Solution of Kepler’s Equation with Machine Precision

Astronomy Reports - An algorithm for the numerical solution of Kepler’s equation with machine precision is presented. The convergence of the iterative sequence of Newton’s method is...     

Solution of Kepler’s equation for rectilinear motion

The properties of the solution of the kinematic equation (Kepler’s equation) presented as a series in powers of a function of time are considered for the case of unperturbed, rectilinear elliptical and rectilinear hyperbolic motion. Kepler’s equation for unperturbed, rectilinear elliptical motion has the form E − sinE = z ³/6, where E is the eccentric anomaly determining the position in the orbit, z ³/6 is the mean anomaly, which is proportional to the time measured from an encounter, and its solution can be represented as a series in powers of z. It is established that the coefficients of the series are positive. The asymptotic for the coefficients in the region of convergence of the series is found, which covers the entire orbit. The series continues to converge over the entire boundary of the circle of convergence. The kinematic equation for unperturbed, rectilinear hyperbolic motion has the form sinhH − H = ζ ³/6, where, as before, ζ ³ is proportional to time. The substitution E = iH, z = iζ reduces one equation to the other. The series for the solution in the hyperbolic case differs from the series for the elliptical solution only in its alternating-sign coefficients. However, the region of convergence covers only part of the orbit in the hyperbolic case.     

Erratum to: Solution of Kepler’s Equation with Machine Precision

Astronomy Reports - An Erratum to this paper has been published: https://doi.org/10.1134/S1063772921330015     

An Elastic Solution for Twin Circular Tunnels’ Stress in Hydrostatic Stress Field

Geotechnical and Geological Engineering - In order to present an elastic solution for twin circular tunnels’ stress distribution in hydrostatic stress field, based on the complex variable...     

Accounting for the influence of the Earth’s sphericity in three-dimensional density modelling

A method for transformation of the three-dimensional regional “flat” density models of the Earth’s crust and upper mantle to the “spherical” models and vice versa is proposed. A computation algorithm and a method of meaningful comparison of the vertical component of the gravity field of both models are presented.     

Development of tomographic technology for the Earth’s crust in the shelf regions

On the basis of experimental data obtained during a comprehensive experiment in Vityaz Bay of the Sea of Japan using onshore laser strainmeters and a low-frequency hydroacoustic emitter generating complex phase-manipulated signals with a central frequency of 33 Hz, we developed the basic principles of contactless tomography of the Earth’s crust in the shelf regions of various seas, including those covered by ice, making it possible to determine efficiently the structure and composition of the upper Earth’s crust under seas.

     

Development of methods for constructing models for intra-year irregularity in the Earth’s rotation

The development of methods for the construction of stochastic, dynamical models for intra-year irregularity of the Earth’s rotation is considered. A correlational model based on harmonically additive and parametrically random, colored and broadband, gravitational-tidal perturbations from the Sun and Moon is developed. One-dimensional and multi-dimensional characteristic functions are found for the case of Gaussian and non-Gaussian colored and broadband fluctuations in the irregularity of the Earth’s rotation. Examples of computer modeling of the irregularity in the Earth’s rotation based on a priori and a posteriori IERS data are presented.     

A correlation model for oscillations of the Earth’s pole with parametric perturbations

A differential correlation model for oscillations of the Earth’s pole is constructed. The model has gravitational-tidal, additive and parametric, slowly varying, harmonic (at the Chandler frequency and double this frequency), and random Gaussian, broadband perturbations. Special attention is paid to the analysis of trends and the amplitude-frequency characteristics of stochastic oscillations of the Earth’s pole. Numerical simulations show that first-approximation equations can be used to estimate the correlation characteristics of oscillations of the Earth’s pole to within 10%. The results of the model are compared with the results of statistical modeling of oscillations at the Chandler frequency. The model represents a base of informational resources for analytical modeling of the motion of the Earth’s pole over intervals of three to five years.     

Analytical Conditions for the Formation of Arctic Amplification in the Earth’s Climate System

Doklady Earth Sciences - Analytical conditions for estimating the contribution of various internal and external factors to the formation of Arctic (polar) amplification under warming of the...     

Measurement of the difference in the Earth’s gravitational potentials with the help of a transportable quantum clock

The results of the Russia’s first ground-based experiment for determination of the difference in the Earth’s gravitational potentials on the basis of the measurement of the gravitational effect of the time delay with the help of a high-stability transportable atomic clock are provided. The reference atomic clock was placed in Moscow oblast, and a transportable quantum clock with an instability of 3 × 10^–15 was placed in the Caucasus Mountains, with a difference in height of the clocks of 1804 m. The measured difference in the gravitational potentials between the positions of the two quantum clocks was (182.0 ± 3.1)10²m²s^-2 at a relative measurement error of no more than 1.7%.     

Tectonic model of seismicity for the northeastern segment of the Amur Plate in the Earth’s two-phased rotation

The spatial distribution of the epicenters and hypocenters is analyzed for earthquakes of 2 ≤ M < 6 that occurred in the northeastern segment of the Amur Plate in two phases of changes in the angular speed of the Earth’s rotation. Groups of seismic events in the magnitude interval of 5 ≤ M < 6 are distinguished in the form of NE-trending seismic clusters regularly alternating along the plane of latitude. The seismic clusters are up to 1500 km long and 180–240 km wide and cover the seismic zones with different geodynamic and seismotectonic conditions of seismicity origination. In terms of the epicentral distributions for earthquakes with 2 ≤ M < 4, seismic activity zones are distinguished; these zones are seen as seimolineaments coupling the Tan Lu seismic zones and the eastern flanks of the latitudinal seismic zones. A scheme of distinguishing the compression and extension zones from the spatial clusters of earthquakes with 5 ≤ M < 6 in two phases of changes in the angular speed of the Earth’s rotation is proposed. This scheme satisfactorily agrees with the model of seismotectonic reconstructions of the compression–extension fields and axes.     

