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Since Kepler, Newton and Huygens in the seventeenth century, geodesy has been concerned with determining the figure, orientation and gravitational field of the Earth. With the beginning of the space age in 1957, a new branch of geodesy was created, satellite geodesy. Only with satellites did geodesy become truly global. Oceans were no longer obstacles and the Earth as a whole could be observed and measured in consistent series of measurements. Of particular interest is the determination of the spatial structures and finally the temporal changes of the Earth's gravitational field. The knowledge of the gravitational field represents the natural bridge to the study of the physics of the Earth's interior, the circulation of our oceans and, more recently, the climate. Today, key findings on climate change are derived from the temporal changes in the gravitational field: on ice mass loss in Greenland and Antarctica, sea level rise and generally on changes in the global water cycle. This has only become possible with dedicated gravity satellite missions opening a method known as satellite gravimetry. In the first forty years of space age, satellite gravimetry was based on the analysis of the orbital motion of satellites. Due to the uneven distribution of observatories over the globe, the initially inaccurate measuring methods and the inadequacies of the evaluation models, the reconstruction of global models of the Earth's gravitational field was a great challenge. The transition from passive satellites for gravity field determination to satellites equipped with special sensor technology, which was initiated in the last decade of the twentieth century, brought decisive progress. In the chronological sequence of the launch of such new satellites, the history, mission objectives and measuring principles of the missions CHAMP, GRACE and GOCE flown since 2000 are outlined and essential scientific results of the individual missions are highlighted. The special features of the GRACE Follow-On Mission, which was launched in 2018, and the plans for a next generation of gravity field missions are also discussed.

The influence of gravitational field on entanglement of bipartite states is investigated based on the recent idea of superposition states of gravitational field. Different from earlier considerations, we study the case where the gravitational field cannot be separated unitarily from the bipartite system in the final stage of the interaction. When the different gravitational field states are orthogonal, entanglement cannot be generated for an initial product state. If the different gravitational field states are non-orthogonal, entanglement can be generated and the amount of generated entanglement depends on an overlap parameter between different gravitational field states. The influence of gravitational field on the transfer of the state through quantum teleportation is also studied, which might lead to an observable effect since the quantum teleportation can be performed using macroscopic object.

Radio tracking of the MESSENGER spacecraft has provided a model of Mercury's gravity field. In the northern hemisphere, several large gravity anomalies, including candidate mass concentrations (mascons), exceed 100 mi Hi-Galileos (mgal). Mercury's northern hemisphere crust is thicker at low latitudes and thinner in the polar region and shows evidence for thinning beneath some impact basins. The low-degree gravity field, combined with planetary spin parameters, yields the moment of inertia CIMR² = 0.353 ± 0.017, where M and R are Mercury's mass and radius, and a ratio of the moment of inertia of Mercury's solid outer shell to that of the planet of CJC = 0.452 ± 0.035. A model for Mercury's radial density distribution consistent with these results includes a solid silicate crust and mantle overlying a solid iron-sulfide layer and an iron-rich liquid outer core and perhaps a solid inner core.

The effect of gravity and proper acceleration on the frequency spectrum of an optical resonator-both rigid or deformable-is considered in the framework of general relativity. The optical resonator is modeled either as a rod of matter connecting two mirrors or as a dielectric rod whose ends function as mirrors. Explicit expressions for the frequency spectrum are derived for the case that it is only perturbed slightly and variations are slow enough to avoid any elastic resonances of the rod. For a deformable resonator, the perturbation of the frequency spectrum depends on the speed of sound in the rod supporting the mirrors. A connection is found to a relativistic concept of rigidity when the speed of sound approaches the speed of light. In contrast, the corresponding result for the assumption of Born rigidity is recovered when the speed of sound becomes infinite. The results presented in this article can be used as the basis for the description of optical and opto-mechanical systems in a curved spacetime. We apply our results to the examples of a uniformly accelerating resonator and an optical resonator in the gravitational field of a small moving sphere. To exemplify the applicability of our approach beyond the framework of linearized gravity, we consider the fictitious situation of an optical resonator falling into a black hole.

