On the phase-change hypothesis of the structure of the Earth

1965 ◽  
Vol 287 (1411) ◽  
pp. 471-493 ◽  
Keyword(s):  
Phase Change ◽  
Average Rate ◽  
Liquid Core ◽  
Rotation Period ◽  
Crystal Form ◽  
Outer Radius ◽  
Gravitational Energy ◽  
The Earth ◽  
Slow Expansion ◽  
Central Regions

The hypothesis that the liquid core of the Earth represents a phase-change at high pressure (and suitable temperature) of the mantle material is further investigated. A more accurate series of two-zone models have been computed, and also a new series of three-zone models. The change of overall radius as between an original all-solid Earth and the present size is shown to be at least 370 km. In the outer regions, greater pressure may be needed with rising temperature to effect the transition to denser crystal form (associated with the 20°-discontinuity), and from this cause acting alone slight expansion of the Earth would result but to an extent less than one-tenth the overall contraction. Epochs of rapid contraction (mountain-building eras) could thus be separated by longer intervals of very slow expansion. The initial liquefaction of the central regions brings about pressure increase at the boundary of the core that renders the Earth unstable in that about 6 per cent of the entire mass liquefies extremely rapidly to cause a sudden collapse of the planet as a whole. The accompanying decrease of outer radius is about 70 km. Thereafter the planet remains thoroughly stable and contracts only slowly. The total contraction to date would have reduced the moment of inertia by a factor about 4/5, and the corresponding reduction in rotation period (through conservation of angular momentum) would be an effect comparable with tidal friction. The contraction also leads to release of gravitational energy at an average rate comparable with that from radioactive sources. An important consequence of the phase-change hypothesis is that the melting-point gradient changes sign after sufficient depth, thereby permitting melting of the central regions to occur at moderate temperatures explicable by a reasonable content of radioactive elements.

On the dynamical theory of the rotation of the earth. II. The effect of precession on the motion of the liquid core

1953 ◽  
Vol 49 (3) ◽  
pp. 498-515 ◽  
Author(s):  
H. Bondi ◽  
R. A. Lyttleton
Keyword(s):  
Angular Velocity ◽  
Liquid Core ◽  
Rotation Period ◽  
Steady Motion ◽  
Dynamical Theory ◽  
Rate Of Change ◽  
Body Motion ◽  
Angular Velocity Vector ◽  
The Core ◽  
The Earth

In an earlier paper of the same general title (1) the possibility that the core of the Earth, in view of its supposed liquid nature, does not partake of the rigid-body motion of the outer shell was discussed with particular reference to the secular diminution of the angular velocity. In addition to this small rate of change of the magnitude of the angular velocity vector of the shell there occur changes in its direction consisting of the precession and nutation, but all the rates of change therein involved are small. The secular retardation takes place with extreme slowness, the nutations involve deviations of the axis with small angular amplitudes, while the precession, though of large angular amplitude, is of very long period compared with the rotation period of the Earth. Accordingly, it may be supposed that the effects of these various changes in the angular velocity can be considered separately in their relation to the motion within the core, and it is the object of this paper to give an account of our investigation into what may be termed for brevity the precession problem. It should perhaps be stated at the outset that the work does not constitute a solution of the problem, which our studies have led us to believe is one of the utmost mathematical difficulty presenting features of an exceptional character in hydro-dynamic theory. After first obtaining the equations of steady motion applicable to the interior, and those applicable to the boundary layer, the solution of the latter equations has been obtained; but in respect of the former equations we have been able to carry the question of the interior motion only as far as showing that no motion representable everywhere by analytic functions and consistent with the boundary conditions is possible. The investigation strongly suggests that no steady-state motion of a permanent character is possible for the interior, though the precise nature of the motion that actually occurs poses a problem of special interest from a hydrodynamic standpoint, but it is one to which we are not able to arrive at any definite answer at present. Without making any progress with the problem thus produced, the paper nevertheless makes clear the inherent difficulties of the problem and also serves to emphasize the inadequacy of any simplified mode of attack assuming classical fluid and resembling, for example, Poincaré's method for the nutation problem adopted by Lamb (3). Thus despite its incompleteness it seemed worth while to publish some account of such progress with these highly interesting questions as we have been able to make.

On the origin of mountains

1963 ◽  
Vol 275 (1360) ◽  
pp. 1-22 ◽  
Keyword(s):  
Liquid Core ◽  
Linear Functions ◽  
Core Material ◽  
Earth Model ◽  
Mathematical Form ◽  
Initial Surface ◽  
Mantle Material ◽  
Spherical Form ◽  
The Earth ◽  
Central Regions

