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The pandemic-induced lock-down has resulted in better air quality and visibility, with reported sightings of Himalayan peaks from hundreds of kilometers away. During the 18th and 19th centuries, good visibility was commonplace and there exist many such accounts by reputable Orientalists such as Henry Colebrooke and Sir William Jones. These sightings invite two questions. Is a line of sight to the peak even possible from the location? Is the location close enough and the intensity of visible light sufficient for the peak to be seen? The present work explores both these aspects in the context of the Himalayan peaks. The effects of the curvature of the Earth and refraction on the perceived heights of several mountains in the eastern Himalayas are considered. The effects of Rayleigh scattering by the atmosphere on the visibility of these peaks are also considered as an extension of the work done by Weisskopf. Some historically significant details and a possible inaccuracy in an account by Sir William Jones are discussed. The approach is pedagogical and is not specialized to the eastern Himalayas, so it can be applied readily to other locations, for example, the Andes in South America.

The article analyses an argument given in Jñānarāja's Siddhāntasundara (ca. 1500) on the shape of the earth according to the Purāṇas. The argument involves the use of the word gola, 'ball, globe,' in the Purāṇas, a Purāṇic statement about the mountain Meru being north of everywhere, and a Purāṇic comparison of the earth to a mirror. The article concludes that Jñānarāja breaks with the Purāṇas as well as the traditional commentaries on these texts, and further suggests that we might have to rethink the dictionary definition of gola.

Global gravity modelling is one of the most important issues in geophysics and geodesy. Because a tesseroid model comprises the curvature of the Earth, the computation methods for the gravitational potential of tesseroids and its first-order derivatives in spherical coordinates are attracting great attention in recent years. In this paper we deal with the numerical evaluation of the radial component of the gravitational attraction generated by tesseroid masses at satellite height with the Gauss-Legendre quadrature (GLQ), the Taylor series expansion (TSE) and the prism approximation (PA) methods. Forward modelling of tesseroids of 1° × 1° and 5′ × 5′ are performed by three computation methods and the comparison between them are made in terms of computational efficiency and accuracy. The numerical results show that the GLQ of order 5 can provide the adequate accuracy for the gravity modelling of 1° × 1° tesseroids at satellite height. The GLQ of order 2 and TSE methods are superior to the PA approach in both computational accuracy and efficiency. The satellite height has important impact on the accuracy of the GLQ and TSE, whereas it has no effect on the PA method. In addition, we developed combined GLQ approach and combined TSE method, respectively, for global gravity modelling based on 1° × 1° and 5′ × 5′ tesseroids. Apart from the synthetic tesseroids, 1° × 1° data from the CRUST1.0 global crustal model and 5′ × 5′ rock-equivalent topographic data from the Earth2014 model are used to validate two combined methods. The numerical results show that these two combined methods can balance the computational accuracy and efficiency.

Identification of when and where moving areas intersect is an important problem in maritime operations and air traffic control. This problem can become particularly complicated when considering large numbers of objects, and when taking into account the curvature of the earth. In this paper, we present an approach to conflict identification as a series of stages where the earlier stages are fast, but may result in a false detection of a conflict. These early stages are used to reduce the number of potential conflict pairs for the later stages, which are slower, but more precise. Our approach is generally applicable to objects moving in piece-wise straight lines on a 2D plane, and we present a specific case where the Mercator Projection is used to transform objects moving along rhumb lines on the earth into straight lines to fit in our approach. We present several examples to demonstrate our methods, as well as to quantify the empirical time complexity by using randomly generated areas.

Quantitative indicators of the influence of the shape of the Earth (ellipsoidal with modern polar compression or spherical) on the characteristics of its irradiation calculated without taking into account the changes in the solar activity have been obtained. The differences between specific energies as well as intensities of irradiation of the ellipsoid (and its parts) and sphere (and its parts) on the average over a tropical year and in its parts (half-years, months) have been revealed. It is found from the data on the Earth’s motion in the interval of the tropical years from 3000 BC to AD 2999 that, at a constant intensity of solar irradiance, each of the two annual irradiation characteristics for the sphere is 0.085% higher than one for the ellipsoid. The monthly irradiation specific energy for the sphere is almost constant, and one for the ellipsoid varies with a scatter 0.043% of the average value (minima at the equinoxes and maxima at the solstices). The monthly irradiation intensities for the sphere and ellipsoid vary synchronously with a scatter 5.98% relative to the average value (the minimum occurs in the third month and the maximum in the ninth). The seasonality for a hemisphere (absolute value of the difference between irradiation specific energies for a hemisphere in different half-year periods) in the case of the ellipsoidal Earth is on the average 0.223% higher than in the case of the spherical Earth. The maximum of differences in the semiannual irradiation specific energies (and similarly in the semiannual irradiation intensities) in the latitudinal zones of the sphere and ellipsoid is positive and the minimum is negative. In the first half of the year, the maximum and minimum are observed in the zones 45°–50° S and 10°–15° N. In the second half of the year, the maximum and minimum are observed in zones 45°–50° N and 10°–15° S. It is found on the basis of the data from 1978–2008 that the differences in the intensities of irradiation of the sphere and ellipsoid are comparable with the variations in the total solar irradiance (TSI) associated with the changes in solar activity. The mean absolute value of the annual TSI anomaly is 14.01% greater than the average annual difference in the intensities of irradiation of the sphere and ellipsoid. The average absolute value of the monthly TSI anomaly exceeds the average monthly difference in the intensities of the irradiation of the sphere and ellipsoid by 23.37%.

