Early and Inner Moon

**6**

*Lunar Science*

**References**

org/abs/1206.0749

[1] ISECG. The Global Exploration Roadmap, January 2018. The International Space Exploration Coordination Group; 2018. https:// www.globalspaceexploration.org/

[2] Crawford IA, Anand M, Cockell CS, Falcke H, Green DA, Jaumann R, et al. Back to the Moon: The scientific rationale for resuming lunar surface exploration. Planetary and Space Science. 2012;**74**(1):3-14. https://arxiv.

[3] Crawford IA. Science enabled by a Moon Village. Earth and Planetary Astrophysics. 2017. arXiv:1706.06698. https://arxiv.org/abs/1706.06698

[4] Khan-Mayberry N. The Lunar Environment: Determining the Health Effects of Exposure to Moon Dusts. PhD Dissertation. Houston, TX: NASA; 2007. https://ntrs.nasa.gov/archive/nasa/casi.

ntrs.nasa.gov/20070006527.pdf

arxiv.org/abs/1206.6328

abs/1410.6865

[5] Linnarsson D, Carpenter J, Fubini B, Gerde P, Karlsson LL, Loftus DJ, et al. Toxicity of lunar dust. Planetary and Space Science. 2012;**74**(1):57-71. https://

[6] Crawford IA. Lunar Resources: A Review. Progress in Physical Geography. 2015;**39**(2):137-167. https://arxiv.org/

Chapter 2

Abstract

1. Introduction

9

Khachay Yuriy

Initial Evolution of the Moon

these sources provides a solution to the problem.

The problem of the origin of the Moon is of fundamental importance to understanding the mechanism of the planetary solar system's formation. It is important to know the mechanism of differentiation of substances in a growing planet. When planets are formed from a cold protoplanetary cloud, the matter of the inner regions of the Earth and the Moon remains at temperatures lower than the melting point of iron. The main volume of the matter of the protoplanet remains in its unmelted state, and its differentiation occurs in the formed planet. In this work, attention is paid to the most important internal sources of energy: the decay energy of shortlived isotopes, the dissipation of tidal friction energy, and thermal energy from accidental deposition of bodies and particles on a growing surface. Accounting for

Keywords: moon matter, composition, age, growth dynamics, numerical models

The study of the evolution of the moon is of exceptional interest for the knowledge of the processes which had occurred and occurred now on the Moon and the Earth, the formation of the internal regions of these bodies. Of undoubted interest is the study of the material composition of the surface of the planetary body nearest to the Earth, which makes it possible to clarify information about the processes in the

Before to the work of descent vehicles on the surface of Mars, the only direct information about the composition of the substance of external bodies was the results of a study of lunar soil, delivered to Earth by Soviet stations Luna-16 and Luna-20 and American astronauts. The geophysical measurements of the gravitational potential and the speed of propagation of seismic waves made it possible to build indirect models of the distribution of the density and velocity sections of the Moon [1]. For the no uniqueness of the geophysical interpretation, estimates of the density distribution and velocity of seismic waves for 1D models of the Moon that are very important for the knowledge of the current state are obtained. The lunar crust is fixed from the surface to a depth of about 55 km. The velocity of longitudinal seismic waves VP increases here to a value of VP = 5.8 km/s at a depth of 25 km, then sharply increases at this depth to a value of 6.8 km/s. Then the velocity increases slowly down to 7 km/s up to the 55 km border. This section is called the crust-mantle transition boundary. The surface is marked by a sharp increase in the velocity of VP to the value of VP = 8.1 km/s. Then there is a decrease in the velocities VP and Vs to values of 7.8 km/s and 4.7 km/s, respectively, at a depth of 300 m. The layer in the depth interval (55–300) in km is called the upper mantle. In the range of

protoplanetary cloud and the early stages of the accumulation of planets.

## Chapter 2 Initial Evolution of the Moon

Khachay Yuriy

## Abstract

The problem of the origin of the Moon is of fundamental importance to understanding the mechanism of the planetary solar system's formation. It is important to know the mechanism of differentiation of substances in a growing planet. When planets are formed from a cold protoplanetary cloud, the matter of the inner regions of the Earth and the Moon remains at temperatures lower than the melting point of iron. The main volume of the matter of the protoplanet remains in its unmelted state, and its differentiation occurs in the formed planet. In this work, attention is paid to the most important internal sources of energy: the decay energy of shortlived isotopes, the dissipation of tidal friction energy, and thermal energy from accidental deposition of bodies and particles on a growing surface. Accounting for these sources provides a solution to the problem.

