Fundamentals of the Thermohydrogravidynamic Theory of the Global Seismotectonic Activity of the Earth

Abstract

The article presents the fundamentals of the cosmic geophysics (representing the deterministic thermohydrogravidynamic theory intended for earthquakes prediction) based on the author's generalized differential formulation of the first law of thermodynamics extending the classical Gibbs' formulation by taking into account (along with the classical infinitesimal change of heat 𝛿𝑄 and the classical infinitesimal change of the internal energy 𝑑𝑈_𝜏) the infinitesimal increment of the macroscopic kinetic energy 𝑑𝐾_𝜏, the infinitesimal increment of the gravitational potential energy 𝑑𝝅_𝜏, the generalized expression for the infinitesimal work 𝛿𝐴_np,𝜕𝜏 done by the nonpotential terrestrial stress forces (determined by the symmetric stress tensor 𝐓) acting on the boundary of the continuum region 𝜏, and the infinitesimal increment 𝑑𝐺 of energy due to the cosmic and terrestrial nonstationary energy gravitational influence on the continuum region 𝜏 during the infinitesimal time 𝑑𝑡. Based on the established generalized differential formulation of the first law of thermodynamics, the author explains the founded cosmic energy gravitational genesis of the strong Chinese 2008 and the strong Japanese 2011 earthquakes.

1. Introduction

The problem of the long-term predictions of the strong earthquakes [1, 2] is the significant problem of the modern geophysics. The analysis of the period 1977–1985 revealed [3] the strongly nonrandom tendencies in the earthquake-induced geodetic changes (owing to the mass redistribution of material inside the Earth) related to the change of the Earth’s rotation and the Earth’s gravitational field. The analysis of the period 1977–1993 (characterized by 11015 major earthquakes) revealed [4] the strong earthquakes’ tendency to increase the Earth’s spin (rotational) energy. The analysis of the same period 1977–1993 revealed [5] “an extremely strong tendency for the earthquakes to decrease the global gravitational energy” confirming the inherent relation of the earthquakes with the transformation of the Earth’s gravitational energy into the seismic wave energy and frictional heat. The previous analysis of the principal geological features of the past 400×10⁶ years revealed [6] the geological evidence for a pulsating gravitation related to periodic variation of the Earth’s radius during the geological evolution of the Earth. The combination of satellite and gravimetric data revealed [7] the free-air anomalies of the Earth’s gravitational field.

It is well known that “the deterministic prediction of the time of origin, hypocentral (or epicentral) location, and magnitude of an impending earthquake is an open scientific problem” [8]. It was conjectured [8] that the possible earthquake prediction and warning must be carried out on a deterministic basis. However, it was pointed out [8] with some regret that the modern “study of the physical conditions that give rise to an earthquake and the processes that precede a seismic rupture of an ordinary event are at a very preliminary stage and, consequently, the techniques of prediction of time of origin, epicentre, and magnitude of an impending earthquake now available are below standard”. The authors [8] argued that “a new strong theoretical scientific effort is necessary to try to understand the physics of the earthquake”. It was conjectured [8] that the present level of knowledge of the geophysical processes “is unable to achieve the objective of a deterministic prediction of an ordinary seismic event, but it certainly will in a more or less distant future tackle the problem with seriousness and avoiding scientifically incorrect, wasteful, and inconclusive shortcuts, as sometimes has been done”. Sgrigna and Conti conjectured [8] that “it will take long time (may be years, tens of years, or centuries) because this approach requires a great cultural, financial, and organizational effort on an international basis”. It was conjectured [8] that a possible contribution to a deterministic earthquake prediction approach is related to observations and physical modelling of earthquake precursors to formulate, in perspective, “a unified theory able to explain the causes of its genesis, and the dynamics, rheology, and microphysics of its preparation, occurrence, postseismic relaxation, and interseismic phases”. It was conjectured [8] that “the study of the physical conditions that give rise to an earthquake and of the processes that precede a seismic rupture is at a very preliminary stage and, consequently, the techniques of prediction available at the moment are below standard”. However, Sgrigna and Conti believe [8] that “it should be better to pursue the deterministic prediction approach even if a reliable deterministic method of earthquake prediction will presumably be available only in the more distant future”.

It was pointed out [9] that the gravity changes (derived from regional gravity monitoring data in China from 1998 to 2005) exhibited noticeable variations before the occurrence of two large earthquakes in 2008 in the areas surrounding Yutian (Xinjiang) and Wenchuan (Sichuan). A recent research [10] by Zhan and his colleagues demonstrated that significant gravity changes were observed before all nine large earthquakes that ruptured within or near mainland China from 2001 to 2008. It was pointed out [9] that the past experience and empirical data showed that “earthquakes typically occur within one to two years after a period of significant gravity changes in the region in question”. It was concluded [9] that the “additional research is needed to remove the subjective nature in the determination of the timeframe of a forecasted earthquake”.

The need of the thermohydrogravidynamic approach [11, 12] is confirmed by previous studies [3–5, 13] and by noticeable variations of gravitational field identified [9, 10] before strong earthquakes in China from 2001 to 2008. The necessity to consider the gravitational field (during the strong earthquakes) is also related to the observations of the slow gravitational [14, 15] ground waves resulting from strong earthquakes and spreading out from the focal regions [16, 17] of earthquakes. Lomnitz pointed out [16] that the gravitational ground waves (related to great earthquakes) “have been regularly reported for many years and remain a controversial subject in earthquake seismology”. Richter presented [18] the detailed analysis of these observations and made the conclusion that “there is almost certainly a real phenomenon of progressing or standing waves seen on soft ground in the meizoseismal areas of great earthquakes”. Lomnitz presented [17] the real evidence of the existence of the slow gravitational waves in sedimentary layers during strong earthquakes. The fundamental connections of the geodynamics, seismicity, and volcanism with gravitation (and the slow gravitational ground waves resulting from strong earthquakes) are presented in the works [19–22].

It was conjectured [23] that the recent destructive earthquakes occurred in Sichuan (China, 2008), Italy (2009), Haiti (2010), Chile (2010), New Zealand (2010), and Japan (2011) “have shown that, in present state, scientific researchers have achieved little or almost nothing in the implementation of short- and medium-term earthquake prediction, which would be useful for disaster mitigation measures”. It was conjectured [23] that “this regrettable situation could be ascribed to the present poor level of achievements in earthquake forecast”. It was pointed out [23] that “although many methods have been claimed to be capable of predicting earthquakes (as numerous presentations on earthquake precursors regularly show at every international meeting), the problem of formulating such predictions in a quantitative, rigorous, and repeatable way is still open”. It was formulated [23] that “another problem of practical implementation of earthquake forecasting could be due to the lack of common understanding and exchange of information between the scientific community and the governmental authorities that are responsible for earthquake damage mitigation in each country: they operate in two different environments, they aim at different tasks, and they generally speak two different languages”. It was pointed out [23] that “the way how seismologists should formulate their forecasts and how they should transfer them to decision-makers and to the public is still a tricky issue”. It was clearly formulated [23] that “the formulation of probabilistic earthquake forecasts with large uncertainties in space and time and very low probability levels is still difficult to be used by decision-making people”. It was conjectured [23] that “in real circumstances the authorities deal with critical problems related to the high cost of evacuating the population from an area where the scientific methods estimate an expected rate of destructive earthquake as one in many thousand days, while they require much more deterministic statements”. In the special issue [23] of the International Journal of Geophysics, Console et al. assessed the status of the art of earthquake forecasts and their applicability. They invited authors “to report methods and case studies that could concretely contribute or, at least seemed promising, to improve the present frustrating situation, regarding the practical use of earthquake forecasts” [23].

In this article, by accepting with gratitude the personal invitation from Dr. Reem Ali and Dr. Radwa Ibrahim (representing the Editorial Office of the International Journal of Geophysics) to submit an article to the special issue on “Geophysical Methods for Environmental Studies”, the author presents the fundamentals of the thermohydrogravidynamic theory intended for deterministic prediction of earthquakes. The thermohydrogravidynamic theory is based on the established generalized differential formulation [11, 12, 24–26] of the first law of thermodynamics (for moving rotating deforming compressible heat-conducting stratified macroscopic continuum region 𝜏 subjected to the nonstationary Newtonian gravity): 𝑑𝑈_𝜏+𝑑𝐾_𝜏+𝑑𝝅_𝜏=𝛿𝑄+𝛿𝐴_np,𝜕𝜏+𝑑𝐺 (1)

extending the classical formulation [27] by taking into account (along with the classical infinitesimal change of heat 𝛿𝑄 and the classical infinitesimal change of the internal energy 𝑑𝑈_𝜏) the infinitesimal increment of the macroscopic kinetic energy 𝑑𝐾_𝜏, the infinitesimal increment of the gravitational potential energy 𝑑𝝅_𝜏, the generalized expression for the infinitesimal work 𝛿𝐴_np,𝜕𝜏 done by the nonpotential terrestrial stress forces (determined by the symmetric stress tensor 𝐓) acting on the boundary of the continuum region 𝜏, and the infinitesimal increment 𝑑𝐺 of energy due to the cosmic and terrestrial nonstationary energy gravitational influence on the continuum region 𝜏 during the infinitesimal time 𝑑𝑡.

In Section 2 we begin by considering the inherent physical incompleteness of the classical expression [28, 29] for the macroscopic kinetic energy per unit mass 𝜀_𝑘 defined (in classical nonequilibrium thermodynamics) as the sum of the macroscopic translational kinetic energy per unit mass 𝜀_𝑡=(1/2)𝐯² of the mass center of a continuum region and the macroscopic internal rotational kinetic energy per unit mass 𝜀_𝑟=(1/2)𝜃𝝎², where 𝐯 is the speed of the mass center of a small continuum region, 𝝎 is an angular velocity of internal rotation [29, 30], and 𝜃 is an inertia moment per unit mass of a small continuum region [28]. The classical de Groot and Mazur expression has inherent physical incompleteness [24, 31] related to the questionable assumption about the rigid-like rotation of a small continuum region. The classical de Groot and Mazur expression [28] does not consider the nonequilibrium component of the macroscopic velocity field related to the velocity shear defined by the rate of strain tensor 𝑒_𝑖𝑗. In Section 2 the macroscopic kinetic energy per unit mass 𝜀_𝑘 is presented [31] as a sum of the macroscopic translational kinetic energy per unit mass 𝜀_𝑡=(1/2)𝐯²of the mass center of a continuum region, the classical macroscopic internal rotational kinetic energy per unit mass 𝜀_𝑟 [28], the new macroscopic internal shear kinetic energy per unit mass 𝜀_𝑠 [31], and the new macroscopic internal kinetic energy of shear-rotational coupling per unit mass 𝜀^coup_𝑠,𝑟 [31] with a small correction. The presented expression for 𝜀_𝑘 and its particular form for homogeneous continuum regions of spherical and cubical shapes generalized [31] the classical de Groot and Mazur expression in classical nonequilibrium thermodynamics [28, 29] by taking into account the new macroscopic internal shear kinetic energy per unit mass 𝜀_𝑠, which expresses the kinetic energy of irreversible dissipative shear motion, and also the new macroscopic internal kinetic energy of shear-rotational coupling per unit mass 𝜀^coup_𝑠,𝑟, which expresses the kinetic energy of local coupling between irreversible dissipative shear and reversible rigid-like rotational macroscopic continuum motions.

Following the “Statistical thermohydrodynamics of irreversible strike-slip-rotational processes” [11] and the “Thermohydrogravidynamics of the Solar System” [12], in Section 2.2 we present the generalized differential formulation of the first law of thermodynamics (in the Galilean frame of reference) for nonequilibrium shear-rotational states of the deformed finite one-component individual continuum (characterized by the symmetric stress tensor 𝐓) region 𝜏 moving in the nonstationary gravitational field. In Section 2.3 we present the generalized differential formulation [11, 12] of the first law of thermodynamics (in the Galilean frame of reference) for nonequilibrium shear-rotational states of the deformed finite individual region 𝜏 of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. We present the generalization [11, 12] of the classical [27] expression 𝛿𝐴_np,𝜕𝜏=−𝛿𝑊=−𝑝𝑑𝑉 by taking into account (for Newtonian continuum) the infinitesimal works 𝛿𝐴_𝑐 and 𝛿𝐴_𝑠, respectively, of acoustic and viscous Newtonian forces acting during the infinitesimal time interval 𝑑𝑡 on the boundary surface 𝜕𝜏 of the individual continuum region 𝜏 bounded by the continuum boundary surface 𝜕𝜏. Based on the generalized differential formulation of the first law of thermodynamics, in Section 2.4 we present the analysis [11, 12] of the gravitational energy mechanism of the gravitational energy supply into the continuum region 𝜏 owing to the local time increase of the potential 𝜓 of the gravitational field inside the continuum region 𝜏 subjected to the nonstationary Newtonian gravitational field.

Following the “Statistical thermohydrodynamics of irreversible strike-slip-rotational processes” [11] and the “Thermohydrogravidynamics of the Solar System” [12], in Section 3 we present the fundamentals of the cosmic energy gravitational genesis of earthquakes. Using the evolution equation of the total mechanical energy of the macroscopic continuum region 𝜏 (of the compressible viscous Newtonian continuum), we demonstrate the physical adequacy [11, 12] of the rotational model [2] of the earthquake focal region for the seismic zone of the Pacific Ring. We present the thermodynamic foundation [11, 12] of the classical deformational (shear) model [1] of the earthquake focal region for the quasi-uniform medium of the Earth’s crust characterized by practically constant viscosity. We present the generalized thermohydrogravidynamic shear-rotational model [11, 12] of the earthquake focal region by taking into account the classical macroscopic rotational kinetic energy [28, 29], the macroscopic nonequilibrium kinetic energies [24, 31], and the external cosmic energy gravitational influences [12, 25, 26] on the focal region of earthquakes.

In Section 4 we present the fundamentals of the cosmic geophysics [12] applicable for the planets of the Solar System. In Section 4.1 we consider the energy gravitational influences on the Earth of the inner planets and the outer planets of the Solar System. In Section 4.1.1 we present the relation for the energy gravitational influences (on the Earth) of the inner and the outer planets in the second approximation of the elliptical orbits of the planets of the Solar System. In Section 4.1.2 we present the evaluation of the relative maximal planetary instantaneous energy gravitational influences on the Earth in the approximation of the circular orbits of the planets of the Solar System. In Section 4.1.3 we present the evaluation of the relative maximal planetary integral energy gravitational influences on the Earth in the approximation of the circular orbits of the planets of the Solar System. In Section 4.2 we consider the energy gravitational influences on the Earth of the Moon. In Section 4.2.1 we present the evaluation of the relative maximal instantaneous energy gravitational influence of the Moon on the Earth in the second approximation of the elliptical orbits of the Earth and the Moon around the combined mass center 𝐶_3,MOON of the Earth and the Moon. In Section 4.2.2. we present the evaluation of the maximal integral energy gravitational influence of the Moon on the Earth in the approximation of the elliptical orbits of the Earth and the Moon around the combined mass center 𝐶_3,MOON of the Earth and the Moon. In Section 4.3 we demonstrate the reality of the cosmic energy gravitational genesis of preparation and triggering of earthquakes owing to the energy gravitational influence on the Earth of the Moon and the planets of the Solar System. In Section 4.3.1 we demonstrate the real cosmic energy gravitational genesis of preparation of earthquakes by considering the energy gravitational influence on the Earth of Venus. In Section 4.3.2 we demonstrate the real cosmic energy gravitational genesis of triggering of the preparing earthquakes. In Section 4.4 we demonstrate the cosmic energy gravitational genesis of the seismotectonic activity induced by the nonstationary cosmic energy gravitational influences on the Earth of the Sun, the Moon, Venus, Jupiter, and Mars. In Section 4.4.1 we present the evaluations of the time periodicities of the maximal (instantaneous and integral) energy gravitational influences on the Earth of the Sun-Moon system, Venus, Jupiter, and Mars. In Section 4.4.2 we present the empirical time periodicities of the seismotectonic activity for various regions of the Earth. In Section 4.4.3 we present the set of the time periodicities of the periodic global seismotectonic (and volcanic) activity and the global climate variability of the Earth induced by the different combinations of the cosmic energy gravitational influences on the Earth of the Sun and the Moon, Venus, Jupiter, and Mars. In Section 4.5 we present the evidence of the cosmic energy gravitational genesis of the strong Chinese 2008 earthquakes. In Section 4.6 we present the evidence of the cosmic energy gravitational genesis of the strongest Japanese earthquakes near the Tokyo region. In Section 5 we present the summary of main results and conclusion.

2. The Generalized Formulation of the First Law of Thermodynamics for Moving Rotating Deforming Compressible Heat-Conducting Macroscopic Individual Continuum Region 𝝉 Subjected to the Nonstationary Newtonian Gravitational Field

2.1. The Generalized Expression for the Macroscopic Kinetic Energy of a Small Continuum Region in Nonequilibrium Thermodynamics

De Groot and Mazur defined the macroscopic kinetic energy per unit mass 𝜀_𝑘 as [28] the sum of the macroscopic translational kinetic energy per unit mass 𝜀_𝑡=(1/2)𝐯² of a continuum region (particle) mass center and the macroscopic internal rotational kinetic energy per unit mass 𝜀_𝑟=(1/2)𝜃𝝎²: 𝜀_𝑘=𝜀_𝑡+𝜀_𝑟=12𝐯²+12𝜃𝝎², (2)

where 𝐯 is the speed of the mass center of a small continuum region, 𝝎 is an angular velocity of internal rotation [29], and 𝜃 is an inertia moment per unit mass of a small continuum region [28]. Gyarmati’s definition [29] of the macroscopic kinetic energy per unit mass is analogous to de Groot and Mazur’s one. The classical de Groot and Mazur’s and Gyarmati’s definition (2) of the macroscopic kinetic energy per unit mass for a shear flows has some inherent physical incompleteness associated with the assumption about the rigid-like rotation of the continuum region with the angular velocity vector 𝝎. This definition is based on the assumption of local thermodynamic equilibrium since it does not consider the nonequilibrium shear component of the macroscopic continuum motion related to the rate of strain tensor 𝑒_𝑖𝑗. However, the assumption of local thermodynamic equilibrium, as noted by de Groot and Mazur [28], may be justified only by reasonable agreement of the experimental results with the theoretical deductions based on this assumption.

Landau and Lifshitz defined [32] the macroscopic internal energy of a small macroscopic continuum region as the difference between the total kinetic energy of the continuum region and kinetic energy of the translational macroscopic motion of the continuum region. According to Landau and Lifshitz’s definition [32] of the macroscopic internal energy, the term (1/2)𝜃𝝎² in the expression (2) is the internal energy of the macroscopic (hydrodynamic) continuum motion. The classical definition [28, 29] of the macroscopic internal rotational kinetic energy per unit mass (1/2)𝜃𝝎² is consistent with the Landau and Lifshitz’s definition of the macroscopic internal energy. We shall use further the Landau and Lifshitz’s definition [32] of the macroscopic internal energy.

Following the works [11, 12, 24–26], we shall present the foundation of the generalized expression for the macroscopic kinetic energy in nonequilibrium thermodynamics. We shall assume that 𝜏 is a small individual continuum region (domain) bounded by the closed continual boundary surface 𝜕𝜏 considered in the three-dimensional Euclidean space with respect to a Cartesian coordinate system 𝐾. We shall consider the small continuum region 𝜏 in a Galilean frame of reference with respect to a Cartesian coordinate system 𝐾 centred at the origin 𝑂 and determined by the axes 𝑋₁, 𝑋₂, and 𝑋₃ (see Figure 1).

Figure 1 

Cartesian coordinate system 𝐾 of a Galilean frame of reference and the continuum region mass center-affixed Lagrangian coordinate system 𝐾′.

The unit normal 𝐾-basis coordinate vectors triad 𝝁_𝟏, 𝝁_𝟐, and 𝝁_𝟑 is taken in the directions of the axes 𝑋₁, 𝑋₂, and 𝑋₃, respectively. The 𝐾-basis vector triad is taken to be right-handed in the order 𝝁_𝟏, 𝝁_𝟐, and 𝝁_𝟑; see Figure 1. 𝐠 is the local gravity acceleration.

An arbitrary point 𝑃 in three-dimensional physical space will be uniquely defined by the position vector 𝐫=𝑋_𝑖𝝁_𝑖≡(𝑋₁,𝑋₂,and𝑋₃) originating at the point 𝑂 and terminating at the point 𝑃. The continuum region-affixed Lagrangian coordinate system 𝐾^′ (with the axes 𝑥₁, 𝑥₂, and 𝑥₃) is centered to the mass center 𝐶 of the continuum region 𝜏. The axes 𝑥₁, 𝑥₂, and 𝑥₃ are taken parallel to the axes 𝑋₁, 𝑋₂, and 𝑋₃, respectively: the axis 𝑥_𝑖 parallel to the axis 𝑋_𝑖, where 𝑖=1,2,3. The unit normal 𝐾^′-basis coordinate vector triad 𝝂₁, 𝝂₂, and 𝝂₃ is taken in the directions of the axes 𝑥₁, 𝑥₂, and 𝑥₃, respectively. The 𝐾^′-basis vector triad is taken to be right-handed in the order 𝝂₁, 𝝂₂, and 𝝂_𝟑. The mathematical differential of the position vector 𝐫, 𝛿𝐫≡𝑥_𝑖𝝂_𝑖≡(𝑥₁,𝑥₂,and𝑥₃), expressed in terms of the coordinates 𝑥_𝑖 (𝑖=1,2,3) in the 𝐾^′-coordinate system, originates at the mass centre 𝐶 of the continuum region 𝜏 and terminates at the arbitrary point 𝑃 of the continuum region.

The position vector 𝐫_𝐜 of the mass center 𝐶 of the continuum region 𝜏 in the 𝐾-coordinate system is given by the following expression: 𝐫_𝐜=1𝑚_𝜏𝜏𝐫𝜌𝑑𝑉, (3) where 𝑚_𝜏=_𝜏𝜌𝑑𝑉 (4) is the mass of the continuum region 𝜏,  𝑑𝑉 is the mathematical differential of physical volume of the continuum region, 𝜌≡𝜌(𝐫,𝑡) is the local macroscopic density of mass distribution, 𝐫 is the position vector of the continuum volume 𝑑𝑉, and  𝑡 is the time. The speed of the mass centre 𝐶 of the continuum region 𝜏 is defined by the following expression: 𝐕_𝑐=𝑑𝐫_𝐜=∭𝑑𝑡^𝜏𝐯𝜌𝑑𝑉𝑚_𝜏, (5) where 𝐯=𝑑𝐫/𝑑𝑡 is the hydrodynamic velocity vector and the operator 𝑑/𝑑𝑡=(𝜕/𝜕𝑡)+𝐯⋅∇ denotes the total derivative following the continuum substance [33]. The relevant three-dimensional fields such as the velocity and the local mass density (and also the first and the second derivatives of the relevant fields) are assumed to vary continuously throughout the entire continuum bulk of the continuum region 𝜏. The instantaneous macroscopic kinetic energy of the continuum region 𝜏 (bounded by the continuum boundary surface 𝜕𝜏) is the sum of the kinetic energies of small parts constituting the continuum region 𝜏 when the number of the parts, 𝑛, tends to infinity and the maximum from their volumes tends to zero [33]: 𝐾_𝜏≡_𝜏𝜌𝐯^𝟐2𝑑𝑉, (6)

where 𝐯 is the local hydrodynamic velocity vector, 𝜌 is the local mass density, and 𝑑𝑉 is the mathematical differential of physical volume of the continuum region. We use the common Riemann’s integral here and everywhere.

For the analysis of the relative continuum motion in the physical space in the vicinity of the position vector 𝐫_𝐜 of the mass centre 𝐶 we have the Taylor series expansion (consistent with the Helmholtz’s theorem [30, 34]) of the hydrodynamic velocity vector 𝐯(𝐫) for each time moment 𝑡: 𝐯𝐫_𝐜+𝛿𝐫=𝐯𝐫_𝐜+𝝎𝐫_𝐜+×𝛿𝐫_3𝑖,𝑗=1𝑒_𝑖𝑗𝐫_𝐜𝛿𝑟_𝑗𝝁_𝑖+12³_{𝑖,𝑗,𝑘=1}𝜕²𝑣_𝑖𝜕𝑋_𝑗𝜕𝑋_𝑘𝛿𝑟_𝑗𝛿𝑟_𝑘𝝁_𝑖+𝐯_res, (7) where 𝐯(𝐫)≡(𝑣₁(𝐫), 𝑣₂(𝐫), 𝑣₃(𝐫)) is the hydrodynamic velocity vector at the position vector 𝐫, 𝛿𝐫≡𝐫−𝐫_𝐜≡(𝛿𝑟₁,𝛿𝑟₂,𝛿𝑟₃)≡(𝑥₁,𝑥₂,𝑥₃) is the differential of the position vector 𝐫, 𝝎(𝐫)≡12(∇×𝐯(𝐫))=𝜔₁,𝜔₂,𝜔₃ (8) is the angular velocity of internal rotation (a half of the vorticity vector) in the 𝐾-coordinate system at the position vector 𝐫, 𝝎_𝑣(𝐫)≡(∇×𝐯(𝐫)) (9) is the local vorticity in the 𝐾-coordinate system at the position vector 𝐫, 𝑒_𝑖𝑗(𝐫)=12𝜕𝑣_𝑖(𝐫)𝜕𝑋_𝑗+𝜕𝑣_𝑗(𝐫)𝜕𝑋_𝑖 (10) is the rate of strain tensor in the 𝐾-coordinate system at the position vector 𝐫, (𝑖,𝑗=1,2,3), ∇≡𝝁_𝟏𝜕𝜕𝑋₁+𝝁₂𝜕𝜕𝑋₂+𝝁₃𝜕𝜕𝑋₃ (11) is the gradient operator, and 𝐯_res=³_𝑖=1𝑤_𝑖𝝁_𝑖 (12)

is the small residual part of the Taylor series expansion (7), where 𝑤_𝑖=𝑂(𝑑³_𝜏), (𝑖=1,2,3), 𝑑_𝜏=sup_{𝐴,𝐵∈𝜕𝜏}(𝐫(𝐴,𝐵))² is the diameter of the continuum region 𝜏 and the vector 𝐫(𝐴,𝐵) originates at point 𝐴 and terminates at point 𝐵 of the surface 𝜕𝜏. The linear of 𝛿𝐫 terms of the Taylor series expansion (7) is presented in the classical form [33].