In the gravity method of exploration geophysics, Bouguer is the name of a specific correction to measurements, and also the name of the anomaly remaining after that correction and other corrections...

Turkish Islamic scientists, who took over the legacy of antiquity, conducted various researches in all areas of science. Earth's shape, rotation and gravitation are one of these issues. In this study, have presented the views of the Islamic scientists in the Middle Ages about the shape, movements and gravity of the Earth. In the Middle Ages, Islamic scientists stated that the shape of the world was a sphere in a period that can be considered very early and they always referred to the Earth as a globe. Turkish Islamic scientists have always mentioned the Earth as ‘al-qura’t al- ard (the Earth globe)’. This statement shows that they have no other thoughts about the shape of the Earth. With this research, it has understood that there is the information that the Earth rotates around its own axis in the works of the medieval Turkish Islamic scientists. In relation to this issue, they stated that the center of the Earth attracts everything like a magnet to itself. Thus, they explained gravity correctly. The effects of Turkish Islamic scientists' thoughts on the shape of the Earth, gravity and the law of motion can be seen in Galileo and Newton.

Epicurus’s opinion on the shape of the earth forms a delicate issue that has animated recent scholarship. Similarly, in the reconstruction of Epicurus’s philosophy by the French polymath Pierre Gassendi (1592–1655), the issue emerged under various guises. Interestingly, the question played a particularly puzzling role in Gassendi’s considerations of Epicurus’s atomic motion. By comparing the unpublished manuscript book De atomis (1636–1637) to the published Animadversiones (1649) and Syntagma philosophicum (1658), I will reveal that, in his quest for an accurate presentation of the original theory, Gassendi considerably modified his initial manuscript reconstruction of Epicurus’s atomic movements and developed an account in which the concept of a plane terrestrial surface was indispensable. At the same time, the article will indicate that Gassendi, from his own point of view, also proposed an alternative to Epicurus’s atomic motion in which he reinterpreted the original concepts in such a way that he could abstract himself from the Epicurean swerve of the atoms.

At a conference on educational physics, someone described a series of multiple-choice problems to test students’ ideas about various mechanical phenomena. One of the problems questioned students’ conceptions about the shape of the Earth’s orbit in the solar system. The question was as follows: Which of the following schematic illustrations corresponds most accurately to the real situation regarding the position of the Sun and the shape of the Earth’s trajectory around the Sun, when viewed from top?

Processes within the Earth shape and influence the surface environment and the emergence and evolution of life. Our Editorial board members outline recent advances and future directions in our attempt to understand the history of our planet and its environment. Processes within the Earth shape and influence the surface environment and the emergence and evolution of life. Our Editorial board members outline recent advances and future directions in our attempt to understand the history of our planet and its environment.

In this work, we aimed to see how changes in the Earth's gravitational field in the context of general relativity modify the radius and age values of the universe in the previously developed distance-determining model. By applying the relationships of light propagating in accelerating systems and gravity to the Earth moving with the Milky Way galaxy in the expanding universe and to the light propagating in the surface gravity of our planet moving in this way, the radius and age of the universe can be determined. For a redshift extent of 3.1415 in the cosmos, we can determine a so-called “short evolving distance” from the Einstein equation. Knowledge of the entire plane angle (2.\(\pi\)) and the deviating angle of a light beam (\(\alpha\)) passing through the Earth’s surface gravity it is possible to determine extreme distances by utilizing the rules of trigonometry. This “long evolving distance,” which is approximately in the range of the radius of the universe, can be transformed into the “evolving time” divided by the speed of light. Based upon this dating method, the universe may have been formed at an average value of 13.7536 billion years. In addition to the cosmos radius and age, different but small changes can be observed in these, taking into account the changes in the Earth's surface gravity associated with its shape and position. These values fall between 13.6879 and 13.8498 billion years, almost completely covering the range given by the analysis of the measurement results of the WMAP and Planck satellites.

