Earth Moments Research Articles

The general relativistic effects on the magnetic moment in Earth’s gravity

It is well known that when a pair of random variables is statistically independent, it has no-correlation (zero covariance, $E[XY] - E[X]E[Y] = 0$), and that the converse is not true. However, if both of these random variables take only two values, no-correlation entails statistical independence. We provide here a general proof. We subsequently examine whether this equivalence property carries over to quantum mechanical systems. A counter-example is explicitly constructed to show that it does not. This observation provides yet another simple theorem separating classical and quantum theories.

The general relativistic effects on the magnetic moment in Earth’s gravity

The general relativistic effects to the anomalous magnetic moment of the electron ${\rm g}_{\rm e}$-2 in the Earth's gravitational field have been examined. The magnetic moment of electrons to be measured on the Earth's surface is evaluated as $\mu_{\rm m}^{\rm eff} \simeq (1\!+\!3\phi/c^2)\,\mu_{\rm m}$ on the basis of the Dirac equation containing the post-Newtonian effects of the general relativity for fermions moving in the Earth's gravitational field. This implies that the anomalous magnetic moment of $10^{-9}$ appears in addition to the radiative corrections in the quantum field theory. This may seem contradictory with the fact of the 12th digit agreement between the experimental value measured on the ground level ${\rm g}_{\rm e(EXP)}$ and the theoretical value calculated in the flat spacetime ${\rm g}_{\rm e(SM)}$. In this paper, we show that the apparent contradiction can be explained consistently with the framework of the general relativity.

Membranes, energetics, and evolution across the prokaryote-eukaryote divide.

The evolution of the eukaryotic cell marked a profound moment in Earth's history, with most of the visible biota coming to rely on intracellular membrane-bound organelles. It has been suggested that this evolutionary transition was critically dependent on the movement of ATP synthesis from the cell surface to mitochondrial membranes and the resultant boost to the energetic capacity of eukaryotic cells. However, contrary to this hypothesis, numerous lines of evidence suggest that eukaryotes are no more bioenergetically efficient than prokaryotes. Thus, although the origin of the mitochondrion was a key event in evolutionary history, there is no reason to think membrane bioenergetics played a direct, causal role in the transition from prokaryotes to eukaryotes and the subsequent explosive diversification of cellular and organismal complexity.

EARTH’S ROTATIONAL DECELERATION: DETERMINATION OF TIDAL FRICTION INDEPENDENT OF TIMESCALES

ABSTRACT This paper determines Earth's rotational deceleration without relying on atomic or ephemeris timescales. Earth's rotation defines the civil time standard called Universal Time (UT). Our previous paper did not examine tidal friction in depth when analyzing the timescale divergence between UT and International Atomic Time (TAI). We examine all available paleontological fossils and deposits for the direct measurements of Earth's past rotation rates, because that record includes all contributing effects. We examine paleontological reports that date Earth's rotation rate using corals, bivalves, brachiopods, rhythmites, and stromatolites. Contributions that vary Earth's moment of inertia, such as continental plate drifts, coastline changes, ice age formations, and viscous glacial rebounds, are superimposed with the secular deceleration. The average deceleration of Earth's rotation rate from all available fossil data is found to be (5.969 ± 1.762) × 10−7 rad yr−2. Our value is 99.8% of the total rotational deceleration determined by Christodoulidis et al., who used artificial satellite data, and our value is 96.6% of the expected tidal friction value obtained by Stephenson and Morrison. Taking the derivative of conserved angular momentum, the predicted lunar orbital deceleration caused by the average rotational deceleration corresponds closely to lunar models. When evaluating the significant time gaps between UT and TAI, Earth's rotational deceleration is a minor contributing factor. Also, the secular deceleration rate is necessary to correctly date ancient astronomical events. We strongly encourage that more ocean paleontological evidence be found to supplement the record to separate the many periodic variations embedded in these data.

