scholarly journals REFINING GEOIDAL HEIGHTS COMPUTED BY FULLY-NORMALISED SPHERICAL HARMONICS COEFFICIENTS OF THE EARTH GRAVITATIONAL MODEL EGM2008 BY ADJUSTMENT OF COMBINED GPS/LEVELLING/GEOID NETWORK

2014 ◽  
Vol 36 (2) ◽  
Author(s):  
Bui Khac Luyen
Keyword(s):  
Spherical Harmonics ◽  
The Earth ◽  
Gravitational Model

scholarly journals The potentially hazardous NEA 2001 BB16

2019 ◽  
Vol 28 (1) ◽  
pp. 180-190
Author(s):  
Ireneusz Wlodarczyk
Keyword(s):  
Mean Value ◽  
Optical Observations ◽  
Uniform Sampling ◽  
The Earth ◽  
22Nd Century ◽  
Lyapunov Time ◽  
Gravitational Model ◽  
Calculated Probability ◽  
Potential Impact ◽  
The Impact

AbstractWe computed the impact solutions of the potentially dangerous Near Earth Asteroid (NEA) 2001 BB16 based on 47 optical observations from January 20.08316 UTC, 2001, through February 09.15740 UTC, 2016, and one radar observation from January 19.90347 UTC, 2016. We used two methods to sample the starting Line of Variation (LOV). First method, called thereafter LOV1, with the uniform sampling of the LOV parameter, out to LOV = 5 computing 3000 virtual asteroids (VAs) on both sides of the LOV, which gives 6001 VAs and propagated their orbits to JD2525000.5 TDT=February 12, 2201. We computed the non-gravitational parameterA2=(34.55±7.38)·10–14 au/d2 for nominal orbit of 2001 BB16 and possible impacts with the Earth until 2201. For potential impact in 2195 we find A2=20.0·10−14 au/d2. With a positive value of A2, 2001 BB16 can be prograde rotator. Moreover, we computed Lyapunov Time (LT) for 2001 BB16, which for all VAs, has a mean value of about 25 y. We showed that impact solutions, including the calculated probability of a possible collision of a 2001 BB16 asteroid with the Earth depends on how to calculate and take into account the appropriate gravitational model, including the number of perturbing massive asteroids. In some complicated cases, it may depend also on the number of clones calculated for a given sigma LOV1. The second method of computing the impact solutions, called thereafter LOV2, is based on a non-uniformly sampling of the LOV. We showed that different methods of sampling the LOV can give different impact solutions, but all computed dates of possible impacts of the asteroid 2001 BB16 with the Earth occur in accordance at the end of the 22nd century.

scholarly journals Geometrical Characterisation of the Mamfe Basin, Cameroon, from the Earth, Gravitational Model (EGM 2008)

2018 ◽  
Vol 7 (1) ◽  
pp. 94
Author(s):  
Anatole Eugene Djieto Lordon ◽  
Mbohlieu YOSSA ◽  
Christopher M Agyingi ◽  
Yves Shandini ◽  
Thierry Stephane Kuisseu
Keyword(s):  
Average Length ◽  
Gravity Data ◽  
Bouguer Anomaly ◽  
Igneous Rocks ◽  
Average Depth ◽  
The Earth ◽  
Gravitational Model ◽  
Petroleum Generation ◽  
Anomaly Map ◽  
Bouguer Anomaly Map

Gravimetric studies using the ETOPO1-corrected high resolution satellite-based EGM2008 gravity data was used to define the surface extent, depth to basement and shape of the Mamfe basin. The Bouguer anomaly map was produced in Surfer 11.0. The Fast Fourier Transformed data was analyzed by spectral analysis to remove the effect of the regional bodies in the study area. The residual anomaly map obtained was compared with the known geology of the study area, and this showed that the gravity highs correspond to the metamorphic and igneous rocks while the gravity lows match with Cretaceous sediments. Three profiles were drawn on the residual anomaly map along which 2D models of the Mamfe basin were drawn. The modeling was completed in Grav2dc v2.06 software which uses the Talwini’s algorithm and the resulting models gave the depth to basement and the shape of the basement along the profiles. After processing and interpretation, it was deduced that the Mamfe basin has an average length and width of 77.6 km and 29.2 km respectively, an average depth to basement of 5 km and an overall U-shape basement. These dimensions (especially the depth) theoretically create the depth and temperature conditions for petroleum generation. 