The mathematical expressions, specifying the estimation of the capabilities of remote sensing satellites with opto-electronic imaging equipment for observing the Earth’s surface with the necessary (predetermined) spatial resolution of the space imaging materials when deflecting the optical axis of on-board information complex, are obtained and substantiated in the article. The validity of the obtained expressions is proved by comparing the results of the solution of the direct and inverse problems of determining the spatial resolution of the space imaging materials when the optical axis of the on-board information complex of remote sensing satellites is deflected. Taking into account the deterioration of spatial resolution depending on the angle of deviation of the on-board information system of the satellites and the curvature of the Earth provides increasing the accuracy in determining the field of regard of satellites when modeling the state and changes in the space situation, as well as assessing their surface capabilities for space imaging. The capabilities of remote sensing satellites with opto-electronic equipment have been evaluated and it has been proved that the space imagery they receive cannot be applied to detect and identify (to the kind, class, type) objects of observation within the entire field of regard of imaging equipment, provided that the claimed spatial resolution of the space imaging materials. Prospects for further research are the refinement solution of practical problems in planning the use of on-board resource of remote sensing satellites, the improvement of algorithms for taking into account the possible observation of domestic territory (objects) with the help of foreign spacecraft’s and improving the methods of estimation of the remote sensing satellites with radar equipment.

As a new potential platform for Earth observation, the Moon which is a natural satellite unique to the Earth has been paid more and more attention for its consistent and continuous observation capability of global-scale and macroscopic geoscience phenomena on Earth. Because of the effect of Earth curvature and the Earth–Moon geometric relationship, the geometric correction of the Moon-based Earth observation (MBEO) hemisphere images is more complex than that required for images obtained from low Earth orbit (LEO) platforms and need to be investigated, such as optical images in visible/thermal infrared frequencies or synthetic aperture radar (SAR) images in microwave frequency. In addition, it is also difficult to find the uniformly distributed geometric correction control points on the entire surface of the Earth caused by sea and cloud cover in most regions. In this paper, a polar coordinate system on a projection surface related to the sublunar point is designed to facilitate the geometric correction of MBEO hemisphere images and to try to compensate for the problems of geometrical distortion caused by the shift of the sublunar point, the curvature of the Earth and the terrain fluctuation. Through simulation analysis, the geometric error characteristics and conditions in which the proposed moon-projection polar coordinate system can be used are discussed. This involves considering the Earth model, the projection method, the offset of the sublunar point and the external error. The results show the validity and adaptation of the proposed polar coordinate system and will support the further study of the geometric correction of MEBO images.

Compared with ground radars, space-borne bistatic radar shows great advantages in higher signal-to-noise ratio (SNR) and robuster security. Firstly, this study establishes space-borne geosynchronous satellite-low orbiting satellite (GEO-LEO) space–time two-dimensional model considering the influence of the rotation of the earth together with the curvature of the earth surface. Then the Doppler trajectory distance dependence of the clutter angle is analysed. Finally, the characteristics of the space-borne GEO-LEO model are proved to be beneficial to clutter suppression and ground-moving targets indication. Simulation results show that the space-borne GEO-LEO model has good performance.

Due to the elliptic shape of Earth's orbit around the Sun, the apparent diameter of the Sun varies throughout the year. With digital analysis of a number of images at a given date, the effect can be demonstrated with a digital single lens reflection camera (DSLR) and a standard telephoto lens even over a few days. In the analysis, the influence of image brightness and atmospheric refraction on apparent Sun size and shape have to be removed to reach the accuracy that is required.

The present study examines the response of the terrestrial magnetosphere to the long-term steady declining trends observed in solar magnetic fields and solar wind micro-turbulence levels since mid-1990's that has been continuing beyond the mini-solar maximum of cycle 24. A detailed analysis of the response of the terrestrial magnetosphere has been carried out by studying the extent and shape of the Earth's magnetopause and bow shock over the past four solar cycles. We estimate sub-solar stand-off distance of the magnetopause and bow shock, and the shape of the magnetopause using numerical as well as empirical models. The computed magnetopause and bow shock stand-off distances have been found to be increasing steadily since around mid-1990's, consistent with the steady declining trend seen in solar magnetic fields and solar wind micro-turbulence levels. Similarly, we find an expansion in the shape of the magnetopause since 1996. The implications of the increasing trend seen in the magnetopause and bow shock stand-off distances are discussed and a forecast of the shape of the magnetopause in 2020, the minimum of cycle 24, has been made. Importantly, we also find two instances between 1968 and 1991 when the magnetopause stand-off distance dropped to values close to 6.6 earth radii, the geostationary orbit, for duration ranging from 9$-$11 hours and one event in 2005, post 1995 when the decline in photospheric fields began. Though there have been no such events since 2005, it represents a clear and present danger to our satellite systems., Accepted for publication in JGR-Space Physics. arXiv admin note: text overlap with arXiv:1709.01407

