Technology of determining the Earth’s gravitational field parameters using gradiometric measurements Part 6. Calculation the components of the gravitational potential tensor in the earth’s spatial rectangular coordinate system

2021 ◽  
Vol 973 (7) ◽  
pp. 2-8
Author(s):  
A.A. Kluykov
Keyword(s):  
Gravitational Field ◽  
Coordinate System ◽  
Gravitational Potential ◽  
Rectangular Coordinate System ◽  
Gradient Tensor ◽  
Gravitational Gradient ◽  
Earth’S Gravitational Field ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Potential Tensor

This is the sixth one in a series of articles describing the technology of determining the Earth’s gravitational field parameters through gradiometric measurements performed with an onboard satellite electrostatic gradiometer. It provides formulas for calculating the components of the gravitational potential tensor in a geocentric spatial rectangular earth coordinate system in order to convert them into a gradiometric one and obtain a free term for the equations of correcting gradiometric measurements when determining the parameters of the Earth’s gravitational field. The components of the gravitational gradient tensor are functions of test masses accelerations measured by accelerometers and relate to the gradiometer coordinate system, while the desired parameters of the Earth’s gravitational field model relate to the Earth’s coordinate one. The components of the gravitational gradient tensor are the second derivatives of the gravitational potential in rectangular coordinates. The calculated values of the gravitational potential tensor components in the earth’s spatial rectangular coordinate system are obtained through double differentiation of the gravitational potential formula. Basing on the obtained formulas, an algorithm and a program in the Fortran algorithmic language were developed. Using this program, experimental calculations were performed, the results of which were compared with the data of the EGG_TRF_2 product.

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

scholarly journals Generalization of the Particle Spin as it Ensues from the Ether Theory

2016 ◽  
Vol 8 (4) ◽  
pp. 58
Author(s):  
David Zareski
Keyword(s):  
Magnetic Field ◽  
Gravitational Field ◽  
Elasticity Theory ◽  
Gravitational Potential ◽  
Electric Charge ◽  
Neutral Particle ◽  
Circularly Polarized ◽  
Ether Theory ◽  
The Magnetic Field ◽  
Potential Tensor

In previous papers we generalized the ether waves associated to photons, to waves generally denoted  , associated to Par(m,e)s, (particles of mass m and electric charge e), and demonstrated that a Par(m,e)s is a superposition   of such waves that forms a small globule moving with the velocity   of this  . That, at a point near to a moving  , the ether velocity  , i.e., the magnetic field H, is of the same form as that of a point of a rotating solid. This is the spin of the Par(m,e)s, in particular, of the electron. Then, we considered the case where e=0 and showed that the perturbation caused by the motion of a Par(m,e)s is also propagated in the ether, and is a propagating gravitational field such that the Newton approximation (NA) is a tensor  Guobtained by applying the Lorenz transformation for Vm,o on the NA of the static gravitational potential of forces Gu,s. It appeared that Gu is also of the form of a Lienard-Wiechert potential tensor Au created by an electric charge.<br />In the present paper, we generalized the above results regarding the spin by showing that the ether elasticity theory implies also that like the electron, the massive neutral particle possesses a spin but much smaller than that of the electron, and that the photon can possess also a spin, when for example it is circularly polarized. In fact, we show that the spin associated to a particle is a vortex in ether which in closed trajectories will take only quantized values.<br /><br />

Analytical Formulas for Underwater and Aerial Object Localization by Gravitational Field and Gravitational Gradient Tensor

2017 ◽  
Vol 14 (9) ◽  
pp. 1557-1560 ◽  
Author(s):  
Jingtian Tang ◽  
Shuanggui Hu ◽  
Zhengyong Ren ◽  
Chaojian Chen ◽  
Xiao Xiao ◽  
...  
Keyword(s):  
Gravitational Field ◽  
Object Localization ◽  
Gradient Tensor ◽  
Gravitational Gradient ◽  
Analytical Formulas

Computation of first and second order derivatives gravity potential in different coordinate systems