Kappa distribution in the presence of a potential energy

AbstractThe present paper develops the theory and formulations of the kappa distributions that describe particle systems characterized by a nonzero potential energy. As yet, kappa distributions were used for the statistical description of the velocity or kinetic energy of particles but not of the potential energy. With the results provided here, it is straightforward to use the developed kappa distributions to describe any particle population of space plasmas subject to a nonnegligible potential energy. Starting from the kappa distribution of the Hamiltonian function, we develop the distributions that describe either the complete phase space or the marginal spaces of positions and velocities. The study shows, among others: (a) The kappa distributions of velocities that describe space plasmas can be vastly different from the standard formulation of the kappa distribution, because of the presence of a potential energy; the correct formulation should be given by the marginal kappa distribution of velocities by integrating the distribution of the Hamiltonian over the potential energy. (b) The long‐standing problem of the divergence of the Boltzmannian exponential distribution for bounded radial potentials is solved using kappa distributions of negative kappa index. (c) Anisotropic distributions of velocities can exist in the presence of a velocity‐dependent potential. (d) A variety of applications, including derivations/verifications of the following: (i) the Jeans', the most frequent, and the maximum radii in spherical/linear gravitational potentials; (ii) the Virial theorem for power law potentials; (iii) the generalized barometric formula, (iv) the plasma density profiles in Saturnian magnetosphere, and (v) the average electron magnetic moment in Earth's magnetotail.

Legal Aspects of Corporate Social Responsibility

We stand at a critical moment in Earth’s history, a time when humanity must choose its future. As the world becomes increasingly interdependent and fragile, the future at once holds great peril and great promise. To move forward we must recognize that in the midst of a magnificent diversity of cultures and life forms we are one human family and one Earth community with a common destiny. Preamble of the Earth Charter1

The astronomical rhythm of Late-Devonian climate change (Kowala section, Holy Cross Mountains, Poland)

Rhythmical alternations between limestone and shales or marls characterize the famous Kowala section, Holy Cross Mountains, Poland. Two intervals of this section were studied for evidence of orbital cyclostratigraphy. The oldest interval spans the Frasnian–Famennian boundary, deposited under one of the hottest greenhouse climates of the Phanerozoic. The youngest interval encompasses the Devonian–Carboniferous (D–C) boundary, a pivotal moment in Earth's climatic history that saw a transition from greenhouse to icehouse. For the Frasnian–Famennian sequence, lithological variations are consistent with 405-kyr and 100-kyr eccentricity forcing and a cyclostratigraphic floating time-scale is presented. The interpretation of observed lithological rhythms as eccentricity cycles is confirmed by amplitude modulation patterns in agreement with astronomical theory and by the recognition of precession cycles in high-resolution stable isotope records. The resulting relative time-scale suggests that ∼800kyr separate the Lower and Upper Kellwasser Events (LKE and UKE, respectively), two periods of anoxia that culminated in massive biodiversity loss at the end of the Frasnian. Th/U and pyrite framboid analyses indicate that during the UKE, oxygen levels remained low for 400kyr and δ13Corg measurements demonstrate that more than 600kyr elapsed before the carbon cycle reached a steady state after a +3‰ UKE excursion. The Famennian–Tournaisian (D–C) interval also reveals eccentricity and precession-related lithological variations. Precession-related alternations clearly demonstrate grouping into 100-kyr bundles. The Famennian part of this interval is characterized by several distinctive anoxic black shales, including the Annulata, Dasberg and Hangenberg shales. Our high-resolution cyclostratigraphic framework indicates that those shales were deposited at 2.2 and 2.4Myr intervals respectively. These durations strongly suggest a link between the long-period (∼2.4Myr) eccentricity cycle and the development of the Annulata, Dasberg and Hangenberg anoxic shales. It is assumed that these black shales form under transgressive conditions, when extremely high eccentricity promoted the collapse of small continental ice-sheets at the most austral latitudes of western Gondwana.

Buckling instabilities of subducted lithosphere beneath the transition zone

A sheet of viscous fluid poured onto a surface buckles periodically to generate a pile of regular folds. Recent tomographic images beneath subduction zones, together with quantitative fluid mechanical scaling laws, suggest that a similar instability can occur when slabs of subducted oceanic lithosphere encounter enhanced resistance to penetration at 660 km depth. Beneath subduction zones such as Central America and Java, the fast anomalies in elastic wavespeed are too broad (≥400 km) to be explained by simple compressional thickening of subducted lithosphere that is initially 50–100 km thick. By contrast, a periodic buckling mechanism can account for both the characteristic ‘wedge’ shapes of the anomalies and for their widths, which agree closely with those predicted by the scaling laws. In addition to providing a plausible physical interpretation of tomographic observations beneath some subduction zones, buckling instabilities have important consequences for other aspects of global geodynamics such as tectonic plate reconstructions, interpretation of geoid anomalies, and modeling of secular changes in Earth's moment of inertia.