Since the earth is spherical in shape, the paths from the satellite to various points on the earth are different in both the distance and atmospheric attenuation. In order to maintain approximately uniform effective signal strength on the surface of the earth, the radiation pattern of the antenna should be formed according to the shape of the earth. Former antennas for earth coverage purpose are either bulky or hard to realize the desired beam pattern within a wide angle range, making them unsuitable for small satellite applications. To solve this problem, this paper presents a novel circularly polarized earth coverage antenna for spaceborne applications. The proposed antenna consists of a specially designed dielectric cylinder, in which a cavity backed patch antenna is embedded. With the help of the concave structure at the top of the dielectric cylinder, the refraction of electromagnetic wave can be controlled, and then the radiation pattern is shaped. A sequentially feeding network is designed to drive the antenna. To verify our concept, a prototype is fabricated and experiments are carried out. The simulation agrees well with the measurement. The earth coverage radiation pattern is obtained within the beamwidth of $\theta $ in the range of ±50° with the measured space axial ratio is better than 3 dB from 4.9 to 5.2 GHz. The measured peak gain is about 3.45 dBic at the elevation angle of ±50°.

The history of geodesy can be traced back to Thales of Miletus (∼600 BC), who developed the concept of geometry, i.e. the measurement of the Earth. Eratosthenes (276–195 BC) recognized the Earth as a sphere and determined its radius. In the 18th century, Isaac Newton postulated an ellipsoidal figure due to the Earth's rotation, and the French Academy of Sciences organized two expeditions to Lapland and the Viceroyalty of Peru to determine the different curvatures of the Earth at the pole and the Equator. The Prussian General Johann Jacob Baeyer (1794–1885) initiated the international arc measurement to observe the irregular figure of the Earth given by an equipotential surface of the gravity field. This led to the foundation of the International Geodetic Association, which was transferred in 1919 to the Section of Geodesy of the International Union of Geodesy and Geophysics. This paper presents the activities from 1919 to 2019, characterized by a continuous broadening from geometric to gravimetric observations, from exclusive solid Earth parameters to atmospheric and hydrospheric effects, and from static to dynamic models. At present, we identify geodesy as the discipline of quantifying global change by geodetic measurements.

The relationship between the proper time of a satellite clock and the geocentric coordinate time (TCG) is discussed in the framework of general relativity. The influences of Earth's shape and the tidal potentials of the Sun and Moon are analyzed for onboard satellite clocks in different orbits. The results show that the influence of the shape of the Earth on Middle Earth Orbit (MEO) satellite reaches the level of 4.0 × 10-15, and it may be the direct cause of the half orbital-period variation in the clock frequency. The influences of the tidal potentials of the Sun and the Moon are relatively smaller. These influences on the frequencies of the clocks in the BeiDou MEO and GEO/IGSO (Geosynchronous Earth Orbit/Inclined Geosynchronous Satellite Orbit) satellites are 5.9 × 10-16 and 1.2 × 10-15, respectively, and appear to be half-day periodic variations. Therefore, these effects need to be considered in the precise assessment and application of BeiDou satellite clocks.

As in many regions of the world, astronomy also played a major role in the ancient Peruvian cultures. However, the discussion of the astronomical relevance of the Nasca geoglyphs is very controversial. A really precise and extensive investigation of astronomical phenomena has not yet taken place, the necessary data were simply missing. In the Nasca project Dresden, these data have been recorded in recent years and stored in an Oracle database. In the very first step, all geoglyphs with an astronomical orientation documented by Maria Reiche were checked and verified. Subsequently, all lines of the entity “straight line” were systematically examined with regard to the celestial bodies of the Sun and bright stars. For this purpose, on the one hand, the ellipsoidal azimuths of all straight lines were calculated and, on the other hand, the elevation angles in relation to the horizon with the help of digital terrain models (DTM) were determined. Corrections for refraction, the curvature of the Earth, visibility and atmospheric disturbances were largely considered. The azimuths of the celestial bodies during the Nasca period were calculated with software developed in-house (theses by students) and compared with those of the lines. As a result, it was possible to establish that there are individual straight lines that are aligned with the Sun and the seven randomly selected bright stars. However, the number of hits found does not justify the theory that the Nasca Pampas are an astronomical calendar system.