2017 ◽  
Vol 925 (7) ◽  
pp. 15-22
Author(s):  
A.A. Kluykov ◽  
S.N. Yashkin
Keyword(s):  
Gravitational Field ◽  
Coordinate System ◽  
Gravity Potential ◽  
Measured Parameter ◽  
Measurement Information ◽  
Sensor Systems ◽  
Coordinate Systems ◽  
Satellite Gravity ◽  
Earth’S Gravitational Field ◽  
Mathematical Processing

The determination of parameters of the Earth’s gravitational field model by using the satellite gravity projects CHAMP, GRACE, GOCE is carried out on the basis of mathematical processing of measurement information of sensor systems installed on board of a spacecraft. Each of these sensor systems realizes its coordinate system, in which measurements are performed. Measured parameters, as a rule, are related to the coordinate system of the sensory system, and the required parameters refer to the Earth’s coordinate system (EFRF). Therefore, to determine the required parameters, it is necessary to perform the conversion of the measured parameter from one system to another. In this paper we obtain formulas that allow us to calculate the first and second derivatives of the gravitational potential in spherical, rectangular and local rectangular coordinate systems. Matrices are also obtained, with the help of which the transformation from one coordinate system to another is carried out. The formulas given in the article are necessary for performing mathematical processing of gradientometric measurements in order to determine the parameters of the Earth’s gravitational field.

scholarly journals Space Gravity Spectroscopy - determination of the Earth’s gravitational field by means of Newton interpolated LEO ephemeris Case studies on dynamic (CHAMP Rapid Science Orbit) and kinematic orbits

2003 ◽  
Vol 1 ◽  
pp. 127-135 ◽  
Author(s):  
T. Reubelt ◽  
G. Austen ◽  
E. W. Grafarend
Keyword(s):  
Gravitational Field ◽  
Gravitational Potential ◽  
Spherical Harmonics ◽  
Interpolation Formula ◽  
Frame Of Reference ◽  
Linear System Of Equations ◽  
Kinematic Orbit ◽  
Fixed Frame ◽  
Earth’S Gravitational Field ◽  
Acceleration Vector

Abstract. An algorithm for the (kinematic) orbit analysis of a Low Earth Orbiting (LEO) GPS tracked satellite to determine the spherical harmonic coefficients of the terrestrial gravitational field is presented. A contribution to existing long wavelength gravity field models is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high accuracy in the range of a few centimeters. To demonstrate the applicability of the proposed method, first results from the analysis of real CHAMP Rapid Science (dynamic) Orbits (RSO) and kinematic orbits are illustrated. In particular, we take advantage of Newton’s Law of Motion which balances the acceleration vector and the gradient of the gravitational potential with respect to an Inertial Frame of Reference (IRF). The satellite’s acceleration vector is determined by means of the second order functional of Newton’s Interpolation Formula from relative satellite ephemeris (baselines) with respect to the IRF. Therefore the satellite ephemeris, which are normally given in a Body fixed Frame of Reference (BRF) have to be transformed into the IRF. Subsequently the Newton interpolated accelerations have to be reduced for disturbing gravitational and non-gravitational accelerations in order to obtain the accelerations caused by the Earth’s gravitational field. For a first insight in real data processing these reductions have been neglected. The gradient of the gravitational potential, conventionally expressed in vector-valued spherical harmonics and given in a Body Fixed Frame of Reference, must be transformed from BRF to IRF by means of the polar motion matrix, the precession-nutation matrices and the Greenwich Siderial Time Angle (GAST). The resulting linear system of equations is solved by means of a least squares adjustment in terms of a Gauss-Markov model in order to estimate the spherical harmonics coefficients of the Earth’s gravitational field.Key words. space gravity spectroscopy, spherical harmonics series expansion, GPS tracked LEO satellites, kinematic

scholarly journals A Universal Generating Algorithm of the Polyhedral Discrete Grid Based on Unit Duplication

2019 ◽  
Vol 8 (3) ◽  
pp. 146 ◽  
Author(s):  
Li Meng ◽  
Xiaochong Tong ◽  
Shuaibo Fan ◽  
Chengqi Cheng ◽  
Bo Chen ◽  
...  
Keyword(s):  
Coordinate System ◽  
Grid Cell ◽  
Triangular Grid ◽  
Grid System ◽  
Rectangular Coordinate System ◽  
Factors Affecting ◽  
Grid Systems ◽  
Starting Point ◽  
Rectangular Coordinates ◽  
Discrete Grid

