, tadpole-shape and horseshoe-shape orbits are well-known. Other 1:1 libration modes different from the elementary ones can exist at moderate or large eccentricities and inclinations. Coorbital objects are not rare in our solar system, for example the Trojans asteroids and the coorbital satellite systems of Saturn. Recently, dozens of coorbital bodies have been identified among the near-Earth asteroids. These coorbital asteroids are believed to transit recurrently between different 1:1 libration modes mainly due to orbital precessions, planetary perturbations, and other possible effects. The Hamiltonian system and the Hill’s three-body problem are two effective approaches to study coorbital motions. To apply the perturbation theory to the Hamiltonian system, standard procedures involve the development of the disturbing function, averaging and normalization, theory of ideal resonance model, secular perturbation theory, etc. Global dynamics of coorbital motion can be revealed by the Hamiltonian approach with a suitable expansion. The Hill’s problem is particularly suitable for the studies on the relative motion of two coorbital bodies during their close encounter. The Hill’s equation derived from the circular restricted three-body problem is well known. However, the general Hill’s problem whose equation of motion takes exactly the same form applies to the non-restricted case where the mass of each body is non-negligible, namely the planetary case. The Hill’s problem can be transformed into a “canonical shape” so that the averaging principle can be applied to construct a secular perturbation theory. Besides the two analytical theories, numerical methods may be consulted, for example the approach of periodic orbit, the surface of section, and the computation of invariant manifolds carried by equilibriums or periodic orbits.

SUMMARY By anatomizing the classic Liouville equation (LE), an alternative theoretical framework is established for the Earth polar motion where the turbulent time derivative of the fluid forcing, Lforce, is eliminated. The observed polar motion is found a lumped signal of two physical variables, one of which is the forced polar motion, mforce, that balances out the fluid forcing through a simple algebraic identity. The second component is the inertial polar motion, minert, which conserves the total angular momentum by the restoring power of the equatorial bulge. Aside from its numerical difficulties, the time derivative dLforce/dt has proved to be a redundant artefact that mixes the physical signals in the solution of the LE. By an analytical appraisal, we find that for non-Chandler forced polar motions, the dLforce/dt term in the excitation function of the classic LE is 100 per cent responsible for the forced term, mforce, and contributes 0.3 per cent of the inertial term, minert. The Chandler signal originates exclusively from the inertial polar motion minert. The artefact dLforce/dt perturbs the true Chandler signal from the classic LE by 0.3 per cent. Anatomy of the LE allows us to show that a linearized mechanism for the Chandler Wobble excitation is valid, since the linearized governing equation for the inertial polar motion minert stands as part of the exact solution of the full nonlinear equation. Anatomized polar motion equations are also considered in the presence of ocean tides, under the lunisolar torque, and on an elastically deformable Earth. Inaccuracies and misinterpretations associated with the classic LE under those circumstances are clarified in formulating the anatomized polar motion equation group.

By transforming a 1D second-order linear oscillator into a 2D first-order polar motion differential equation, it can be shown that the finite smoothness (i.e., the presence of jump in finite order derivatives) of the applied Newtonian forcing constitutes the sufficient and necessary condition for instantaneous excitation of free eigen-mode. This condition can be met by forcing functions originated from turbulent and multiphase fluid motions. Sub-macroscopic transition time associated with astatic elastic deformation limits the physical smoothness of the applied forcing for the Earth's polar motion. Eigen-modes can also be excited by an infinitely smooth forcing that has a finite domain of non-zero values. The eigen-period serves as a macroscopic timescale to characterize the inertia of a linear oscillator. If a zero mean irregular forcing of finite smoothness exhibits a high degree of randomness and the timescale is much shorter than the eigen-period, then for negligible damping the eigen-waveform will increase in proportion to the squareroot of time, while the waveform distortion is statistically a constant. As a result, the pattern of distinctive eigen-oscillation will dominate the forced solution for longer enough duration.

We study the secular behaviour of a test particle orbiting a dominant central body and perturbed by a Miyamoto–Nagai (MN) disc. We derive a quadrupole-level secular Hamiltonian of this system, which involves a dimensionless parameter η that is used to characterize the flattening of MN disc. (The smaller η, the flatter disc; and η = 0 gives the infinitely thin Kuzmin disc.) We find that, in the quadrupole approximation, the perturbation of the MN disc can give rise to the von Zeipel–Lidov–Kozai (ZLK)-like dynamics and depending on the value of η the dynamics has two different manifestations: (i) when η < 1/3, the test particle’s behaviour is similar to that described in the classical ZLK problem. In particular, as η increases from 0 to 1/3, the critical inclination for the large eccentricity oscillations decreases from $26{_{.}^{\circ}}56$ to 0°, ; (ii) when η > 1/3, the orbital evolution of the test particle and the phase-space morphology are opposite to the classical ZLK case. This leads to a striking result that the test particle cannot remain on a near-coplanar orbit if its eccentricity is sufficiently large. However, as η increases further the ZLK-like dynamics would be gradually suppressed by the spherical term in the Hamiltonian. We also survey the global secular dynamics numerically in which the quadrupole approximation is no longer valid.