Green’s functions (GFs) for elastic deformation due to unit slip on the fault plane comprise an essential tool for estimating earthquake rupture and underground preparation processes. These estimation results are often applied to generate important information for public such as seismic and tsunami hazard assessments. So, it is important to minimize the distortion of the estimation results on the numerical models used for calculating GFs to guarantee assessment reliability. For this purpose, we here calculated GFs based on a numerical model that is of high fidelity to obtain realistic topography and subsurface structural models of the Earth. We targeted two well-known subduction zones in Japan, the Nankai Trough and the Japan Trench. For these subduction zones, databases for realistic topography and subsurface structural models of the Earth are available in the “Japan integrated velocity structure model version 1”, which was proposed for earthquake hazard assessments conducted by the Japanese government. Furthermore, we eliminated the inconsistency in processing calculated GFs and space geodetic observation data for surface displacements, which is often overlooked, by using the same coordinate system. The ellipsoidal shape of the Earth, which is often approximated with a projected plane or a spherical shape, was also incorporated by faithfully following the definitions of the coordinate systems in Geodetic Reference System 1980, which is the global standard for space geodesy. To calculate elastic GFs based on such high-fidelity subduction zone databases with the ellipsoidal shape of the Earth, we introduced the finite element (FE) method. In the FE meshes, the resolution of the topography and subsurface structure is the same as that of the original databases. Recent development of the state-of-the-art computation techniques for the rapid calculation of crustal deformation using large-scale FE models allows for GF calculation based on such a high-fidelity model. However, it is generally not easy to perform such calculations. Thus, we released a library for the GFs calculated with 1-km grid spacing on the ground surface in this study to the geoscience community on a web server, aiming to contribute more reliable seismic and tsunami hazard assessment.

The Earth’s shape concept develops as consecutive cognitive problems (e.g., the location of people and trees on the spherical Earth) are gradually resolved. Establishing the order of problem solving may be important for the organisation of teaching situations. This study attempted to determine the sequence of problems to be resolved based on tasks included in the EARTH2 test. The study covered a group of 444 children between 5 and 10 years of age. It captured the order in which children solve cognitive problems on the way to constructing a science-like concept. The test results were compared with previous studies. The importance of cultural influences connected to significant differences (24%) in test results was emphasised. Attention was drawn to the problem of the consistency of the mental model approach highlighted in the literature. The analysis of the individual sets of answers provided a high level of consistency of indications referring to the same model (36%), emphasising the importance of the concept of mental models.

This paper proposes a new wide-beam bifilar helix antenna with earth coverage beam for satellite communication. Former helix antennas for micro-satellites are focused on narrow-beam high-gain purpose and hard to realize the desired beam pattern within a wide angle range, making them unsuitable for earth coverage applications. This novel antenna realizes the uniform illumination of the earth's surface with miniaturized structure. (The earth coverage radiation pattern is obtained within the beamwidth of θ in the range of ±56°). According to the calculation, the antenna is modeled and simulated. The simulation results show good agreement.

Between the eighteenth and nineteenth centuries, the Via Appia Antica was the site of intense geodetic activity, serving as the baseline for the map of the Papal States and the study about the shape of the Earth. Eventually this work was connected to the determination of the Prime Meridian of Italy, which served as the basis of the new map of Italy after its unification. In recent years, new discoveries have allowed us to reevaluate this historical and scientific heritage.

Some of the most powerful spatial analysis software solutions (Oracle, Google Earth Engine, PostgreSQL + PostGIS, etc.) are currently performing geometric calculations directly on the ellipsoid (a quadratic surface that models the earth shape), with a double purpose: to attain a high degree of accuracy and to allow the full management of large areas of territory (countries or even continents). It is well known that both objectives are impossible to achieve by means of the traditional approach using local mathematical projections and Cartesian coordinates. This paper demonstrates in a quantitative methodological way that most of the spatial analysis software products make important deviations in calculations regarding to geodesics, being the users unaware of the magnitude of these inaccuracies, which can easily reach meters depending on the distance. This is due to the use of ellipsoid calculations in an approximate way (e.g., using a sphere instead of an ellipsoid). This paper presents the implementation of two algorithms that solve with high accuracy (less than 100 nm) and efficiently (few iterations) two basic geometric calculations on the ellipsoid that are essential to build more complex spatial operators: the intersection of two geodesics and the minimum distance from a point to a geodesic.

In this essay, Giovanbattista Tusa explores the geophilosophical possibility of rethinking the figure of the earth in twentieth-century Western philosophical thought and suggests new opportunities for thinking that open up with the twenty-first century. On the one hand, “Earth” as a Western concept has been reduced to an exhaustible resource—an endangered planet condemned to its own ending. On the other hand, another continent seems to have emerged in contemporary philosophical thought in reaction to this brutal relationship with the planet—“Earth” as a dark, impenetrable and indestructible reserve, the last resource of thought. A materialization of an unconditional power, this Earth seems to reproduce the original need for a wild, ultimate refuge for the philosophical thought of the twentieth century. To this archaic Earth of thought, whose survival seems to depend on preserving itself untouched, and untouchable, the author confronts another possibility, the geo-anarchy of other earths—the material insurrection of planetary energies.