Substituting formula (7) into the formula (6) and integrating by parts, then we obtain the following expression [24, 31]: 𝐾_𝜏=𝐾_𝑡+𝐾_𝑟+𝐾_𝑠+𝐾^coup_𝑠,𝑟+𝐾_res=12𝑚_𝜏𝐕²_𝑐+12³_𝑖,𝑘=1𝐼_𝑖𝑘𝜔_𝑖𝐫_𝐜𝜔_𝑘𝐫_𝐜+12³_{𝑖,𝑗,𝑘=1}𝐽_𝑗𝑘𝑒_𝑖𝑗𝐫_𝐜𝑒_𝑖𝑘𝐫_𝐜+³_{𝑖,𝑗,𝑘,𝑚=1}𝜀_𝑖𝑗𝑘𝐽_𝑗𝑚𝜔_𝑖𝐫_𝐜𝑒_𝑘𝑚𝐫_𝐜+𝐾_res, (13) where 𝑚_𝜏 is the mass of the continuum region 𝜏 and 𝐼_𝑖𝑘 is the 𝑖𝑘-component of the classical inertia tensor depending on the mass distribution in the continuum region 𝜏 under consideration: 𝐼_𝑖𝑘=_𝜏,𝐾′𝛿_𝑖𝑘³_𝑗=1𝑥²_𝑗−𝑥_𝑖𝑥_𝑘𝜌𝑑𝑉, (14) where 𝑥_𝑖, 𝑥_𝑘 are the 𝑖, 𝑘-components of the vector 𝛿𝐫, respectively, in the 𝐾^′-coordinate system, 𝛿_𝑖𝑘 is the Kronecker delta tensor, 𝜀_𝑖𝑗𝑘 is the third-order permutation symbol, and 𝐽_𝑗𝑘 is the 𝑗,  𝑘-component classical centrifugal tensor depending on the mass distribution in the continuum region 𝜏 under consideration: 𝐽_𝑗𝑘=_𝜏,𝐾′𝑥_𝑗𝑥_𝑘𝜌𝑑𝑉, (15)

𝐾_res=𝑂(𝑑⁷_𝜏) is a small residual part of the macroscopic kinetic energy after substituting the Taylor series expansion (7) into formula (6).

Formula (13) states that the macroscopic kinetic energy 𝐾_𝜏 of the small continuum region 𝜏 is the sum of the macroscopic translational kinetic energy 𝐾_𝑡 of the continuum region 𝜏𝐾_𝑡=12𝑚_𝜏𝐕²_𝐜, (16) the macroscopic internal rotational kinetic energy 𝐾_𝑟 of the continuum region 𝜏𝐾_𝑟=12³_𝑖,𝑘=1𝐼_𝑖𝑘𝜔_𝑖𝐫_𝐜𝜔_𝑘𝐫_𝐜≡12𝐼_𝑖𝑘𝜔_𝑖𝐫_𝐜𝜔_𝑘𝐫_𝐜, (17) the macroscopic internal shear kinetic energy 𝐾_𝑠 of the continuum region 𝜏𝐾_𝑠=12³_{𝑖,𝑗,𝑘=1}𝐽_𝑗𝑘𝑒_𝑖𝑗𝐫_𝐜𝑒_𝑖𝑘𝐫_𝐜≡12𝐽_𝑗𝑘𝑒_𝑖𝑗𝐫_𝐜𝑒_𝑖𝑘𝐫_𝐜, (18) and the macroscopic kinetic energy of shear-rotational coupling 𝐾^coup_𝑠,𝑟 of the continuum region 𝜏𝐾^coup_𝑠,𝑟=³_{𝑖,𝑗,𝑘,𝑚=1}𝜀_𝑖𝑗𝑘𝐽_𝑗𝑚𝜔_𝑖𝐫_𝐜𝑒_𝑘𝑚𝐫_𝐜≡𝜀_𝑖𝑗𝑘𝐽_𝑗𝑚𝜔_𝑖𝐫_𝐜𝑒_𝑘𝑚𝐫_𝐜. (19)

The macroscopic internal rotational kinetic energy 𝐾_𝑟 is the classical [28, 29] kinetic energy of reversible (equilibrium) rigid-like macroscopic rotational continuum motion. The macroscopic internal shear kinetic energy 𝐾_𝑠 expresses the kinetic energy of irreversible (nonequilibrium) shear continuum motion related to the rate of strain tensor 𝑒_𝑖𝑗. The macroscopic internal kinetic energy of the shear-rotational coupling 𝐾^coup_𝑠,𝑟 expresses the kinetic energy of the local coupling between irreversible deformation and reversible rigid-like rotation.

The deduced expression (13) for 𝐾_𝜏 confirms the postulate [35] that the velocity shear (𝑒_𝑖𝑗≠0) represents an additional energy source taking into account the Evans, Hanley, and Hess’s extended formulation [35] of the first law of thermodynamics for nonequilibrium deformed states of continuum motion. The energies 𝐾_𝑟, 𝐾_𝑠, 𝐾^coup_𝑠,𝑟, and 𝐾_res are the Galilean invariants with respect to different inertial 𝐾-coordinate systems.

We obtained [31] from (13) the following expression for the macroscopic kinetic energy per unit mass 𝜀_𝑘=𝐾_𝜏/𝑚_𝜏: 𝜀_𝑘=𝜀_𝑡+𝜀_𝑟+𝜀_𝑠+𝜀^coup_𝑠,𝑟+𝜀_res=12𝐕^𝟐_𝐜+12³_𝑖,𝑘=1𝜃_𝑖𝑘𝜔_𝑖𝜔_𝑘+12³_{𝑖,𝑗,𝑘=1}𝛽_𝑗𝑘𝑒_𝑖𝑗𝑒_𝑖𝑘+³_{𝑖,𝑗,𝑘,𝑚=1}𝜀_𝑖𝑗𝑘𝛽_𝑗𝑚𝜔_𝑖𝑒_𝑘𝑚+𝜀_res, (20) where 𝜃_𝑖𝑘=𝐼_𝑖𝑘𝑚_𝜏=𝐼_𝑖𝑘∭_𝜏()𝜌𝑑𝑉𝑖,𝑘=1,2,3 (21) is the 𝑖𝑘-component of the classical inertia tensor per unit mass of the continuum region 𝜏, 𝛽_𝑖𝑘=𝐽_𝑖𝑘𝑚_𝜏=𝐽_𝑖𝑘∭_𝜏()𝜌𝑑𝑉𝑖,𝑘=1,2,3 (22) is the 𝑖𝑘-component of the classical centrifugal tensor per unit mass of the continuum region 𝜏, 𝜀_𝑡=𝐾_𝑡𝑚_𝜏=12𝐕^𝟐_𝐜 (23) is the macroscopic translational kinetic energy per unit mass of the continuum region 𝜏 (moving as a whole at speed 𝐕_𝐜 of the mass center of the continuum region 𝜏), 𝜀_𝑟=𝐾_𝑟𝑚_𝜏=12𝜃_𝑖𝑘𝜔_𝑖𝜔_𝑘 (24) is the macroscopic internal rotational kinetic energy per unit mass of the continuum region 𝜏, 𝜀_𝑠=𝐾_𝑠𝑚_𝜏=12𝛽_𝑗𝑘𝑒_𝑖𝑗𝑒_𝑖𝑘 (25) is the macroscopic internal shear kinetic energy per unit mass of the continuum region 𝜏, 𝜀^coup_𝑠,𝑟=𝐾^coup_𝑠,𝑟𝑚_𝜏=𝜀_𝑖𝑗𝑘𝛽_𝑗𝑚𝜔_𝑖𝑒_𝑘𝑚 (26)

is the macroscopic internal kinetic energy of the shear-rotational coupling per unit mass (of the continuum region 𝜏), and 𝜀_res=𝑂(𝑑⁴_𝜏) is the residual correction. The energies 𝜀_𝑟, 𝜀_𝑠, 𝜀^coup_𝑠,𝑟, and 𝜀_res are the Galilean invariants with respect to different inertial 𝐾-coordinate systems. We have 𝜀_𝑟=𝑂(𝑑²_𝜏), 𝜀_𝑠=𝑂(𝑑²_𝜏), 𝜀^coup_𝑠,𝑟=𝑂(𝑑²_𝜏), and 𝜀_res=𝑂(𝑑⁴_𝜏), when 𝑑_𝜏→0, where 𝑑_𝜏 is the defined diameter of the continuum region 𝜏.

For a homogeneous continuum region of simple form (sphere or cube) we have 𝐼_𝑖𝑘=𝐼𝛿_𝑖𝑘,𝐽_𝑗𝑘=𝐽𝛿_𝑗𝑘. (27) Formula (17) for the macroscopic internal rotational kinetic energy 𝐾_𝑟 is reduced to the classical expression [29] 𝐾_𝑟=12𝐼𝝎², (28) where 𝝎²=𝜔²₁+𝜔²₂+𝜔²₃. Formula (18) for the macroscopic internal shear kinetic energy 𝐾_𝑠 is reduced to the expression [24, 31] 𝐾_𝑠=12𝐽𝑒_𝑖𝑗𝑒_𝑖𝑗≡12𝐽𝑒_𝑖𝑗², (29) which is proportional to the local kinetic energy dissipation rate per unit mass 𝜀_dis=2𝜈(𝑒_𝑖𝑗)² in an incompressible viscous Newtonian continuum, where 𝜈 is the molecular viscosity. The macroscopic internal kinetic energy of shear-rotational coupling 𝐾^coup_𝑠,𝑟 vanishes for the homogeneous continuum region 𝜏 of the form of the sphere or cube. The macroscopic kinetic energy 𝐾_𝜏 for the homogeneous continuum region 𝜏 of the shape of sphere or cube is given by following expression [24, 31] 𝐾_𝜏=12𝑚_𝜏𝐕²_𝐜+12𝐼𝝎²+12𝐽𝑒_𝑖𝑗²+𝐾_res. (30) Hence, the macroscopic kinetic energy per unit mass 𝜀_𝑘 for the homogeneous continuum sphere or cube 𝜏 is expressed as the sum of explicit terms [24, 31] 𝜀_𝑘=12𝐕^𝟐_𝐜+12𝜃𝝎^𝟐+12𝛽𝑒_𝑖𝑗²+𝜀_res, (31)

where 𝜀_𝑡=(1/2)𝐕²_𝐜 is the macroscopic translational kinetic energy per unit mass of the continuum region 𝜏; 𝜃=𝐼/𝑚_𝜏; 𝛽=𝐽/𝑚_𝜏; 𝜀_𝑟=(1/2)𝜃𝝎² is the classical [28, 29] macroscopic internal rotational kinetic energy per unit mass of the continuum region 𝜏; 𝜀_𝑠=(1/2)𝛽(𝑒_𝑖𝑗)² is the macroscopic internal shear kinetic energy per unit mass of the homogeneous continuum sphere or cube 𝜏 [24, 31].

We have the following expression for the macroscopic internal kinetic energy 𝐾_int of the homogeneous continuum region 𝜏 of the shape of sphere or cube [24, 31] 𝐾_int=12𝐼𝝎²+12𝐽𝑒_𝑖𝑗²+𝐾_res. (32) The macroscopic internal kinetic energy per unit mass 𝜀_int for the homogeneous continuum region 𝜏 of the shape of sphere or cube is given by the sum of explicit terms [24, 31]: 𝜀_int=12𝜃𝝎^𝟐+12𝛽𝑒_𝑖𝑗²+𝜀_res. (33)

Compare formula (31) with the de Groot and Mazur’s definition (2). Expression (31) is reduced to de Groot and Mazur’s definition (2) under condition 𝑒_𝑖𝑗=0𝑖,𝑗=1,2,3 (34)

of local thermodynamic equilibrium. Therefore, we can conclude that the definition (2) of the macroscopic kinetic energy per unit mass 𝜀_𝑘 in classical nonequilibrium thermodynamics [28, 29] is based on the assumption 𝑒_𝑖𝑗=0 of local thermodynamic equilibrium [24, 31, 35].

The obtained formula (20) for 𝜀_𝑘 and its particular form (31) (obtained for homogeneous continuum regions of spherical and cubical shapes) generalized [24, 31] the classical de Groot and Mazur expression (2) in classical nonequilibrium thermodynamics [28, 29] by taking into account the irreversible dissipative shear component of the macroscopic continuum motion related to the rate of strain tensor 𝑒_𝑖𝑗. The expression (20) for 𝜀_𝑘 contains the new macroscopic internal shear kinetic energy per unit mass 𝜀_𝑠, which expresses the kinetic energy of irreversible dissipative shear motion, and also the new macroscopic internal kinetic energy of the shear-rotational coupling per unit mass 𝜀^coup_𝑠,𝑟, which expresses the kinetic energy of local coupling between irreversible dissipative shear and reversible rigid-like rotational macroscopic continuum motions.

The macroscopic internal shear kinetic energy per unit mass (for homogeneous continuum regions of spherical and cubical shapes) 𝜀_𝑠=12𝛽𝑒_𝑖𝑗² (35) is proportional to the kinetic energy viscous dissipation rate per unit mass 𝜀_dis,𝑠=2𝜈𝑒_𝑖𝑗² (36) in an incompressible viscous Newtonian continuum characterized by the kinematic viscosity 𝜈. We have shown [24] that the proportionality 𝜀_𝑠∼𝜀_dis,𝑠=2𝜈𝑒_𝑖𝑗² (37) is the basis of the established association [36, 37] between a structure and an order (and, hence, the associated macroscopic kinetic energy), on the one hand, and irreversible dissipation, on the other hand, for the dissipative structures of turbulence in viscous Newtonian fluids.

2.2. The Generalized Differential Formulation of the First Law of Thermodynamics (in the Galilean Frame of Reference) for Nonequilibrium Shear-Rotational States of the Deformed One-Component Individual Finite Continuum Region (Characterized by the Symmetric Stress Tensor T) Moving in the Nonstationary Newtonian Gravitational Field

Following the works [11, 12, 25, 26], we shall present the foundation of the generalized differential formulation of the first law of thermodynamics (in the Galilean frame of reference) for nonequilibrium shear-rotational states of the deformed finite one-component individual continuum region (characterized by the symmetric stress tensor 𝐓) moving in the nonstationary Newtonian gravitational field. We shall consider the deformed finite one-component individual continuum region in nonequilibrium shear-rotational states characterized by the following condition: 𝑒_𝑖𝑗≠0𝑖,𝑗=1,2,3. (38) Considering the graphical methods in the thermodynamics of fluids, Gibbs [27] formulated the first law of thermodynamics for the fluid body (fluid region) as follows (in Gibbs’ designations): 𝑑𝜀=𝑑𝐻−𝑑𝑊, (39)

where 𝑑𝜀 is the differential of the internal thermal energy of the fluid body, 𝑑𝐻 is the differential change of heat across the boundary of the fluid body related to the thermal molecular conductivity (associated with the corresponding external or internal heat fluxes), and 𝑑𝑊=𝑝𝑑𝑉 is the differential work produced by the considered fluid body on its surroundings (surrounding fluid) under the differential change 𝑑𝑉 of the fluid region (of volume 𝑉) characterized by the thermodynamic pressure 𝑝.

The formulation [32] of the first law of thermodynamics for the general thermodynamic system (material region) is given by the equivalent form (in Landau’s and Lifshitz’s designations [32]) 𝑑𝐸=𝑑𝑄−𝑝𝑑𝑉, (40)

where 𝑑𝐴=−𝑝𝑑𝑉 is the differential work produced by the surroundings (surroundings of the thermodynamic system) on the thermodynamic system under the differential change 𝑑𝑉 of volume 𝑉 of the thermodynamic system characterized by the thermodynamic pressure 𝑝;  𝑑𝑄 is the differential heat transfer (across the boundary of the thermodynamic system) related to the thermal interaction of the thermodynamic system and the surroundings (surrounding environment); 𝐸 is the energy of the thermodynamic system, which should contain (as supposed [32]) the kinetic energy of the macroscopic continuum motion.

We shall use the differential formulation of the first law of thermodynamics [28] for the specific volume 𝜗=1/𝜌 of the compressible viscous one-component deformed continuum with no chemical reactions: 𝑑𝑢=𝑑𝑡𝑑𝑞𝑑𝑡−𝑝𝑑𝜗𝑑𝑡−𝜗𝚷∶Grad𝐯, (41) where 𝑢 is the specific (per unit mass) internal thermal energy, 𝑝 is the thermodynamic pressure, 𝚷 is the viscous stress tensor, 𝐯 is the hydrodynamic velocity of the continuum macrodifferential element [28], and 𝑑𝑞 is the differential change of heat across the boundary of the continuum region (of unit mass) related to the thermal molecular conductivity described by the heat equation [28]: 𝜌𝑑𝑞𝑑𝑡=−div𝐉_𝑞, (42) where 𝐉_𝑞 is the heat flux [28]. The viscous stress tensor 𝚷 is taken from the decomposition of the pressure tensor 𝐏 [28]: 𝚸=𝑝𝜹+𝚷, (43)

where 𝜹 is the Kronecker delta tensor.

Considering the Newtonian viscous stress tensor 𝐏^𝑣≡𝚷 of the compressible viscous Newtonian continuum with the components [29] Π_𝑖𝑗=23𝜈𝜌−𝜂_𝑣div𝐯𝛿_𝑖𝑗−2𝜈𝜌𝑒_𝑖𝑗, (44) the differential formulation (41) of the first law of thermodynamics (for the continuum region (of unit mass) of the compressible viscous Newtonian one-component deformed continuum with no chemical reactions) can be rewritten as follows: 𝑑𝑢=𝑑𝑡𝑑𝑞𝑑𝑡−𝑝𝑑𝜗+𝜈𝑑𝑡²−23𝜈(div𝐯)²+2𝜈𝑒_𝑖𝑗², (45) where 𝜈=𝜂/𝜌 is the coefficient of the molecular kinematic (first) viscosity and 𝜈₂=𝜂_𝑣/𝜌 is the coefficient of the molecular volume (second) viscosity [38]. The first and the second terms in the right-hand side of relation (45) are analogous to the corresponding respective first and second terms in the right-hand side of the classical formulations (39) and (40). The third term in the right-hand side of relation (45) 𝑑𝑞_𝑖,𝑐=𝜂_𝜈𝜌−23𝜈(div𝐯)²𝑑𝑡 (46) is related to the “internal” heat induced during the time interval 𝑑𝑡 by viscous-compressible irreversibility [24]. The fourth term in the right-hand side of relation (45) 𝑑𝑞_𝑖,𝑠=2𝜈𝑒_𝑖𝑗²𝑑𝑡 (47)

is related to the “internal” heat induced during the time interval 𝑑𝑡 by viscous-shear irreversibility [24]. The differential formulation (45) of the first law of thermodynamics (for the continuum element of the compressible viscous Newtonian one-component deformed continuum with no chemical reactions) takes into account (in addition to the classical terms) the viscous-compressible irreversibility and viscous-shear irreversibility inside the continuum element of the compressible viscous Newtonian one-component deformed continuum with no chemical reactions.

Using the differential formulation (41) of the first law of thermodynamics [28] for the total derivative 𝑑𝑢/𝑑𝑡 (following the liquid substance) of the specific (per unit mass) internal thermal energy 𝑢 of a compressible viscous one-component deformed continuum with no chemical reactions, the heat equation (42) [28], the general equation of continuum movement [29] 𝑑𝐯=1𝑑𝑡𝜌div𝐓+𝐠 (48) for the deformed continuum characterized by the symmetric stress tensor 𝐓 of general form (in particular, with the components [29] 𝑇_𝑖𝑗2=−𝑝+3𝜈𝜌−𝜂_𝑣div𝐯𝛿_𝑖𝑗+2𝜈𝜌𝑒_𝑖𝑗 (49) for the compressible viscous Newtonian one-component continuum) and taking into account the time variations of the potential 𝜓 of the nonstationary gravitational field (characterized by the local gravity acceleration vector 𝐠=−∇𝜓) inside an arbitrary finite macroscopic individual continuum region 𝜏, we derived [11, 12] the generalized differential formulation (for the Galilean frame of reference) of the first law of thermodynamics (for moving rotating deforming compressible heat-conducting stratified macroscopic continuum region 𝜏 subjected to the nonstationary Newtonian gravitational field): 𝑑𝐾_𝜏+𝑈_𝜏+𝝅_𝜏=𝑑𝑡_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧−𝑑𝑡_𝜕𝜏𝐉_𝑞⋅𝐧𝑑Ω_𝐧+𝑑𝑡_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉, (50) where 𝛿𝐴_np,𝜕𝜏=𝑑𝑡_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧 (51) is the differential work done during the infinitesimal time interval 𝑑𝑡 by nonpotential stress forces (pressure, compressible, and viscous forces for Newtonian continuum) acting on the boundary surface 𝜕𝜏 of the continuum region 𝜏; 𝑑Ω_𝐧 is the differential element (of the boundary surface 𝜕𝜏 of the continuum region 𝜏) characterized by the external normal unit vector 𝐧; 𝐭=𝐧⋅𝐓 is the stress vector [29], 𝐓=−𝐏 [29], where 𝐏 is the pressure tensor characterized (in particular, for the model of the compressible viscous Newtonian continuum characterized by the coefficients of kinematic viscosity 𝜈 and the volume viscosity 𝜂_𝑣) by components (obtained from(49)) 𝑃_𝑖𝑗=2𝑝+3𝜈𝜌−𝜂_𝑣div𝐯𝛿_𝑖𝑗−2𝜈𝜌𝑒_𝑖𝑗;𝛿𝑄=−𝑑𝑡_𝜕𝜏𝐉_𝑞⋅𝐧𝑑Ω_𝐧 (52) is the differential change of heat of the macroscopic individual continuum region 𝜏 related to the thermal molecular conductivity of heat across the boundary 𝜕𝜏 of the continuum region 𝜏, where 𝐉_𝑞 is the heat flux [28] (across the element 𝑑Ω_𝐧 of the continuum boundary surface 𝜕𝜏); 𝜋_𝜏≡_𝜏𝜓𝜌𝑑𝑉 (53) is the macroscopic potential energy (of the macroscopic individual continuum region 𝜏) related to the nonstationary potential 𝜓 of the gravitational field; 𝑈_𝜏≡_𝜏𝑢𝜌𝑑𝑉 (54) is the classical microscopic internal thermal energy of the macroscopic individual continuum region 𝜏; 𝐾_𝜏=_𝜏𝜌𝐯^𝟐2𝑑𝑉 (55)

is the instantaneous macroscopic kinetic energy of the macroscopic individual continuum region 𝜏. The instantaneous macroscopic kinetic energy 𝐾_𝜏 is given by the relation (13) [24, 31] for the small macroscopic individual continuum region 𝜏.

The generalized differential formulation (50) of the first law of thermodynamics can be rewritten as follows [11, 12]: 𝑑𝑈_𝜏+𝑑𝐾_𝜏+𝑑𝝅_𝜏=𝛿𝑄+𝛿𝐴_np,𝜕𝜏+𝑑𝐺 (56) extending the classical [12] formulations (39) and (40) 𝑑𝑈=𝛿𝑄−𝑝𝑑𝑉,𝑑𝜀≡𝑑𝑈,−𝛿𝑊=−𝑝𝑑𝑉 (57) by taking into account (along with the classical infinitesimal change of heat 𝛿𝑄 and the classical infinitesimal change of the internal energy 𝑑𝑈_𝜏≡𝑑𝑈) the infinitesimal increment of the macroscopic kinetic energy 𝑑𝐾_𝜏, the infinitesimal increment of the gravitational potential energy 𝑑𝝅_𝜏, the generalized infinitesimal work 𝛿𝐴_np,𝜕𝜏 done on the continuum region 𝜏 by the surroundings of 𝜏, and the infinitesimal amount 𝑑𝐺 of energy [11, 12] 𝑑𝐺=𝑑𝑡_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉 (58)

added (or lost) as the result of the Newtonian nonstationary gravitational energy influence on the continuum region 𝜏 during the infinitesimal time interval 𝑑𝑡.

The generalized differential formulation (50) of the first law of thermodynamics can be rewritten as follows [11, 12]: 𝑑𝐸_𝜏=𝑑𝑑𝑡𝐾𝑑𝑡^𝜏+𝑈_𝜏+𝝅_𝜏=_𝜏12𝐯²=+𝑢+𝜓𝜌𝑑𝑉_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧−_𝜕𝜏𝐉_𝑞⋅𝐧𝑑Ω_𝐧+_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉. (59)

The equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics take into account the following factors:(1)the classical heat thermal molecular conductivity (across the boundary 𝜕𝜏 of the macroscopic continuum region 𝜏) related to the classical infinitesimal change of heat 𝛿𝑄: 𝛿𝑄=−𝑑𝑡_𝜕𝜏𝐉_𝑞⋅𝐧𝑑Ω_𝐧, (60)(2)the classical infinitesimal change of the internal energy 𝑑𝑈_𝜏 of the macroscopic continuum region 𝜏: 𝑑𝑈_𝜏≡𝑑_𝜏𝑢𝜌𝑑𝑉, (61)(3)the established [11, 12] infinitesimal increment of the macroscopic kinetic energy 𝑑𝐾_𝜏 of the macroscopic continuum region 𝜏: 𝑑𝐾_𝜏=𝑑_𝜏𝜌𝐯^𝟐2𝑑𝑉, (62)(4)the established [11, 12] infinitesimal increment of the gravitational potential energy 𝑑𝝅_𝜏 of the macroscopic continuum region 𝜏: 𝑑𝝅_𝜏=𝑑_𝜏𝜓𝜌𝑑𝑉, (63)(5)the established [11, 12] generalized infinitesimal work 𝛿𝐴_np,𝜕𝜏 done on the macroscopic continuum region 𝜏 by the surroundings of 𝜏: 𝛿𝐴_np,𝜕𝜏=𝑑𝑡_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧, (64)(6)the established [11, 12] infinitesimal amount 𝑑𝐺 of energy added (or lost) as the result of the Newtonian nonstationary gravitational energy influence on the macroscopic continuum region 𝜏 during the infinitesimal time interval 𝑑𝑡: 𝑑𝐺=𝑑𝑡_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉. (65)

The generalized differential formulations (50), (56), and (59) of the first law of thermodynamics (given for the Galilean frame of reference) are valid for nonequilibrium shear-rotational states of the deformed finite individual continuum region (characterized by the symmetric stress tensor 𝐓 in the general equation (48) of continuum movement [29]) moving in the nonstationary gravitational field. The generalized differential formulations (50) and (56) of the first law of thermodynamics [11, 12] are the generalizations of the classical formulations (39) and (40) of the first law of thermodynamics taking into account: (1) the generalized expression (51) for the differential work 𝛿𝐴_np,𝜕𝜏 done during the infinitesimal time interval 𝑑𝑡 by nonpotential stress forces acting on the boundary surface 𝜕𝜏 of the individual continuum region 𝜏 and (2) the time variations of the potential 𝜓 of the nonstationary gravitational field inside the individual continuum region 𝜏 due to the deformation of the individual continuum region 𝜏 and due to the external (terrestrial and cosmic) gravitational influence on the individual continuum region 𝜏 moving in the total (internal + external) nonstationary gravitational field.