Estimation of the frictional dissipation of lunisolar tide

Lunisolar tidal friction and variation in the earth's moment of inertia are the major factors for the long-term change in the length of day (lod). This article uses the latest geophysical and paleontological clock data to study and compare numerically the lunisolar tidal friction, variation in the terrestrial moment of inertia and long-term change in the lod for the past 1500 Ma. Two important conclusions are obtained: 1) the variation in J2 can not be explained solely by the rotational deformation of the earth, which shows that gravitational differentiation still exists in the earth; 2) tidal friction was much greater a few hundred million years ago than it is now. If it is assumed that tidal dissipation is inversely proportional to the cube of the distance between the earth and the moon, then the theoretical results of the numbers of days in a tropical year and in a synodic month accord rather well with the paleontological clock data.

Evolution of the Earth's principal axes and moments of inertia: the canonical form of solution

Harmonic coefficients of the 2nd degree are separated into the invariant quantitative (the 2nd-degree variance) and the qualitative (the standardized harmonic coefficients) characteristics of the behavior of the potential V2(t). On this basis the evolution of the Earth's dynamical figure is described as a solution of the time-dependent eigenvalues–eigenvectors problem in the canonical form. Such a canonical quadratic form is defined only by temporal variations of the harmonic coefficients and always remains finite, even within an infinite time interval. An additional condition for the correction or the determination of temporal variations of the 2nd degree is obtained. Temporal variations of the fully normalized sectorial harmonic coefficients are estimated in addition to ˙C¯20, ˙C¯21, and ˙S¯21 of the EGM96 gravity model. In addition, a non-linear hyperbolic model for C¯2m(t), S¯2m(t) is constructed. The trigonometric form of the hyperbolic model leads to the consideration of the potential V2(ψ) instead of V2(t) within the closed interval −π/2≤ψ≤+π/2. Thus, it is possible to evaluate the global trend of V2(t), the Earth's principal axes and the differences of the moments of inertia within the whole infinite time interval.

Geological constraints on the Precambrian history of Earth's rotation and the Moon's orbit

Over the past decade the analysis of sedimentary cyclic rhythmites of tidal origin, i.e., stacked thin beds or laminae usually of sandstone, siltstone, and mudstone that display periodic variations in thickness reflecting a strong tidal influence on sedimentation, has provided information on Earth's paleorotation and the evolving lunar orbit for Precambrian time (before 540 Ma). Depositional environments of tidal rhythmites range from estuarine to tidal delta, with a wave‐protected, distal ebb tidal delta setting being particularly favorable for the deposition and preservation of long, detailed rhythmite records. The potential sediment load of nearshore tidal currents and the effectiveness of the tide as an agent of sediment entrainment and deposition are related directly to tidal range (or maximum tidal height) and consequent current speed. Hence the thickness of successive laminae deposited by tidal currents can be a proxy tidal record, with paleotidal and paleorotational values being determined by analysis of measured records of lamina and cycle thickness. The validity of the findings can be investigated by testing the primary, observed values for internal self‐consistency through application of the laws of celestial mechanics. Paleotidal and paleorotational values provided by late Neoproterozoic (∼620 Ma) tidal rhythmites in South Australia are validated by these tests and indicate 13.1±0.1 synodic (lunar) months/yr, 400±7 solar days/yr, a length of day of 21.9±0.4 h, and a relative Earth‐Moon distance a/a0 of 0.965±0.005. The mean rate of lunar recession since that time is 2.17±0.31 cm/yr, which is little more than half the present rate of lunar recession of 3.82±0.07 cm/yr obtained by lunar laser ranging. The late Neoproterozoic data militate against significant overall change in Earth's moment of inertia and radius at least since 620 Ma. Cyclicity displayed by Paleoproterozoic (2450 Ma) banded iron formation in Western Australia may record tidal influences on the discharge and/or dispersal of submarine hydrothermal plumes and suggests 14.5±0.5 synodic months/yr and a/a0 = 0.906±0.029. The combined rhythmite data give a mean rate of lunar recession of 1.24±0.71 cm/yr during most of the Proterozoic (2450–620 Ma), suggesting that a close approach of the Moon did not occur during earlier time. Concentrated study of Precambrian tidal rhythmites promises to illuminate the evolving dynamics of the early Earth‐Moon system and may permit the lunar orbit to be traced back to near the time of the Moon's origin.