The method for calculating the downward solar energy falling to the Earth without an atmosphere is developed. The calculations take into account the ellipsoidal shape of the Earth and variations of the orbital motion and the rotation of the Earth produced by the Moon and planets. The insolation integrals were prepared for Earth’s surface latitudinal strips and slices of tropical years from 3000 BC to AD 2999.

In this paper the material culture and representative archaeological artefacts are used to address the question either if the spherical shape of the Earth was an exclusive knowledge or if it was known by the many in the Roman imperial time.

A “gravity tunnel” is the name given to a fictitious deep shaft drilled inside the Earth so that objects dropped from the surface of the Earth would free fall without ever touching the walls. It is well known that because of the rotation of the Earth, such tunnels are not straight lines but instead they emerge westward of the antipodal point, when the Earth is approximated as a rotating sphere. In this article, we determine the shape of gravity tunnels by taking into account the polar flattening of the Earth resulting from its rotation. The Earth is described as a McLaurin spheroid, an exact equilibrium shape for a rotating homogeneous deformable body that provides a fair description of the actual shape of the Earth. It turns out that the gravitational force acting on an object located inside the spheroid has a simple form (it is harmonic), so that it is straightforward to compute analytically the free fall trajectories. This study follows a procedure presented several times in this journal and elsewhere, i.e., the trajectory is first computed in the geocentric (non-rotating) frame, and it is then analysed in the terrestrial (rotating) frame. We find that when the flattening of the Earth is taken into account, gravity tunnels have no exit: an object dropped from a point of the surface (other than on the equator or a pole) never reaches the surface again, unless the flattening has very specific (and unnatural) values. We also compute the deviations from the vertical for short falls and compare them to standard eastwards and southwards deviation expressions obtained with other modelizations of the gravity of the rotating Earth, in particular, for a rotating spherical body.

A generalized theory of the figures of the Earth’s interior to a third-order precision of ellipticity is proposed in accompanying paper in which all the odd degree and nonzero order spherical harmonic terms are included. As both the direct and indirect contributions of the asymmetric crust are included, this theory makes a significant improvement for calculating the asymmetric equilibrium figures of the real Earth comparing with the traditional theories which can only deal with the ideal symmetric Earth. The principal moments of inertia (PMOI: A, B, C) and global dynamical flattening (H) are important quantities in studying the rotating Earth. Precession and gravity observations give observation value of H ( $$H_{\mathrm{obs}} \approx 1/305.4559$$ ) with very high precision, while its theoretical calculated value ( $$H_{\mathrm{theory}} \approx 1/308.5$$ ) from traditional theories and a starting symmetric Earth model (like PREM model) is about $$1\%$$ less than $$H_{\mathrm{obs}}$$ . Using the new theory in accompanying paper and replacing the homogeneous outermost crust and oceanic layers in PREM with CRUST1.0 model, we recalculate the equilibrium figures of the Earth’s interior and finally get new values of PMOI and $$H_{\mathrm{theory}}$$ ( $${\approx } \,1/304.7167$$ ) whose consistency with $$H_{\mathrm{obs}}$$ are significantly improved to 0.24%. Furthermore, the asymmetric figures of some interesting boundaries, like inner core boundary, core-mantle boundary, are also given as by-products of this work as these boundaries’ figures are key input for studies of their topographic effect on global rotation and geodynamics, like nutation, normal modes, especially like free core nutation.

Traditionally a laterally homogeneous and spherical base Earth model (e.g., the PREM model) is considered as input when computing the Earth’s equipotential surfaces, which are then resulted to be in symmetric shape. However, the Earth, known with a complex distribution of interior material and density, especially in the upper mantle and the crust, cannot be treated as a symmetric sphere. Recently, a CRUST1.0 model of crust layer is published and well accepted. But the effect caused by the asymmetric crust (and mantle) on equilibrium figures of the Earth cannot be analyzed by the traditional theories. A generalized theory of the figure of the Earth to third-order precision is firstly proposed in this paper, as well as the iterative calculation strategy to solve the complex equation system. In order to validate this generalized theory, the degeneration of this generalized theory with the PREM model is made and is compared with traditional theories, and it is shown that the result of this generalized theory, after degeneration, is consistent very well with traditional theory. Meanwhile, the effect (including both the direct and indirect effects) of the crust layer, from the CRUST1.0 model, on the figures of equipotential surfaces of the Earth’s interior, as well as their effects on the global dynamics flattening, will be presented as an application of this theory in accompanying paper.