2.3. The Generalized Differential Formulation of the First Law of Thermodynamics (in the Galilean Frame of Reference) for Nonequilibrium Shear-Rotational States of the Deformed Finite Individual Region of the Compressible Viscous Newtonian One-Component Continuum Moving in the Nonstationary Gravitational Field

There is evidence [39] that the rocks of the Earth’s crust at protracted loadings may be considered as fluids characterized by the very high viscosity. According to the classical viewpoint [39], the local mechanism of creation of the earthquakes is related to the release of the accumulated potential energy of the elastic deformation during the sudden local break (i.e., the discontinuous shear) of the Earth’s crust (or the sudden increase of fluidity in the local region of the Earth’s crust) accompanied by viscous relaxation and generation of seismic waves. It was conjectured [40] that “more punctual and refined methods of the mathematical analysis are obligatory” for “the practical assessment of the seismic hazard”.

Taking into account the established [31] conception of the macroscopic internal shear kinetic energy (per unit mass) 𝜀_𝑠 related to the rate of medium deformation (i.e., with the rate of strain tensor 𝑒_𝑖𝑗=𝑑𝜀_𝑖𝑗/𝑑𝑡, where 𝜀_𝑖𝑗 is the deformation tensor [30]), we have elucidated [41] from the viewpoint of nonequilibrium thermodynamics the mechanism of generation of seismic waves from the separate deformed finite zone of the Earth’s crust. The proportionality (37) takes place also for deformed compressible finite region of the Earth’s crust for sudden rise of fluidity (in a local region of the Earth’s crust) related to the local sudden medium deformation in the separate seismic zones of the seismic activity. Taking into account the established [31] proportionality (37), we have assumed [31] that the accumulated potential energy of the elastic deformation (related to the deformation tensor 𝜀_𝑖𝑗) converts to the macroscopic internal shear kinetic energy 𝐾_𝑠 (related to the rate of strain tensor 𝑒_𝑖𝑗) in the seismic zone simultaneously with the damping of 𝐾_𝑠 by viscous dissipation and radiation of seismic waves during several oscillations. In Section 3 we shall evaluate this mechanism on the basis of the generalized differential formulation (50) of the first law of thermodynamics for nonequilibrium shear-rotational states of the deformed finite individual continuum region (characterized by the symmetric stress tensor 𝐓) moving in the nonstationary gravitational field.

Following the works [11, 12, 25, 26], we shall present the foundation of the generalized differential formulation of the first law of thermodynamics for nonequilibrium shear-rotational states of the deformed finite individual region of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. The generalized differential formulation (50) of the first law of thermodynamics (formulated for the Galilean frame of reference) is valid for arbitrary symmetric stress tensor 𝐓, in particular for nonequilibrium shear-rotational states of the deformed finite individual region of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. The coefficient of molecular kinematic (first, shear) viscosity 𝜈=𝜂/𝜌 and the coefficient of molecular volume (second) viscosity 𝜈₂=𝜂_𝑣/𝜌 are assumed to vary for each time moment 𝑡 as an arbitrary continuous functions of the Cartesian space (three-dimensional) coordinates.

The differential work 𝛿𝐴_np,𝜕𝜏 for the Newtonian symmetric stress tensor 𝐓 (characterized by the components (49)) is presented by three explicit terms [11, 12]: 𝛿𝐴_np,𝜕𝜏=𝛿𝐴_𝑝+𝛿𝐴_𝑐+𝛿𝐴_𝑠=−𝑑𝑡_𝜕𝜏𝑝(𝐯⋅𝐧)𝑑Ω_𝐧−𝑑𝑡_𝜕𝜏23𝜂−𝜂_𝑣div𝐯(𝐯⋅𝐧)𝑑Ω_𝐧+𝑑𝑡_𝜕𝜏2𝜂𝑣_𝛽𝑛_𝛼𝑒_𝛼𝛽𝑑Ω_𝐧, (66) where 𝛿𝐴_𝑝=−𝑑𝑡_𝜕𝜏𝑝(𝐯⋅𝐧)𝑑Ω_𝐧 (67) is the differential work of the hydrodynamic pressure forces acting on the boundary surface 𝜕𝜏 of the individual continuum region 𝜏 (bounded by the continuum boundary surface 𝜕𝜏) during the infinitesimal time interval 𝑑𝑡; 𝛿𝐴_𝑐=−𝑑𝑡_𝜕𝜏23𝜂−𝜂_𝑣div𝐯(𝐯⋅𝐧)𝑑Ω_𝐧 (68) is the differential work (related to the combined effects of the acoustic compressibility, molecular kinematic viscosity, and molecular volume viscosity) of the acoustic (compressible) pressure forces acting on the boundary surface 𝜕𝜏 of the individual continuum region 𝜏 during the infinitesimal time interval 𝑑𝑡; 𝛿𝐴_𝑠=𝑑𝑡_𝜕𝜏2𝜂𝑣_𝛽𝑛_𝛼𝑒_𝛼𝛽𝑑Ω_𝐧 (69)

is the differential work of the viscous Newtonian forces (related to the combined effect of the velocity shear, that is, the deformation of the continuum region 𝜏, and the molecular kinematic viscosity) acting on the boundary surface 𝜕𝜏 of the individual continuum region 𝜏 during the infinitesimal time interval 𝑑𝑡.

Along with (45) the differential formulation of the first law of thermodynamics [28] for the total derivative 𝑑𝑢/𝑑𝑡 (following the continuum substance) of the internal thermal energy per unit mass 𝑢 of the one-component deformed continuum with no chemical reactions, the thermohydrodynamic theory [28] contains additionally the equations of the mass and momentum balances: 𝜕𝜌𝜌𝜕𝑡=−div𝜌𝐯,𝑑𝐯1𝑑𝑡=−Grad𝑝+𝜂Δ𝐯+3𝜂+𝜂_𝜈Graddiv𝐯. (70)(71)

The generalized differential formulation (50) of the first law of thermodynamics (together with the generalized differential work 𝛿𝐴_np,𝜕𝜏 given by the expression (66)) is valid for nonequilibrium shear-rotational states of the deformed finite individual region of the compressible viscous Newtonian one-component continuum moving in the nonstationary gravity field. The coefficient of molecular kinematic (first, shear) viscosity 𝜈=𝜂/𝜌 and the coefficient of molecular volume (second) viscosity 𝜈₂=𝜂_𝑣/𝜌 are assumed to vary for each time moment 𝑡 as an arbitrary continuous functions of Cartesian space (three-dimensional) coordinates.

The generalized differential formulation (50) takes into account the dependences of the hydrodynamic pressure on the hydrodynamic vorticity 𝝎_𝑣 and on the rate of strain tensor 𝑒_𝑖𝑗 by means of the component 𝛿𝐴_𝑝 (in the expression (66) for 𝛿𝐴_np,𝜕𝜏) given by the expression (67). The presence of the third term 𝛿𝐴_𝑠 (given by the expression (69) and related to the combined effect of the molecular kinematic viscosity and the deformation of the continuum region 𝜏 defined by the rate of strain tensor 𝑒_𝛼𝛽) in the expression (66) for 𝛿𝐴_np,𝜕𝜏 generalizes essentially the classical formulations (39) and (40) of the first law of thermodynamics by taking into account the differential work of the viscous Newtonian forces acting on the boundary continuum surface 𝜕𝜏 of the individual continuum region 𝜏.

The general equation (48) of continuum movement [29] for the compressible viscous Newtonian one-component continuum is reduced to the following equation: 𝜌𝑑𝐯+1𝑑𝑡=−Grad𝑝+𝜂Δ𝐯3𝜂+𝜂_𝜈+2Graddiv𝐯Grad𝜂⋅𝐞−div𝐯Grad3𝜂−𝜂_𝜈+𝐠, (72)

where (Grad𝜂)⋅𝐞 is the internal multiplication of the vector (Grad𝜂) and the rate of strain tensor 𝐞 in accordance with the corresponding definition [29]. Equation (72) generalizes the Navier-Stokes equation (71) (given for 𝐠=0) by taking into account the dependences of the coefficient of molecular kinematic viscosity 𝜈=𝜂/𝜌 and the coefficient of molecular volume viscosity 𝜈₂=𝜂_𝑣/𝜌 on the space (three-dimensional) Cartesian coordinates.

The relevant example for illustration of the significance of the term 𝛿𝐴_𝑠 (in the expression (66) for the differential work 𝛿𝐴_np,𝜕𝜏) is related to the thermodynamic consideration [12] of the processes of the energy exchange [42] between the oceans and the lithosphere of the Earth. According to the expression (69) for the term 𝛿𝐴_𝑠, the energy exchange between the oceans (and the atmosphere) and the lithosphere of the Earth is possible only under the presence of the medium acoustic compressibility (i.e., div𝐯≠0) and the medium deformations (i.e., 𝑒_𝛼𝛽≠0) in the boundary regions of fluid (in the oceans), air (in the atmosphere), and the compressible deformed lithosphere of the Earth. According to the generalized expression (66) for the differential work 𝛿𝐴_np,𝜕𝜏, the energy exchange between the oceans (and the atmosphere) and the lithosphere of the Earth is impossible for nondeformed (𝑒_𝛼𝛽=0) and noncompressible (div𝐯=0) lithosphere.

We have the evolution equation for the total mechanical energy (𝐾_𝜏+𝝅_𝜏) of the deformed finite individual macroscopic continuum region 𝜏 [11, 12]: 𝑑𝐾𝑑𝑡^𝜏+𝝅_𝜏=𝑑𝑑𝑡_𝜏12𝐯²=+𝜓𝜌𝑑𝑉_𝜏+𝑝div𝐯𝑑𝑉_𝜏23𝜂−𝜂_𝑣(div𝐯)²𝑑𝑉−_𝜏2𝜈𝑒_𝑖𝑗²+𝜌𝑑𝑉_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧+_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉 (73) obtained from the generalized differential formulation (50) for the compressible viscous Newtonian one-component continuum moving in the nonstationary gravitational field. In Section 3 we shall use the evolution equation (73) of the total mechanical energy to found the rotational, shear, and the shear-rotational models of the earthquake focal region.

2.4. Cosmic and Terrestrial Energy Gravitational Genesis of the Seismotectonic (and Volcanic) Activity of the Earth Induced by the Combined Cosmic and Terrestrial Nonstationary Energy Gravitational Influences on an Arbitrary Individual Continuum Region 𝜏 (of the Earth) and by the Nonpotential Terrestrial Stress Forces Acting on the Boundary Surface 𝝏𝜏 of the Individual Continuum Region 𝜏

Following the works [11, 12], we shall present the physical mechanisms of the energy fluxes to the continuum region 𝜏 related to preparation of earthquakes. The equivalent generalized differential formulations (50) and (59) of the first law of thermodynamics show that the nonstationary gravitational field (related to the nonstationary gravitational potential 𝜓) gives the following gravitational energy power: 𝑊_gr(𝜏)=_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉=𝑑𝐺𝑑𝑡 (74)

associated with the gravitational energy power of the total combined (external cosmic and terrestrial and internal related to the macroscopic continuum region 𝜏) gravitational field. If the macroscopic continuum region 𝜏 is not very large, consequently, it cannot induce the significant time variations to the potential 𝜓 of the gravity field inside the continuum region 𝜏. According to the equivalent generalized differential formulations (50) and (59) of the first law of thermodynamics and to the evolution equation (73) for the total mechanical energy (𝐾_𝜏+𝝅_𝜏) of the deformed finite individual macroscopic continuum region 𝜏, the energy power of the gravitational field may produce the fractures in the continuum region 𝜏. We shall consider this aspect in Section 3.

The generalized differential formulation (59) of the first law of thermodynamics and the expression (74) for the gravitational energy power 𝑊_gr(𝜏) show that the local time increase of the potential 𝜓 of the gravitational field is the gravitational energy mechanism of the gravitational energy supply into the continuum region 𝜏. Really, the local time increase of the potential 𝜓 of the gravitational field inside the continuum region 𝜏 (𝜕𝜓/𝜕𝑡>0) supplies the gravitational energy into the continuum region 𝜏. Consequently, according to the generalized differential formulation (59) and to the evolution equation (73), the total energy (𝐾_𝜏+𝑈_𝜏+𝝅_𝜏) of the continuum region 𝜏 and the total mechanical energy (𝐾_𝜏+𝝅_𝜏) of the continuum region 𝜏 are increased if 𝜕𝜓/𝜕𝑡>0.

According to the generalized differential formulation (59) of the first law of thermodynamics and to the evolution equation (73), the gravitational energy supply into the continuum region 𝜏 may induce the formation of fractures in the continuum region 𝜏 related to the production of earthquake. This conclusion corresponds to the observations [1, 3–5, 9, 10, 13] of the identified anomalous variations of the gravitational field before strong earthquakes.

According to the generalized differential formulation (59) of the first law of thermodynamics and to the evolution equation (73), the supply of energy into the continuum region 𝜏 may occur also by means of the work 𝐴_np,𝜕𝜏=^𝑡_𝑡0𝑑𝑡_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧 (75)done by nonpotential stress forces (pressure, compressible, and viscous forces for Newtonian continuum) acting on the boundary surface 𝜕𝜏 of the continuum region 𝜏 during the time interval (𝑡−𝑡₀).

The considered mechanisms of the energy supply to the Earth’s macroscopic continuum region 𝜏 should result in the irreversible process of the splits formation in the rocks related to the generation of the high-frequency acoustic waves from the focal continuum region 𝜏 before the earthquake. Taking this into account, the sum 𝛿𝐴_𝑐+𝛿𝐴_𝑠 in the expression (45) is interpreted [11, 12] as the energy flux (related to the compressible and viscous forces acting on the boundary surface 𝜕𝜏 of the continuum region 𝜏) [38] 𝛿𝐹_vis,𝑐=𝛿𝐴_𝑐+𝛿𝐴_𝑠 (76)

directed across the boundary 𝜕𝜏 (see Figures 2 and 3) of the continuum region 𝜏.

Figure 2 

The macroscopic continuum region 𝜏 containing two subsystems 𝜏1 and 𝜏2 interacting on the surface 𝐹1(𝜏) of the tangential jump of the continuum velocity.

Figure 3 

The macroscopic continuum region 𝜏 consisting of the subsystems 𝜏int and 𝜏ext interacting on the surface 𝜕𝜏𝑖 of the continuum velocity jump.

The considered mechanisms of the energy supply to the Earth’s macroscopic continuum region 𝜏 should result in the significant increase of the energy flux 𝛿𝐹_vis,𝑐 of the geo-acoustic energy from the focal region 𝜏 before the earthquake. The deduced conclusion is in a good agreement with the results of the detailed experimental studies [43].

3. Generalized Thermohydrogravidynamic Shear-Rotational and Classical Shear and Rotational Models of the Earthquake Focal Region

3.1. The Generalized Thermohydrogravidynamic Shear-Rotational and the Classical Shear (Deformational) Models of the Earthquake Focal Region Based on the Generalized Differential Formulation of the First Law of Thermodynamics

Following the works [11, 12], we shall present the foundation of the generalized thermohydrogravidynamic shear-rotational model of the earthquake focal region based on the generalized differential formulation (59) of the first law of thermodynamics. Using the evolution equation (73) of the total mechanical energy of the subsystem 𝜏 (the macroscopic continuum region 𝜏) of the Earth, we shall show now that the formation of fractures (modeling by the jumps of the continuum velocity on some surfaces) is related to irreversible dissipation of the macroscopic kinetic energy and the corresponding increase of entropy. We consider at the beginning the analysis of formation of the main line flat fracture (associated with the surface 𝐹₁(𝜏) of the continuum velocity jump) inside the macroscopic continuum region 𝜏 (bounded by the closed surface 𝜕𝜏). The macroscopic continuum region 𝜏 may be divided into two subsystems 𝜏₁ and 𝜏₂ by continuing mentally the surface 𝐹₁(𝜏) by means of surface 𝑅₁(𝜏) crossing the surface 𝜕𝜏 of the macroscopic region 𝜏. The surface of the subsystem 𝜏₁ consists of the surface (𝜕𝜏)₁ (which is the part of the surface 𝜕𝜏) and the surfaces 𝐹₁(𝜏) and 𝑅₁(𝜏). The surface of the subsystem 𝜏₂ consists of the surface (𝜕𝜏)₂ (which is the part of the surface 𝜕𝜏) and the surfaces 𝐹₁(𝜏) and 𝑅₁(𝜏).

Using the formulation (73), we have the evolution equations for the total mechanical energy of the macroscopic subsystems 𝜏₁ and 𝜏₂: 𝑑𝐾𝑑𝑡^𝜏1+𝝅_𝜏1=𝑑𝑑𝑡_𝜏112𝐯²=+𝜓𝜌𝑑𝑉_𝜏1+𝑝div𝐯𝑑𝑉_𝜏123𝜂−𝜂_𝑣(div𝐯)²−𝑑𝑉_𝜏12𝜈𝑒_𝑖𝑗²+𝜌𝑑𝑉_(𝜕𝜏)1(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧+_𝐹1(𝜏)𝐯₁𝜏₁⋅𝜻₁⋅𝐓𝑑Σ_𝜻1+_𝑅1(𝜏)𝐯₁𝜏₁⋅𝜻₁⋅𝐓𝑑Σ_𝜻1+_𝜏1𝜕𝜓𝑑𝜕𝑡𝜌𝑑𝑉,𝐾𝑑𝑡^𝜏2+𝝅_𝜏2=𝑑𝑑𝑡_𝜏212𝐯²=+𝜓𝜌𝑑𝑉_𝜏2𝑝div𝐯𝑑𝑉+_𝜏223𝜂−𝜂_𝑣(div𝐯)²−𝑑𝑉_𝜏22𝜈𝑒_𝑖𝑗²+𝜌𝑑𝑉_(𝜕𝜏)2(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧−_𝐹1(𝜏)𝐯₁𝜏₂⋅𝜻₁⋅𝐓𝑑Σ_−𝜻1−_𝑅1(𝜏)𝐯₁𝜏₂⋅𝜻₁⋅𝐓𝑑Σ_−𝜻1+_𝜏2𝜕𝜓𝜕𝑡𝜌𝑑𝑉, (77)where 𝜻₁ is the external unit normal vector of the surface (of the subsystem 𝜏₁) presented by surfaces 𝐹₁(𝜏) and 𝑅₁(𝜏) and −𝜻₁ is the external unit normal vector of the surface (of the subsystem 𝜏₂) presented also by surfaces 𝐹₁(𝜏) and 𝑅₁(𝜏). Adding (77) (by using the equality 𝑑Σ_𝜻1=𝑑Σ_−𝜻1 of the elements of area of surfaces 𝐹₁(𝜏) and 𝑅₁(𝜏)) we get the evolution equation for the total mechanical energy (𝐾_𝜏+𝝅_𝜏)=(𝐾_𝜏1+𝐾_𝜏2+𝝅_𝜏1+𝝅_𝜏2) of the macroscopic region 𝜏 consisting of subsystems 𝜏₁ and 𝜏₂ interacting on the surface 𝐹₁(𝜏) of the tangential jump of the continuum velocity: 𝑑𝐾𝑑𝑡^𝜏+𝝅_𝜏=𝑑𝑑𝑡_𝜏12𝐯²=+𝜓𝜌𝑑𝑉_𝜏+𝑝div𝐯𝑑𝑉_𝜏23𝜂−𝜂_𝑣(div𝐯)²−𝑑𝑉_𝜏2𝜈𝑒_𝑖𝑗²+𝜌𝑑𝑉_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧+_𝐹1(𝜏)𝐯₁𝜏₁−𝐯₁𝜏₂⋅𝜻₁⋅𝐓𝑑Σ_𝜻1+_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉, (78)

where 𝐯₁(𝜏₁) is the vector of the continuum velocity on the surface 𝐹₁(𝜏) in the subsystem 𝜏₁, and 𝐯₁(𝜏₂) is the vector of the continuum velocity on the surface 𝐹₁(𝜏) in the subsystem 𝜏₂.

The evolution equation (78) takes into account the total mechanical energy (𝐾_𝜏+𝝅_𝜏) of the macroscopic region 𝜏 consisting of subsystems 𝜏₁ and 𝜏₂ interacting on the surface 𝐹₁(𝜏) of the tangential jump of the continuum velocity. The first term in the right-hand side (of (78)) describes the evolution of the total mechanical energy of the macroscopic continuum region 𝜏 due to the continuum reversible compressibility; the second and third terms express the dissipation of the macroscopic kinetic energy by means of the irreversible continuum compressibility and the velocity shear. The forms of three primary terms in the right-hand side (of (78)) are related to the considered model of the compressible viscous Newtonian continuum. The fourth, fifth, and sixth terms in the right-hand side (of (78)) are the universal terms for arbitrary model of continuum characterized by symmetrical stress tensor 𝐓. The fourth term expresses the power 𝑊_np,𝜕𝜏=𝛿𝐴_np,𝜕𝜏=𝑑𝑡_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧 (79)

of external (for the continuum region 𝜏) nonpotential stress forces acting on the boundary surface 𝜕𝜏 of the macroscopic continuum region 𝜏. The fifth term expresses the power of external (for the continuum region 𝜏) forces on different sides of the surface 𝐹_𝑖(𝜏) characterized by the velocity jump during the fracture formation. The sixth term in (78) presents the power of the total mechanical energy added (or lost) as the result of the Newtonian nonstationary gravitational energy influence on the macroscopic continuum region 𝜏 related to variations of the potential 𝜓 of the combined gravitational field in the continuum region 𝜏.

Consider (78) for one continuum velocity jump on the nonstationary surface 𝐹₁(𝜏) during the time interval (𝑡,𝑡+Δ𝑡). Taking into account the form of the fifth term on the right-hand side of the evolution equation (78), we obtained [11, 12] the expression for the work 𝛿𝐴_{np,𝐹1(𝜏)} (done during the time interval (𝑡,𝑡+Δ𝑡) by the external (for the continuum region 𝜏) nonpotential stress forces acting on different sides of the velocity jump on the surface 𝐹₁(𝜏)): 𝛿𝐴_{np,𝐹1(𝜏)}=_𝑡^𝑡+Δ𝑡_𝐹1(𝜏)𝐯₁𝜏₁−𝐯₁𝜏₂⋅𝜁₁⋅𝐓𝑑Σ_𝜁1𝑑𝑡, (80) which is reduced to the following expression: 𝛿𝐴_{np,𝐹1(𝜏)}=_{𝐹1(𝜏)𝑡}^𝑡+Δ𝑡𝐯₁𝜏₁−𝐯₁𝜏₂⋅𝜻₁⋅𝐓𝑑𝑡𝑑Σ_𝜻1. (81)

To test the formula (81), let us calculate the energy 𝛿𝐴_np,ΔΣ, which dissipates during formation of the surface dislocation on the small surface ΔΣ during the time interval (𝑡,𝑡+Δ𝑡). Using the theorem about the average value and integrating the internal integral on time, we obtained from relation (81) for 𝐹₁(𝜏)=ΔΣ the following relation [11, 12]: 𝛿𝐴_np,ΔΣ=_ΔΣ𝐰𝜻₁,𝑡+Δ𝑡−𝐰−𝜻₁,𝑡+Δ𝑡⋅𝜻₁⋅𝐓𝑑Σ_𝜻1, (82) where ⟨(𝜻₁⋅𝐓)⟩ is the average value of the stress vector for the element of area 𝑑Σ_𝜻1 of the two-side surface ΔΣ, and 𝐰(𝜻₁,𝑡+Δ𝑡) and 𝐰(−𝜻₁,𝑡+Δ𝑡) are the vectors of the continuum displacement on different sides of the element of area 𝑑Σ_𝜻1 of the two-side surface ΔΣ in the points characterized by normal unit vectors 𝜻₁ and −𝜻₁. Using the obvious expression for “linear” time average ⟨(𝜻₁⋅𝐓)⟩𝜻₁=1⋅𝐓2𝐩𝜻₁,𝑡−𝐩−𝜻₁,𝑡+Δ𝑡 (83) as the arithmetical average of the values of the stress vectors 𝐩 on the different sides from the surface of the jump of the continuum velocity, we obtained [11, 12] the expression for the elementary work of the external nonpotential stress forces on the two-side surface ΔΣ of dislocation: 𝛿𝐴_np,ΔΣ=12_ΔΣ𝐰𝜻₁,𝑡+Δ𝑡−𝐰−𝜻₁⋅𝐩𝜻,𝑡+Δ𝑡₁,𝑡+𝐩−𝜻₁,𝑡+Δ𝑡𝑑Σ_𝜻1. (84) This expression was obtained [44] in the frame of the classical linear approach to formation of surface dislocations in rigid compressible continuum on the small area of surface ΔΣ. It is clear that the assumption (83) is valid only for weak tangential jumps of the continuum displacement. Consequently, we can consider the expression (80) as the natural nonlinear generalization of expression (84) for arbitrary surface 𝐹₁(𝜏) of dislocation and for strong tangential jumps of the continuum displacement on the surface 𝐹₁(𝜏) of dislocation. The work (80) of the external (for the continuum region 𝜏) nonpotential stress forces should be negative. The sufficient energy 𝛿𝐸_{𝑑,𝐹1(𝜏)} needed for formation of the surface 𝐹₁(𝜏) of dislocation is equal to the work of the internal forces in the macroscopic continuum region 𝜏. The energy 𝛿𝐸_{𝑑,𝐹1(𝜏)} should be positive and equal to the expression (80) with the sign “−”: 𝛿𝐸_{𝑑,𝐹1(𝜏)}=−𝛿𝐴_{np,𝐹1(𝜏)}=−_𝑡^𝑡+Δ𝑡_𝐹1(𝜏)𝐯_𝟏𝜏₁−𝐯_𝟏𝜏₂⋅𝜻₁⋅𝐓𝑑Σ_𝜻1𝑑𝑡>0. (85)

The formulae (80), (84), and (85) are obtained (taking into account the generalized differential formulation (50) of the first law of thermodynamics) for the model of continuum characterized by an arbitrary symmetrical stress tensor 𝐓.