We have detected in Cassini spacecraft data the signature of the periodic tidal stresses within Titan, driven by the eccentricity (e = 0.028) of its 16-day orbit around Saturn. Precise measurements of the acceleration of Cassini during six close flybys between 2006 and 2011 have revealed that Titan responds to the variable tidal field exerted by Saturn with periodic changes of its quadrupole gravity, at about 4% of the static value. Two independent determinations of the corresponding degree-2 Love number yield k 2 = 0.589 ± 0.150 and k 2 = 0.637 ± 0.224 (2σ). Such a large response to the tidal field requires that Titan's interior be deformable over time scales of the orbital period, in a way that is consistent with a global ocean at depth.

The paper presents the principle of operation, the main components and the results of the work of the software created in the Sternberg Astronomical Institute of Moscow State University. The PC is designed for processing of large volumes of space geodetic data. The developed software was used to process inter-satellite measurements of a space-based constellation intended to measure the parameters of the Earth’s gravitational field (EGF). The experimental option of the software enables working with both simulated data and real data of GRACE and GRACE Follow-on missions. This experimental version was used to recover the EGF parameters on real GRACE and GRACE-FO mission data. Solutions were developed for every month within the measurement time intervals from 2010 to 2021, as well as for extended time intervals of 4.3 and 7.6 years. A comparison of the obtained solutions with the results of the EGF recovery obtained by other researchers is presented.

A new model of multi-temperatures for a generalized micropolar thermoelastic medium has been established in this paper. A medium is affected by a gravitational field and two types of mechanical strip load (continuous load and impact load). The technique called Laplace Fourier transform has been utilized to obtain the analytical expressions of variables under deliberation. The numerical and graphical illustration of the results has been carried out to indicate the differences among one temperature model, the classical dual-temperature model, and the hyperbolic dual-temperature model upon the Lord and Shulman theory. Also, in the case of Coupled Theory (CT) and Lord and Shulman theory (L-S), we discussed the effect of the gravitational field and mechanical strip load. The most significant points are highlighted. The current investigation has led us to deduce some particular cases of special interest. When it comes to heat conduction's new general model then this study will be extremely beneficial in developing a better understanding of the ingrained features.

In this paper, we consider the attraction field of a rigid body close to a dynamically symmetric one. The question of how to approximate this attractive field by the attractive field of three homogeneous balls so that the series expansion of the gravitational potential for these fields coincides up to terms of the fifth order of smallness is studied. As an example, the three-ball approximation as applied to asteroid (433) Eros is considered.

A description of the dynamical response of uniformly trapped Bose-Einstein condensates (BECs) to oscillating external gravitational fields is developed, with the inclusion of damping. Two different effects that can lead to the creation of phonons in the BEC are identified; direct driving and parametric driving. Additionally, the oscillating gravitational field couples phonon modes, which can lead to the transition of excitations between modes. The special case of the gravitational field of a small, oscillating sphere located closely to the BEC is considered. It is shown that measurement of the effects may be possible for oscillating source masses down to the milligram scale, with a signal to noise ratio of the order of 10. To this end, noise terms and variations of experimental parameters are discussed and generic experimental parameters are given for specific atom species. The results of this article suggest the utility of BECs as sensors for the gravitational field of very small oscillating objects which may help pave the way towards gravity experiments with masses in the quantum regime.

Exact computation of the gravitational field and gravitational gradient tensor for a general mass body is a core routine to model the density structure of the Earth. In this study, we report on the existence of closed-form solutions of the gravitational potential, gravitational field and gravitational gradient tensor for a general polyhedral mass body with a polynomial density function of arbitrary non-negative integer orders that can simultaneously vary in both horizontal and vertical directions. Our closed-form solutions of the gravitational potential and the gravitational field are singularity-free, which implies that the observation sites can have arbitrary geometric relationships with polyhedral mass source bodies. However, weak logarithmic singularities exist on the edges of polyhedra for the gravitational gradient tensor. A simple prismatic mass body with polynomial density contrast varying in the vertical direction and a complicated dodecahedral mass body with quartic-order density contrasts were tested to verify the accuracy of the newly derived closed-form solutions. For the gravitational potential, gravitational fields and gradient tensors, our closed-form solutions are in excellent agreement with previously published analytical solutions and Gaussian numerical quadrature solutions.