Keywords: moon matter, composition, age, growth dynamics, numerical models

## 1. Introduction

The study of the evolution of the moon is of exceptional interest for the knowledge of the processes which had occurred and occurred now on the Moon and the Earth, the formation of the internal regions of these bodies. Of undoubted interest is the study of the material composition of the surface of the planetary body nearest to the Earth, which makes it possible to clarify information about the processes in the protoplanetary cloud and the early stages of the accumulation of planets.

Before to the work of descent vehicles on the surface of Mars, the only direct information about the composition of the substance of external bodies was the results of a study of lunar soil, delivered to Earth by Soviet stations Luna-16 and Luna-20 and American astronauts. The geophysical measurements of the gravitational potential and the speed of propagation of seismic waves made it possible to build indirect models of the distribution of the density and velocity sections of the Moon [1]. For the no uniqueness of the geophysical interpretation, estimates of the density distribution and velocity of seismic waves for 1D models of the Moon that are very important for the knowledge of the current state are obtained. The lunar crust is fixed from the surface to a depth of about 55 km. The velocity of longitudinal seismic waves VP increases here to a value of VP = 5.8 km/s at a depth of 25 km, then sharply increases at this depth to a value of 6.8 km/s. Then the velocity increases slowly down to 7 km/s up to the 55 km border. This section is called the crust-mantle transition boundary. The surface is marked by a sharp increase in the velocity of VP to the value of VP = 8.1 km/s. Then there is a decrease in the velocities VP and Vs to values of 7.8 km/s and 4.7 km/s, respectively, at a depth of 300 m. The layer in the depth interval (55–300) in km is called the upper mantle. In the range of 300–800 km, called the average mantle, the velocity is reduced to VP = 7.5 km/s and VS = (3.6–4.0) km/s.

formation of the Moon as a result of the mega impact, the Earth's collision with a space body of planetary size (with mass of Mars or more). A number of geochemical contradictions that are incompatible with the mega-impact hypothesis are consid-

We were faced with the task: using the model of accumulation of the terrestrial planets proposed in papers [16–18], to conduct numerical simulations of the temperature distribution in the interior of the planet for successively increasing with time values of the body radius in the 3D environment and in contrast to the results [18], explore the features of the evolution of primary heterogeneities depending on the rate of accumulation of the second largest body in the "feeding" zone of a growing Earth in a 3D process model. In works [16–18], the proposed model of the heterogeneous accumulation of the Earth, based on a two-stage mechanism for the formation of the pre-planetary of the planet, is presented. It assumes that in the first stage, primary pre-planets are formed, the central part of which consists of the most high-temperature condensates close in composition to the CAI—Ca-Al inclusions found in the Allende meteorite. The middle envelope of these pre-planets is represented by an iron-nickel material, which condenses from the gaseous phase following high-aluminum condensates. In the process of growth of primary preplanets, they were heated, as a result of the decay of short-lived radioactive isotopes, the main of which is 26Al with a half decay of <sup>τ</sup> = 7.38 <sup>10</sup><sup>5</sup> years [19]. The ratio of 26Al/27Al in protoplanetary matter is estimated to be 5 <sup>10</sup><sup>5</sup> [20]. With this content of 26Al, as the mass of the pre-planetary grows, the temperature of their central regions increases, and in the center of the pre-planetary with a radius of more than 200 km, it can reach 2200 K [17]. This is quite enough to melt the Ca-Al material in the central part of the core, the melting point of which is 1830 K [1], and the iron-nickel mixture in its middle envelope. The outer envelope of the preplanetary, which transfers heat to the space, will remain solid. Further development of the process of formation of the planets is as follows. In accordance with the accumulation model of Safronov [21], the number of cores formed at the initial stage of the agglomeration process of condensation products is large, and they will often collide with each other. The collision of two pre-planetary with similar sizes, molten aluminosilicate core and middle envelope, folded with iron and a solid silicate outer envelope will lead to their destruction. Medium, molten envelopes in a collision will merge, forming a new pre-planet, the core of which consists of an iron-nickel alloy. The substance of the aluminosilicate core of the primary core will be extruded from their centers and thrown into the feeding zone for new core formed as a result of the collision. The outer hard envelope, the lower part of which could consist of a substance close to the composition of pallasites or ordinary chondrites, will be destroyed, and part of the fragments will also be thrown out of the feeding area of the growing planet. In this way, the metallic core of the Earth and the Moon is formed, and the separation of their chemical reservoirs of the core

2. Mathematical modeling of the temperature distribution in the moon

The model [21] on the formation of planets and their satellites from a protoplanetary cloud is used as the initial one. The results of numerical modeling

obtained by us [16, 17] showed that already at an early stage of the Earth's accumulation process, heat release during decay by short-lived naturally radioactive elements and above all 26Al is enough to exceed in the protoplanetary dimensions (50–100 km); a molten central region and a relatively thin, solid, predominantly

ered by Galimov in [7].