The macroscopic internal shear kinetic energy (𝐾_𝑠)_𝜏1 (of the subsystem 𝜏₁), the macroscopic internal rotational kinetic energy (𝐾_𝑟)_𝜏1 (of the subsystem 𝜏₁), and the macroscopic kinetic energy of shear-rotational coupling (𝐾^coup_𝑠,𝑟)_𝜏1 (of the subsystem 𝜏₁) are the significant components of the macroscopic internal shear-rotational kinetic energy (𝐾_𝑠-𝑟)_𝜏1 [11, 12, 24, 31]: 𝐾_{𝑠-𝑟𝜏1}=𝐾_𝑟𝜏1+𝐾_𝑠𝜏1+𝐾^coup_{𝑠,𝑟𝜏1}, (86)

taken into account (along with the classical internal thermal energy 𝑈_𝜏1 of the macroscopic continuum region 𝜏₁, the macroscopic potential energy 𝝅_𝜏1 of the macroscopic continuum region 𝜏₁, and the macroscopic translational kinetic energy (𝐾_𝑡)_𝜏1=(1/2)𝑚_𝜏1(𝐕_𝑐)²_𝜏1 of the continuum region 𝜏₁ (of a mass 𝑚_𝜏1) moving as a whole at speed equal to the speed (𝐕_𝑐)_𝜏1 of the center of mass of the continuum region 𝜏₁) in the generalized differential formulation (50) of the first law of thermodynamics for the macroscopic continuum region 𝜏₁.

The macroscopic internal shear kinetic energy (𝐾_𝑠)_𝜏2 (of the subsystem 𝜏₂), the macroscopic internal rotational kinetic energy (𝐾_𝑟)_𝜏2, and the macroscopic kinetic energy of shear-rotational coupling (𝐾^coup_𝑠,𝑟)_𝜏2 are the significant components of the macroscopic internal shear-rotational kinetic energy (𝐾_𝑠-𝑟)_𝜏2 [11, 12, 24, 31]: 𝐾_{𝑠-𝑟𝜏2}=𝐾_𝑟𝜏2+𝐾_𝑠𝜏2+𝐾^coup_{𝑠,𝑟𝜏2}, (87)

taken into account (along with the classical internal thermal energy 𝑈_𝜏2, the macroscopic potential energy 𝝅_𝜏2, and the macroscopic translational kinetic energy (𝐾_𝑡)_𝜏2=(1/2)𝑚_𝜏2(𝐕_𝑐)²_𝜏2 of the continuum region 𝜏₂ (of a mass 𝑚_𝜏2) moving as a whole at the speed (𝐕_𝑐)_𝜏2 of the center of mass of the continuum region 𝜏₂) in the generalized differential formulation (50) of the first law of thermodynamics for the macroscopic continuum region 𝜏₂.

The macroscopic internal shear kinetic energy (𝐾_𝑠)_𝜏1 (of the subsystem 𝜏₁), the macroscopic internal rotational kinetic energy (𝐾_𝑟)_𝜏1, the macroscopic kinetic energy of shear-rotational coupling (𝐾^coup_𝑠,𝑟)_𝜏1, the macroscopic translational kinetic energy (𝐾_𝑡)_𝜏1=(1/2)𝑚_𝜏1(𝐕_𝑐)²_𝜏1, the macroscopic potential energy 𝝅_𝜏1, the macroscopic internal shear kinetic energy (𝐾_𝑠)_𝜏2 (of the subsystem 𝜏₂), the macroscopic internal rotational kinetic energy (𝐾_𝑟)_𝜏2, the macroscopic kinetic energy of shear-rotational coupling (𝐾^coup_𝑠,𝑟)_𝜏2, the macroscopic translational kinetic energy (𝐾_𝑡)_𝜏2=(1/2)𝑚_𝜏2(𝐕_𝑐)²_𝜏2, and the macroscopic potential energy 𝝅_𝜏2 are the significant energy components taken into account in the presented thermohydrogravidynamic shear-rotational model described by the evolution equation (78) for the total mechanical energy (𝐾_𝜏+𝝅_𝜏) of the macroscopic region 𝜏 consisting of interacting subsystems 𝜏₁ and 𝜏₂.

3.2. The Rotational Model of the Earthquake Focal Region Based on the Generalized Differential Formulation of the First Law of Thermodynamics

Following the works [11, 12], we shall present the foundation of the rotational model [2] of the earthquake focal region for the seismic zone of the Pacific Ring. It was noted [2] that the studies of the dislocation models of the focal regions of strong earthquakes showed the bad correspondence with the model of flat endless dislocation in the uniform continuum [45–47]. The analysis [2] showed that the conditions exist to realize the rotational mechanism related to the rotation of the geoblocks by means of the stress forces related to the Earth rotation in the vicinity of the seismic zone of the Pacific Ring. It was noted [2] that the rotational mechanism can be more real compared to the conventional mechanism related to the formation of the main line flat fracture inside the focal region.

Let us consider the energy thermodynamic analysis of the rotational mechanism [2] of the earthquake focal region, related to formation of the circular continuum velocity jump revealed in the form of circular dislocation after relaxation of the seismic process in the earthquake focal region. The developed and tested (in this section) mathematical formalism of description of the main line flat fracture may be generalized on the closed surfaces of the continuum velocity jumps. Following the rotational model [2] of the earthquake focal region, we consider the separate geoblock 𝜏_int of the seismic zone. If the external influences of the nonstationary gravitational forces (on the geoblock 𝜏_int) and the nonpotential stress forces (on the boundary 𝜕𝜏_𝑖 of the geoblock 𝜏_int) exceed the certain critical value then the geoblock may rotate and slip relative to the surrounding fine plastic layer (subsystem) 𝜏_ext with the tangential continuum velocity jump on the boundary surface 𝜕𝜏_𝑖 of the geoblock 𝜏_int. We assume that fine plastic layer (subsystem) 𝜏_ext is limited by external surface 𝜕𝜏 of the considered thermodynamic system 𝜏 consisting of the macroscopic subsystems 𝜏_int and 𝜏_ext.

Using the evolution equation (73) of the total mechanical energy of the subsystem 𝜏, we obtained [11, 12] the evolution equations for the total mechanical energy of the macroscopic subsystems 𝜏_int and 𝜏_ext: 𝑑𝐾𝑑𝑡^𝜏int+𝝅_𝜏int=𝑑𝑑𝑡_𝜏int12𝐯²=+𝜓𝜌𝑑𝑉_𝜏int𝑝div𝐯𝑑𝑉+_𝜏int23𝜂−𝜂_𝑣(div𝐯)²𝑑𝑉−_𝜏int2𝜈𝑒_𝑖𝑗²+𝜌𝑑𝑉_𝜕𝜏𝑖𝐯_int𝜕𝜏_𝑖⋅(𝐦⋅𝐓)𝑑Σ_𝐦+_𝜏int𝜕𝜓𝑑𝜕𝑡𝜌𝑑𝑉,𝐾𝑑𝑡^𝜏ext+𝝅_𝜏ext=𝑑𝑑𝑡_𝜏ext12𝐯²=+𝜓𝜌𝑑𝑉_𝜏ext+𝑝div𝐯𝑑𝑉_𝜏ext23𝜂−𝜂_𝑣(div𝐯)²−𝑑𝑉_𝜏ext2𝜈𝑒_𝑖𝑗²+𝜌𝑑𝑉_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧−_𝜕𝜏𝑖𝐯_ext𝜕𝜏_𝑖⋅(𝐦⋅𝐓)𝑑Σ_−𝐦+_𝜏ext𝜕𝜓𝜕𝑡𝜌𝑑𝑉, (88)

where 𝐦 is the external unit normal vector of the surface 𝜕𝜏_𝑖 of the subsystem 𝜏_int, −𝐦 is the internal unit normal vector of the surface 𝜕𝜏_𝑖, 𝐧 is the external unit normal vector of the surface 𝜕𝜏, 𝐯_int(𝜕𝜏_𝑖) are the velocities vectors on the inner side of the surface 𝜕𝜏_𝑖 in the subsystem 𝜏_int, and 𝐯_ext(𝜕𝜏_𝑖) are the velocities vectors on the outer side of the surface 𝜕𝜏_𝑖 in the subsystem 𝜏_ext.

Adding the evolution equation (88) and using the condition of equality 𝑑Σ_𝐦=𝑑Σ_−𝐦 of the area elements of the surface 𝜕𝜏_𝑖, we get the evolution equation for the total mechanical energy of the macroscopic continuum region 𝜏 consisting of the subsystems 𝜏_int and 𝜏_ext interacting on the surface 𝜕𝜏_𝑖 of the continuum velocity jump: 𝑑𝐾𝑑𝑡^𝜏+𝝅_𝜏=𝑑𝑑𝑡_𝜏12𝐯²=+𝜓𝜌𝑑𝑉_𝜏+𝑝div𝐯𝑑𝑉_𝜏23𝜂−𝜂_𝑣(div𝐯)²−𝑑𝑉_𝜏2𝜈𝑒_𝑖𝑗²+𝜌𝑑𝑉_𝜕𝜏(𝐯⋅(𝐧⋅𝐓))𝑑Ω_𝐧+_𝜕𝜏𝑖𝐯_int𝜕𝜏_𝑖−𝐯_ext𝜕𝜏_𝑖⋅(𝐦⋅𝐓)𝑑Σ_𝐦+_𝜏𝜕𝜓𝜕𝑡𝜌𝑑𝑉. (89)

Equation (89) is analogous to (78). The energy needed for formation of the continuum velocities jumps (on the surfaces 𝐹₁(𝜏) and 𝜕𝜏_𝑖) is related to the penultimate terms in the right-hand sides of (78) and (89). Similar to expression (85), we have the expression for the sufficient energy 𝛿𝐸_{𝑑,𝜕𝜏𝑖} needed to rotate and slip (for the subsystem 𝜏_int) during the time interval (𝑡,𝑡+Δ𝑡) relative to the surrounding fine plastic layer (subsystem) 𝜏_ext (with the tangential continuum velocity jump (𝐯_int(𝜕𝜏_𝑖)−𝐯_ext(𝜕𝜏_𝑖)) on the boundary surface 𝜕𝜏_𝑖 of the geoblock 𝜏_int): 𝛿𝐸_{𝑑,𝜕𝜏𝑖}=−𝛿𝐴_np,𝜕𝜏i=−_𝑡^𝑡+Δ𝑡_𝜕𝜏𝑖𝐯_int𝜕𝜏_𝑖−𝐯_ext𝜕𝜏_𝑖⋅(𝐦⋅𝐓)𝑑Σ_𝐦𝑑𝑡>0. (90)

Taking into account the information [2] that the critical continuum stresses (required for rotation of the geoblock 𝜏_int weakly coupled with the surrounding plastic layer 𝜏_ext) are less than the critical continuum stresses required to split the mountain rock by forming the main line flat fracture, we concluded [11, 12] that the required energy 𝛿𝐸_{𝑑,𝜕𝜏𝑖} (given by the expression (90)) is less than the required energy 𝛿𝐸_{𝑑,𝐹1(𝜏)} (given by the expression (85)) if the displacements of the rock continuum on different sides of the analyzed different jumps of the continuum displacements (the closed dislocation and the main line flat fracture) have the same order of magnitude and the ratio of the surfaces area of the closed dislocation to the surfaces area of the main line flat fracture does not exceed 10.

This thermodynamic energy consideration showed [11, 12] the preferable realization of the rotational motion of the geoblock 𝜏_int (under condition that exists the surrounding plastic layer around the geoblock 𝜏_int) as compared with formation of the of the main line flat fracture inside the geoblock 𝜏_int. This result explains the rotational motions of the geoblocks in the seismic zone of the Pacific Ring [2] and the vortical structures of the lithosphere of Earth [48] and the lithospheres of the planets [49] of the Solar System.

Using of the generalized differential formulation (50) of the first law of thermodynamics for the macroscopic continuum region 𝜏 of the quasi-uniform medium of the Earth’s crust characterized by practically constant viscosity, we obtained [11, 12] the thermodynamic foundation of the classical deformational (shear) model [1] of the earthquake focal region for the quasi-uniform medium of the Earth’s crust characterized by practically constant viscosity.

4. Fundamentals of the Cosmic Geophysics

4.1. The Energy Gravitational Influences on the Earth of the Planets of the Solar System

4.1.1. The Instantaneous Energy Gravitational Influences on the Earth of the Planets of the Solar System in the Approximation of the Elliptical Orbits of the Planets

We shall consider the movements of the Earth 𝜏₃ and the outer (or inner) planet 𝜏_𝑖 in the ecliptic plane 𝑋𝑍 (see Figure 4) around the Sun 𝜏_0,0 in the approximation of the elliptical orbits of the planets. The mass center 𝐶₃ of the Earth 𝜏₃, the mass center 𝑂 of the Sun, and the mass center 𝐶_𝑖 of the inner (𝑖=1,2) and the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖 are located on the direct coordinate axis 𝑋 at a certain initial time moment 𝑡=0 characterized by the minimal distance between the mass center 𝐶_𝑖 of the inner (𝑖=1,2) and the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖 and the mass center 𝐶₃ of the Earth 𝜏₃. The fixed mass center 𝑂 of the Sun is considered as the right focus of the elliptical orbits of the inner (𝑖=1,2) planet 𝜏_𝑖, the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖, and the Earth 𝜏₃.

Figure 4 

The geometric sketch of circulation of the outer planet 𝜏𝑖 (Mars or Jupiter, Saturn, Uranus, Neptune, and Pluto) and the Earth 𝜏3 around the mass center 𝑂 of the Sun.

We have the following relations: 𝑟_𝑖𝜑_𝑖(𝑡)=𝑝_𝑖1+𝑒_𝑖cos𝜑_𝑖(𝑡),(),𝑟𝑖=1,2,4,5,6,7,8,9₃𝜑₃(𝑡=𝑝)³1+𝑒₃cos𝜑₃(𝑡) (91)(92)

for the distance 𝑟_𝑖(𝜑_𝑖(𝑡)) between the mass center 𝑂 of the Sun and the mass center 𝐶_𝑖 of the inner (𝑖=1,2) or the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖 and for the distance 𝑟₃(𝜑₃(𝑡)) between the mass center 𝑂 of the Sun and the mass center 𝐶₃ of the Earth 𝜏₃. Here 𝑝_𝑖 and 𝑒_𝑖 are the focal parameters and the eccentricity, respectively, of the elliptical orbit of the inner (𝑖=1,2) and the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖 and 𝑝₃ and 𝑒₃ are the focal parameters and the eccentricity, respectively, of the elliptical Earth’s orbit. We have 𝜑_𝑖(0)=0 (𝑖=1,2,3,4,5,6,7,8,9) for the initial time moment 𝑡=0.

We shall consider the gravitational potential 𝜓_3𝑖(𝐶₃,int,ext,𝑡,𝑟₃(𝜑₃(𝑡))) created by the inner (𝑖=1,2) or the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖 in the mass center 𝐶₃ (of the Earth 𝜏₃): 𝜓_3𝑖𝐶₃,int,ext,𝑡,𝑟₃𝜑₃(𝑡)𝑀=−𝛾^𝑖𝑟_3𝑖𝐶₃,𝐶_𝑖,,𝑡 (93) where 𝑟_3𝑖(𝐶₃,𝐶_𝑖,𝑡) is the distance between the mass center 𝐶₃ of the Earth 𝜏₃ and the mass center 𝐶_𝑖 of the inner (𝑖=1,2) or the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖. We find the distance 𝑟_3𝑖(𝐶₃,𝐶_𝑖,𝑡) for the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖 from the following relation: 𝑟_3𝑖𝐶₃,𝐶_𝑖,𝑡²=𝑟₃𝜑₃(𝑡)²+𝑟_𝑖𝜑_𝑖(𝑡)²−𝑟₃𝜑₃(𝑡)𝑟_𝑖𝜑_𝑖(𝑡)cos𝜑₃(𝑡)−𝜑_𝑖(𝑡), (94) which is valid also for the inner planet 𝜏_𝑖 (𝑖=1,2). The relation (93) can be rewritten as follows: 𝜓_3𝑖𝐶₃,int,ext,𝑡,𝑟₃𝜑₃(𝑡)=−𝛾𝑀_𝑖×𝑟₃𝜑₃²+𝑟_𝑖𝜑_𝑖(𝑡)²−2𝑟₃𝜑₃𝑟_𝑖𝜑_𝑖(𝑡)cos𝜑₃−𝜑_𝑖(𝑡)^−1/2. (95)

We obtained [25, 26] the expression for the partial derivative of the gravitational potential (95): 𝜕𝜓𝜕𝑡^3𝑖𝐶₃,int,ext,𝑡,𝑟₃𝜑₃(𝑡)=𝛾𝑀_𝑖𝑟_𝑖𝜑_𝑖(𝑡)𝑑𝜑_𝑖(𝑡)×𝑟𝑑𝑡₃𝜑₃(𝑡)²+𝑟_𝑖𝜑_𝑖(𝑡)²−2𝑟₃𝜑₃(𝑡)𝑟_𝑖𝜑_𝑖(𝑡)cos𝜑₃(𝑡)−𝜑_𝑖(𝑡)^−3/2×𝑟₃𝜑₃(𝑡)sin𝜑_𝑖(𝑡)−𝜑₃(𝑡+𝑒)^𝑖𝑟_𝑖𝜑_𝑖(𝑡)sin𝜑_𝑖(𝑡)1+𝑒_𝑖cos𝜑_𝑖(𝑡)−𝑒_𝑖𝑟₃𝜑₃(𝑡)sin𝜑_𝑖(𝑡)cos𝜑_𝑖(𝑡)−𝜑₃(𝑡)1+𝑒_𝑖cos𝜑_𝑖(𝑡), (96)

where the distances 𝑟_𝑖(𝜑_𝑖(𝑡)) and 𝑟₃(𝜑₃(𝑡)) are given by the relations (91) and (92), respectively.

The expression (96) is reduced to the following expression [12]: 𝜕𝜓𝜕𝑡^3𝑖𝐶₃=𝜕,int𝜓𝜕𝑡^3𝑖𝐶₃=,ext𝛾𝑀_𝑖𝑅_𝑂3𝑅_𝑂𝑖𝜔_𝑖sin𝜔_𝑖−𝜔₃𝑡𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜔_𝑖−𝜔₃𝑡^3/2 (97)

under the following conditions: 𝑒_𝑖=0,  𝑒₃=0, and 𝑑𝜑_𝑖(𝑡)/𝑑𝑡=𝜔_𝑖 corresponding to the circular orbits of the planet 𝜏_𝑖 (𝑖=1,2,4,5,6,7,8,9) and the Earth 𝜏₃.

The first term in the figured brackets of the expression (96) gives the principal contribution to the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐶₃,int,ext,𝑡,𝑟₃(𝜑₃(𝑡))). The expression (96) contains the additional two small terms (vanishing at 𝑒_𝑖→0 and 𝑒₃→0) related to the eccentricities 𝑒_𝑖 and 𝑒₃ of the elliptical orbits of the planet 𝜏_𝑖 (𝑖=1,2,4,5,6,7,8,9) and the Earth 𝜏₃, respectively.

The combined maximal contribution of these additional two terms is of the order 𝑂𝑒_𝑖,𝑒₃𝜕max𝜓𝜕𝑡^3𝑖𝐶₃,int (98) for the inner (𝑖=1,2) planet 𝜏_𝑖 and of the order 𝑂𝑒_𝑖,𝑒₃𝜕max𝜓𝜕𝑡^3𝑖𝐶₃,ext (99)

for the outer (𝑖=4,5,6,7,8,9) planet 𝜏_𝑖. Consequently, the contribution of the first term in the figured brackets of the expression (96) is 𝑂(1/𝑒_𝑖,1/𝑒₃) times larger than the contribution of the additional two new terms related to the eccentricities 𝑒_𝑖 and 𝑒₃. Using the maximal eccentricity 𝑒₁=0.206 of Mercury’s orbit, we have that the contribution of the first term in the figured brackets of the expression (96) is approximately 5 times larger than the contribution of the additional two new terms (in the figured brackets of the expression (96)) related to the eccentricities 𝑒₁=0.206 and 𝑒₃=0.017 of the elliptical orbits of Mercury 𝜏₁ and the Earth 𝜏₃, respectively. We have that the first term (in the figured brackets of the expression (96)) is significantly larger than the contribution of the additional two new terms for the other planets (of the Solar System) having the small eccentricities of the elliptical orbits.

Thus, the obtained [12, 25, 26] evaluation (97) (for the circular orbits of the planets) of the relative maximal energy gravitational influences on the Earth (of the planets of the Solar System) may be considered as the first sound approximation for the evaluation of the relative maximal energy gravitational influences of the inner (𝑖=1,2) and the outer (𝑖=4,5,6,7,8,9) planets on the Earth.

4.1.2. The Evaluation of the Relative Maximal Planetary Instantaneous Energy Gravitational Influences on the Earth in the Approximation of the Circular Orbits of the Planets of the Solar System

Following the monograph [12], we shall consider the movement of the Earth 𝜏₃ and the inner planet 𝜏_𝑖 around the Sun 𝜏_0,0 in the first approximation of the circular orbits of the planets. The planets revolve in the ecliptic plane 𝑋𝑍 (see Figures 5 and 6). The mass center of the Sun is located at the fixed point 𝑂 of the origin of the coordinate system. The mass center 𝐶₃ of the Earth, the mass center 𝑂 of the Sun, and the mass center 𝐶_𝑖 of the inner planet 𝜏_𝑖 are located on the direct coordinate axis 𝑋 at a certain initial time moment 𝑡=0 characterized by the minimal distance between the Earth and the inner planet 𝜏_𝑖 (see Figure 5).

Figure 5 

The initial (𝑡=0) planetary configuration characterized by the opposition of the inner planet 𝜏𝑖 (Mercury or Venus) and the Earth 𝜏3.

Figure 6 

The initial planetary configuration of the outer planet 𝜏𝑖 (Mars or Jupiter, Saturn, Uranus, Neptune, and Pluto) and the Earth 𝜏3.

We have the following expressions for the angles 𝜑_𝑖 and 𝜑₃: 𝜑_𝑖(𝑡)=𝜔_𝑖𝑡=2𝜋𝑇_𝑖𝜑𝑡,₃(𝑡)=𝜔₃𝑡=2𝜋𝑇₃𝑡, (100)

which describe the positions of the mass centers of the planet 𝜏_𝑖 and the Earth 𝜏₃ during the time 𝑡.

We shall consider the gravitational potential 𝜓_3𝑖(𝐶₃,int)𝜓_3𝑖𝐶₃𝑀,int=−𝛾^𝑖𝑑_3𝑖𝐶₃ (101) created by the inner planet 𝜏_𝑖 in the mass center 𝐶₃ of the Earth. We have the distance 𝑑_3𝑖(𝐶₃) between the mass center 𝐶_𝑖 of the inner planet 𝜏_𝑖 and the mass center 𝐶₃ of the Earth (see Figure 7): 𝑑²_3𝑖𝐶₃=𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜔_𝑖𝑡−𝜑₃. (102) The relation (101) can be rewritten as follows [12]: 𝜓_3𝑖𝐶₃,int=−𝛾𝑀_𝑖𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜔_𝑖𝑡−𝜑₃. (103) We derived [12] the expression for the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐶₃,int): 𝜕𝜓𝜕𝑡^3𝑖𝐶₃=,int𝛾𝑀_𝑖𝑅_𝑂3𝑅_𝑂𝑖𝜔_𝑖sin𝜔_𝑖−𝜔₃𝑡𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜔_𝑖−𝜔₃𝑡^3/2, (104)which is reduced to zero for the time moments 𝑡^∗_𝑛(𝑖,3)=(1/2)(𝑇_𝑖𝑇₃/(𝑇₃−𝑇_𝑖))𝑛 (for 𝑖=1,2; 𝑛=0,1,2,…), when the mass centers of the Sun, the Earth, and the inner planet 𝜏_𝑖 are located on the direct line.

Figure 7 

The geometric sketch of circulations of the Earth 𝜏3 and the inner planet 𝜏𝑖 (Mercury or Venus) around the mass center 𝑂 of the Sun.

We have (for configuration of the Earth and the outer planet 𝜏_𝑖 shown on Figure 8) the distance 𝑑_𝑖3(𝐶₃) between the mass center 𝐶_𝑖 of the planet 𝜏_𝑖 and the mass center 𝐶₃ of the Earth: 𝑑²_3𝑖𝐶₃=𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜔₃−𝜔_𝑖𝑡. (105) The gravitational potential created by the outer planet 𝜏_𝑖 at the point 𝐶₃ (for configuration shown on Figure 8) is given by the following expression: 𝜓_3𝑖𝐶₃,ext=−𝛾𝑀_𝑖𝑑_3𝑖𝐶₃=−𝛾𝑀_𝑖𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜑₃−𝜔_𝑖𝑡. (106)

Figure 8 

The geometric sketch of circulation of the outer planet 𝜏𝑖 (Mars or Jupiter, Saturn, Uranus, Neptune, and Pluto) and Earth 𝜏3 around the mass center 𝑂 of the Sun.

