Purpose. To build a three-dimensional function of the mass distribution of the Earth's interior according to the parameters (Stokes constant to the second order inclusive) of the external gravitational field of the Earth without considering the minimum deviation from its known density models in geophysics. Methodology. The classic methods of constructing mass distribution use only the Stoke's constants zero and second orders. In iterative methods of determining the distribution models the reference model of density is taken for zero approximation which is agreed upon by Stoke's constants up to the second order inclusive. Further, the coefficients of potential expansion to a certain order are taken into account, but their contribution to the function of mass density does not investigate. This research provides an attempt to obtain such an estimation. The proposed method is approximate, but in the iterative process a function of the density is not only used, but also its derivatives. Bringing the order moments of density toward the controlled values (values that are defined on the surface of a sphere) makes it possible to analyze the process of successive approximations. Results. In contrast to the second-order model, which describes the global gross irregularities, the obtained distribution function gives a detailed picture of the placement density anomalies (deviation of three-dimensional functions from the average on the sphere-"izoden"). Analysis of maps at different depths (2891 km core-mantle, 5,150 km of the inner-outer core) allow making preliminary conclusions about global redistribution of mass due to the rotating component of gravity across the radius: its dilution along the axis of rotation and accumulation of rejecting it. This is particularly evident for the equatorial regions. On the contrary, there is minimum deviation in the polar regions of the Earth, which also have their own justification since the value of the rotation force decreases when moving away from the equator. The function of mass distribution, which is constructed using the proposed method, describes the mass distribution better. Originality. This research is in contrast to the classical results which have been obtained from the Adams-William's equations for the derivatives of the density of one variable (depth), and make attempted to obtain derivatives using Cartesian coordinates. Using the gravitational field parameters up to second order increases the order of approximation of the distribution function of the masses of three variables from two to four through the possibility of restoring the planet's mass distribution by its derivatives. At the same time, in contrast to previous research, geophysical information accumulated in the reference PREM model is used, therefore, features of the internal structure are taken into account. Practical significance. The received function of mass distribution of the Earth can be used as a zero-order approximation when used in the presented algorithm Stokes constant of higher order. Their applicat...

Анотація. У даній статті створено робочий набір даних про забруднення атмосферного повітря у Львівській області в програмі Excel. Опрацьовано результати найбільших підприємств-забруднювачів атмосферного повітря Львівщини. Створено інтерактивну карту забруднення атмосферного повітря Львівщини у програмному середовищі ArcGis Online. Виконано аналіз екологічної якості території забруднення атмосферного повітря Львівської області. Зроблено відповідні висновки. Ключові слова: інтерактивна карта, екологічні карти, атмосферне повітря, викиди шкідливих речовин, забруднюючі речовини, стаціонарні джерела забруднення, геокодування, програмне середовище ArcGis Online. Creation of interaCtiVe map of atmospheriC air pollution of lViV regionsummary. This article creates a working set of data on atmospheric air pollution in the Lviv region in the Excel program. The results of the largest enterprises of atmospheric air pollutants in Lviv region have been worked out. Created an interactive map of atmospheric air pollution of Lviv region in ArcGIS Online software environment. An analysis of the environmental quality of the air pollution region of the Lviv region is carried out. Appropriate conclusions are made.

Formulas are derived for the calculation of the potential of bodies, which surface is a sphere or an ellipsoid, and the distribution function has a special form: a piecewise continuous one-dimensional function and a three-dimensional mass distribution. For each of these cases, formulas to calculate both external and internal potentials are derived. With their help, further the expressions are given for calculation of the potential (gravitational) energy of the masses of such bodies and their corresponding distributions. For spherical bodies, the exact and approximate relations for determining the energy are provided, which makes it possible to compare the iterative process and the possibility of its application to an ellipsoid. The described technique has been tested by a specific numerical example.

The expressions of spherical functions and their derivatives in a Cartesian coordinate system are obtained. In opposite to the representation of polynomials in a spherical coordinate system, the derived recurrence relations make it possible to use them in the description of physical processes, and the obtained formulae for derivatives of spherical functions within the sphere allow obtaining the solutions to the problems of mathematical physics for spherical bodies in a Cartesian coordinate system. This approach has its advantages precisely in the applied problems. For example, for the determination of the artificial satellites orbits, it is necessary to represent the external potential of gravitation and its derivatives for the GPS systems in a Cartesian coordinate system. Investigation of the internal structure of Earth and astrometric studies of processes in galaxies are associated with the study of internal potential, and, consequently, there is a necessity for its presentation in the Cartesian coordinates.

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