Non-Singular Cross-Track Derivatives of the Gravitational Potential Using Rotated Spherical Harmonics

1992 ◽  
pp. 349-357
Author(s):  
Nico Sneeuw
Keyword(s):  
Gravitational Potential ◽  
Spherical Harmonics ◽  
Cross Track ◽  
Derivatives Of

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 Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals On some expressions for the radial and axial components of the magnetic force in the interior of solenoids of circular cross-section

1899 ◽  
Vol 64 (402-411) ◽  
pp. 192-202 ◽  
Keyword(s):  
Cross Section ◽  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Magnetic Force ◽  
Circular Cross Section ◽  
Total Force ◽  
Harmonic Method ◽  
Zonal Harmonics ◽  
Derivatives Of ◽  
Spherical Harmonic Method

In the present paper, certain expressions are arrived at, in terms of zonal spherical harmonics and their first derivatives, by which the values of the two components of the magnetic force may be calculated for any point in the interior of a coil, and hence the total force may be found both in magnitude and direction. The resulting series suffer from the well-known defect in the spherical harmonic method, in that they are not very rapidly converging for points near the boundary of the space for which they apply. A table of the values of the first derivatives of the first seven zonal harmonics is added.

Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies

Geophysics ◽  
1979 ◽  
Vol 44 (4) ◽  
pp. 730-741 ◽  
Author(s):  
M. Okabe
Keyword(s):  
Gravitational Potential ◽  
Computing Time ◽  
Gravity Anomalies ◽  
Magnetic Anomalies ◽  
Two Dimensions ◽  
First Derivative ◽  
Second Derivatives ◽  
Analytical Expressions ◽  
Derivatives Of ◽  
The Magnetic Field

Complete analytical expressions for the first and second derivatives of the gravitational potential in arbitrary directions due to a homogeneous polyhedral body composed of polygonal facets are developed, by applying the divergence theorem definitively. Not only finite but also infinite rectangular prisms then are treated. The gravity anomalies due to a uniform polygon are similarly described in two dimensions. The magnetic potential due to a uniformly magnetized body is directly derived from the first derivative of the gravitational potential in a given direction. The rule for translating the second derivative of the gravitational potential into the magnetic field component is also described. The necessary procedures for practical computer programming are discussed in detail, in order to avoid singularities and to save computing time.

scholarly journals Effects of Dipole Magnetic Fields on Axisymmetric p-Mode Pulsations

2002 ◽  
Vol 185 ◽  
pp. 294-295
Author(s):  
Hideyuki Saio ◽  
Alfred Gautschy
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Angular Dependence ◽  
Spherical Harmonics ◽  
Field Interaction ◽  
Dipole Magnetic Field ◽  
Derivatives Of

We investigated nonradial pulsations in the presence of a dipole magnetic field in a non-rotating 1.7 M⊙ ZAMS star. Formally, like in the case of pulsation-rotation coupling (Lee & Saio, 1986), the angular dependence of the pulsations is expanded into a series of spherical harmonics of different latitudinal degrees l. To start with, we considered only axisymmetric (m = 0) modes under the adiabatic and the Cowling approximations. In contrast to previous studies of pulsation-magnetic field interaction (Dziembowski & Goode, 1996; Bigot et al., 2000; Cunha & Gough, 2000), we retained the latitudinal derivatives of the perturbed quantities.

scholarly journals ВЫЧИСЛЕНИЕ ВЫСШИХ ПРОИЗВОДНЫХ ГРАВИТАЦИОННОГО ПОТЕНЦИАЛА НА ОСНОВЕ F-АППРОКСИМАЦИИ

2016 ◽  
Author(s):  
И.А. Керимов
Keyword(s):  
Gravitational Field ◽  
Approximate Method ◽  
Gravitational Potential ◽  
Basic Data ◽  
Gravity Potential ◽  
Irregular Grid ◽  
Field Transformation ◽  
Gravimetric Data ◽  
Derivatives Of

В статье рассмотрен метод трансформации гравитационного поля (определение различных компонент потенциала силы тяжести) на аппроксимационной основе. Разработанный автором метод и компьютерные технологии F-аппроксимации позволяют вычислять горизонтальные и вертикальные  производные гравитационного потенциала для исходных данных, заданных как на регулярной, так и на нерегулярной сети. Метод апробирован на модельных и фактических гравиметрических данных This article considers an approximate method of the gravitational field transformation (the determination of different components of the gravity potential). The method developed by the author and computer technologies of the F-approximation permit the calculation of horizontal and vertical derivatives of the gravitational potential for basic data defined both on a regular and on an irregular grid. The method is tested on model and actual gravimetric data.

Sign in / Sign up

Export Citation Format

Share Document