We derived [12] the expression of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐶₃,ext) for the expression (106): 𝜕𝜓𝜕𝑡^3𝑖𝐶₃,ext=−𝛾𝑀_𝑖𝑅_𝑂3𝑅_𝑂𝑖𝜔_𝑖sin𝜔₃−𝜔_𝑖𝑡𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜔₃−𝜔_𝑖𝑡^3/2. (107)

We used [25, 26] the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑀(𝐶₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑀(𝐶₃,int) of the gravitational potential 𝜓_3𝑀(𝐶₃,int) created by Mercury at the mass center 𝐶₃ of the Earth) as a scale of the energy gravitational influence of the planets of the Solar System on the Earth in the considered first approximation of the circular orbits of the planets. To evaluate the relative energy gravitational influence of the inner planet 𝜏_𝑖 (Mercury or Venus) at the mass center 𝐶₃ of the Earth, we considered [25, 26] the ratio 𝑓(𝑖,𝐶₃) of the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑖(𝐶₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐶₃,int) of the gravitational potential 𝜓_3𝑖(𝐶₃,int) created by the inner planet 𝜏_𝑖 at the point 𝐶₃) and the maximal positive value max(𝜕𝜓₃₁(𝐶₃,int)/𝜕𝑡) (of the partial derivative 𝜕𝜓₃₁(𝐶₃,int)/𝜕𝑡 of the gravitational potential 𝜓₃₁(𝐶₃,int)≡𝜓_3𝑀(𝐶₃,int) created by Mercury at the mass center 𝐶₃ of the Earth): 𝑓𝑖,𝐶₃=max𝜕𝜓_3𝑖𝐶₃,int/𝜕𝑡max𝜕𝜓_3𝑀𝐶₃,(),int/𝜕𝑡𝑖=1,2. (108)

We obtained [25, 26] from expression (108) the obvious value 𝑓(1,𝐶₃)=1 for Mercury 𝜏₁. Using the formula (108) for Venus (𝑖=2), we calculated [25, 26] the numerical value 𝑓(2,𝐶₃)=37.69807434 for the following numerical values [35]: the mass 𝑀_𝑀=0.06𝑀₃ of Mercury, where 𝑀₃ is the mass of the Earth, the mass 𝑀_𝑉=𝑀₂=0.82𝑀₃ of Venus, the time period 𝑇₃=365.3 days of the Earth’s circulation around the Sun, the time period 𝑇_𝑀=88 days of Mercury’s circulation around the Sun, the time period 𝑇_𝑉=𝑇₂=224.7 days of the Venusian circulation around the Sun, the average radius 𝑅_𝑂𝑀=𝑅_𝑂1=57.85⋅10⁶km of Mercury’s orbit, around the Sun, the average radius 𝑅_𝑂3=149.6⋅10⁶km of the Earth’s orbit, and the average radius 𝑅_𝑂𝑉=𝑅_𝑂2=108.1⋅10⁶km of Venusian orbit around the Sun. The power of the maximal energy gravitational Venusian influence (on the unit mass at the mass center 𝐶₃ of the Earth) is 𝑓(2,𝐶₃)=37.69807434 times larger than the power of the maximal energy gravitational influence of Mercury (on the unit mass at the mass center 𝐶₃ of the Earth).

To evaluate the relative energy gravitational influence on the Earth of the outer planet 𝜏_𝑖 at the mass center 𝐶₃ of the Earth, we considered [25, 26] the ratio 𝑓(𝑖,𝐶₃) (for 𝑖=4,5,6,7,8,9) of the maximal value max(𝜕/𝜕𝑡)𝜓_3𝑖(𝐶₃, ext) of the partial derivative of the gravitational potential 𝜓_3𝑖(𝐶₃, ext) (created by the outer planet 𝜏_𝑖 at the mass center 𝐶₃ of the Earth) and the maximal value max(𝜕/𝜕𝑡)𝜓_3𝑀(𝐶₃,int) of the partial derivative of the gravitational potential 𝜓_3𝑀(𝐷₃,int) (created by Mercury at the mass center 𝐶₃ of the Earth): 𝑓𝑖,𝐶₃=max𝜕𝜓_3𝑖𝐶₃,ext/𝜕𝑡max𝜕𝜓_3𝑀𝐶₃,(),int/𝜕𝑡𝑖=4,5,6,7,8,9. (109)

Using the formula (109), we calculated [25, 26] the following numerical values: 𝑓(4,𝐶₃)=0.67441034 (for Mars 𝜏₄, 𝑖=4), 𝑓(5,𝐶₃)=7.41055774 (for Jupiter 𝜏₅, 𝑖=5), 𝑓(6,𝐶₃)=0.24601009 (for Saturn 𝜏₆, 𝑖=6), 𝑓(7,𝐶₃)=0.00319056 (for Uranus 𝜏₇, 𝑖=7), 𝑓(8,𝐶₃)=0.00077565 (for Neptune 𝜏₈, 𝑖=8), and 𝑓(9,𝐶₃)=    3.4813⋅10⁻⁸ (for Pluto 𝜏₉, 𝑖=9). We used the following additional planetary numerical values [50]: the mass 𝑀_MARS=𝑀₄=0.11𝑀₃ of Mars, the time period 𝑇_MARS=𝑇₄=687 days of Mars’ circulation around the Sun, the average radius 𝑅_𝑂MARS=𝑅_𝑂4=227.7⋅10⁶km of Mars’ orbit, the mass 𝑀_𝐽=𝑀₅=318𝑀₃ of Jupiter, the time period 𝑇_𝐽=𝑇₅=4332 days of Jupiter’s circulation around the Sun, and the average radius 𝑅_𝑂𝐽=𝑅_𝑂5=777.6⋅10⁶km of Jupiter’s orbit, the time period 𝑇_SAT=𝑇₆=10759 days of Saturn’s circulation around the Sun, the mass 𝑀_SAT=𝑀₆=95.2𝑀₃ of Saturn, and the average radius 𝑅_𝑂SAT=𝑅_𝑂6=1426⋅10⁶km of Saturn’s orbit, the mass 𝑀_𝑈=𝑀₇=14.6𝑀₃ of Uranus, the time period 𝑇_𝑈=𝑇₇=30685 days of Uranus’ circulation around the Sun, and the average radius 𝑅_𝑂𝑈=𝑅_𝑂7=2868⋅10⁶km of Uranus’ orbit, the mass 𝑀_𝑁=𝑀₈=17.2𝑀₃ of Neptune, the average radius 𝑅_𝑂𝑁=𝑅_𝑂8=4497⋅10⁶km of Neptune’s orbit, and the time period 𝑇_𝑁=𝑇₈=60189 days of Neptune’s circulation around the Sun, the mass 𝑀_𝑃=𝑀₉=0.002𝑀₃ of Pluto, the time period 𝑇_𝑃=𝑇₉=90465 days of Pluto’s circulation around the Sun, and the average radius 𝑅_𝑂𝑃=𝑅_𝑂9=5900⋅10⁶km of Pluto’s orbit.

Taking into account the calculated powers of the maximal energy gravitational influences of the planets on the unit mass of the Earth (at the mass center 𝐶₃ of the Earth), we obtained [25, 26] the following numerical sequence of the nondimensional relative maximal powers of the planetary energy gravitational influences on the unit mass of the Earth (at the mass center 𝐶₃ of the Earth): 𝑓(2,𝐶₃)=37.69807434 (for Venus), 𝑓(5,𝐶₃)=7.41055774 (for Jupiter), 𝑓(1,𝐶₃)=1 (for Mercury), 𝑓(4,𝐶₃)=0.67441034 (for Mars), 𝑓(6,𝐶₃)=0.24601009 (for Saturn), 𝑓(7,𝐶₃)=0.00319056 (for Uranus), 𝑓(8,𝐶₃)=0.00077565 (for Neptune), and 𝑓(9,𝐶₃)=3.4813⋅10⁻⁸ (for Pluto).

To evaluate the relative energy gravitational influence of the inner planets 𝜏_𝑖 (Mercury and Venus) and the outer planets 𝜏_𝑖 (Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto) at the surface point 𝐷₃ (which is the intersection of the direct line (connecting the mass center 𝑂 of the Sun and the mass center 𝐶₃ of the Earth) with the surface of the Earth), we obtained [12] the gravitational potential 𝜓_3𝑖(𝐷₃,int): 𝜓_3𝑖𝐷₃,int=−𝛾𝑀_𝑖𝑅_𝑂3−𝑅₃²+𝑅²_𝑂𝑖−2𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖cos𝜔_𝑖𝑡−𝜑₃ (110) created by the inner planet 𝜏_𝑖 in the surface point 𝐷₃ of the Earth. We derived the expression for the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐷₃,int) of the gravitational potential (110): 𝜕𝜓𝜕𝑡^3𝑖𝐷₃=,int𝛾𝑀_𝑖𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖𝜔_𝑖sin𝜔_𝑖−𝜔₃𝑡𝑅_𝑂3−𝑅₃²+𝑅²_𝑂𝑖−2𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖cos𝜔_𝑖−𝜔₃𝑡^3/2. (111) The gravitational potential created by the outer planet 𝜏_𝑖 at the surface point 𝐷₃ (for configuration shown on Figure 8) is given by the following expression [12]: 𝜓_3𝑖𝐷₃,ext=−𝛾𝑀_𝑖𝑑_3𝑖𝐷₃=−𝛾𝑀_𝑖𝑅_𝑂3−𝑅₃²+𝑅²_𝑂𝑖−2𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖cos𝜑₃−𝜔_𝑖𝑡. (112)

We derived [12] the expression of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐷₃,ext) of the gravitational potential (112): 𝜕𝜓𝜕𝑡^3𝑖𝐷₃,ext=−𝛾𝑀_𝑖𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖𝜔_𝑖sin𝜔₃−𝜔_𝑖𝑡×𝑅_𝑂3−𝑅₃²+𝑅²_𝑂𝑖−2𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖cos𝜔₃−𝜔_𝑖𝑡^3/2−1, (113)

which is reduced to the relation (111) owing to the equalities sin(𝜔_𝑖−𝜔₃)𝑡=−sin(𝜔₃−𝜔_𝑖)𝑡 and cos(𝜔_𝑖−𝜔₃)𝑡=cos(𝜔₃−𝜔_𝑖)𝑡. However, we take into account that the expression (113) is given for the outer planet 𝜏_𝑖, but the expression (111) is given for the inner planet 𝜏_𝑖.

We used [12, 25, 26] the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) of the gravitational potential 𝜓_3𝑀(𝐷₃,int) created by Mercury at the surface point 𝐷₃ of the Earth) as a scale of the energy gravitational influence of the planets of the Solar System on the Earth in the considered first approximation of the circular orbits of the planets.

To evaluate the relative energy gravitational influence of the inner planet 𝜏_𝑖 (𝑖=1,2) at the surface point 𝐷₃, we considered [12, 25, 26] the ratio 𝑓(𝑖,𝐷₃) of the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑖(𝐷₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐷₃,int) of the gravitational potential 𝜓_3𝑖(𝐷₃,int) created by the inner planet 𝜏_𝑖 at the point 𝐷₃) and the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) of the gravitational potential 𝜓₃₁(𝐷₃,int)≡𝜓_3𝑀(𝐷₃,int) created by Mercury at the surface point 𝐷₃ of the Earth): 𝑓𝑖,𝐷₃=max𝜕𝜓_3𝑖𝐷₃,int/𝜕𝑡max𝜕𝜓_3𝑀𝐷₃,(),int/𝜕𝑡𝑖=1,2. (114)

To evaluate the relative energy gravitational influence of the outer planet 𝜏_𝑖 (𝑖=4,5,6,7,8,9) on the surface point 𝐷₃ of the Earth, we considered [12, 25, 26] the ratio𝑓(𝑖,𝐷₃) (for 𝑖=4,5,6,7,8,9) of the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑖(𝐷₃,ext) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑖(𝐷₃,ext) of the gravitational potential 𝜓_3𝑖(𝐷₃,ext) created by the outer planet 𝜏_𝑖 at the surface point 𝐷₃) and the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) of the gravitational potential 𝜓_3𝑀(𝐷₃,int) created by Mercury at surface point 𝐷₃): 𝑓𝑖,𝐷₃=max𝜕𝜓_3𝑖𝐷₃,ext/𝜕𝑡max𝜕𝜓_3𝑀𝐷₃,(),int/𝜕𝑡𝑖=4,5,6,7,8,9. (115)

Using the formulae (114) and (115), we calculated [25, 26] (for the average radius 𝑅₃=6371km of the Earth and the planetary numerical values [50]) the following numerical values (corrected slightly the previous numerical values of 𝑓(𝑖,𝐷₃) [9]): 𝑓(2,𝐷₃)=37.70428085 (for Venus), 𝑓(5,𝐷₃)=7.40926122 (for Jupiter), 𝑓(1,𝐷₃)=1 (for Mercury), 𝑓(4,𝐷₃)=0.67420160 (for Mars), 𝑓(6,𝐷₃)=0.24596865 (for Saturn), 𝑓(7,𝐷₃)=0.00319004 (for Uranus), 𝑓(8,𝐷₃)=0.00077552 (for Neptune), and 𝑓(9,𝐷₃)=3.4807⋅10⁻⁸ (for Pluto).

Taking into account the calculated powers of the maximal energy gravitational influences of the planets on the unit mass of the Earth (at the mass center 𝐶₃ of the Earth and at the surface point 𝐷₃ of the Earth), we obtained [12, 25, 26] the following order of signification of the planets of the Solar System (Venus, Jupiter, Mercury, Mars, Saturn, Uranus, Neptune, and Pluto) with respect to the planetary power of the maximal energy gravitational influences on the unit mass of the Earth.

To evaluate the relative energy gravitational influence of the inner planet 𝜏_𝑖 at the surface point 𝐷₃ and at the mass center 𝐶₃ of the Earth, we considered [25, 26] the ratio 𝑞_𝑖(𝐷₃,𝐶₃): 𝑞_𝑖𝐷₃,𝐶₃=max𝜕𝜓_3𝑖𝐷₃,int/𝜕𝑡max𝜕𝜓_3𝑖𝐶₃,(),int/𝜕𝑡𝑖=1,2. (116)

To evaluate the relative energy gravitational influence of the outer planet 𝜏_𝑖 at the surface point 𝐷₃ and at the mass center 𝐶₃ of the Earth, we considered [25, 26] the ratio 𝑞_𝑖(𝐷₃,𝐶₃): 𝑞_𝑖𝐷₃,𝐶₃=max𝜕𝜓_3𝑖𝐷₃,ext/𝜕𝑡max𝜕𝜓_3𝑖𝐶₃,(),ext/𝜕𝑡𝑖=4,5,6,7,8,9. (117)

Using the formula (116), we calculated [25, 26] the following numerical values: 𝑞₁(𝐷₃,𝐶₃)=1.000123023 (for Mercury) and 𝑞₂(𝐷₃,𝐶₃)=1.000287771 (for Venus). Using the formula (117), we calculated [25, 26] the following numerical values: 𝑞₄(𝐷₃,𝐶₃)=0.999813318 (for Mars), 𝑞₅(𝐷₃,𝐶₃)=0.999948084 (for Jupiter), 𝑞₆(𝐷₃,𝐶₃)=0.999954640 (for Saturn), 𝑞₇(𝐷₃,𝐶₃)=0.999956727 (for Uranus), 𝑞₈(𝐷₃,𝐶₃)=0.999957084 (for Neptune), and 𝑞₉(𝐷₃,𝐶₃)=0.999957263 (for Pluto).

The revealed [25, 26] small difference of the maximal energy gravitational influence of each planet at the surface point 𝐷₃ and at the mass center 𝐶₃ of the Earth must lead to the small difference of the combined maximal energy gravitational influences of the planets of the Solar System at the points 𝐶₃ and 𝐷₃ of the Earth. It was recognized [25, 26] that the small difference of the combined planetary maximal energy gravitational influences at the surface point 𝐷₃ and at the mass center 𝐶₃ of the Earth can contribute to the following related geophysical phenomena: the small oscillatory motion of the rigid kernel of the Earth relative to the fluid kernel of the Earth; the small oscillation of the Earth’s pole (i.e., the Chandler’s wobble of the Earth’s pole); the small oscillations of the boundary of the Pacific Ocean (i.e., the seismic zone of the Pacific Ring); the oscillations, rotations, and deformations of the geoblocks weakly coupled with the surrounding plastic layers in all seismic zones of the Earth and the formation of fractures related to the strong earthquakes and the planetary cataclysms.

4.1.3. The Evaluation of the Relative Maximal Planetary Integral Energy Gravitational Influences on the Earth in the Approximation of the Circular Orbits of the Planets of the Solar System

We assume that 𝜑₀₃ and 𝜑_0𝑖 are the initial phases of the Earth 𝜏₃ and the planet 𝜏_𝑖, respectively. Consequently, the positions of the center of the Earth 𝜏₃ and the center of the planet 𝜏_𝑖 (inner or outer) for the time moment 𝑡 are given (instead of the relation (100)) by the following expressions: 𝜑_𝑖(𝑡)=𝜔_𝑖𝑡=2𝜋𝑇_𝑖𝑡+𝜑_0𝑖,𝜑₃(𝑡)=𝜔₃𝑡=2𝜋𝑇₃𝑡+𝜑₀₃. (118)

Taking into account the initial phases 𝜑₀₃ and 𝜑_0𝑖, the expressions (104), (107), (111), and (113) can be generalized as follows: 𝜕𝜓𝜕𝑡^3𝑖𝐶₃=𝜕,int𝜓𝜕𝑡^3𝑖𝐶₃=𝜕,ext𝜓𝜕𝑡^3𝑖𝐶₃=𝛾𝑀_𝑖𝑅_𝑂3𝑅_𝑂𝑖𝜔_𝑖sin𝜔_𝑖−𝜔₃𝑡+𝜑_0𝑖−𝜑₀₃𝑅²_𝑂3+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖cos𝜔_𝑖−𝜔₃𝑡+𝜑_0𝑖−𝜑₀₃^3/2,𝜕𝜓𝜕𝑡^3𝑖𝐷₃=𝜕,int𝜓𝜕𝑡^3𝑖𝐷₃=𝜕,ext𝜓𝜕𝑡^3𝑖𝐷₃=𝛾𝑀_𝑖𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖𝜔_𝑖×sin𝜔_𝑖−𝜔₃𝑡+𝜑_0𝑖−𝜑₀₃×𝑅_𝑂3−𝑅₃²+𝑅²_𝑂𝑖−2𝑅_𝑂3−𝑅₃𝑅_𝑂𝑖×cos𝜔_𝑖−𝜔₃𝑡+𝜑_0𝑖−𝜑₀₃^3/2−1. (119)(120)

We obtained [12] the integral energy gravitational influence Δ_𝑔𝐸₃(𝜏_𝑖,𝜑_0𝑖,𝜑₀₃,𝑡,𝑡₀) on the Earth 𝜏₃ owing to the nonstationary instantaneous energy gravitational influence of the planet 𝜏_𝑖 (inner or outer) during the time interval (𝑡₀,𝑡): Δ_𝑔𝐸₃𝜏_𝑖,𝜑_0𝑖,𝜑₀₃,𝑡,𝑡₀=^𝑡_𝑡0𝜏3𝜕𝜓_3𝑖𝐶₃𝜕𝑡^′𝜌𝑑𝑉𝑑𝑡^′≈𝑀₃^𝑡_𝑡0𝜕𝜓_3𝑖𝐶₃𝜕𝑡^′𝑑𝑡^′, (121) where 𝑀₃ is the mass of the Earth. Substituting (119) into (121), we obtained [12] Δ_𝑔𝐸₃𝜏_𝑖,𝜑_0𝑖,𝜑₀₃,𝑡,𝑡₀=𝑀₃^𝑡_𝑡0𝛾𝑀_𝑖𝑅_𝑂3𝑅_𝑂𝑖𝜔_𝑖sin𝜔_𝑖−𝜔₃𝑡^′+𝜑_0𝑖−𝜑₀₃×𝑅_𝑂3²+𝑅²_𝑂𝑖−2𝑅_𝑂3𝑅_𝑂𝑖×cos𝜔_𝑖−𝜔₃𝑡^′+𝜑_0𝑖−𝜑₀₃^3/2−1𝑑𝑡^′. (122) The result of integration of the expression (122) is given by the relation [12]: Δ_𝑔𝐸₃𝜏_𝑖,𝜑_0𝑖,𝜑₀₃,𝑡,𝑡₀=2𝛼_𝑖𝜔_𝑖−𝜔₃𝜒_𝑖1𝛽_𝑖−𝜒_𝑖cos𝜔_𝑖−𝜔₃𝑡₀+𝜑_0𝑖−𝜑₀₃^1/2−1𝛽_𝑖−𝜒_𝑖cos𝜔_𝑖−𝜔₃𝑡+𝜑_0𝑖−𝜑₀₃^1/2, (123) characterized by the following coefficients: 𝛼_𝑖=𝛾𝑀₃𝑀_𝑖𝑅_𝑂3𝑅_𝑂𝑖𝜔_𝑖,𝛽_𝑖=𝑅_𝑂3²+𝑅_𝑂𝑖²,𝜒_𝑖=2𝑅_𝑂3𝑅_𝑂𝑖. (124)

Using the initial phases 𝜑₀₃=0 and 𝜑_0𝑖=0 for the initial time moment 𝑡₀=0, the expression (123) gives the more simple relation [12]: Δ_𝑔𝐸₃𝜏_𝑖=,0,0,𝑡,02𝛼_𝑖𝜔_𝑖−𝜔₃𝜒_𝑖1𝛽_𝑖−𝜒_𝑖^1/2−1𝛽_𝑖−𝜒_𝑖cos𝜔_𝑖−𝜔₃𝑡^1/2 (125) used for calculation of the maximal integral energy gravitational influence of the planet 𝜏_𝑖 (inner or outer) on the Earth 𝜏₃. Consider the expression (125) by taking into account that the mass center 𝐶_𝑖 of the inner planet 𝜏_𝑖, the mass center 𝐶₃ of the Earth 𝜏₃, and the mass center 𝑂 of the Sun are located on the axis 𝑋 for the initial time moment 𝑡₀=0 as it is shown on Figure 6. Considering the time duration 𝑡^∗₁()=1𝑖,32𝑇_𝑖𝑇₃𝑇₃−𝑇_𝑖,(𝑖=1,2) (126) we obtain that the mass center 𝐶_𝑖 of the inner planet 𝜏_𝑖, the mass center 𝐶₃ of the Earth 𝜏₃, and the mass center 𝑂 of the Sun will be localized again on the same straight line and the distance between the mass centers of the inner planet 𝜏_𝑖, and the Earth 𝜏₃ will be maximal. We have from relation (125) the positive integral energy gravitational influence on the Earth’s continuum during the time 𝑡^∗₁(𝑖,3): Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡^∗₁()=𝑖,3,02𝛼_𝑖𝜔_𝑖−𝜔₃𝜒_𝑖1𝑅_𝑂3−𝑅_𝑂𝑖−1𝑅_𝑂3+𝑅_𝑂𝑖>0. (127) We obtained [12] from relation (125) that the integral energy gravitational influence on the Earth of the inner planet 𝜏_𝑖 is equal to the zero Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡^∗₂(𝑖,3),0=0 (128) during the time duration 𝑡^∗₂()=𝑇𝑖,3^𝑖𝑇₃𝑇₃−𝑇_𝑖,(𝑖=1,2) (129)

when the distance between the mass centers of the inner planet 𝜏_𝑖 and the Earth 𝜏₃ will be minimal.

We tested [12] the relation (125) for the outer planet 𝜏_𝑖 and the Earth 𝜏₃ assuming that the mass centers of the outer planet 𝜏_𝑖 and the Earth 𝜏₃ and the mass center 𝑂 of the Sun are located on the axis 𝑋 for the initial time moment 𝑡₀=0 as it is shown on Figure 6. Considering the time duration 𝑡^∗₁()=13,𝑖2𝑇_𝑖𝑇₃𝑇_𝑖−𝑇₃,(𝑖=4,5,6,7,8,9) (130) we have that the mass center of the outer planet 𝜏_𝑖, the mass center 𝐶₃ of the Earth 𝜏₃, and the mass center 𝑂 of the Sun will again be located on the same straight line, and the distance between the mass centers of the outer planet 𝜏_𝑖 and the Earth 𝜏₃ will be maximal. We obtained [12] from relation (125) the negative integral energy gravitational influence on the Earth of outer planet 𝜏_𝑖 during the time 𝑡^∗₁(3,𝑖): Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡^∗₁()=3,𝑖,02𝛼_𝑖𝜔_𝑖−𝜔₃𝜒_𝑖1𝑅_𝑂𝑖−𝑅_𝑂3−1𝑅_𝑂𝑖+𝑅_𝑂3<0. (131)

Considering the time duration 𝑡^∗₂()=𝑇3,𝑖^𝑖𝑇₃𝑇_𝑖−𝑇₃, (132) we obtained [12] from relation (125) the zero integral energy gravitational influence on the Earth from the outer planet 𝜏_𝑖 during the time 𝑡^∗₂(3,𝑖): Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡^∗₂(3,𝑖),0=0. (133)

Using the expression (124), we established [12] that the expressions (127) and (131) give the following extreme values: max_𝑡Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡,0=Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡^∗₁(𝑖,3),0=2𝛾𝑀₃𝑀_𝑖𝑅_𝑂𝑖𝑇₃𝑅²_𝑂3−𝑅²_𝑂𝑖𝑇₃−𝑇_𝑖(),>0,𝑖=1,2min_𝑡Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡,0=Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡^∗₁(3,𝑖),0=2𝛾𝑀₃𝑀_𝑖𝑅_𝑂3𝑇₃𝑅²_𝑂𝑖−𝑅²_𝑂3𝑇₃−𝑇_𝑖()<0,𝑖=4,5,6,7,8,9 (134)(135)

of the integral energy gravitational influences (respectively, the positive maximal integral energy gravitational influence from the inner planet 𝜏_𝑖 and the negative minimal integral energy gravitational influence from the outer planet 𝜏_𝑖) for the given initial phases 𝜑₀₃=0 and 𝜑_0𝑖=0 (for the initial time moment 𝑡₀=0) corresponding to the initial configurations shown on Figures 5 and 6.

Using relation (134), we obtained [12] for Mercury (𝑖=1) and for Venus (𝑖=2) the following expressions of the maximal positive integral energy gravitational influences on the Earth: max_𝑡Δ_𝑔𝐸₃𝜏₁,0,0,𝑡,0=Δ_𝑔𝐸₃𝜏₁,0,0,𝑡^∗₁(1,3),0=2𝛾𝑀₃𝑀₁𝑅_𝑂1𝑇₃𝑅²_𝑂3−𝑅²_𝑂1𝑇₃−𝑇₁>0,𝑖=1,max_𝑡Δ_𝑔𝐸₃𝜏₂,0,0,𝑡,0=Δ_𝑔𝐸₃𝜏₂,0,0,𝑡^∗₁(2,3),0=2𝛾𝑀₃𝑀₂𝑅_𝑂2𝑇₃𝑅²_𝑂3−𝑅²_𝑂2𝑇₃−𝑇₂>0,𝑖=2. (136)(137)

Considering the mass 𝑚_𝜏 of the macroscopic continuum region near the surface point 𝐷₃ (instead of the mass 𝑀₃ of the Earth) in relations (136) and (137), we obtained [12] the following expressions for the positive integral energy gravitational influences of Mercury (𝑖=1) and Venus (𝑖=2) on the macroscopic continuum region of the mass 𝑚_𝜏 near the surface point 𝐷₃ of the Earth: max_𝑡Δ_𝑔𝐸₃𝜏₁,𝐷₃,𝑚_𝜏,0,0,𝑡,0=Δ_𝑔𝐸₃𝜏₁,𝐷₃,𝑚_𝜏,0,0,𝑡^∗₁(1,3),0=2𝛾𝑚_𝜏𝑀₁𝑅_𝑂1𝑇₃𝑅²_𝑂3−𝑅²_𝑂1𝑇₃−𝑇₁>0,𝑖=1,max_𝑡Δ_𝑔𝐸₃𝜏₂,𝐷₃,𝑚_𝜏,0,0,𝑡,0=Δ_𝑔𝐸₃𝜏₂,𝐷₃,𝑚_𝜏,0,0,𝑡^∗₁(2,3),0=2𝛾𝑚_𝜏𝑀₂𝑅_𝑂2𝑇₃𝑅²_𝑂3−𝑅²_𝑂2𝑇₃−𝑇₂>0,𝑖=2. (138)(139)

We shall use the expression (136) as a measuring unit for evaluations of the maximal absolute values of the integral energy gravitational influences on the Earth of the planets of the Solar System and the Moon.

