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     

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     

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.     

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.     

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.     

Investigation of gravimetric records at non-tidal frequencies

¶rt;m nmaumuu uma a nuu amma. aa, m ua u ua um ¶rt; nm am ¶rt;au u nuu uma. aumaa mu ¶rt; u a au nu u mau. aa ua u ua . n¶rt; nnau anum¶rt; aamumu aa nuu . aa, m nm a¶rt; u um a aau a amm 56°/h, ma aa a au mau. aamuam au u m u.

---

---

Presented at the meeting of Working Group 3.3. of the KAPG (Prague, November 1975).     

Temporal variations of the polar moment of inertia and the second-degree geopotential

In order to study the geodynamic behaviour of the Earth over short (elastic Earth) and long (almost perfectly liquid Earth) geological periodic variations, the changes of the moment of inertia are decomposed into two parts: the first, described by a volume integral, explains the effect of the density variations, while the second gives the impact of the surface variations using a surface integral. It is shown that both components have physical significance, but their contribution is different in case of short (lunisolar) and long (connected to secular despinning) periods.     

Characterization of iron meteorites by scanning electron microscopy,X-ray diffraction,magnetization measurements,and Mössbauer spectroscopy: Gibeon IVA

Gibeon IVA iron meteorite fragment was characterized using optical microscopy, scanning electron microscopy (SEM) with energy dispersive spectroscopy (EDS), X-ray diffraction (XRD), magnetization measurements, and Mössbauer spectroscopy. Optical microscopy and SEM made on the polished section of the meteorite, show the presence of α-Fe(Ni, Co) and γ-Fe(Ni, Co) phases and plessite structures. There are no troilite inclusions observed in the studied section. EDS studies indicate some variations in the Ni concentrations: (i) within the α-Fe(Ni, Co) phase in the range ~5.0 ± 0.1 – ~7.5 ± 0.1 at% and (ii) within the γ-Fe(Ni, Co) phase in the range ~26.0 ± 0.2 – ~36.1 ± 0.2 at%. The latter Ni concentration range indicates the presence of small amount of the paramagnetic γ-phase in addition to the ferromagnetic γ-phase. EDS also shows that Ni content in two plessite structures is varying in the range ~16–37 at%, which can indicate the presence of only the α₂-Fe(Ni, Co) and γ-Fe(Ni, Co) phases in the duplex plessite structure. This may be a result of the γ-phase decomposition with the incomplete martensitic transformation: γ → α₂ + γ due to a faster cooling rate. XRD indicates the presence of ~1.3 wt% of the γ-Fe(Ni, Co) phase in Gibeon VIA. The saturation magnetization moment of 185(2) emu g⁻¹ obtained also confirms the presence of phases with low and high Ni concentrations. The most appropriate fit of the Gibeon IVA Mössbauer spectrum demonstrates the presence of five magnetic sextets and one paramagnetic singlet which are assigned to the ferromagnetic α₂-Fe(Ni, Co), α-Fe(Ni, Co), γ-Fe(Ni, Co), and paramagnetic γ-Fe(Ni, Co) phases. The relative average Fe contents in these phases are: 13.4% in the α₂-Fe(Ni, Co) phase, 78.3% in the α-Fe(Ni, Co) phase, and 8.3% in the ferromagnetic and paramagnetic γ-Fe(Ni, Co) phases.