The Mars’ rotational rate, or equivalently its length-of-day (LOD), varies slightly due to global atmospheric changes. And the atmospheric dust cycle is considered to be one key process impacting the Mars’ atmosphere. Lewis and Barker (2005) demonstrated the close link of the semidiurnal tidal amplitude with the atmospheric dust content. In consideration of tidal effects in the Mars’ variable rotation, possible connection may exist between the Mars’ semidiurnal LOD amplitude variation and atmospheric dust cycles. In this study, the Mars’ LOD variations which are influenced by global atmosphere, and the atmospheric dust cycle index (ADCI) are computed based on eight realistic scenarios of the Martian Years (MY) 24–31 of the Martian Climate Database. The semidiurnal LOD amplitude change and the ADCI are found to be correlated at the 99% significance level, and they present similar time-varying wavelet spectral structures. In mature stages of global dust storms in MYs 25 and 28, relatively high LOD and ADCI values occurred synchronously across the time scales, ranging from a few months’ high frequency to seasonal and over one year’s lower frequency band. The close relation between the semidiurnal LOD amplitude variation and the ADCI reflects the strong coupling between the solid Mars and surficial atmosphere system, which relates primarily to the semidiurnal atmospheric tide.

In this paper, the averaged Hamiltonian of a nonrestricted hierarchical triple system truncated at the third order is investigated. First, each secular resonant term is studied. For the well-studied secular quadrupole theory, it is analytically reformulated in a different manner in our work. The resonance width is numerically determined and displayed on the root 1 - e(1)(2) - root 1 - e(2)(2) plane (also denoted as the (e) over tilde (1) - (e) over tilde (2) plane). In terms of the octupole terms, we show that for a near-planar configuration of the system, considerable variations of both the eccentricities of the inner and outer orbits can be generated by a single resonant term. The resonance width for every secular resonant angle from the octupole terms is also numerically determined and displayed on the (e) over tilde (1) - (e) over tilde (2) plane. The results show that an orbit flip with a near-perpendicular initial mutual inclination is possible for each secular resonance. By displaying the resonance widths of different resonant terms on the same (e) over tilde (1) - (e) over tilde (2) plane, we intuitively show the overlap of different secular resonances. Then, the full averaged Hamiltonian with both quadrupole and octupole terms is investigated using the Poincare surface of section, with a special focus on the orbit flip. For the cases we exploited, we find that the near-planar flip of the inner orbit can be either regular or chaotic while the outer orbit flip is generally chaotic.

The secular behavior of an orbit under gravitational perturbation due to a two-dimensional uniform disk is studied in this paper, through analytical and numerical approaches. We develop the secular approximation of this problem and obtain the averaged Hamiltonian for this system first. We find that when the ratio of the semimajor axes of the inner orbit and the disk radius takes a very small value (MUCH LESS-THAN1), and if the inclination between the inner orbit and the disk is greater than the critical value of 30 degrees, the inner orbit will undergo the (classical) Lidov-Kozai resonance in which variations of eccentricity and inclination are usually very large and the system has two equilibrium points at omega = pi/2, 3 pi/2 (omega is the argument of perihelion). The critical value will slightly drop to about 27 degrees as the ratio increases to 0.4. However, the secular resonances will not occur for the outer orbit and the variations of the eccentricity and inclination are small. When the ratio of the orbit and the disk radius is nearly 1, there are many more complicated Lidov-Kozai resonance types which lead to orbital behaviors that are different from the classical Lidov-Kozai case. In these resonances, the system has more equilibrium points which could appear at omega = 0,pi/2,pi, 3 pi/2, and even other values of omega. The variations of eccentricity and inclination become relatively moderate, moreover, and in some cases the orbit can be maintained at a highly inclined state. In addition, an analysis shows that a Kuzmin disk can also lead to the (classical) Lidov-Kozai resonance, and the critical inclination is also 30 degrees.

The angle between planetary spin and the normal direction of an orbital plane is supposed to reveal a range of information about the associated planetary formation and evolution. Since the orbit's eccentricity and inclination oscillate periodically in a hierarchical triple body and tidal friction makes the spin parallel to the normal orientation of the orbital plane with a short timescale in an isolated binary system, we focus on the comprehensive effect of third body perturbation and tidal mechanism on the angle. Firstly, we extend the Hut tidal model (1981) to the general spatial case, adopting the equilibrium tide and weak friction hypothesis with constant delay time, which is suitable for arbitrary eccentricity and any angle v between the planetary spin and normal orientation of the orbital plane. Furthermore, under the constraint of angular momentum conservation, the equations of orbital and ratational motion are given. Secondly, considering the coupled effects of tidal dissipation and third body perturbation, and adopting the quadrupole approximation as the third body perturbation effect, a comprehensive model is established by this work. Finally, we find that the ultimate evolution depends on the timescales of the third body and tidal friction. When the timescale of the third body is much shorter than that of tidal friction, the angle v will oscillate for a long time, even over the whole evolution; when the timescale of the third body is observably larger than that of the tidal friction, the system may enter stable states, with the angle v decaying to zero ultimately, and some cases may have a stable inclination beyond the critical value of Lidov-Kozai resonance. In addition, these dynamical evolutions depend on the initial values of the orbital elements and may aid in understanding the characteristics of the orbits of exoplanets.

In response to the primary electric field originated by the fluid-core-dynamo, electric charges accumulate at the boundaries of a solid conducting D″ layer due to the poor circuitry conditions until the irrotational primary electric field is neutralized. The neutralization undoes the internal induction excited by the primary electric field and, as a result, eliminates the secondary toroidal magnetic field inside the D″. The primary magnetic field generated by the fluid-core-dynamo gives rise to a sustainable secondary poloidal magnetic field inside the conducting D″ layer. We introduce the general kinematic boundary conditions that are valid with different induction processes and parameter jumps across the core mantle boundary. We find by dimensional analysis that the strength of the secondary poloidal magnetic field is much weaker, by a factor of ∼10−2, than the irrotational primary. These findings indicate that the D″ is in effect an insulator for the geodynamo regardless of its material composition. Any mechanism for the core-mantle magnetic coupling should take this fact into consideration.