The present review article is devoted to the consideration of the causes of the practically crisis state of study and unsuccessful attempts to explain the solar–terrestrial relationships. It substantiates the need to take into account the endogenous activity of the Earth, (in addition to solar activity and fluxes of galactic cosmic rays). Solar activity and fluxes of galactic cosmic rays contribute to their respective roles and share of external impact on the Earth. When all three external factors of influence on the Earth are taken into account, mechanisms, cyclicity, energetics, polar asymmetry, synchronism, variations in the shape of the Earth, the level of the ocean, gravity, seismicity and volcanism become understandable, as well as the previously unexplained events and processes of manifestation solar–terrestrial relationships. Endogenous activity of the Earth is caused by the gravitational actions of the Moon, the Sun and other planets in the process of barycentric motion of the solar system in the gravitational field of the Galaxy. It is shown that, under external influence on the solar system, the Earth’s shells respond to it synchronously with the responses of the observed layers of the Sun’s atmosphere.

The paper focuses on the comparison of metrics of three most common reference surfaces of the Earth used in geodesy (excluding the plane which also belongs to reference surfaces used in geodesy when dealing with small areas): a sphere, an ellipsoid of revolution and a triaxial ellipsoid. The two latter surfaces are treated in a more detailed way. First, the mathematical form of the metric tensors using three types of coordinates is derived and the lengths of meridian and parallel arcs between the two types of ellipsoids are compared. Three kinds of parallels, according to the type of latitude, can be defined on a triaxial ellipsoid. We show that two types of parallels are spatial curves and one is represented by ellipses. The differences of curvature of both kinds of ellipsoid are analysed using the normal curvature radii. Priority of the chosen triaxial ellipsoid is documented by its better fit with respect to the high-degree geoid model EIGEN6c4 computed up to degree and order 2160.

The aim of this paper is to study the size, shape, and orientation of the asteroid Vesta based on the Dawn spacecraft observations. In this line, three main reference surfaces are defined; the geoid, reference ellipsoid, and best-fit ellipsoid. To be consistent with International Astronomical Union (IAU) standards, all the computations are done in the Claudia Double–Prime coordinate system. The geoid and its potential value W 0 are computed by fitting an equipotential surface to the shape of Vesta in a least-squares sense regarding the topographic bias effect. We find that the topographic bias effect on the geoid potential value as well as the geoidal and topographic heights is significant. The geoid potential value W 0 is estimated equal to 68709 ± 24 m2/s2. The reference ellipsoid is defined as an equipotential ellipsoid which best fits the geoid. The reference ellipsoid is computed based on the fundamental geodetic constants which define a geodetic reference system (GRS). The semi-axes of the triaxial reference ellipsoid are found to be equal to 280413 ± 104 m, 274572 ± 102 m, and 231253 ± 86 m with the equatorial semi-major axis longitude 8.29°E, while the semi-axes of the biaxial reference ellipsoid are 278556 ± 117 m and 229921 ± 76 m. The results show that the geoidal heights with respect to the triaxial reference ellipsoid are significantly different from the geoidal heights with respect to the biaxial reference ellipsoid. The parameters of the best-fit ellipsoid are estimated by fitting geometrical ellipsoids with various degrees of freedom to Vesta's shape in a least-squares sense. We report the semi-axes of the general best-fit ellipsoid with 9 degrees of freedom equal to 284562 ± 75 m, 277248 ± 73 m, and 226405 ± 57 m. Regarding the orientation of Vesta, we find that the angle between the equatorial semi-major axis of the general best-fit ellipsoid and the X-axis of the coordinate system is 9.17 ∘ ± 0.47 , and the angle between its polar axis and the Z-axis is equal to 0.63 ∘ ± 0.05 . Furthermore, the spherical harmonic coefficients of the shape model of Vesta up to degree 180 are computed, giving the ability to estimate the parameters of the general best-fit ellipsoid. The significant differences observed between the parameters of the general best-fit ellipsoid derived from the spherical harmonic coefficients of the shape model, and those derived from the fitting are due to the large polar and equatorial flattenings of Vesta. We estimate that the offset of the center of figure from the center of mass is 1350 ± 53 m, and the mean radius of Vesta is 261403 ± 39 m.

Geophysical models and their interpretations of several processes of interest, such as sea-level rise, post-seismic relaxation, and glacial isostatic adjustment, are inter-twined with the need to realize the International Terrestrial Reference Frame. However, this realization needs to take into account the geocenter motion, that is, the motion of the Center of Figure of the Earth surface, due to, for example, deformation of the surface by earthquakes or hydrological loading effects. Usually there is also a discrepancy, known as the network effect, between the theoretically convenient Center of Figure and the physically accessible Center of Network frames, because of unavoidable factors such as uneven station distribution, lack of stations in the oceans, disparity in the coverage between the two hemispheres, and the existence of tectonically deforming zones. Here, we develop a method to estimate the magnitude of the network effect, that is, the error introduced by the incomplete sampling of the Earth surface, in measuring the geocenter motion, for a network of space geodetic stations of a fixed size N. For this purpose we use, as our proposed estimate, the standard deviations of the changes in Helmert parameters measured by a random network of the same size N. We show that our estimate scales as 1/N, and give an explicit formula for it in terms of the vector spherical harmonics expansion of the displacement field. In a complementary paper we apply this formalism to co-seismic displacements and elastic deformations due to surface water movements.