Motion of a satellite in the earth’s gravitational field

1960 ◽  
Vol 254 (1276) ◽  
pp. 48-65 ◽  
Keyword(s):  
Gravitational Field ◽  
Spherical Harmonics ◽  
Main Part ◽  
Equations Of Motion ◽  
Orbital Elements ◽  
Partial Derivatives ◽  
Rates Of Change ◽  
Zonal Harmonics ◽  
The Earth ◽  
Secular Terms

The equations of motion of a satellite are given in a general form, account being taken of the precession and nutation of the earth. The main part of the paper deals with the motion arising from the gravitational field of the earth, expressed as a general expansion in spherical harmonics. By evaluating the partial derivatives in Lagrange’s planetary equations, • expressions are obtained for the rates of change of the orbital elements. Particular consideration is given to the form of the expressions for the secular terms arising from the first four zonal harmonics.

Modelling the core magnetic field of the Earth

1982 ◽  
Vol 306 (1492) ◽  
pp. 179-191 ◽  
Keyword(s):  
Magnetic Field ◽  
Spherical Harmonics ◽  
Secular Variation ◽  
Long Wavelength ◽  
Core Field ◽  
The Core ◽  
The Earth ◽  
High Degree ◽  
Current Systems ◽  
The Magnetic Field

The magnetic field generated in the core of the Earth is often represented by spherical harmonics of the magnetic potential. It has been found from looking at the equations of spherical harmonics, and from studying the values of the spherical harmonic coefficients derived from data from Magsat, that this is an unsatisfactory way of representing the core field. Harmonics of high degree are characterized by generally shorter wavelength expressions on the surface of the Earth, but also contain very long wavelength features as well. Thus if it is thought that the higher degree harmonics are produced by magnetizations within the crust of the Earth, these magnetizations have to be capable of producing very long wavelength signals. Since it is impossible to produce very long wavelength signals of sufficient amplitude by using crustal magnetizations of reasonable intensity, the separation of core and crustal sources by using spherical harmonics is not ideal. We suggest that a better way is to use radial off-centre dipoles located within the core of the Earth. These have several advantages. Firstly, they can be thought of as modelling real physical current systems within the core of the Earth. Secondly, it can be shown that off-centred dipoles, if located deep within the core, are more effective at removing long wavelength signals of potential or field than can be achieved by using spherical harmonics. The disadvantage is that it is much more difficult to compute the positions and strengths of the off-centred dipole fields, and much less easy to manipulate their effects (such as upward and downward continuation). But we believe, along with Cox and Alldredge & Hurwitz, that the understanding that we might obtain of the Earth’s magnetic field by using physically reasonable models rather than mathematically convenient models is very important. We discuss some of the radial dipole models that have been proposed for the nondipole portion of the Earth’s field to arrive at a model that agrees with observations of secular variation and excursions.

scholarly journals Libration of the Moon: shape of the Earth and motion of the ecliptic plane

1986 ◽  
Vol 114 ◽  
pp. 141-144
Author(s):  
M. Moons
Keyword(s):  
Gravity Field ◽  
Spherical Harmonics ◽  
Fourth Order ◽  
Center Of Mass ◽  
Ecliptic Plane ◽  
The Sun ◽  
Lunar Gravity ◽  
The Moon ◽  
Lunar Gravity Field ◽  
The Earth