Pressure effects in ferromagnetic manganites and carrier scattering by disordered magnetic rare earths

The influence of hydrostatic pressure on the transport properties of (La1-xRx)0.67Ca0.33MnO3 ( ,Tb) ferromagnetic manganites is investigated. The enhancement of the Curie temperature TC under pressure agrees with previous data. In the paramagnetic range, the resistivity can be represented by a Mott localisation law, with a characteristic temperature T0 decreasing with pressure. The variation of TC with pressure is compared to the effect induced by replacing La by a magnetic rare earth in (La1-xRx)0.67Sr0.33MnO3 manganites ( , ..., Tm). The main effect is not related to the decrease of the mean radius of the cation, but to an additional scattering by the magnetic moment of the rare earth.

Magnetic interactions between Copper and RE in RE Ba2Cu3O7-δ

The antiferromagnetic ordering temperatures of the rare earth (RE) moments in RE Ba2 Cu3O7-gd, obtained from specific heat measurements, are roughly in agreement with the de Gennes factors, i.e., the ordering mechanism can be mainly seen as an indirect spin-spin exchange. However, the oxygen dependence of TN is found to be reversed for the light rare earths compared to the heavy rare earths. As origin for this systematic observation an indirect interaction between the 3d-moments of copper and the 4f-moments of rare earths is discussed as a second order effect. Such an interaction is supported by measurements of the 4f relaxation behavior on the Nd 1: 2: 3 cuprates by inelastic magnetic neutron scattering. Here, the usual thermally driven increase of the magnetic relaxation rate is suppressed up to about 80 K. This correlates with the appearance of a spin gap found by Rossat- Mignod in YBa2Cu3O7 and therefore the 3d-4f coupling can be understood as an interaction of the 4f moments with a spin-fluctuation exchange in the CuO2 planes. Furthermore, the quasielastic magnetic response has a Gaussian contribution at temperatures below 100 K, i.e., much above the long ranged ordering temperatures TN. Magnon-like excitations appear already at slightly larger temperatures than TN. In addition the paramagnetic inelastic spectra show only little dependence of the crystal field scheme on the oxygen concentration.

Comment On ‘The Earth's Equatorial Principal Axes and Moments of Inertia’ By H. S. Liu and B. F. Chao

Recently Liu & Chao (1991) (hereafter cited as ‘L&C’) published a research note stating that prior studies about the orientation of the principal axes of the Earth ‘fell short in providing explicit derivation and expressions’ and that discussions on the subject matter were made, ‘usually in passing’. One can only interpret these remarks as inferring that no previous investigator has derived and published values for the angle between the first principal axis of the Earth and a ‘geographical’ (terrestrial would be a more appropriate word) reference frame. This inquiry appears to be the final objective of their analysis from which everything else evolves, as their summary emphatically demonstrates. The basic theory (equations 2a-2e in L&C) to which the authors parenthetically add ‘complete detailed derivation of the above with explicit normalization factors’ while directing the reader to another reference written by one of the co-authors, was fully elaborated in the seminal book by Hotine (1969, p. 161), well before the availability of reliable artificial satellite data. The coverage of the subject in this treatise does not fall under the author’s imprecise criticism of ‘skimpy, if treated at all’. Hotine arrives at very general equations which are presented, coincidentally, in complex form, the notation ultimately favoured by L&C. Later Bursa (1977) formulated, rather exhaustively, the mathematical problem addressed by L&C with complete explicit derivations and final angular expressions to determine the Earth’s principal directions. This was followed by another short and revealing contribution (Bursa 1983a) (hereafter referred to as ‘B83A’) which is crucial in this context. In fact, equation (6) in B83A [and also in Bursa (1983b)l is practically identical to the central equation in L&C (number Sb), except for an elementary switch to complex notation. It is difficult to understand why L&C also fail to mention the most obvious and rigorous alternative for determining this and other related dynamical parameters: diagonalization of the Earth’s 3 x 3 symmetric inertia tensor by using well known eigentheory procedures. At the core of the problem is the straightforward relationship between the inertia tensor of any arbitrary body and the normalized coefficients of degree 2 and order 2 (i.e. C,,, S,,, n=2, m = 0, . . . , n) of its spherical harmonic expansion of the gravitational potential. By diagonalizing the inertia matrix, not only the orientation of the principal axes (normalized eigenvectors) with respect to a coefficient-implied terrestrial coordinate system can be obtained, but also, as a by-product, the three principal moments of inertia