Considering the ratio of the extreme value max_𝑡Δ_𝑔𝐸₃(𝜏_𝑖,0,0,𝑡,0) (given by expression (134)) and the maximal positive integral energy gravitational influence max_𝑡Δ_𝑔𝐸₃(𝜏₁,0,0,𝑡,0) (given by the expression (136)) of Mercury on the Earth, we obtained [12] the relative values 𝑠(𝑖) of the maximal integral energy gravitational influences on the Earth of the inner planets: 𝑠(𝑖)=max_𝑡Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡,0max_𝑡Δ_𝑔𝐸₃𝜏₁=𝑀,0,0,𝑡,0_𝑖𝑀₁𝑅_𝑂𝑖𝑅²_𝑂3−𝑅²_𝑂1𝑇₃−𝑇₁𝑅_𝑂1𝑅²_𝑂3−𝑅²_𝑂𝑖𝑇₃−𝑇_𝑖,𝑖=1,2. (140)

We have the obvious value 𝑠(1)=1 for Mercury (𝑖=1). We calculated [12] the value 𝑠(2)=89.6409 for Venus (𝑖=2) based on the planetary numerical values [50] of the average radii of the orbits, the time periods of circulations around the Sun, and the masses of the Earth, Venus, and Mercury.

Since the values given by the expression (135) are negative for the outer planets, we used the absolute (positive) value min_𝑡Δ_𝑔𝐸₃𝜏_𝑖=Δ,0,0,𝑡,0_𝑔𝐸₃𝜏_𝑖,0,0,𝑡^∗₁(3,𝑖),0=2𝛾𝑀₃𝑀_𝑖𝑅_𝑂3𝑇₃𝑅²_𝑂𝑖−𝑅²_𝑂3𝑇_𝑖−𝑇₃>0,𝑖=4,5,6,7,8,9. (141)

Using the expressions (141) and (136), we obtained [12] the relative values 𝑠(𝑖) of the maximal integral energy gravitational influences on the Earth of the outer planets of the Solar System (𝑖=4,5,6,7,8,9): 𝑠(𝑖)=min_𝑡Δ_𝑔𝐸₃𝜏_𝑖,0,0,𝑡,0max_𝑡Δ_𝑔𝐸₃𝜏₁=𝑀,0,0,𝑡,0_𝑖𝑀₁𝑅_𝑂3𝑅²_𝑂3−𝑅²_𝑂1𝑇₃−𝑇₁𝑅_𝑂1𝑅²_𝑂𝑖−𝑅²_𝑂3𝑇_𝑖−𝑇₃,𝑖=4,5,6,7,8,9. (142)

Based on the planetary numerical values [50], we calculated the following numerical values [9]: 𝑠(4)=2.6396 for Mars (𝑖=4), 𝑠(5)=31.319 for Jupiter (𝑖=5), 𝑠(6)=1.036 for Saturn (𝑖=6), 𝑠(7)=0.0133 for Uranus (𝑖=7), 𝑠(8)=0.003229 for Neptune (𝑖=8), and 𝑠(9)=1.4495⋅10⁻⁷ for Pluto (𝑖=9).

Taking into account the calculated relative values 𝑠(𝑖), we obtained [9] the following order of signification of the planets of the Solar System: Venus (𝑠(2)=89.6409), Jupiter (𝑠(5)=31.319), Mars (𝑠(4)=2.6396), Saturn (𝑠(6)=1.036), Mercury (𝑠(1)=1), Uranus (𝑠(7)=0.0133), Neptune (𝑠(8)=0.003229), and Pluto (𝑠(9)=1.4495⋅10⁻⁷) with respect to the established significance of the planetary maximal integral energy gravitational influences on the Earth.

We established [12] that Venus and Jupiter induce the main maximal integral energy gravitational influences on the Earth. Mars, Saturn, and Mercury induce the combined maximal integral energy gravitational influence on the Earth, which is one order of the magnitude smaller than the maximal integral energy gravitational influence of Jupiter. The maximal integral energy gravitational influences on the Earth of Uranus, Neptune, and Pluto are two, three, and seven orders of the magnitude, respectively, smaller than the maximal integral energy gravitational influence of Mercury.

4.2. The Energy Gravitational Influences on the Earth of the Moon

4.2.1. The Evaluation of the Relative Maximal Instantaneous Energy Gravitational Influence of the Moon on the Earth in the Second Approximation of the Elliptical Orbits of the Earth and the Moon around the Combined Mass Center 𝐶_3,MOON of the Earth and the Moon

We shall consider the movement of the Earth 𝜏₃ and the Moon along the elliptical orbits around the combined mass center 𝐶_3,MOON of the Earth and the Moon. We have the following relations (see Figure 9): 𝑟_MOON𝜑_MOON(𝑡)=𝑝_MOON1+𝑒_MOONcos𝜑_MOON(𝑡),𝑟_𝐸𝜑_𝐸(𝑡)=𝑟_𝐸𝜑_MOON(𝑡)=𝑝+𝜋_𝐸1−𝑒_𝐸cos𝜑_MOON(𝑡)=𝑝+𝜋_𝐸1+𝑒_𝐸cos𝜑_MOON(𝑡) (143)(144) for the distance 𝑟_MOON(𝜑_MOON(𝑡)) between the combined mass center 𝐶_3,MOON and the mass center 𝐶_MOON of the Moon and for the distance 𝑟_𝐸(𝜑_𝐸(𝑡)) between the combined mass center 𝐶_3,MOON and the mass center 𝐶₃ of the Earth 𝜏₃. We have the focal parameter 𝑝_MOON and the eccentricity 𝑒_MOON of the elliptical orbit of the Moon. 𝑝_𝐸 and 𝑒_𝐸 (𝑒_𝐸=𝑒_MOON) are the focal parameter and the eccentricity, respectively, of the elliptical Earth’s orbit around the combined mass center 𝐶_3,MOON. We have the expression for the distance 𝑑_3𝑀(𝐸_𝑀)𝑑_3𝑀𝐸_𝑀=𝑝_MOON1+𝑒_MOONcos𝜑_MOON(𝑡)+𝑝_𝐸1+𝑒_𝐸cos𝜑_MOON(𝑡)−𝑅₃ (145) between the mass center 𝐶_MOON of the Moon and the point 𝐸_𝑀, which is the intersection of the direct line (connecting the mass center 𝐶_MOON of the Moon and the mass center 𝐶₃ of the Earth 𝜏₃) with the surface of the Earth 𝜏₃. We have the expression for the gravitational potential 𝜓_3,MOON(𝐸_𝑀) created by the Moon at the point 𝐸_𝑀 of the Earth 𝜏₃: 𝜓_3,MOON𝐸_𝑀𝑀=−𝛾^MOON𝑑_3𝑀𝐸_𝑀. (146)

Figure 9 

The geometric sketch of circulation of the mass center 𝐶3 of the Earth 𝜏3 and the mass center 𝐶MOON of the Moon around the combined mass center 𝐶3,MOON of the Earth-Moon system.

We obtained [25, 26] from the relation (146) the expression for the partial derivative (𝜕/𝜕𝑡)𝜓_3,MOON(𝐸_𝑀) of the gravitational potential 𝜓_3,MOON(𝐸_𝑀) created by the Moon at the point 𝐸_𝑀 of the Earth 𝜏₃: 𝜕𝜓𝜕𝑡^3,MOON𝐸_𝑀=𝛾𝑀_MOON𝑒_MOON𝑝_MOONsin𝜑_MOON(𝑡)𝑝_MOON+𝑝_𝐸−𝑅₃1−𝑒_MOONcos𝜑_MOON(𝑡)²×𝑑𝜑_MOON(𝑡).𝑑𝑡 (147)

We have the expression for the distance 𝑑_3𝑀(𝐶₃)𝑑_3𝑀𝐶₃=𝑝_MOON1+𝑒_MOONcos𝜑_MOON(𝑡)+𝑝_𝐸1+𝑒_𝐸cos𝜑_MOON(t) (148) between the mass center 𝐶_MOON of the Moon and the mass center 𝐶₃ of the Earth 𝜏₃. We have the expression for the gravitational potential 𝜓_3,MOON(𝐶₃) created by the Moon at the mass center 𝐶₃  of the Earth 𝜏₃: 𝜓_3,MOON𝐶₃𝑀=−𝛾^MOON𝑑_3𝑀𝐶₃. (149) We obtained [25, 26] from the relation (149) the expression for the partial derivative: 𝜕𝜓𝜕𝑡^3,MOON𝐶₃=𝛾𝑀_MOON𝑒_MOON𝑝_MOONsin𝜑_MOON(𝑡)𝑝_MOON+𝑝_𝐸²×𝑑𝜑_MOON(𝑡).𝑑𝑡 (150)

We can obtain the focal parameter 𝑝_MOON of the elliptical orbit of the Moon in terms of the average distance 𝑅_3𝑀 (between the mass centers of the Earth and the Moon) and the eccentricity 𝑒_MOON=𝑒_𝐸 of the elliptical orbits of the Moon and the Earth around the combined mass center 𝐶_3,MOON of the Earth and the Moon. Using the relation (143), we have the relation for the large semiaxis 𝑎_MOON of the elliptical orbit of the Moon: 2𝑎_MOON=𝑟_MOON(0)+𝑟_MOON(𝜋)=𝑝_MOON1+𝑒_MOON+𝑝_MOON1−𝑒_MOON=2𝑝_MOON1−𝑒²_MOON, (151) which gives the relation for the focal parameter 𝑝_MOON of the elliptical orbit of the Moon: 𝑝_MOON=1−𝑒²_MOON𝑎_MOON. (152) We can obtain the large semiaxis 𝑎_MOON of the elliptical orbit of the Moon in terms of the average distance 𝑅_3𝑀 (between the mass centers of the Earth and the Moon) and the eccentricity 𝑒_MOON of the elliptical orbit of the Moon. Defining the average distance 𝑅_3𝑀 (between the mass centers of the Earth and the Moon) as the average arithmetic value of the large semiaxis 𝑎_MOON and the small semiaxis 𝑏_MOON of the elliptical orbit of the Moon 𝑅_3𝑀=𝑎_MOON+𝑏_MOON2 (153) and using the definition of the eccentricity 𝑒_MOON of the elliptical orbit of the Moon 𝑒_MOON=𝑐_MOON𝑎_MOON=𝑎²_MOON−𝑏²_MOON𝑎_MOON, (154) we obtained [25, 26] the relation for the average distance 𝑅_3𝑀: 𝑅_3𝑀=𝑎_MOON1+1−𝑒²_MOON2, (155) which leads to the relation for the large semiaxis 𝑎_MOON: 𝑎_MOON=2𝑅_3𝑀1+1−𝑒²_MOON. (156)

Using the relations (152) and (156), we obtained [25, 26] the relation for the focal parameter 𝑝_MOON of the elliptical orbit of the of the Moon: 𝑝_MOON=2𝑅_3𝑀1−𝑒²_MOON1+1−𝑒²_MOON. (157) Using the relation (157) and the relations 𝑝_MOON=𝑝_MOON,𝐸𝑀₃𝑀₃+𝑀_MOON,𝑝_𝐸=𝑝_MOON,𝐸𝑀_MOON𝑀₃+𝑀_MOON, (158) the relation (150) can be rewritten as follows [25, 26]: 𝜕𝜓𝜕𝑡^3,MOON𝐶₃=𝛾𝑀_MOON𝑒_MOON1+1−𝑒²_MOONsin𝜑_MOON(𝑡)2𝑅_3𝑀1−𝑒²_MOON𝑀_MOON𝑀₃+1²×𝑑𝜑_MOON(𝑡).𝑑𝑡 (159)

Taking sin(𝜑_MOON(𝑡))=1 and equating evaluating 𝑑𝜑_MOON(𝑡)/𝑑𝑡 to 𝜔_MOON (for the corresponding hypothetical circular orbits of the Earth and the Moon), we obtained [25, 26] the characteristic maximal positive value of the partial derivative (𝜕/𝜕𝑡)𝜓_3,MOON(𝐶₃): 𝜕char.max.pos.𝜓𝜕𝑡^3,MOON𝐶₃=,secondapproximation𝛾𝑀_MOON𝑒_MOON1+1−𝑒²_MOON𝜔_MOON2𝑅_3𝑀1−𝑒²_MOON𝑀_MOON/𝑀₃+1². (160)

To evaluate (in the second approximation) the relative power of the energy gravitational influence of the Moon on the Earth, we obtained the ratio 𝑓_MOONM(𝐶₃,secondapproximation) of the maximal characteristic maximal positive value char.max.pos.(𝜕/𝜕𝑡)𝜓_3,MOON(𝐶₃,secondapproximation) and the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑀(𝐶₃,int): 𝑓_MOONM𝐶₃=()𝜓,secondapproximationchar.max.pos.𝜕/𝜕𝑡_3,MOON𝐶₃,secondapproximationmax(𝜕/𝜕𝑡)𝜓_3𝑀𝐶₃.,int (161)

The corresponding numerical value 𝑓_MOONM(𝐶₃,secondapproximation)=19.44083 was calculated [25, 26] by taking into account the following numerical values: 𝑒_MOON=0.05, 𝑀_MOON=𝑀₃/81,  𝑀_𝑀=0.06𝑀₃, 𝑇_MOON=29.5306 days, and 𝑇_𝑀=88 days.

The calculated numerical value 𝑓_MOONM(𝐶₃,secondapproximation) is 19.44083 times larger than the maximal power of the energy gravitational influence (on the unit mass at the mass center 𝐶₃ of the Earth) of Mercury moving around the mass center 𝑂 of the Sun along the hypothetical circular orbit.

Taking into account the calculated [25, 26] nondimensional maximal instantaneous energy gravitational influences on the unit mass of the Earth at the mass center 𝐶₃ of the Earth: 𝑓(2,𝐶₃)=37.69807434 (for Venus), 𝑓_MOONM(𝐶₃,secondapproximation)=19.44083404 (for the Moon), 𝑓(5,𝐶₃)=7.41055774 (for Jupiter), 𝑓(1,𝐶₃)=1 (for Mercury), 𝑓(4,𝐶₃)=0.67441034 (for Mars), 𝑓(6,𝐶₃)=0.24601009 (for Saturn), 𝑓(7,𝐶₃)=0.00319056 (for Uranus), 𝑓(8,𝐶₃)=0.00077565 (for Neptune), and 𝑓(9,𝐶₃)=3.4813⋅10⁻⁸ (for Pluto), we have the following order of significance (in the frame of the considered second approximation) of the Moon and the planets of the Solar System: Venus, Moon, Jupiter, Mercury, Mars, Saturn, Uranus, Neptune, and Pluto. The obtained numerical sequence reveals the main instantaneous energy gravitational influences on the Earth of Venus, Moon, Jupiter, Mercury, and Mars, which determine (in collection) the main combined instantaneous energy gravitational influence on the Earth (not taking into account the instantaneous energy gravitational influences of the Sun and our Galaxy).

4.2.2. The Evaluation of the Maximal Integral Energy Gravitational Influence of the Moon on the Earth in the Approximation of the Elliptical Orbits of the Earth and the Moon around the Combined Mass Center 𝐶_3,MOON of the Earth and the Moon

We evaluated [25, 26] the maximal integral energy gravitational influence of the Moon on the Earth in the approximation of the elliptical orbits of the Earth and the Moon around the combined mass center 𝐶_3,MOON of the Earth and the Moon. We have the integral energy gravitational influence of the Moon on the Earth 𝜏₃ (characterized by the mass 𝑀₃) during the time interval (𝑡₀,𝑡): Δ_𝑔𝐸₃Moon,𝜑_MOON𝑡₀,𝑡,𝑡₀=^𝑡_𝑡0𝜏3𝜕𝜓_3,MOON𝐶₃𝜕𝑡^′𝜌𝑑𝑉𝑑𝑡^′≈𝑀₃^𝑡_𝑡0𝜕𝜓_3,MOON𝐶₃𝜕𝑡^′𝑑𝑡^′. (162) Substituting the expression (159) into (162) and integrating, we obtained [25, 26] the following analytical relation: Δ_𝑔𝐸₃Moon,𝜑_MOON𝑡₀,𝑡,𝑡₀=−𝛾𝑀₃𝑀_MOON𝑒_MOON1+1−𝑒²_MOON×cos𝜑_MOON(𝑡)−cos𝜑_MOON𝑡₀×2𝑅_3𝑀1−𝑒²_MOON𝑀_MOON𝑀₃+1²⁻¹. (163)

Considering the following phases: 𝜑_MOON(𝑡)=𝜋 and 𝜑_MOON(𝑡₀)=0, we have from relation (163) the maximal positive value of the integral energy gravitational influence of the Moon on the Earth: max_𝑡Δ_𝑔𝐸₃Moon,𝜑_MOON𝑡₀,𝑡,𝑡₀=𝛾𝑀₃𝑀_MOON𝑒_MOON1+1−𝑒²_MOON𝑅_3𝑀1−𝑒²_MOON𝑀_MOON𝑀₃+1². (164)

To evaluate the maximal integral energy gravitational influence of the Moon on the Earth, we consider the ratio 𝑠(Moon,secondapproximation) of the maximal positive value (164) and the maximal positive value (136) (of the integral energy gravitational influence of Mercury on the Earth): 𝑠=Moon,secondapproximationmax_𝑡Δ_𝑔𝐸₃Moon,𝜑_MOON𝑡₀,𝑡,𝑡₀max_𝑡Δ_𝑔𝐸₃𝜏₁=𝑒,0,0,𝑡,0_MOON1+1−𝑒²_MOON𝑀_MOON×𝑅²_𝑂3−𝑅²_𝑂1𝑇₃−𝑇₁×1−𝑒²_MOON𝑀_MOON𝑀₃+1²𝑀₁𝑅_3𝑀𝑅_𝑂1𝑇₃⁻¹. (165)

Using the relation (165), we calculated [25, 26] the numerical value 𝑠(Moon,secondapproximation)=13.0693 for the following numerical values: the eccentricity 𝑒_MOON=0.05 of the elliptical orbits of the Moon and the Earth around the combined mass center 𝐶_3,MOON, the mass 𝑀₁=𝑀_𝑀=0.06𝑀₃ of Mercury, the mass 𝑀_MOON=𝑀₃/81 of the Moon, the average distance 𝑅_3𝑀=384400km between the mass centers of the Earth and the Moon, the time period 𝑇₃=365.3 days of the Earth’s circulation around the Sun, the time period 𝑇₁=𝑇_𝑀=88 days of Mercury’s circulation around the Sun, the average radius 𝑅_OM=𝑅_𝑂1=57.85⋅10⁶km of Mercury’s orbit around the Sun, and the average radius 𝑅_𝑂3=149.6⋅10⁶km of the Earth’s orbit around the Sun.

Thus, considering the aspect of the cosmic gravitational preparation of the strong earthquakes, we have demonstrated the Venusian (𝑠(2)=89.6409) and Jupiter’s (𝑠(5)=31.319) energy gravitational predominance [12] in supplying of the cosmic planetary gravitational energy to the focal region of the preparing earthquakes. We have demonstrated the significant maximal integral energy gravitational influence of the Moon (𝑠(Moon,secondapproximation)=13.0693) [25, 26] on the Earth. Venus, Jupiter, and the Moon induce the main combined (planetary and lunar) integral energy gravitational influence on the Earth. The combined maximal integral energy gravitational influence on the Earth of Mars (𝑠(4)=2.6396), Saturn (𝑠(6)=1.036), and Mercury (𝑠(1)=1) is one order of the magnitude smaller than the maximal integral energy gravitational influence of Venus. The combined maximal integral energy gravitational influence on the Earth of Uranus (𝑠(7)=0.0133), Neptune (𝑠(8)=0.003229), and Pluto (𝑠(9)=1.4495⋅10⁻⁷) is two orders of the magnitude smaller (i.e., negligible) than the maximal integral energy gravitational influence of Mercury.

4.3. The Demonstration of the Reality of the Cosmic Energy Gravitational Genesis of Preparation and Triggering of Earthquakes

4.3.1. The Demonstration of the Real Cosmic Energy Gravitational Genesis of Preparation of Earthquakes

Using the formula (139), we evaluated [12, 25, 26] the numerical value 𝐸_𝑔(𝜏₂,𝐷₃,𝑚_𝜏) of the integral energy gravitational influence of Venus on the macroscopic continuum region 𝜏 (the focal region of the preparing earthquake) of mass 𝑚_𝜏 near the surface point 𝐷₃ of the Earth during the time 𝑇_𝑔(2)=𝑡^∗₁()=13,22𝑇₂𝑇₃𝑇₂−𝑇₃=291.902days (166) of the energy gravitational influence of Venus on the macroscopic continuum region 𝜏 of the Earth. Using the expression (139) for the maximal positive integral energy gravitational influence 𝐸_𝑔(𝜏₂,𝐷₃,𝑚_𝜏) of Venus (𝑖=2) on the macroscopic continuum region 𝜏 of mass 𝑚_𝜏 near the surface point 𝐷₃ of the Earth, we obtained the obvious estimation for the value 𝐸_𝑔(𝜏₂,𝐷₃,𝑚_𝜏): 𝐸_𝑔𝜏₂,𝐷₃,𝑚_𝜏=2𝛾𝑚_𝜏𝑀₂𝑅_𝑂2𝑇₃𝑅²_𝑂3−𝑅²_𝑂2𝑇₃−𝑇₂=2𝛾𝑙_𝜏³𝜌_𝜏𝑀₂𝑅_𝑂2𝑇₃𝑅²_𝑂3−𝑅²_𝑂2𝑇₃−𝑇₂>0, (167) where the final expression for the estimation 𝐸_𝑔(𝜏₂,𝐷₃,𝑚_𝜏) is given for the focal region of the cubical form characterized by the size 𝑙_𝜏 of the cube. Considering the following numerical values: 𝑙_𝜏=10 km, 𝜌_𝜏=5000 kg/m³ (the average density of the cubical focal region), and 𝛾=6.67⋅10⁻¹¹J⋅m/kg² (the gravitational constant) and using the following known [50] parameters of the Solar System: 𝑇_𝑉=𝑇₂=224.7 days, 𝑅_𝑂𝑉=𝑅_𝑂2=108.1⋅10⁶ km, and 𝑀_𝑉=𝑀₂=0.82𝑀₃, 𝑀₃=6⋅10²⁴ kg, 𝑇₃=365.3 days, we calculated [12, 25, 26] from the expression (167) the numerical value 𝐸_𝑔𝜏₂,𝐷₃,𝑚_𝜏=8.619⋅10¹⁹J, (168) which is close to the change “Δ𝑊≈10²⁰ J” [2] of the rotational kinetic energy of the Earth during the strongest earthquakes. The order of magnitude of the estimation (168) for the value 𝐸_𝑔(𝜏₂,𝐷₃,𝑚_𝜏) is consistent with the earlier estimation of the seismotectonic energy 𝐸_ST [2], which can discharge in the focal region of the strongest earthquakes: 𝐸_ST𝜏₂,𝐷₃,𝑚_𝜏≤𝐸_𝑔𝜏₂,𝐷₃,𝑚_𝜏. (169)

It was pointed out [2] that the coincidence of the values 𝐸_ST and Δ𝑊 is not the casual fact: it is the indication that the strongest earthquake can be considered as the energy quantum corresponding to the regular change of the rotational regime of the Earth. Using the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics applied for the focal region 𝜏 of the preparing earthquake, we found strictly the equality of the orders of the magnitude of the values 𝐸_ST and Δ𝑊 for the strongest earthquakes. Consequently, the coincidence of the orders of the magnitude of the values 𝐸_ST, Δ𝑊, and 𝐸_𝑔(𝜏₂,𝐷₃,𝑚_𝜏) is the indication that the regular changes of the rotational regime of the Earth are related to the regular discharges of the accumulated potential energy (in the different focal regions of earthquakes) supplying of the cosmic gravitational energy influences of the planets of the Solar System, the Sun, and the Moon.

Thus, based on the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics used for the Earth’s macroscopic continuum region 𝜏 (the focal region of the preparing earthquake), we evaluate the reality of the cosmic energy gravitational genesis of the preparation of the earthquakes.

4.3.2. The Demonstration of the Real Cosmic Energy Gravitational Genesis of Triggering of the Preparing Earthquakes

We evaluated [12] the relative (normalized on the maximal integral energy gravitational influence of Mercury 𝜏₁ on the Earth) average integral energy gravitational planetary influences corresponding to the time duration 𝑇_MOON/2 of the maximal integral energy gravitational influence of the Moon on the Earth. We take into account the time durations of the maximal integral energy gravitational influences on the Earth of the inner planets (Mercury 𝜏₁ and Venus 𝜏₂) 𝑇_𝑔(𝑖)=𝑡^∗₁()=1𝑖,32𝑇_𝑖𝑇₃𝑇₃−𝑇_𝑖,(𝑖=1,2), (170) which are the time durations of supplying of the cosmic planetary gravitational energy from the inner planets (𝑖=1,2) to the focal region of the preparing earthquakes. We take into account the time durations of the maximal integral energy gravitational influences on the Earth of the outer planets (Mars 𝜏₄, Jupiter 𝜏₅, Saturn 𝜏₆, Uranus 𝜏₇, Neptune 𝜏₈, and Pluto 𝜏₉) 𝑇_𝑔(𝑖)=𝑡^∗₁()=13,𝑖2𝑇_𝑖𝑇₃𝑇_𝑖−𝑇₃,(𝑖=4,5,6,7,8,9) (171)which are the time durations of supplying of the cosmic planetary gravitational energy from the outer planets (𝑖=4,5,6,7,8,9) to the focal region of the preparing earthquakes.