Based on the analysis of the problems in the generation algorithm of discrete grid systems domestically and abroad, a new universal algorithm for the unit duplication of a polyhedral discrete grid is proposed, and its core is “simple unit replication + effective region restriction”. First, the grid coordinate system and the corresponding spatial rectangular coordinate system are established to determine the rectangular coordinates of any grid cell node. Then, the type of the subdivision grid system to be calculated is determined to identify the three key factors affecting the grid types, which are the position of the starting point, the length of the starting edge, and the direction of the starting edge. On this basis, the effective boundary of a multiscale grid can be determined and the grid coordinates of a multiscale grid can be obtained. A one-to-one correspondence between the multiscale grids and subdivision types can be established. Through the appropriate rotation, translation and scaling of the multiscale grid, the node coordinates of a single triangular grid system are calculated, and the relationships between the nodes of different levels are established. Finally, this paper takes a hexagonal grid as an example to carry out the experiment verifications by converting a single triangular grid system (plane) directly to a single triangular grid with a positive icosahedral surface to generate a positive icosahedral surface grid. The experimental results show that the algorithm has good universality and can generate the multiscale grid of an arbitrary grid configuration by adjusting the corresponding starting transformation parameters.

The characteristics of transforming coordinates from the geocentric system WGS-84 for the Mercator projection under low latitudes conditions

2018 ◽  
Vol 940 (10) ◽  
pp. 2-6
Author(s):  
J.A. Younes ◽  
M.G. Mustafin
Keyword(s):  
Coordinate System ◽  
Satellite System ◽  
Programming Algorithm ◽  
Low Latitude ◽  
Model Calculations ◽  
Mercator Projection ◽  
Low Latitudes ◽  
Rectangular Coordinates ◽  
Global Navigation Satellite ◽  
Coordination Methods

The issue of calculating the plane rectangular coordinates using the data obtained by the satellite observations during the creation of the geodetic networks is discussed in the article. The peculiarity of these works is in conversion of the coordinates into the Mercator projection, while the plane coordinate system on the base of Gauss-Kruger projection is used in Russia. When using the technology of global navigation satellite system, this task is relevant for any point (area) of the Earth due to a fundamentally different approach in determining the coordinates. The fact is that satellite determinations are much more precise than the ground coordination methods (triangulation and others). In addition, the conversion to the zonal coordinate system is associated with errors; the value at present can prove to be completely critical. The expediency of using the Mercator projection in the topographic and geodetic works production at low latitudes is shown numerically on the basis of model calculations. To convert the coordinates from the geocentric system with the Mercator projection, a programming algorithm which is widely used in Russia was chosen. For its application under low-latitude conditions, the modification of known formulas to be used in Saudi Arabia is implemented.

scholarly journals Comment on “Measurement of quantum states of neutrons in the Earth’s gravitational field”

2003 ◽  
Vol 68 (10) ◽  
Author(s):  
Johan Hansson ◽  
David Olevik ◽  
Christian Türk ◽  
Hanna Wiklund
Keyword(s):  
Gravitational Field ◽  
Quantum States ◽  
Earth’S Gravitational Field

The spin-zero Duffin-Kemmer-Petiau equation in a cosmic-string space-time with the Cornell interaction

2016 ◽  
Vol 31 (36) ◽  
pp. 1650191 ◽  
Author(s):  
M. de Montigny ◽  
M. Hosseinpour ◽  
H. Hassanabadi
Keyword(s):  
Gravitational Field ◽  
Coordinate System ◽  
Cosmic String ◽  
Topological Defects ◽  
Space Time ◽  
Dkp Equation

In this paper, we study the covariant Duffin-Kemmer-Petiau (DKP) equation in the cosmic-string space-time and consider the interaction of a DKP field with the gravitational field produced by topological defects in order to examine the influence of topology on this system. We solve the spin-zero DKP oscillator in the presence of the Cornell interaction with a rotating coordinate system in an exact analytical manner for nodeless and one-node states by proposing a proper ansatz solution.

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