Corrigendum to ‘Origin of terrestrial heat flow‘

When my paper entitled 'Origin of terrestrial heat flow' (Vacquier 1991) was in press, R. P. Von Herzen called my attention to the paper 'Energetics of core formation' by Francis Birch (1965) in which the release of gravitational potential energy by unmixing of the iron and silicate constituting the primordial Earth was calculated to be 400calg-' of the mixture. Birch revised this figure to 600 cal g-' (Flasar & Birch 1973). Multiplying 600 cal g-' by the mass of the Earth gives a total energy production of 15 X lo3 J. The Earth's moment of inertia and seismic speeds require that the iron of the outer core be alloyed with a light element. In the present theory this element has to be oxygen coming as FeO from the silicates of the primordial mantle. The outer core contains 12.5wt. per cent, corresponding to the composition Fe,O. From this composition and the mass of the outer core we find that 100 x 10 of FeO has accreted to the core, or a little more than half of its present mass of 189 X 10g. The energy released by this process is (100/189) X (15 x lo3) = 7.94 x lo3 J. Instead of 11.3 x lo3 in Vacquier (1991). The Flasar and Birch elegant derivation of the gravitational energy released is more accurate than mine, and will be used in the calculations given below.

Tidal rhythmites: Key to the history of the Earth's rotation and the lunar orbit.

The recent recognition of cyclically laminated tidal rhythmites provides a new approach to tracing the dynamic history of the Earth-Moon system. Late Proterozoic (-5650 Ma) elastic rhythmites in South Australia represent an unsurpassed palaeotidal record of -560 years' duration that provides numerous palaeorotational parameters. At -5650 Ma there were 13.1+-0.1 lunar months/year, 400+-7 solar days/year, and 30.5+-0.5 solar days/lunar month. The lunar apsides and lunar nodal cycles were then 9.7+-0.1 years and 19.5+-0.5 years, respectively. The indicated mean Earth-Moon distance of 58.28+-0.30 Earth radii at -5650 Ma gives a mean rate of lunar retreat of 1.95+-0.29 cm/year since that time, about half the present rate of lunar retreat of 3.7+-0.2 cm/year obtained by lunar laser ranging. The rhythmite data imply a substantial obliquity of the ecliptic at -5650 Ma, and indicate virtually no overall change in the Earth's moment of inertia, which militates against significant Earth expansion since -5650 Ma. Early Proterozoic (-52, 500 Ma) cyclic banded iron-formation in Western Australia, that may record submarine fumarolic activity triggered by earth tides, suggests -514.5+-0.5 lunar months/year and a mean Earth-Moon distance of -554.6 Earth radii at -52, 500 Ma. The combined rhythmite data suggest a mean rate of lunar retreat of -51.27 cm/year during the Proterozoic (-52, 500-650 Ma); the indicated increasing mean rate of lunar retreat since -52, 500 Ma is consistent with increasing oceanic tidal dissipation as the Earth's rotation slows. A close approach of the Moon during earlier time is uncertain. Continued study of tidal rhythmites promises to further illuminate the evolving dynamics of the Earth-Moon system.