Very accurate theories of the libration of the Moon have been recently built by Migus (1980), Eckhardt (1981, 1982) and Moons (1982, 1984). All of them take into account the perturbation due to the Earth and the Sun on the motion of a rigid Moon about its center of mass. Additional perturbations (influence of the planets, shape of the Earth, elasticity of the Moon, …) are also often included.We present here the perturbations due to the shape of the Earth and the motion of the ecliptic plane on our theory which already contains planetary perturbations. This theory is completely analytical with respect to the harmonic coefficients of the lunar gravity field which is expanded in spherical harmonics up to the fourth order. The ELP 2000 solution (Chapront and Chapront-Touzé, 1983) supplies us with the motion of the center of mass of the Moon.

scholarly journals Geophysical models of the surface global vorticity vector

1988 ◽  
Vol 128 ◽  
pp. 411-411
Author(s):  
Erik W. Grafarend
Keyword(s):  
Spherical Harmonics ◽  
Earth Rotation ◽  
Polar Motion ◽  
Deformable Body ◽  
Earth Model ◽  
Order Zero ◽  
The Earth ◽  
Vorticity Vector ◽  
Ellipsoid Of Revolution ◽  
Length Of The Day

Within the framework of Newtonian kinematics the local vorticity vector is introduced and averaged with respect to global earth geometry, namely the ellipsoid of revolution. For a deformable body like the earth the global vorticity vector is defined as the earth rotation. A decomposition of the Lagrangean displacement and of the Lagrangean vorticity vector into vector spherical harmonics, namely into spheroidal and toroidal parts, proves that the global vorticity vector only contains toroidal coefficients of degree and order one (polar motion) and toroidal coefficients of degree one and order zero (length of the day) in the case of an ellipsoidal earth. Once we assume an earth model of type ellipsoid of revolution the earth rotation is also slightly dependent on the ellipsoidal flattening and the radial derivative of the spheroidal coefficients of degree two and order one.

scholarly journals Note on the representation of the earth's surface by means of spherical harmonics of the first three degrees

1908 ◽  
Vol 80 (542) ◽  
pp. 553-556
Keyword(s):  
Spherical Harmonics ◽  
Spherical Surface ◽  
Equipotential Surface ◽  
The Earth ◽  
First Approximation ◽  
Equipotential Surfaces ◽  
Gravitational Stability ◽  
Type V ◽  
Diurnal Rotation ◽  
Equal Density

In my paper on “The Gravitational Stability of the Earth,” dynamical arguments were adduced in favour of the hypothesis that the distribution of density within the earth is such that the surfaces of equal density present, in addition to the inequalities depending upon the diurnal rotation, other inequalities which can be specified by spherical harmonics of the first, second, and third degrees. If this is the case, the surface of the earth, by which I mean the surface of the lithosphere, should present corresponding inequalities, and so also should the equipotential surfaces. Analytically, if the density ρ is given by an equation of the form ρ = f 0 ( r ) + ϵ 1 f 1 ( r )S 1 + ϵ 2 f 2 ( r )S 2 + ϵ 3 f 3 ( r )S 3 , (1) where f 0 ( r ), f 1 ( r ), . . . are functions of the distance r from the centre, S 1 , S 2 , S 3 are spherical surface harmonics of degrees indicated by the suffixes, and ϵ 1 , ϵ 2 , ϵ 3 are small coefficients, then the surface should have an equation of the form r = a + α 1 S 1 + α 2 S 2 + α 3 S 3 , (2) where a and α 1 , α 2 , α 3 are constants, and the α 's are small. The elevations and depressions of the lithosphere should be, at least in their main features, expressible by a formula of this type. The actual elevations and depressions are difficult to determine, because all that can be found by observation is the amount of elevation above, or depression below, a particular equipotential surface, the geoid , or the surface of the ocean, continued beneath the continents. For a first approximation the potential due to such a distribution of density as is expressed by (1) within a surface expressed by (2) would be given by formulæ of the type V = F 0 ( r ) + β 1 F 1 ( r )S 1 + β 2 F 2 ( r )S 2 + β 3 F 3 ( r )S 3 , ( r < a )

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