We defined [12] and calculated the relative values 𝑠(𝑖) (normalized on the maximal integral energy gravitational influence of Mercury (𝑖=1) on the Earth) of the maximal integral energy gravitational influences on the Earth of the planets of the Solar System (𝑖=1,2,4,5,6,7,8,9). We evaluated [12] the relative (normalized on the maximal integral energy gravitational influence of Mercury (𝑖=1) on the Earth) average values 𝑒(𝑖)𝑒(𝑖)=𝑠(𝑖)0.5𝑇_MOON𝑇_𝑔(𝑖),(𝑖=1,2,4,5,6,7,8,9) (172)of the integral energy gravitational influences on the Earth of the planets of the Solar System corresponding to the time duration 𝑇_MOON/2 of the maximal integral energy gravitational influence of the Moon on the Earth.

Using the expression (170) for the time 𝑇_𝑔(𝑖) of supplying of the cosmic planetary gravitational energy from the inner planets (𝑖=1,2), the expression (171) for the time 𝑇_𝑔(𝑖) of supplying of the cosmic planetary gravitational energy from the outer planets (𝑖=4,5,6,7,8,9) and the expression (172) for 𝑒(𝑖), we calculated the following numerical values [12]: 𝑇_𝑔(1)=57.96 days and 𝑒(1)=0.2547 (which is one order of magnitude smaller than 𝑠(Moon,secondapproximation)=13.0693) for Mercury; 𝑇_𝑔(2)=291.902 days and 𝑒(2)=4.5342 (which is smaller than 𝑠(Moon,secondapproximation)=13.0693) for Venus; 𝑇_𝑔(4)=390.0545 days and 𝑒(4)=0.0999 (which is significantly smaller than 𝑠(Moon,secondapproximation)=13.0693) for Mars; 𝑇_𝑔(5)=199.4705 days and 𝑒(5)=2.3182 (which is smaller significantly than 𝑠(Moon,secondapproximation)=13.0693) for Jupiter; 𝑇_𝑔(6)=189.069 days and 𝑒(6)=0.0809 (which is significantly smaller than 𝑠(Moon,secondapproximation)=13.0693) for Saturn; 𝑇_𝑔(7)=184.8506 days and 𝑒(7)=0.001066 (which is four orders of the magnitude smaller than 𝑠(Moon,secondapproximation)=13.0693) for Uranus; 𝑇_𝑔(8)=183.7653 days and 𝑒(8)=0.0002594 (which is four-five orders of the magnitude smaller than 𝑠(Moon,secondapproximation)=13.0693) for Neptune; 𝑇_𝑔(9)=183.3905 days and 𝑒(9)=1.1671⋅10⁻⁸ (which is nine orders of the magnitude smaller than 𝑠(Moon,secondapproximation)=13.0693) for Pluto.

In Section 4.2 we have demonstrated the following order of significance of the planets of the Solar System and the Moon for the cosmic gravitational preparation of the strong earthquakes: Venus (𝑠(2)=89.6409), Jupiter (𝑠(5)=31.319), the Moon (𝑠(Moon,secondapproximation)=13.0693), Mars (𝑠(4)=2.6396), Saturn (𝑠(6)=1.036), Mercury (𝑠(1)=1), Uranus (𝑠(7)=0.0133), Neptune (𝑠(8)=0.003229), and Pluto (𝑠(9)=1.4495⋅10⁻⁷). We have the different order of significance of the planets of the Solar System and the Moon related to the defined relative average values 𝑒(𝑖): the Moon (𝑠(Moon,secondapproximation)=13.0693), Venus (𝑒(2)=4.5342), Jupiter (𝑒(5)=2.3182), Mercury (𝑒(1)=0.2547), Mars (𝑒(4)=0.0999), Saturn (𝑒(6)=0.0809), Uranus (𝑒(7)=0.001066), Neptune (𝑒(8)=0.0002594), and Pluto (𝑒(9)=1.1671⋅10⁻⁸).

Taking into account the obtained [12] numerical values 𝑒(𝑖) for the planets of the Solar System and the numerical value 𝑠(Moon,secondapproximation)=13.0693 for the Moon [25, 26], we can recognize the established [25, 26] predominant significance of the Moon (along with the minor significance of Venus, Jupiter, and Mercury) as the predominant lunar cosmic trigger mechanism of earthquakes preparing by the combined (planetary and lunar) integral energy gravitational influences on the Earth of Venus, Jupiter, the Moon, Mars, and Mercury.

It was suggested in [51] the hypothesis that the Chandler’s wobble of the Earth’s pole can be generated by the motion of the rigid kernel of the Earth induced by the disturbances in the Sun-Earth-Moon system. The presented results of Sections 4.1, 4.2, and 4.3 support the stated [25, 26] conclusion that the related geophysical phenomena (the small oscillatory motion of the rigid kernel of the Earth relative to the fluid kernel of the Earth [51]; the small oscillation of the Earth’s pole (i.e., the Chandler’s wobble of the Earth’s pole); the small oscillations of the boundary of the Pacific Ocean (i.e., the seismic zone of the Pacific Ring [2]); the oscillations [42], rotations, and deformations [1, 2] of the geoblocks weakly coupled with the surrounding plastic layers in all seismic zones of the Earth and the formation of fractures related to the strong earthquakes [1–5, 9–12, 23] and the planetary cataclysms [25, 26]) are induced by the combined nonstationary cosmic energy gravitational influence on the Earth of the planets of the Solar System, the Sun, and the Moon.

4.4. The Cosmic Energy Gravitational Genesis of the Time Periodicities of the Seismotectonic Activity Induced by the Nonstationary Cosmic Energy Gravitational Influences on the Earth of the Sun, the Moon, Venus, Jupiter, and Mars

4.4.1. The Time Periodicities of the Maximal (Instantaneous and Integral) Energy Gravitational Influences on the Earth of the Sun and the Moon, Venus, Jupiter, and Mars

(1)  The Time Periodicities of the Maximal (Instantaneous and Integral) Energy Gravitational Influences on the Earth of the Sun-Moon System. If the configuration of the Sun and the Moon is characterized at any time moment by the maximal combined (instantaneous or integral) energy gravitational influence on the Earth, then the Sun and the Moon will have the recurrence of the same configuration after different integer numbers of circulations (𝑗_MOON,3 circulations of the Moon around the Earth and 𝑚_3,MOON circulations of the Earth around the Sun) to satisfy the following condition [12]: 𝑗_MOON,3𝑇_MOON=𝑚_3,MOON𝑇₃. (173)

Following the known method [52], we present the ratio 𝑇₃/𝑇_MOON by the following mathematical fraction: 𝑇₃𝑇_MOON=365.329.5306=12+1093281295306=12+12+11+12+12+11+8+458.1204 (174)

Considering the different approximation of the ratio 𝑇₃/𝑇_MOON, we obtained [12] the successive approximations for the time periodicities: 𝑇_𝑆-MOON,31𝑇=3years,_𝑆-MOON,32𝑇=8years,_𝑆-MOON,33𝑇=19years,_𝑆-MOON,34𝑇=27years,_𝑆-MOON,35=235years (175) of recurrence of the maximal (instantaneous and integral) combined energy gravitational influences of the Sun and the Moon on the Earth in the first, second, third, fourth, and fifth approximations, respectively. We can verify that the time periodicity 𝑇_𝑆-MOON,33=𝑇_𝑆-MOON,31+𝑇_𝑆-MOON,32=11years (176)

may be considered approximately as the third approximation of recurrence of the maximal (instantaneous and integral) energy gravitational influences of the Sun-Moon system on the Earth.

(2)  The Time Periodicities of the Maximal (Instantaneous and Integral) Energy Gravitational Influences on the Earth of Venus. If the configuration of the Earth and Venus is characterized at any time moment by the maximal (instantaneous or integral) energy gravitational influence on the Earth, then the Earth and Venus will have the recurrence of the same configuration (in the frame of the real elliptical orbits of the Earth and Venus) after 𝑘_𝑉,3 circulations of Venus around the Sun and 𝑚_3,𝑉 circulations of the Earth around the Sun to satisfy the following condition [12]: 𝑘_𝑉,3𝑇_𝑉=𝑚_3,𝑉𝑇₃. (177) Following the known method [52], we present the ratio 𝑇₃/𝑇_𝑉 by the following mathematical fraction: 𝑇₃𝑇_𝑉=365.3224.7=1+140612247=1+11+11+11+12+.280 (178)

Considering the different approximation of the ratio 𝑇₃/𝑇_𝑉, we obtained the successive approximations for the time periodicities [12]: 𝑇_𝑉,31𝑇=3years,_𝑉,32=8years (179) of recurrence of the maximal (instantaneous and integral) energy gravitational influences of Venus on the Earth in the first and second approximations, respectively. We can verify that the time periodicity 𝑇_𝑉,33=𝑇_𝑉,31+𝑇_𝑉,32=11years (180)

may be considered as the third approximation of recurrence of the maximal (instantaneous and integral) energy gravitational influences of Venus on the Earth.

(3)  The Time Periodicities of the Maximal (Instantaneous and Integral) Energy Gravitational Influences on the Earth of Jupiter. If the configuration of the Earth and Jupiter is characterized at any time moment by the maximal (instantaneous or integral) energy gravitational influences on the Earth, then the Earth and Jupiter will have the recurrence of the same configuration (in the frame of the real elliptical orbits of the Earth and Jupiter) after 𝑛_𝐽,3 circulations of Jupiter around the Sun and 𝑚_3,𝐽 circulations of the Earth around the Sun to satisfy the condition [12] 𝑛_𝐽,3𝑇_𝐽=𝑚_3,𝐽𝑇₃. (181) Following the known method [52], we present the ratio 𝑇_𝐽/𝑇₃ by the following mathematical fraction: 𝑇_𝐽𝑇₃=43321365.3=11+11+16+12+24.41 (182)

Considering the different approximation of the ratio 𝑇_𝐽/𝑇₃, we obtained the successive approximations for the time periodicities [12]: 𝑇_𝐽,31𝑇=11years,_𝐽,32𝑇=12years,_𝐽,33=83years (183)(184)(185)

of recurrence of the maximal (instantaneous and integral) energy gravitational influences of Jupiter on the Earth in the first, second, and third approximations, respectively.

(4)  The Time Periodicities of the Maximal (Instantaneous and Integral) Energy Gravitational Influences on the Earth of Mars. If the configuration of the Earth and Mars is characterized at any time moment by the maximal (instantaneous or integral) energy gravitational influence on the Earth, then the Earth and Mars will have the recurrence of the same configuration after 𝑔_MARS,3 circulations of Mars around the Sun and 𝑚_3,MARS circulations of the Earth around the Sun to satisfy the following condition [12]: 𝑔_MARS,3𝑇_MARS=𝑚_3,MARS𝑇₃. (186)

Following the known method [52], we present the ratio 𝑇_MARS/𝑇₃ by the mathematical fraction: 𝑇_MARS𝑇₃=687.0365.3=1+321713653=1+11+17+12+11+1+47.59 (187)

Considering the different approximation of the ratio 𝑇_MARS/𝑇₃, we obtained the successive approximations for the time periodicities [12]: 𝑇_MARS,31𝑇=15years,_MARS,32𝑇=32years,_MARS,33=47years (188) of recurrence of the maximal (instantaneous or integral) energy gravitational influences of Mars on the Earth in the first, second, and third approximations, respectively.

4.4.2. Empirical Time Periodicities of the Seismotectonic Activity of the Earth

It was pointed out [1] that the sinusoidal “saw-like” form of the graphic dependence of growth and recession of the seismotectonic activization of the separate geological structure is related to the following empirical time periodicities: 11years,22years,44years,88years,352years. (189)(190)(191)(192)(193)

It was pointed out [2] that the strongest earthquakes in the all boundary region of the Pacific Ocean are characterized by the established tendency for recurrence on average once during the following time period [53–61]: 𝑇_𝑟=100±50=50÷150years. (194) It was pointed out [2] also that the close values for recurrence of the strongest earthquakes were established for different seismic belts of the Earth: 90÷140 years for the Caucasus [62] and 150 years for the Anatolian fault zone [63]. We present the data of the monograph [2] concerning to the recurrence of the strongest earthquakes in different regions of the seismic zone of the Pacific Ring [1, 64, 65]: 90±40=50÷130years---Kamchatka,130±50=80÷180years---Japan,110±50=60÷160years---Peru,100±50=50÷150years---Aleutians. (195)(196)(197)(198)

The empirical range of the time periodicities [66] was revealed: 250÷300years (199)

for recurrence of the strongest earthquakes in Turkey.

The empirical range of the time periodicities [67] was revealed (as a result of analysis of Chinese earthquakes during the time Δ𝑡=2200 years [2] of long observations): 240÷280years (200)

for recurrence of the strongest earthquakes in China.

Based on the data presented in the monograph [2] concerning the recurrence of the strongest earthquakes in different regions of the Earth, Dr. A. V. Vikulin made the valid conclusion that the seismic processes have the global nature for the Earth. In the next section we shall explain the considered empirical time periodicities by the different combinations of the cosmic energy gravitational influences on the Earth of the Sun-Moon system, Venus, Jupiter, and Mars.

4.4.3. The Time Periodicities of the Periodic Global Seismotectonic (and Volcanic) Activity and the Global Climate Variability of the Earth Induced by the Different Combinations of the Cosmic Energy Gravitational Influences of the Sun-Moon System, Venus, Jupiter, and Mars

We have shown [12] that the periodic recurrence (characterized by the time periodicity 𝑇_energy) of the maximal integral energy gravitational influences on the Earth (defined by the planetary combination of the Sun-Moon system and the arbitrary combination of the planets: Venus, Mars, and Jupiter) leads (according to the generalized differential formulations (50) applied for the Earth) to the periodic recurrence of the maximal tectonic activity (characterized by the same time periodicity 𝑇_tec=𝑇_energy) of the geospheres of the Earth and each geoblock of Earth. We have shown [12] that the periodic recurrence (characterized by the time periodicity 𝑇_tec=𝑇_energy) of the maximal tectonic (and volcanic) activity of the geospheres of the Earth and each geoblock of the Earth (defined by the combination of the Sun-Moon system and the arbitrary combinations of the planets: Venus, Mars, and Jupiter) leads to the periodic recurrence (characterized by the time periodicity 𝑇_tec=𝑇_energy) of the maximal concentration of the atmospheric greenhouse gases owing to the periodic increase (characterized by the time periodicity 𝑇_tec=𝑇_energy) of the output of the greenhouse gases related to the periodic tectonic-volcanic activization of the Earth. We have shown [12] that the periodic increase (characterized by the time periodicity 𝑇_tec=𝑇_energy) of the average planetary concentration of the atmospheric greenhouse gases leads (as a consequence of the greenhouse effect produced by the gravity-induced periodic tectonic-volcanic activization accompanied by increase of the atmospheric greenhouse gases) to the periodic global planetary warming related to the increase (characterized by the time periodicity 𝑇_tec=𝑇_energy) of temperature of the atmosphere-ocean system of the Earth. The periodic decrease (characterized by the time periodicity 𝑇_tec=𝑇_energy) of the average planetary concentration of the atmospheric greenhouse gases leads (as a consequence of the decreased greenhouse effect) to the periodic global planetary cooling [12] related to the fall (characterized by the time periodicity 𝑇_tec=𝑇_energy) of temperature of the atmosphere-ocean system of the Earth. The time periodicity 𝑇_energy of the periodic recurrence of the maximal integral energy gravitational influences on the Earth (defined by the combination of the Sun-Moon system and the arbitrary combinations of the planets: Venus, Mars, and Jupiter) corresponds to different two time periodicities of the Earth’s climate variability [12]: the first time periodicity 𝑇_clim1=𝑇_tec=𝑇_energy (related to the periodic tectonic-volcanic activity of the geospheres of the Earth induced by the cosmic nonstationary combined energy gravitational influences on the Earth of the Sun-Moon system, Venus, Mars, and Jupiter) and the second time periodicity 𝑇_clim2=𝑇_endog=𝑇_energy/2 related to the periodic tectonic-endogenous heating (of the geospheres of the Earth, the atmosphere, and the oceans of the Earth) induced by the periodic continuum deformation (characterized by the time periodicity 𝑇_energy) owing to the periodic cosmic nonstationary combined energy gravitational influences (characterized by the time periodicity 𝑇_energy) on the Earth of the Sun-Moon system, Venus, Mars, and Jupiter.

Using the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics [12] applied for the Earth and the evaluated successive approximations for the time periodicities of the periodic recurrence of the maximal (instantaneous and integral) energy gravitational influences on the Earth of the Sun-Moon system, Venus, Jupiter, and Mars, we founded [12] the sets of the global seismotectonic and volcanic periodicities 𝑇_tec (of the periodic global gravity-induced seismotectonic and volcanic activities) and the global climatic periodicities 𝑇_clim1 (of the periodic global gravity-induced climate variability and the global variability of the quantities of the fresh water and glacial ice resources): 𝑇_tec=𝑇_clim1=𝑇_energy=𝑇_𝑆-MOON,3^𝑙𝑜_𝑖×𝑇_𝑉,3^𝑙2_𝑗×𝑇_MARS,3^𝑙4_𝑘×𝑇_𝐽,3^𝑙5_𝑛, (201)

determined by the multiplications of various successive time periodicities related to the different combinations of the following integer numbers: 𝑖=1,2,3,4,5; 𝑗=1,2; 𝑘=1,2,3; 𝑛=1,2,3; 𝑙_𝑜=0,1; 𝑙₂=0,1; 𝑙₄=0,1; 𝑙₅=0,1. The global seismotectonic and volcanic periodicities 𝑇_tec and the global climatic periodicities 𝑇_clim1 are related to the periodic recurrence of the maximal combined integral energy gravitational influences on the Earth induced by the different combinations of the cosmic nonstationary energy gravitational influences of the Sun-Moon system, Venus, Jupiter, and Mars.

By comparing the combined time periodicities (obtained from the expression (201)) with the empirical time periodicities of the seismotectonic activity of the Earth submitted in Section 4.4.2, we established [12] that the empirical time periodicities (of the seismotectonic activity of the Earth) may be satisfactory approximated by the expression (201) with the different combinations of the various integer numbers.

The calculated time periodicity 24 years (given by 3×8 years determined by the Sun-Moon system and Venus) is close to empirical time periodicity 22 years given by (190). The calculated average time periodicity 34.5 years (of the evaluated time range 33÷36=3×(11÷12) years [12, 25, 26] of the Earth’s periodic seismotectonic and volcanic activity and the global climate variability determined by the combined cosmic nonstationary energy gravitational influence on the Earth of the Sun-Moon system, Venus, and Jupiter) is very close to the empirical time periodicity 35 years [68] of the sesmotectonic activity of various regions of the seismic belts around the Pacific Ocean (the Pacific Ring). The calculated time periodicity 45 years (given by 3×15 years determined by the Sun-Moon system, Venus, and Mars) is close to the empirical time periodicity 44 years given by (191).

The empirical time periodicity 88 years (given by (192)) is equal to the same time periodicity 88 years (given by 8×11 years determined by the Sun-Moon system, Venus, and Jupiter). Since the ratio 88 years/ 𝑇_MARS=46.786 in close to 47, we concluded [25, 26] that the time periodicity 88 years is determined also by Mars.

The calculated time periodicities 88÷96 years (given by 8×(11÷12) years determined by the Sun-Moon system, Venus, and Jupiter) get into the four empirical ranges of the time periodicities: 𝑇_𝑟=100±50 years (given by (194)), 90±40 years (given by (195)), 130±50 years (given by (196)), 110±50 years (given by (197)), and 100±50 years given by (198). The calculated time periodicity 96 years (given by 3×32 years determined by the Sun-Moon system, Venus, and Mars) gets into the four empirical ranges of the time periodicities: 𝑇_𝑟=100±50 years (given by (194)), 90±40 years (given by (195)), 130±50 years (given by (196)), 110±50 years (given by (197)), and 100±50 years given by (198). The calculated range 99÷108 years (given by 3×3×(11÷12) years determined by the Sun-Moon system, Venus, and Jupiter) gets into the empirical ranges of the time periodicities: 𝑇_𝑟=100±50 years given by (194), 90±40 years (given by (195)), 130±50 years (given by (196)), 110±50 years (given by (197)), and 100±50 years given by (198). The calculated time periodicity 120 years (given by 8×15 years determined by the Sun-Moon system, Venus, and Mars) gets into the four empirical ranges of the time periodicities: 𝑇_𝑟=100±50 years (given by (194)), 90±40 years (given by (195)), 130±50 years (given by (196)), 110±50 years (given by (197)), and 100±50 years given by (198). The calculated time periodicity 135 years (given by 3×3×15 years determined by the Sun-Moon system, Venus, and Mars) gets into the empirical range 𝑇_𝑟=100±50years given by (194). The calculated time periodicity 152 years (given by 152=19×8 determined by the Sun-Moon system and Venus) gets into the empirical range 130±50 years (given by (196)) and gets into the empirical range 110±50 years given by (197). The calculated range 165÷180 years (given by 15×(11÷12) years determined by the Sun-Moon system, Mars, and Jupiter) gets into the empirical range 130±50 years given by (196). The calculated time periodicity 249 years (given by 249=3×83 years determined by the Sun-Moon system, Venus, and Jupiter) is near the lower value of the empirical range 250÷300 years (given by (199)) and gets into the empirical range 240÷280years given by (200). The calculated time periodicity 285 years (given by 19×15 years determined by the Sun-Moon system and Mars) is close to the upper value of the empirical range 240÷280 years given by (200). The calculated time periodicity 285 years (given by 19×15 years determined by the Sun-Moon system and Mars) gets into the empirical range 250÷300 years given by (199). The calculated range of the time periodicities 264÷288 years (given by 3×8×(11÷12) years determined by the Sun-Moon system, Venus, and Jupiter) gets approximately into the empirical range 240÷280 years given by (200). The calculated time periodicity 352 years (given by 32×11 years determined by the Sun-Moon system, Mars, and Jupiter) is equal to the empirical time periodicity 352 years given by (193).

4.5. Cosmic Energy Gravitational Genesis of the Chinese 2008 Earthquakes

It was forecasted in advance the single Chinese Haicheng 1975 earthquake using the warning omens. The nearest (to the Haicheng 1975 earthquake) Chinese Luhuo 1973 and Tang Shan 1976 earthquakes were not forecasted using the warning omens. The modern Chinese Bachu 2003, Ruichang 2005, and Yanjin 2006 earthquakes were not forecasted also using the warning omens. Using the results of previous sections and the presented set of the Chinese earthquakes (Luhuo 1973, Haicheng 1975, and Tang Shan 1976; Bachu 2003, Ruichang 2005, and Yanjin 2006), we predicted (Simonenko, 2007) the nearest (to the Yanjin 2006 earthquake) Chinese 2008 earthquakes by considering the following arguments.

The difference equal to 3 years between the times of the Luhuo 1973 and Tang Shan 1976 earthquakes means that the Tang Shan 1976 earthquake may be induced by the integral energy gravitational influence of the Sun-Moon system or Venus (characterized by the same time periodicity 3 years of recurrence of the maximal integral energy gravitational influence on the Earth in the first approximation) or by the combined integral energy gravitational influence on the Earth of the Sun-Moon system and Venus. If the Tang Shan 1976 earthquake was induced by the integral energy gravitational influence of the Sun-Moon system then the Bachu 2003 earthquake can be explained by the recurrence of the integral energy gravitational influence of the Sun-Moon system (characterized by the time periodicity 27 years of recurrence of the maximal integral energy gravitational influence on the Earth in the fourth approximation): 1976+27=2003---Bachu2003earthquake. (202) The Tang Shan 1976 earthquake can be induced by the participation of the integral energy gravitational influence of Venus (characterized by the time periodicity 8 years of recurrence of the maximal integral energy gravitational influence on the Earth in the second approximation) since it is easily to explain the Ruichang 2005 earthquake by the recurrence of the maximal integral energy gravitational influence of Venus on the Earth. Considering [12] 1973+𝑖_𝑉8=2005, (203) which should be solved for integer number 𝑖_𝑉, we obtained [12] the integer number 𝑖_𝑉=4 confirming the Venusian genesis of the Tang Shan 1976 and the Ruichang 2005 earthquakes: 1973+4×8=2005---Ruichang2005earthquake. (204) Taking into account the Chinese Yanjin 2006 earthquake (explained [12] by the sum 1973+3×11) and considering the Venusian genesis of the Tang Shan 1976 earthquake, we obtained the time of the Chinese 2008 earthquake from the following inequality [12]: 0<1976+𝑗_𝑉8−2006<𝑇_𝑉,31=3, (205) which should be solved for integer number 𝑗_𝑉. The obtained solution 𝑗_𝑉=4 founded the time of the several (owing to the participation of Mars characterized by the time periodicity 32 years of recurrence of the maximal integral energy gravitational influence on the Earth in the second approximation) distinct Chinese 2008 earthquakes [12]: 1976+4×8=2008. (206) The occurrence of two large earthquakes in 2008 in the areas surrounding Yutian (Xinjiang) and Wenchuan (Sichuan) confirmed the proposed [12, 25, 26] cosmic energy gravitational genesis of the strong Chinese earthquakes.

4.6. Cosmic Energy Gravitational Genesis of the Strongest Japanese Earthquakes near the Tokyo Region

We have shown [12, 25, 26] the cosmic energy gravitational genesis of the strongest (𝑀≥7.9) Japanese earthquakes near Tokyo region and southwest from Tokyo. To confirm the proposed cosmic energy gravitational genesis of the strongest (𝑀≥7.9) Japanese earthquakes, we present in Table 1 the time periods 𝑇₁ (given in years) of recurrence of the strongest Japanese earthquakes [2] and the obtained [12, 25, 26] corresponding time periodicities (given in years) determined by the following planetary combinations.

Table 1 

The time periods 𝑇₁ (given in years) of recurrence of the strongest Japanese earthquakes [2] and the obtained [12, 25, 26] corresponding time periodicities (given in years) determined by the following cosmic combinations.