Growth rhythms, evolution of the Earth's interior, and origin of the Metazoa

Growth patterns preserved in the accretionary skeletons of fossils provide the only known method of directly measuring the rate of the Earth's rotation in the distant past. From seasonal and tidal growth patterns of fossils, one can determine the number of days per year and per month, respectively, in the distant past. Together, these values can be used to distinguish the effects of moment of inertia changes on the length of day from those of tidal friction. When the Metazoan accretionary skeleton originated in the Late Precambrian-Cambrian, the length of day determined from fossils was approximately 19 hr. This value requires that density differentiation of the Earth was essentially complete well the end of the Precambrian. The growing length of day, as well as prior differentiation of oxygenated outer layers (atmosphere, hydrosphere, and crust) from the Earth's dense layers within, were prerequisites for the origin of the Metazoa. Circadia (=approximatelly 24 hr) rhythms in living Metazoa do not readily adapt to environemtal cycles less than about 19hr. Prokaryotes generally lack circadian rhythms because their generation times are less than a day; prokaryotes were well-adapted to Precambrian days less than 19 hr duration, as well as to oxygen-poor environments. As the length of day increased to 19 hr or more during the Late Precambrian, eukaryotes with life spans substantially longer than a day (and consequently with an ability to postpone energyusage beyond a day) evolved. During the Phanerozoic, moment of inertia changes were relatively small, so that lunar tidal friction became the most important cause of changing length of day. However, some researchers believe that even the former may have left an imprint on fossil growth patterns. This conclusion is difficult to confirm, given the uncertainties of growth pattern analyses. But facies-by-facies comparisons of growth patterns can help reduce this uncertainty: presumed tidal growth patterns should change systematically with depth of habitat, for example. Preliminary analyses for Late Ordovician brachiopods from Indiana suggest that this approach will be productive, and may help evaluate the suggestion that the Late Ordovician-Silurian was a time of unusual evolution of the Earth's moment of inertia during the Phanerozoic.

NMR Study of GdRh4B4 and SmRh4B4

Pulsed NMR study of 11 B has been performed in GdRh 4 B 4 and SmRh 4 B 4 at frequencies of 1–10 MHz and at 50 mK–300 K. From the isotropic Knight shift, we obtained the conduction electron spin polarization at B site. T 1 in paramagnetic state is explained by the dipolar and RKKY coupling between 11 B and the rare earth (RE) moments which are modulated by the RE-RE exchange interaction and the s-f interaction. Below the magnetic transition temperature both the line width and T 1 increase, which is associated with the magnetic ordering. In SmRh 4 B 4 NMR measurement was made in antiferromagnetic and superconducting states. Appreciable difference in T 1 was not observed between normal and superconducting state.

The Rise of Global Mean Sea Level as an Indication of Climate Change

Rising mean sea level, it is proposed, is a significant indicator of global climate change. The principal factors that can have contributed to the observed increases of global mean sea level in recent decades are thermal expansion of the oceans and the discharge of polar ice sheets. Calculations indicate that thermal expansion cannot be the sole factor responsible for the observed rise in sea level over the last 40 years; significant discharges of polar ice must also be occurring. Global warming, due in some degree presumably to increasing atmospheric carbon dioxide, has been opposed by the extraction of heat necessary to melt the discharged ice. During the past 40 years more than 50,000 cubic kilometers of ice has been discharged and has melted, reducing the surface warming that might otherwise have occurred by as much as a factor of 2. The transfer of mass from the polar regions to a thin spherical shell covering all the oceans should have increased the earth's moment of inertia and correspondingly reduced the speed of rotation by about 1.5 parts in 10(8). This accounts for about three quarters of the observed fractional reduction in the earth's angular velocity since 1940. Monitoring of global mean sea level, ocean surface temperatures, and the earth's speed of rotation should be complemented by monitoring of the polar ice sheets, as is now possible by satellite altimetry. All parts of the puzzle need to be examined in order that a consistent picture emerge.

Continental drift and the rotation of the earth-a new and critical evaluation

Sconzo has proposed that the continental drift of a land mass like Greenland can make a detect-able change in the Earth's rotation rate through a change in the Earth's moment of inertia. The correction of theoretical and numerical errors in his paper results in a change in the length of the day of only 3.5×10−8 s cy−1, a rate which is 6 orders of magnitude less than the value he gives. Continental drift does not appear to cause an important change in the length of the day.