Salts in the clay playas of China’s arid regions: gone with the wind

In China’s arid regions, aeolian salts are transported along with dust aerosols, but their contents and the emission rates are poorly understood. Using samples with undisturbed surfaces and artificially disrupted surfaces (to simulate human disturbances such as reclamation, planting, trampling by animals, and other activities), the transport rates for windblown material and the salt contents of clay playa surfaces were investigated in a wind tunnel. The results showed that under wind velocities ranging from 8 to 22 m/s, wind erosion rates ranged between 0.009 and 0.066 g/m²/s for the undisturbed surfaces and between 0.011 and 0.048 g/m²/s for the artificially disrupted playa surfaces. Although the amount of wind-transported material that was captured was too small to allow an analysis of individual salt contents using the available equipment, electrical conductivity measurements and correlation analysis revealed mean total salt contents of 9.59 g/kg in the transported materials collected above the sealed playa surfaces versus 3.38 g/kg in materials from the same surfaces that were artificially cracked after testing the sealed surfaces. Given the large areas containing clay playas in China’s arid regions, these results will improve our understanding of aeolian salt transport in arid China and will facilitate interpretation of the sources of aeolian salts.     

Masses of carbon in the Earth’s hydrosphere

Recent data were summarized on the concentration and mass of inorganic and organic carbon in reservoirs of the Earth’s hydrosphere. We compared carbon masses and accumulation conditions in the surface hydrosphere and waters of the sedimentary shell and proportions between carbonate, dissolved, and suspended particulate organic carbon. It was shown that the total masses of carbon in the surface hydrosphere and in the waters of the sedimentary shell are approximately equal to 80 × 10¹⁸ g C at an organic to carbonate carbon ratio of 1 : 36 and 1 : 43, respectively. Three main forms of organic compounds in the ocean (living organisms, suspended particles, and dissolved species) occur in the proportion 1 : 13 : 250 and form the pyramid of masses 4 × 10¹⁵ g, 50 × 10¹⁵ g, and 1000 × 10¹⁵ g C_org. The descending sequence of the organic to carbonate carbon ratio in water, ocean (1 : 36) > glaciers (1 : 8) > lakes (1 : 2) > rivers (1 : 0.6) > wetlands (1 : 0.3), is in general consistent with an increase in the same direction in the mean concentrations of organic matter: 0.77 mg C_org/L in the ocean, 0.7 mg C_org/L in glaciers, 6–30 mg C_org/L in lakes, 15 mg C_org/L in rivers, and 75 mg C_org/L in wetlands. Both the mean concentrations and masses of dissolved organic matter in the pore waters of oceanic sediments and in the waters of the sedimentary shell are similar: 36–37 mg/L and 5 × 10¹⁸ and 5.6 × 10¹⁸ g, respectively. The mass of carbonate carbon in the pore waters of the ocean, (19–33) × 10¹⁸ g, is comparable with its mass in the water column, 38.1 × 10¹⁸ g.     

An indicator based approach for assessing household’s perceptions of landslide risk in Murree hills of Pakistan

Natural Hazards - The Murree region in Pakistan is frequently affected by landslides and causes damages to lands, houses, life lines and even loss of livestock and human life. We have attempted in...     

Breakthrough in the China’s Separation Technology for Extraction of Rare Earth Elements

Separation technology of rare earth elements （REEs）, as the critical step in the separation process, had long been fraught with technical difficulty. A research project conducted by Baotou Shibo Rare Earth Extraction ＆ Equipment Co. Ltd., Baotou REE Research Institute, and Baotou Steel ＆ REE Group Hi-Tech Co. Ltd （Inner Mongolia）, has successfully solved the problem using a centrifugal extractor and advanced techniques to achieve a key breakthrough.     

A survey-based analysis of the public’s willingness for disaster relief in China

Natural Hazards - Meteorological disasters frequently occur in China and around the world. These natural hazards can cause huge economic losses and threaten the personal safety of citizens. The...     

UNESCO’s role in the rehabilitation of Bamiyan in Afghanistan

This article is complementary to Claudio Margottinis technical article Instability and geotechnical problems of the Buddha niches and surrounding cliff in Bamiyan Valley, central Afghanistan, published in Vol. 1, issue 1 of this journal. The present article provides background information on the framework of UNESCOs strategies for the safeguarding of Afghanistans cultural heritage and, in particular, the Bamiyan site, the mechanisms for the development of operational projects, capacity building of national specialists, UNESCOs partnerships and its fund-raising activities for the successful implementation of projects. The Division of Cultural Heritage is closely involved in ICL, therefore, it was natural to include UNESCOs project for the safeguarding of the Bamiyan site in the framework of IPL. It was approved as IPL project C101-3 titled The geo-morphological instability of the Buddha niches and surroundings cliffs in the Bamiyan Valley.The author is program specialist in charge of Asia at UNESCOs Division of Cultural Heritage and member of ICL