The hypothesis is adopted that the Earth began as an entirely solid body and gradually became melted in its central regions. The incompressibilities of both the liquid and solid regions are linear functions of the pressure, and this enables an integral pressure-density relation to be found and also the effective uncom pressed densities of the different regions. The equations for hydrostatic equilibrium can be reduced to a standard mathematical form, while Earth-models can be related to their solution by homologies with factors depending on the mass and physical constants of the material. An Earth of purely mantle-material and solid throughout would have radius 1.043 times the present radius, while allowance for lower uncompressed density of the outer layers increases this to 1.056 R E . This implies an initial surface area nearly 60 million square kilometres in excess of the present area. The liquid core-material is more compressible than the solid mantle-material at the pressures prevailing deep within the Earth, and it results that as the size of the core gradually increases, the composite Earth-model decreases in overall radius. The extreme lower limit of size corresponding to an entirely molten Earth would be 0.846 R E . The slow contraction of the entire Earth will gradually build up shearing stresses at and near the surface that the material there will eventually be unable to withstand, and periods of surface folding and thrusting will occur intermittently to relieve these stresses. Down to a depth of a few kilometres, less potential energy would be involved in piling up material against gravity than would be required in compressing it horizontally to maintain perfectly spherical form. The Earth would therefore prefer to buckle at the surface thereby remaining as uncompressed as possible at its outer parts. The theory suggests that Venus will have developed in much the same way as the Earth. On the other hand, because of the much lower pressures within the Moon, Mercury, and Mars, these objects are still solid throughout, and if melting ever occurs it would probably result in their expansion, not in contraction. Accordingly, thrusted and folded mountains would not be expected to be found on these bodies.

Dynamical theory of the rotation of the Earth

1986 ◽  
Vol 408 (1835) ◽  
pp. 267-275 ◽  
Keyword(s):  
Average Rate ◽  
Liquid Core ◽  
Angular Acceleration ◽  
Dynamical Theory ◽  
Rate Of Change ◽  
Moment Of Inertia ◽  
Iron Core ◽  
The Past ◽  
The Earth ◽  
The Moment

From recent values of improved accuracy of the apparent secular accelerations v and v' of the Moon and Sun, the lunar and solar tidal couples N and N' can be found. The appropriate dynamical theory shows that the moment of inertia of the Earth, C , has been diminishing at an average rate of -1.67 x 10 27 cm 2 g s -1 during the past 3300 years, giving rise to a non-tidal angular acceleration ω ∙ i = 1.52 x 10 -22 s -2 in addition to the retardations of ω resulting from the lunar and solar couples. The intrinsic couple associated with Ċ , the time-rate of change of C , is considerably greater than the solar tidal couple on all values of v and v' so far determined. For an initially all-solid Earth, use of known seismic data shows that the moment of inertia has decreased during the past 3 Ga at an average rate of -1.72 x 10 27 cm 2 g s -1 since a liquid core first began to form, a figure in close agreement with the value based on ancienteclipse data. On the time-honoured hypothesis that the core has resulted from iron separating downwards in an originally homogeneous Earth, the rate of decrease of C is -0.873 x 10 27 cm 2 g s -1 , only about one-half of that based on ancient-eclipse data, while if applied to these data the ratio N / N' = 11.35, which is more than twice the theoretical ratio on any tidal hypothesis. These results show that the iron-core theory is physically unacceptable.

Correlation between large-scale motions in the liquid core of the Earth and geomagnetic jerks, earthquakes, and variations in the Earth’s length of day

2016 ◽  
Vol 467 (1) ◽  
pp. 280-283 ◽  
Author(s):  
M. B. Gokhberg ◽  
E. V. Olshanskaya ◽  
O. G. Chkhetiani ◽  
S. L. Shalimov ◽  
O. M. Barsukov
Keyword(s):  
Large Scale ◽  
Liquid Core ◽  
Length Of Day ◽  
The Earth ◽  
Geomagnetic Jerks

scholarly journals Nutation in Space and Diurnal Nutation in the Case of an Elastic Earth

1979 ◽  
Vol 82 ◽  
pp. 169-174 ◽  
Author(s):  
Nicole Capitaine
Keyword(s):  
Rotation Axis ◽  
Liquid Core ◽  
Elastic Earth ◽  
The Earth ◽  
Earth’S Interior ◽  
Earth's Interior

In order to improve the representation of nutation, the effect of elasticity of the Earth on the nutation in space and diurnal nutation of the terrestrial rotation axis is considered and its amplitude is evaluated for the principal terms. The choice between several methods taking this effect into account is discussed. A comparison with the effect induced on nutation by the existence of a liquid core in the Earth's interior shows that the consideration of elasticity alone cannot give any amelioration in the representation of nutation.

scholarly journals Accretion of the Moon from non-canonical discs

2014 ◽  
Vol 372 (2024) ◽  
pp. 20130256 ◽  
Author(s):  
J. Salmon ◽  
R. M Canup
Keyword(s):  
Angular Momentum ◽  
Rotation Period ◽  
Major Axis ◽  
The Moon ◽  
Large Excess ◽  
Semi Major Axis ◽  
Earth’S Mantle ◽  
The Earth ◽  
Post Impact ◽  
Impact Simulations

