The spacetime gravitational field of a deformable body

The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents: the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces. The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution, emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus surface Love-Shida functions (h(r),l(r) versus h ^′(r),l ^′(r)) and Love functions k(r) versus k ^′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential, proving an excellent agreement with experimental results. Received: December 1995 / Accepted: 1 February 1997     

Dependence of the Love numbers upon the inner structure of the Earth and comparison of theoretical models with results of measurements

Summary The question this paper is examining is the following: to what extent are the Love numbers dependent on certain characteristics of the inner structure of the Earth? It has been proven — on the basis of calculations carried out by the author-that these quantities are only in a small degree dependent on the density values measured on the surface of the Earth and on the selection of the density function in the mantle of the Earth. On the other hand the value of Love numbersh, k andl is considerably influenced by the assumptions made about the core of the Earth, namely by the position of the boundary between the core and the mantle and by the magnitude of the rigidity coefficient presumed in the core in the vicinity of the core-mantle boundary.The results of the calculations are compared with those mean values of Love numbers obtained from the data of stations operating at different places of the Earth. By reason of this it can be assumed that the core of the Earth has, in the vicinity of the core-mantle boundary, a coefficient of effective rigidity of the order of 10¹⁰ dyn/cm², if the core-mantle boundary is placed at the relative Earth radius of 0.545 from the centre of the Earth.     

Comment on the paper “Fast tidal cycling and the origin of life” by Richard Lathe

We show that the fast tidal cycling postulated by Lathe [Lathe, R., 2004. Icarus 168, 18-22] is not a plausible mechanism to explain the origin of life on Earth about 3.9 Ga ago. The value of LOD at this remote epoch was probably comprised between 15 to 17 h, and the Earth-Moon distance was only about 20% smaller than nowadays, implying that the tidal frequencies and amplitudes were not so dramatically different from the present ones as stated in Lathe's paper.     

Correlations of natural radionuclides in sediment from Danube

The correlation between specific activities of some natural radionuclides (²³⁸U, ²²⁶Ra, ²³²Th, ⁴⁰K) measured in sediment taken from river bottom was studied. The sediment was taken from the Serbian part of the Danube River. Good correlation between some of the isotopes is observed, so that their specific activity ratios are spread over a lower range than specific activities themselves. This suggests that evaluation of specific activity ratios of some natural radionuclides could be a more sensitive method for the determination of increased levels of some of them than the straightforward analysis of specific activities.     

The temporal variation of the spherical and Cartesian multipoles of the gravity field: the generalized MacCullagh representation

 The Cartesian moments of the mass density of a gravitating body and the spherical harmonic coefficients of its gravitational field are related in a peculiar way. In particular, the products of inertia can be expressed by the spherical harmonic coefficients of the gravitational potential as was derived by MacCullagh for a rigid body. Here the MacCullagh formulae are extended to a deformable body which is restricted to radial symmetry in order to apply the Love–Shida hypothesis. The mass conservation law allows a representation of the incremental mass density by the respective excitation function. A representation of an arbitrary Cartesian monome is always possible by sums of solid spherical harmonics multiplied by powers of the radius. Introducing these representations into the definition of the Cartesian moments, an extension of the MacCullagh formulae is obtained. In particular, for excitation functions with a vanishing harmonic coefficient of degree zero, the (diagonal) incremental moments of inertia also can be represented by the excitation coefficients. Four types of excitation functions are considered, namely: (1) tidal excitation; (2) loading potential; (3) centrifugal potential; and (4) transverse surface stress. One application of the results could be model computation of the length-of-day variations and polar motion, which depend on the moments of inertia. Received: 27 July 1999 / Accepted: 24 May 2000     

Low altitude observations of the energetic electrons in the outer radiation belt during isolated substorms

The low energy (1–20 keV) detector registering particles onboard the polar-orbiting low altitude (~ 850 km) DMSP-F2 and -F3 satellites also records high energy electrons penetrating the detector walls. Thus we can study the dynamics of this electron population at L = 3.5, during isolated periods of magnetospheric substorms identified by the indices of auroral electrojet (AE), geomagnetic (K_p) and ring current (D_st). Temporal changes in the electron flux during the substorms are observed to be an additional contribution riding over the top of the pre-storm (or geomagnetically quiet-time) electron population ; the duration of the interval of intensity variation is observed to be about the same as that of the enhancement of the AE index. This indicates the temporal response of the outer radiation belt to the substorm activity, since the observation was made in the “horns” of the outer radiation belt. The observed enhanced radiation at low altitude may associate with the instantaneous increase and/or dumping of the outer radiation belt energetic electrons during each isolated substorm activity.     

Origin, weathering, and geochemical composition of loess in southwestern Hungary

Loess geochemistry generally reflects paleo-weathering conditions and it can be used to determine the average composition of the upper continental crust (UCC). In this study, major and trace element concentrations were analyzed on loess samples from southwestern Hungary to determine the factors influencing their chemical compositions and to propose new average loess compositions. All studied loess samples had nearly uniform chemical composition, suggesting similar alteration history of these deposits. Chemical Index of Alteration values (58–69) suggested a weak to moderate degree of weathering in a felsic source area. Typical non-steady state weathering conditions were shown on the Al₂O₃–CaO + Na₂O–K₂O patterns, indicating active tectonism of the Alpine–Carpathian system during the Pleistocene. Whole-rock element budgets were controlled by heavy minerals derived from a felsic magmatic or reworked sedimentary provenance. Geochemical parameters indicated that dust particles must have been recycled and well homogenized during fluvial and eolian transport processes.     

Changes in Hungary's agrarian sector with special reference to the northern hill region

The Hungarian agriculture and food industry has changed greatly since 1989 and this sector is now in crisis. This essay will present an overview of the most important economic and social problems which are partly the heritage of the socialist system, but which also stem from the change in the political system. It will also deal with the specific problems of the transition in the northern hill region of Nógrad where the physical conditions for agriculture are relatively unfavourable. In 1970s and 1980s state farms and cooperatives tried to compensate for the low demand for labour in agriculture by setting up subsidiary enterprises, which in many cases were the only source of local jobs. Many other people in the rural areas commuted to work in factories in the towns as industry and construction became the largest sector in the county's economy (41.3 percent in 1990 compared with 19.8 in agriculture). However, many of the former commuters are now unemployed and the cooperatives have withdrawn from subsidiary enterprises. So while an unstable situation has developed in Hungarian agriculture generally, through changes in structure and ownership, the problems faced by the less-developed agrarian regions of northern Hungary have been accentuated. Further research is needed to explore the real nature of the crisis in order to make provision for remedial action that best meets the local situation.