Initial Evolution of the Moon

DOI: http://dx.doi.org/10.5772/intechopen.84171

and the mantle takes place.

11

at the stage of its accumulation

In the lower part of the mantle, transverse waves are not recorded. At a depth of about 1500 km, the mantle-core transition region is fixed, and in the second one, there is a sharp decrease in the propagation velocity of the volume seismic waves to VP = (4–5) km/s [2–5]. In spite of the no uniqueness of the geophysical interpretation, very important results of the density distribution and velocity of seismic waves for 1D models of the Moon had been obtained. The scheme of the seismic velocity model is presented in [1]. The nature of the propagation of seismic waves is very different from that observed in the Earth: the amplitude of oscillations increases sharply, and the decline is observed for (1–4) hours. The lunar crust differs significantly from the earth's crust in its elastic and viscous characteristics. The quality factor (the inverse of the attenuation coefficient) is estimated at 5000, while for the Earth, this estimate is in the range of 100–1000. The seismic activity of the moon is much smaller, about 1015 erg/year, whereas on Earth about 1024 erg/year.

Based on the mineralogical study of the delivered collection of lunar samples, it was established that, unlike the Earth, an early and extensive differentiation of matter took place on the Moon. At the same time, at the early stage of the formation of the Moon, the fractional crystallization of the substance, which formed, possibly, the entire Moon, followed by partial melting of the upper envelope, with a capacity of at least 250 km, occurred. Lunar magma was formed during the entire time of its cooling [6]. The question of the origin and composition of the substance of the moon is of fundamental importance for understanding the processes of formation of the planets of the solar system. Before the occurring of the mega-impact hypothesis, three main mechanisms for the formation of the Moon had been discussed in the literature: (1) the hypothesis of separation of the moon from the earth, (2) the capture hypothesis, and (3) hypothesis of the co-formation or co-accretion of the Earth and the Moon. The disadvantages of these hypotheses are considered in [7]. The idea of separating the substance of the Moon from the Earth was proposed by Darwin in 1880 [8]. Its inconsistency with the laws of celestial mechanics is considered in [9]. As the authors of [9] note, in the event of a rotational instability, which causes the separation of a part of a substance from a rotating body, a smooth separation of a satellite from the main body is impossible. The substance ejected as a result of rotational instability either flies away or returns back [9].

Later Ringwood [10] attempted to modify Darwin's hypothesis by assuming that the material from the Earth's mantle was ejected into the Moon's orbit by strikes of large meteorites. The hypothesis of a joint formation of the Earth and the Moon is considered in [11]. Schmidt [11] assumed that the Moon had accumulated in the vicinity of the growing Earth from a near-Earth swarm of bodies, which was continuously increased from a protoplanetary cloud. As the authors of [12] note, "the hypothesis of O. Yu. Schmidt is based on processes that necessary must take place during the accumulation of the Earth and from the mechanical point of view it seems to be the most promising [12]." However, within the framework of this model, it was not possible to explain the difference in the chemical composition of the Earth and the Moon. The main problem that arose in connection with the thermal state of the Moon is the need to substantiate the source of the early differentiation of matter at the stage of their accumulation. To overcome the difficulties of the three main hypotheses of lunar formation, the mega-impact hypothesis [13, 14] was proposed.

In our work, the task was set to construct a numerical model of the Moon accumulation process, in which we would be able to reconcile the experimental obtained data. Taylor [15] believes that if the Earth-Moon system is unique, it is possible that its genesis is unusual and this unusual variant is the hypothesis of the

### Initial Evolution of the Moon DOI: http://dx.doi.org/10.5772/intechopen.84171

300–800 km, called the average mantle, the velocity is reduced to VP = 7.5 km/s and

much smaller, about 1015 erg/year, whereas on Earth about 1024 erg/year.

result of rotational instability either flies away or returns back [9].