Impacts that leave the Earth–Moon system with a large excess in angular momentum have recently been advocated as a means of generating a protolunar disc with a composition that is nearly identical to that of the Earth's mantle. We here investigate the accretion of the Moon from discs generated by such ‘non-canonical’ impacts, which are typically more compact than discs produced by canonical impacts and have a higher fraction of their mass initially located inside the Roche limit. Our model predicts a similar overall accretional history for both canonical and non-canonical discs, with the Moon forming in three consecutive steps over hundreds of years. However, we find that, to yield a lunar-mass Moon, the more compact non-canonical discs must initially be more massive than implied by prior estimates, and only a few of the discs produced by impact simulations to date appear to meet this condition. Non-canonical impacts require that capture of the Moon into the evection resonance with the Sun reduced the Earth–Moon angular momentum by a factor of 2 or more. We find that the Moon's semi-major axis at the end of its accretion is approximately 7 R ⊕ , which is comparable to the location of the evection resonance for a post-impact Earth with a 2.5 h rotation period in the absence of a disc. Thus, the dynamics of the Moon's assembly may directly affect its ability to be captured into the resonance.

Constraints on core composition from shock-waves data

1982 ◽  
Vol 306 (1492) ◽  
pp. 37-47 ◽  
Keyword(s):  
Equations Of State ◽  
Liquid Core ◽  
Light Element ◽  
Outer Core ◽  
Atomic Ratio ◽  
Core Material ◽  
Solar Photosphere ◽  
The Core ◽  
The Earth ◽  
Core Composition

Seismic data demonstrate that the density of the liquid core is some 8-10 % less than pure iron. Equations of state of Fe-Si, C, FeS 2 , FeS, KFeS 2 and FeO, over the pressure interval 133-364 GPa and a range of possible core temperatures (3500- 5000 K), can be used to place constraints on the cosmochemically plausible light element constituents of the core (Si, C, S, K and O ). The seismically derived density profile allows from 14 to 20 % Si (by mass) in the outer core. The inclusion of Si, or possibly G (up to 11 %), in the core is possible if the Earth accreted inhomogeneously within a region of the solar nebulae in which a C :0 (atomic) ratio of about 1 existed, compared with a G : O ratio of 0.6 for the present solar photosphere. In contrast, homogeneous accretion permits Si, but not C, to enter the core by means of reduction of silicates to metallic Fe-Si core material during the late stages of the accumulation of the Earth. The data from the equation of state for the iron sulphides allow up to 9-13 % S in the core. This composition would provide the entire Earth with a S:Si ratio in the range 0.14-0.3, comparable with meteoritic and cosmic abundances. Shock-wave data for KFeS 2 give little evidence for an electronic phase change from 4s to 3d orbitals, which has been suggested to occur in K, and allow the Earth to store a cosmic abundance of K in the metallic core.

An exact solution for the gravity curvature (Bullard B) correction

Geophysics ◽  
1991 ◽  
Vol 56 (8) ◽  
pp. 1179-1184 ◽  
Author(s):  
T. R. LaFehr
Keyword(s):  
Routine Data ◽  
Exact Formula ◽  
Outer Radius ◽  
Physical Significance ◽  
Spherical Cap ◽  
Curvature Correction ◽  
The Earth ◽  
Infinite Slab ◽  
High Elevations ◽  
Closed Form Formula

The complete Bouguer reduction includes, in addition to the simple Bouguer slab correction (Bullard A), both curvature (Bullard B) and terrain (Bullard C) corrections. A new closed‐form formula for the curvature correction is derived for which the calculated values differ from those published by Swick by more than 0.5 mGal for high elevations. These corrections reduce those of an infinite slab (Bullard A) to that of a spherical cap having a surface radius of 166.7 km. The spherical cap produces a lesser effect than the infinite slab because of the “truncation” of that part of the slab above the earth and extending to infinity, but it produces a greater effect than the slab because of subslab earth resulting from curvature. The physical significance of the correction lies in the combination of these two differences, which are each a function of elevation. The Bullard B surface radius (166.7 km: outer radius of the Hayford‐Bowie system) is reaffirmed by the exact formula to be appropriate for exploration surveys. Three series approximations are presented and compared, but the exact Bullard B formula is very efficient and easy to program for routine data processing.

scholarly journals Influence of the Solid Inner Core and Compressibility of the Fluid Core on the Earth Nutation

1991 ◽  
Vol 127 ◽  
pp. 250-253
Author(s):  
Sergei Diakonov
Keyword(s):  
Infinite System ◽  
Inner Core ◽  
Equations Of Motion ◽  
Liquid Core ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Low Frequency Oscillations ◽  
The Earth ◽  
Fluid Core ◽  
Harmonic Representation

While calculating low frequency oscillations of the Earth liquid core spherical harmonic representation of the deformation field is usually used [1-3]:Substitution of (1) into the equations of motion gives an infinite system of differential equations for scalar functions Sɭm and Tɭm . Approximate solutions of such a system are obtained by truncating of the system. But results of [4] show that sometimes such method divergences.

Sign in / Sign up

Export Citation Format

Share Document