Taking into account the time periodicity 83 years (given by (185)) and the year 1927 AD of Jupiter’s opposition with the Earth, respectively, and the time periodicity 88 years = 8 × 11 years (determined by the Sun-Moon system, Venus, Jupiter, and Mars) and the year 1923 AD of the last strongest Japanese earthquake in the Tokyo region, we predicted [25, 26] “the time range 2010÷2011AD(1927+83÷1923+88) (207)

of the next sufficiently strong Japanese earthquake near the Tokyo region”.

The powerful 6.6-magnitude (on Richter scale) Japanese earthquake (that occurred on 14 March, 2010) near Tokyo (with the epicenter in the Fukushima Prefecture) gets into the predicted time range 2010÷2011 AD [25, 26]. The powerful 6.8-magnitude (on Richter scale) Japanese earthquake (that occurred on 11 March, 2011) near Tokyo gets also into the predicted time range 2010÷2011 AD [25, 26]. The occurrence of the strong Japanese earthquakes on 14 March, 2010, and on 11 March, 2011, confirmed the previous predictions [1, 25, 26] and the proposed cosmic energy gravitational genesis [12, 25, 26] of the strongest Japanese earthquakes.

5. Summary of Main Results and Conclusion

We have developed in this article the fundamentals of the thermohydrogravidynamic theory intended for the long-term deterministic predictions of the strong earthquakes. The deterministic thermohydrogravidynamic theory is based on the established generalized formulation [11, 12] of the first law of thermodynamics for moving rotating deforming compressible heat-conducting stratified macroscopic continuum region 𝜏 subjected to the nonstationary Newtonian gravitational field.

Using the classical continuum-mechanical theoretical approach [29, 33], we have presented in Section 2.1 the generalized expression [24, 31] for the macroscopic kinetic energy of a small continuum region. The generalized expression [24, 31] is the generalization of the classical expression [28, 29] in classical nonequilibrium thermodynamics for the macroscopic kinetic energy per unit mass of a small macroscopic continuum region (considered in a stratified three-dimensional shear flow) by taking into account the irreversible shear component of the hydrodynamic velocity field related to the rate of strain tensor 𝑒_𝑖𝑗. The macroscopic kinetic energy (of the small macroscopic continuum region) is presented as a sum of the macroscopic translational kinetic energy and three Galilean invariants: the classical macroscopic internal rotational kinetic energy [28, 29], the established macroscopic internal shear kinetic energy [24, 31], and the established macroscopic internal kinetic energy of shear-rotational coupling [24, 31] with a small correction. The generalized formula (20) for the macroscopic kinetic energy per unit mass 𝜀_𝑘 and its particular form (31) for homogeneous continuum regions of spherical and cubical shapes generalize the classical de Groot and Mazur expression (2) in classical nonequilibrium thermodynamics [28, 29] by taking into account the established [24, 31] macroscopic internal shear kinetic energy per unit mass 𝜀_𝑠, which expresses the kinetic energy of irreversible dissipative shear motion, and also the established [24, 31] macroscopic internal kinetic energy of shear-rotational coupling per unit mass 𝜀^coup_𝑠,𝑟, which expresses the kinetic energy of local coupling between irreversible dissipative shear and reversible rigid-like rotational macroscopic fluid motions. The generalized expression (20) confirms the postulate [35] that the velocity shear (𝑒_𝑖𝑗≠0) represents an additional energy source.

Based on the postulates of thermodynamics [27, 28], continuum mechanics [29, 30] and hydrodynamics [33, 38], we have presented in Section 2.2 the equivalent generalized differential formulations (50), (56), and (59) (given for the Galilean frame of reference) of the first law of thermodynamics [11, 12] for nonequilibrium shear-rotational states of the deformed finite one-component individual continuum region (characterized by the symmetric stress tensor 𝐓) moving in the nonstationary Newtonian gravitational field. The equivalent generalized differential formulations (50), (56), and (59) are valid for moving rotating deforming compressible heat-conducting stratified macroscopic continuum region 𝜏 subjected to the nonstationary Newtonian gravity. The generalized differential formulation (56) 𝑑𝑈_𝜏+𝑑𝐾_𝜏+𝑑𝝅_𝜏=𝛿𝑄+𝛿𝐴_np,𝜕𝜏+𝑑𝐺 (208) generalizes the classical [27, 32] formulations (39) and (40): 𝑑𝑈=𝛿𝑄−𝑝𝑑𝑉,𝑑𝜀≡𝑑𝑈≡𝑑𝑈_𝜏,−𝛿𝑊=−𝑝𝑑𝑉 (209)

by taking into account (along with the classical infinitesimal change of heat 𝛿𝑄 and the classical infinitesimal change of the internal energy 𝑑𝑈_𝜏≡𝑑𝑈) the infinitesimal increment of the macroscopic kinetic energy 𝑑𝐾_𝜏, the infinitesimal increment of the gravitational potential energy 𝑑𝝅_𝜏, the generalized expression [11, 12, 24] for the infinitesimal work 𝛿𝐴_np,𝜕𝜏 done on the continuum region 𝜏 by the surroundings of 𝜏, and the additional [11, 12] infinitesimal amount 𝑑𝐺 of energy added (or lost) as the result of the Newtonian nonstationary energy gravitational influence on the continuum region 𝜏 during the infinitesimal time interval 𝑑𝑡. The presented (in Section 2.3) generalized expression [11, 12] for infinitesimal work 𝛿𝐴_np,𝜕𝜏 (done on the continuum region 𝜏 by the surroundings of 𝜏) generalizes the classical [27] expression 𝛿𝐴_np,𝜕𝜏=−𝛿𝑊=−𝑝𝑑𝑉 by taking into account (for Newtonian continuum) the infinitesimal work 𝛿𝐴_𝑐 (given by expression (68)) of the acoustic forces and the infinitesimal work 𝛿𝐴_𝑠 (given by expression (69)) of the viscous forces acting during the infinitesimal time interval 𝑑𝑡 on the boundary surface 𝜕𝜏 of the individual continuum region 𝜏.

Based on the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics and the obtained expression (74) for the gravitational energy power 𝑊_gr(𝜏), we have presented in Section 2.4 the established [11, 12] gravitational energy mechanism of the gravitational energy supply into the continuum region 𝜏 owing to the local time increase of the potential 𝜓 of the gravitational field inside the continuum region 𝜏 subjected to the nonstationary Newtonian gravitational field. The revealed gravitational mechanism [11, 12] of the gravitational energy supply into the continuum region 𝜏 is consistent with the empirical findings [1, 3–5, 9, 10] related to the anomalous variations of the gravitational field before the strong earthquakes. Based on the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics, we have presented the evidence [12, 25] about the significant increase of the energy flux 𝛿𝐹_vis,𝑐 (given by expression (76)) of the geoacoustic energy from the focal region 𝜏 before the earthquake in a good agreement with the results of the detailed experimental studies [43].

We have presented in Section 3 the evidence [12, 25] of the cosmic energy gravitational genesis of earthquakes. Using the evolution equation (73) (deduced from the generalized differential formulation (50) of the first law of thermodynamics) of the total mechanical energy of the macroscopic continuum region 𝜏 (of the compressible viscous Newtonian continuum), we have presented in Section 3.1 the thermodynamic foundation of the classical deformational (shear) model [1] of the earthquake focal region for the quasi-uniform medium of the Earth’s crust characterized by practically constant viscosity. Based on the generalized differential formulation (50) of the first law of thermodynamics for the macroscopic continuum region 𝜏, we have presented in Section 3.1 the generalized thermohydrogravidynamic shear-rotational model [11, 12] of the earthquake focal region by taking into account the classical macroscopic rotational kinetic energy [28, 29], the macroscopic nonequilibrium kinetic energies [24, 31], and the external cosmic energy gravitational influences on the focal region of the preparing earthquakes. We have presented in Section 3.2 the evidence [11, 12] of the physical adequacy of the rotational model [2] of the earthquake focal region for the seismic zone of the Pacific Ring.

We have presented in Section 4 the fundamentals of the cosmic geophysics [12, 25, 26]. We have presented in Section 4.1 the consideration of the energy gravitational influences on the Earth of the inner planets and the outer planets of the Solar System. We have presented in Section 4.1.1 the relation (96) for the energy gravitational influences (on the Earth) of the inner and the outer planets in the second approximation of the elliptical orbits of the planets of the Solar System. We have presented in Section 4.1.2 the evaluation of the relative maximal planetary instantaneous energy gravitational influences on the Earth in the approximation of the circular orbits of the planets of the Solar System. Considering the maximal positive value max(𝜕/𝜕𝑡)𝜓₃₁(𝐶₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓₃₁(𝐶₃,int) of the gravitational potential 𝜓₃₁(𝐶₃,int) created by Mercury at the mass center 𝐶₃ of the Earth) as a scale of the instantaneous energy gravitational influence of the planets of the Solar System on the Earth (in the considered first approximation of the circular orbits of the planets), we have presented in Section 4.1.2 the following numerical sequence of the nondimensional relative maximal powers of the planetary energy gravitational influences (on the unit mass of the Earth at the mass center 𝐶₃ of the Earth): 𝑓(2,𝐶₃)=37.69807434 (for Venus), 𝑓(5,𝐶₃)=7.41055774 (for Jupiter), 𝑓(1,𝐶₃)=1 (for Mercury), 𝑓(4,𝐶₃)=0.67441034 (for Mars), 𝑓(6,𝐶₃)=0.24601009 (for Saturn), 𝑓(7,𝐶₃)=0.00319056 (for Uranus), 𝑓(8,𝐶₃)=0.00077565 (for Neptune), and 𝑓(9,𝐶₃)=3.4813⋅10⁻⁸ (for Pluto). Considering the maximal positive value max(𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓_3𝑀(𝐷₃,int) of the gravitational potential 𝜓_3𝑀(𝐷₃,int) created by Mercury at the surface point 𝐷₃ of the Earth) as a scale of the energy gravitational influence of the planets of the Solar System on the Earth (in the considered first approximation of the circular orbits of the planets), we have presented in Section 4.1.2 the following numerical sequence (corrected slightly the previous numerical values [12]) of the nondimensional relative maximal powers of the planetary energy gravitational influences (on the unit mass of the Earth at the surface point 𝐷₃ of the Earth): 𝑓(2,𝐷₃)=37.70428085 (for Venus), 𝑓(5,𝐷₃)=7.40926122 (for Jupiter), 𝑓(1,𝐷₃)=1 (for Mercury), 𝑓(4,𝐷₃)=0.67420160 (for Mars), 𝑓(6,𝐷₃)=0.24596865 (for Saturn), 𝑓(7,𝐷₃)=0.00319004 (for Uranus), 𝑓(8,𝐷₃)=0.00077552 (for Neptune), and 𝑓(9,𝐷₃)=3.4807⋅10⁻⁸ (for Pluto). We have recognized [25, 26] that the revealed small difference of the combined planetary maximal energy gravitational influences at the surface point 𝐷₃ and at the mass center 𝐶₃ of the Earth can explain the following related geophysical phenomena: the small oscillatory motion of the rigid kernel of the Earth relative to the fluid kernel of the Earth; the small oscillation of the Earth’s pole (i.e., the Chandler’s wobble of the Earth’s pole); the small oscillations of the boundary of the Pacific Ocean (i.e., the seismic zone of the Pacific Ring); the oscillations, rotations, and deformations of the geoblocks weakly coupled with the surrounding plastic layers in all seismic zones of the Earth and the formation of fractures related to the strong earthquakes and the planetary cataclysms. We have presented in Section 4.1.3 the evaluation of the relative maximal planetary integral energy gravitational influences on the Earth in the approximation of the circular orbits of the planets of the Solar System. Considering the maximal positive integral energy gravitational influence max_𝑡Δ_𝑔𝐸₃(𝜏₁,0,0,𝑡,0) of Mercury (given by the expression (136)) on the Earth as a scale of the integral energy gravitational influence of the planets of the Solar System on the Earth (in the considered first approximation of the circular orbits of the planets), we have presented in Section 4.1.3 the following order of signification (with respect to the established planetary maximal integral energy gravitational influences on the Earth) of the planets of the Solar System: Venus (𝑠(2)=89.6409), Jupiter (𝑠(5)=31.319), Mars (𝑠(4)=2.6396), Saturn (𝑠(6)=1.036), Mercury (𝑠(1)=1), Uranus (𝑠(7)=0.0133), Neptune (𝑠(8)=0.003229), and Pluto (𝑠(9)=1.4495⋅10⁻⁷), which gives the predominant combined planetary integral energy gravitational influence on the Earth of Venus and Jupiter.

We have presented in Section 4.2 the consideration of the lunar energy gravitational influences on the Earth. We have presented in Section 4.2.1 the evaluation of the relative maximal instantaneous energy gravitational influence of the Moon on the Earth in the second approximation of the elliptical orbits of the Earth and the Moon around the combined mass center 𝐶_3,MOON of the Earth and the Moon. Considering the maximal positive value max(𝜕/𝜕𝑡)𝜓₃₁(𝐶₃,int) (of the partial derivative (𝜕/𝜕𝑡)𝜓₃₁(𝐶₃,int) of the gravitational potential 𝜓₃₁(𝐶₃,int) created by Mercury at the mass center 𝐶₃ of the Earth) as a scale for evaluation of the maximal instantaneous energy gravitational influence of the Moon on the Earth, we have presented the nondimensional numerical value 𝑓_MOONM(𝐶₃,secondapproximation)=19.44083, which means that the power of the maximal energy gravitational influence of the Moon (on the unit mass of the Earth at the mass center 𝐶₃ of the Earth) is 19.44083 times larger than the maximal power of the energy gravitational influence (on the unit mass at the mass center 𝐶₃ of the Earth) of Mercury moving around the mass center 𝑂 of the Sun along the hypothetical circular orbit. Taking into account the calculated [25, 26] nondimensional maximal instantaneous energy gravitational influences on the unit mass of the Earth at the mass center 𝐶₃ of the Earth, we established the following order of significance (with respect to the established planetary and lunar maximal instantaneous energy gravitational influences on the Earth) of the Moon and the planets of the Solar System: Venus, the Moon, Jupiter, Mercury, Mars, Saturn, Uranus, Neptune, and Pluto, which reveals the main instantaneous energy gravitational influences on the Earth of Venus, the Moon, Jupiter, Mercury, and Mars determining (in collection) the main combined instantaneous energy gravitational influence on the Earth (not taking into account the instantaneous energy gravitational influences of the Sun and our Galaxy).

We have presented in Section 4.2.2 the evaluation of the maximal integral energy gravitational influence of the Moon on the Earth in the approximation of the elliptical orbits of the Earth and the Moon around the combined mass center 𝐶_3,MOON. Considering the maximal positive integral energy gravitational influence max_𝑡Δ_𝑔𝐸₃(𝜏₁,0,0,𝑡,0) of Mercury (given by the expression (136)) on the Earth as a scale for evaluation of the maximal integral energy gravitational influence of the Moon on the Earth (in the considered first approximation of the circular orbits of the planets and in the approximation of the elliptical orbits of the Earth and the Moon around the combined mass center 𝐶_3,MOON), we established that the maximal positive integral energy gravitational influence of the Moon on the Earth is 𝑠(Moon,secondapproximation)=13.0693 times larger than the maximal positive integral energy gravitational influence of Mercury on the Earth. Considering the aspect of the cosmic gravitational preparation of the strong earthquakes, we established the Venusian (𝑠(2)=89.6409 [12]), Jupiter’s (𝑠(5)=31.319 [12]), and the Moon’s (𝑠(Moon,secondapproximation)=13.0693 [25, 26]) energy gravitational predominance in supplying of the cosmic planetary and lunar gravitational energy to the focal region of the preparing earthquakes. By considering the energy gravitational influence on the Earth’s macroscopic continuum region 𝜏 of Venus, we have presented in Section 4.3.1 the evidence (based on the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics used for the macroscopic continuum region 𝜏 considered as the focal region of the preparing earthquake) of the real cosmic energy gravitational genesis of preparation of earthquakes. Using the expression (167) for the maximal positive integral energy gravitational influence 𝐸_𝑔(𝜏₂,𝐷₃,𝑚_𝜏) of Venus on the macroscopic continuum region 𝜏 of mass 𝑚_𝜏 near the surface point 𝐷₃ of the Earth, we have presented in Section 4.3.1 the evidence [12] that the regular changes of the rotational regime of the Earth [2] are related to the regular discharges of the accumulated potential energy (in the focal region of earthquakes) supplying of the cosmic energy gravitational influences of the planets of the Solar System, the Sun, and the Moon. We have presented in Section 4.3.2 the evidence of the cosmic energy gravitational genesis of triggering of the preparing earthquakes. By considering the calculated relative average values 𝑒(𝑖) of the integral energy gravitational influences (corresponding to the time duration 𝑇_MOON/2 of the maximal integral energy gravitational influence of the Moon on the Earth) on the Earth of the planets of the Solar System and the Moon [12, 25]: 𝑠(Moon,secondapproximation)=13.0693 (for the Moon), 𝑒(2)=4.5342 (for Venus), 𝑒(5)=2.3182 (for Jupiter), 𝑒(1)=0.2547 (for Mercury), 𝑒(4)=0.0999 (for Mars), 𝑒(6)=0.0809 (for Saturn), 𝑒(7)=0.001066 (for Uranus), 𝑒(8)=0.0002594 (for Neptune), and 𝑒(9)=1.1671⋅10⁻⁸ (for Pluto), we established [25] the exceptional predominant significance of the cosmic energy gravitational influence of the Moon (along with the established [12] minor significance of Venus, Jupiter, and Mercury) as the predominant cosmic trigger mechanism of discharge (on the final stage of the preparation of the earthquakes) of the accumulated potential energy (in the focal region of earthquakes) supplying of the cosmic gravitational energy influences of the planets of the Solar System and the Moon. The considered results of Sections 4.1, 4.2, and 4.3 have demonstrated the cosmic energy gravitational genesis (related to the combined nonstationary cosmic energy gravitational influence of the planets of the Solar System, the Sun, and the Moon) of the following correlated geophysical phenomena: the small oscillations [2] of the boundary of the Pacific Ocean (i.e., the seismic zone of the Pacific Ring); the small oscillatory motion of the rigid kernel of the Earth relative to the fluid kernel of the Earth [51]; the Chandler’s wobble of the Earth’s pole; the oscillations [42], rotations, and deformations [1, 2] of the geoblocks weakly coupled with the surrounding plastic layers in all seismic zones of the Earth; the formation of the planetary fractures related to the strong earthquakes [1, 3–5, 9, 10, 23] and the planetary cataclysms [25, 26].

We have demonstrated in Section 4.4 the cosmic energy gravitational genesis [12] of the time periodicities of the global seismotectonic activity of the Earth induced by the nonstationary cosmic energy gravitational influences on the Earth of the Sun-Moon system, Venus, Jupiter, and Mars. We have presented in Section 4.4.1 the time periodicities [12] of the maximal (instantaneous and integral) energy gravitational influences on the Earth of the Sun and the Moon, Venus, Jupiter, and Mars. We have presented in Section 4.4.1(1) the successive approximations of the following time periodicities: (𝑇_𝑆-MOON,3)₁=3 years, (𝑇_𝑆-MOON,3)₂=8 years, (𝑇_𝑆-MOON,3)₃=11 years and (𝑇_𝑆-MOON,3)₃=19 years, (𝑇_𝑆-MOON,3)₄=27 years, and (𝑇_𝑆-MOON,3)₅=235 years of recurrence of the maximal (instantaneous and integral) combined energy gravitational influences of the Sun-Moon system on the Earth in the first, second, third, fourth, and fifth approximations, respectively. We have presented in Section 4.4.1(2) the successive approximations of the following time periodicities: (𝑇_𝑉,3)₁=3 years, (𝑇_𝑉,3)₂=8 years, and (𝑇_𝑉,3)₃=11 years of recurrence of the maximal (instantaneous and integral) energy gravitational influences of Venus on the Earth in the first, second, and third approximations, respectively. We have presented in Section 4.4.1(3) the successive approximations of the following time periodicities: (𝑇_𝐽,3)₁=11 years, (𝑇_𝐽,3)₂=12 years, and (𝑇_𝐽,3)₃=83 years of recurrence of the maximal (instantaneous and integral) energy gravitational influences of Jupiter on the Earth in the first, second, and third approximations, respectively. We have presented in Section 4.4.1(4) the successive approximations of the following time periodicities: (𝑇_MARS,3)₁=15 years, (𝑇_MARS,3)₂=32 years, and (𝑇_MARS,3)₃=47 years of recurrence of the maximal (instantaneous or integral) energy gravitational influences of Mars on the Earth in the first, second, and third approximations, respectively. We have presented in Section 4.4.2 the empirical time periodicities [1, 2, 53–67] of the seismotectonic activity for different regions of the Earth.

Based on the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics [12, 25] used for the Earth and using the evaluated (in Section 4.4.1) successive approximations for the time periodicities of the periodic recurrence of the maximal energy gravitational influences on the Earth of the Sun-Moon system, Venus, Mars, and Jupiter, we have presented in Section 4.4.3 the following sets of the global seismotectonic and volcanic periodicities 𝑇_tec and the global climatic periodicities 𝑇_clim1 of the Earth: 𝑇_tec=𝑇_clim1=𝑇_energy=𝑇_𝑆-MOON,3^𝑙𝑜_𝑖×𝑇_𝑉,3^𝑙2_𝑗×𝑇_MARS,3^𝑙4_𝑘×𝑇_𝐽,3^𝑙5_𝑛, (210)

determined by the successive global periodicities 𝑇_energy (defined by the multiplications of various successive time periodicities related to the different combinations of the following integer numbers: 𝑖=1,2,3,4,5; 𝑗=1,2; 𝑘=1,2,3; 𝑛=1,2,3; 𝑙_𝑜=0,1; 𝑙₂=0,1; 𝑙₄=0,1; 𝑙₅=0,1) of recurrence of the maximal combined energy gravitational influences on the Earth of the different combined combinations of the cosmic nonstationary energy gravitational influences on the Earth of the Sun-Moon system, Venus, Mars, and Jupiter. We have shown that the considered empirical time periodicities [1, 2, 53–67] of the seismotectonic activity may be satisfactory approximated by the different combined combinations of the cosmic nonstationary energy gravitational influences on the Earth of the Sun-Moon system, Venus, Mars, and Jupiter.

We have presented in Section 4.5 the evidence of the cosmic energy gravitational genesis of the Chinese 2008 earthquakes. Using the established time periodicities [12] of the maximal (instantaneous and integral) energy gravitational influences on the Earth of the Sun-Moon system, Venus, Mars, and Jupiter, and considering the presented set of the Chinese earthquakes (Luhuo 1973, Haicheng 1975, and Tang Shan 1976; Bachu 2003, Ruichang 2005, and Yanjin 2006), we predicted [12] the nearest year (2008) of the next Chinese 2008 earthquakes. The proposed cosmic energy gravitational genesis of the predicted [12] strong Chinese 2008 earthquakes was confirmed by the occurrence of two large earthquakes in 2008 in the areas surrounding Yutian (Xinjiang) and Wenchuan (Sichuan). The proposed cosmic energy gravitational genesis of the predicted [12] strong Chinese 2008 earthquakes was also confirmed by the noticeable gravity variations [9] (derived from regional gravity monitoring data in China from 1998 to 2005) before the occurrence of two large earthquakes in 2008 in the areas surrounding Yutian (Xinjiang) and Wenchuan (Sichuan). According to the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics [12, 25] and the presented results of Section 2.4, the noticeable gravity variations (mentioned in articles [1, 3–5, 9, 13]) are related to supply of the gravitational energy into the focal region before the occurrence of earthquake. According to the equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics [12, 25], the noticeable gravity variations in the focal region is the necessary (but not sufficient) condition for preparation of earthquake. A recent research [10] by Zhan and his colleagues confirmed this conclusion demonstrating that significant gravity changes were observed before all nine large earthquakes in China from 2001 to 2008. It is clear that the presented equivalent generalized differential formulations (50), (56), and (59) of the first law of thermodynamics [12, 25] can give the mathematical basis to “remove the subjective nature in the determination of the timeframe of a forecasted earthquake” [9]. This conclusion is in agreement with the demonstrated (in Section 4.6) cosmic energy gravitational genesis of the strongest Japanese earthquakes near the Tokyo region. “The time range 2010 ÷ 2011 AD of the next sufficiently strong Japanese earthquake near the Tokyo region” [25, 26] was predicted based on the established time periodicities (𝑇_𝐽,3)₃=83 years (determined by Jupiter) and 88 years = 8 × 11 years (determined by the Sun-Moon system, Venus, Jupiter, and Mars) and taking into account the year 1923 AD of the strongest Japanese earthquake near the Tokyo region. The proposed [12, 25, 26] cosmic energy gravitational genesis of the strongest Japanese earthquakes was confirmed by occurrence of the strong Japanese earthquakes on 14 March, 2010, and on 11 March, 2011.

The time periodicity 88 years (of the global seismotectonic and volcanic activity and the global climate variability related to recurrence of the maximal combined energy gravitational influences on the Earth of the Sun-Moon system, Venus, Mars, and Jupiter [12, 25, 26]) is in good agreement with the estimated (based on the spectral Fourier analysis) climatic time periodicity 88 years [69] obtained from the studies of sediments from Siberian and Mongolian lakes. This good agreement (of the independent experimental seismotectonic [1] and the climatic [69] periodicity 88 years with the global seismotectonic, volcanic, and climatic periodicity 88 years [12, 25, 26]) is the additional confirmation of the validity of the thermohydrogravidynamic theory [11, 12, 25, 26] of the seismotectonic, volcanic, and climatic activity of the Earth.

The founded generalized differential formulations (50), (56), and (59) of the first law of thermodynamics [12, 25] can be used as the thermohydrogravidynamic basis for rapid development of the urgent technologies of the long-term deterministic predictions of the strong earthquakes in order to sustain the stable evolutionary development, the survival, greatness, and cosmic dignity of the humankind in the present and forthcoming epochs of the critical surrounding cosmic, seismotectonic, volcanic, and climatic conditions of the human existence on the Earth.

Acknowledgments

The author thanks the Academic Editor Dr. Umberta Tinivella with gratitude for the editorial corrections to improve the final text of the article. The author thanks a reviewer for bringing to the author’s attention the additional interesting information, which was used with gratitude for the correction of the article. The author thanks Prof. V. A. Abramov and Dr. Sergey V. Smirnov for helpful discussions and Dr. Yuri Panarin and Amrtatjuti V. Sereda for the editorship of the first version of the article in English.