An important scientific problem of the seventeenth and eighteenth centuries was to determine the figure of the earth. This problem attracted the attention of many notable people, one of whom was Roger Joseph Boscovich a Jesuit priest and natural scientist who formulated—and gave an algorithm for solving—linearly constrained optimization problems whose minimand is the sum of the absolute values of the errors, i.e., a special linear regression model. After first recounting some of the rich history of this development, the present expository note focuses on the method of Boscovich, exploring its interpretation as a variant of the simplex algorithm for linear programming applied to problems of that form.

It is known from observations that the center of mass of the Moon does not coincide with the geometric center of its figure, and the line connecting these two centers is not aligned with the direction toward the center of the Earth, instead deviating toward the Southeast. This stationary deviation of the axis of the inertia ellipsoid of the Moon to the South of the direction toward the Earth is analyzed. A system of five linear differential equations describing the physical libration of the Moon in latitude is considered, and these equations are derived using a new vector method taking into account perturbations from the Earth and partly from the Sun. The characteristic equation of this system is obtained, and all five oscillation frequencies are found. Special attention is paid to the fifth (zero) frequency, for which the solution of the latitude libration equations are stationary and represents a previously unknown additional motion of the rotational axis of theMoon in a cone with a small opening angle. In contrast to the astronomical precession of the Earth, the rotation of the angular-velocity vector is in the positive direction (counter-clockwise), with the period T 3 = 27.32 days. On this basis, this phenomenon has been named “quasi-precession.” This quasi-precession leads to a stationary inclination of the major axis of the inertia ellipsoid of theMoon to the South (for an observer on Earth), making it possible to explain one component of the observed deviation of the center of mass of the Moon from the direction toward the Earth. The opening angle of the quasiprecession cone is approximately 0.834″.

For more than a decade, the global network of GPS stations whose measurements are part of the International GPS Service (IGS) have been recording cyclic variations in the radius vector of the geodetic ellipsoid with a period of one year and amplitude of ~10 mm. The analysis of the figure of the Earth carried out by us shows that the observed variations in the vertical component of the Earth’s surface displacements can induce small changes in the flattening of the Earth’s figure which are, in turn, caused by the instability of the Earth’s rotation. The variations in the angular velocity and flattening of the Earth change the kinetic energy of the Earth’s rotation. The additional energy is ~1021 J. The emerging variations in the flattening of the Earth’s ellipsoid lead to changes in the surface area of the Earth’s figure, cause the development of deformations in rocks, accumulation of damage, activation of seismotectonic processes, and preparation of earthquakes. It is shown that earthquakes can be caused by the instability of the Earth’s rotation which induces pulsations in the shape of the Earth and leads to the development of alternating-sign deformations in the Earth’s solid shell.

Planetary dynamics of century-long Earth shape changes is studied based on the approximation of the physical Earth surface by a biaxial ellipsoid. Trends in century-long changes of Earthshape parameters were revealed from the analysis of the following data: modern GNSS measurements on permanent stations in the Northern and Southern hemispheres and paleoreconstruction models of continental and ocean surfaces.

As an indication of the Earth's mass distribution, the principal moments of inertia (PMOI, i.e., A, B, C ) of the Earth are the basic parameters in studies of the global dynamics of the earth, like earth nutation, and the geophysics. From the aspect of observation, the PMOI can be calculated from the spherical coefficients of observed gravity field. In this paper, the PMOI are calculated directly according to its definition with the figures of the Earth's interior derived by a generalized theory of the hydrostatic equilibrium figure of the Earth. We obtain that the angle between the principal axis of the maximum moment of PMOI and the rotational axis is 0.184°, which means that the other two principal axes are very closely in the equatorial plane. Meanwhile, B-A is 1.60 × 10 −5 MR 2 , and the global dynamical flattening ( H ) is calculated to be 3.29587 × 10 −3 , which is 0.67% different from the latest observation derived value H obs (3.273795 × 10 −3 ) (Petit and Luzum, 2010), and this is a significant improvement from the 1.1% difference between the value of H derived from traditional theories of the figure of the Earth and the value of H obs . It shows that we can calculate the PMOI and H with an appropriate accuracy by a generalized theory of the hydrostatic equilibrium figure of the Earth.

This paper presents a method using a two-position system to determine the spatial coordinates of a moving radiation source from one bearing measurement and relative energy measurements of the received signal taking into account the effect of its reflection from the Earth surface. The passive location problem is solved for low elevation angles taking into account the curvature of the Earth and using a polynomial model of source motion. An algorithm for the parametric identification of this model using a measurement sample of increasing size in a regularized formulation is developed, and the accuracy characteristics of the method are analyzed.