Later Ringwood [10] attempted to modify Darwin's hypothesis by assuming that the material from the Earth's mantle was ejected into the Moon's orbit by strikes of large meteorites. The hypothesis of a joint formation of the Earth and the Moon is considered in [11]. Schmidt [11] assumed that the Moon had accumulated in the vicinity of the growing Earth from a near-Earth swarm of bodies, which was continuously increased from a protoplanetary cloud. As the authors of [12] note, "the hypothesis of O. Yu. Schmidt is based on processes that necessary must take place during the accumulation of the Earth and from the mechanical point of view it seems to be the most promising [12]." However, within the framework of this model, it was not possible to explain the difference in the chemical composition of the Earth and the Moon. The main problem that arose in connection with the thermal state of the Moon is the need to substantiate the source of the early differentiation of matter at the stage of their accumulation. To overcome the difficulties of the three main hypotheses of lunar formation, the mega-impact hypothesis

In our work, the task was set to construct a numerical model of the Moon accumulation process, in which we would be able to reconcile the experimental obtained data. Taylor [15] believes that if the Earth-Moon system is unique, it is possible that its genesis is unusual and this unusual variant is the hypothesis of the

Based on the mineralogical study of the delivered collection of lunar samples, it was established that, unlike the Earth, an early and extensive differentiation of matter took place on the Moon. At the same time, at the early stage of the formation of the Moon, the fractional crystallization of the substance, which formed, possibly, the entire Moon, followed by partial melting of the upper envelope, with a capacity of at least 250 km, occurred. Lunar magma was formed during the entire time of its cooling [6]. The question of the origin and composition of the substance of the moon is of fundamental importance for understanding the processes of formation of the planets of the solar system. Before the occurring of the mega-impact hypothesis, three main mechanisms for the formation of the Moon had been discussed in the literature: (1) the hypothesis of separation of the moon from the earth, (2) the capture hypothesis, and (3) hypothesis of the co-formation or co-accretion of the Earth and the Moon. The disadvantages of these hypotheses are considered in [7]. The idea of separating the substance of the Moon from the Earth was proposed by Darwin in 1880 [8]. Its inconsistency with the laws of celestial mechanics is considered in [9]. As the authors of [9] note, in the event of a rotational instability, which causes the separation of a part of a substance from a rotating body, a smooth separation of a satellite from the main body is impossible. The substance ejected as a

In the lower part of the mantle, transverse waves are not recorded. At a depth of about 1500 km, the mantle-core transition region is fixed, and in the second one, there is a sharp decrease in the propagation velocity of the volume seismic waves to VP = (4–5) km/s [2–5]. In spite of the no uniqueness of the geophysical interpretation, very important results of the density distribution and velocity of seismic waves for 1D models of the Moon had been obtained. The scheme of the seismic velocity model is presented in [1]. The nature of the propagation of seismic waves is very different from that observed in the Earth: the amplitude of oscillations increases sharply, and the decline is observed for (1–4) hours. The lunar crust differs significantly from the earth's crust in its elastic and viscous characteristics. The quality factor (the inverse of the attenuation coefficient) is estimated at 5000, while for the Earth, this estimate is in the range of 100–1000. The seismic activity of the moon is

VS = (3.6–4.0) km/s.

Lunar Science

[13, 14] was proposed.

10

formation of the Moon as a result of the mega impact, the Earth's collision with a space body of planetary size (with mass of Mars or more). A number of geochemical contradictions that are incompatible with the mega-impact hypothesis are considered by Galimov in [7].

We were faced with the task: using the model of accumulation of the terrestrial planets proposed in papers [16–18], to conduct numerical simulations of the temperature distribution in the interior of the planet for successively increasing with time values of the body radius in the 3D environment and in contrast to the results [18], explore the features of the evolution of primary heterogeneities depending on the rate of accumulation of the second largest body in the "feeding" zone of a growing Earth in a 3D process model. In works [16–18], the proposed model of the heterogeneous accumulation of the Earth, based on a two-stage mechanism for the formation of the pre-planetary of the planet, is presented. It assumes that in the first stage, primary pre-planets are formed, the central part of which consists of the most high-temperature condensates close in composition to the CAI—Ca-Al inclusions found in the Allende meteorite. The middle envelope of these pre-planets is represented by an iron-nickel material, which condenses from the gaseous phase following high-aluminum condensates. In the process of growth of primary preplanets, they were heated, as a result of the decay of short-lived radioactive isotopes, the main of which is 26Al with a half decay of <sup>τ</sup> = 7.38 <sup>10</sup><sup>5</sup> years [19]. The ratio of 26Al/27Al in protoplanetary matter is estimated to be 5 <sup>10</sup><sup>5</sup> [20]. With this content of 26Al, as the mass of the pre-planetary grows, the temperature of their central regions increases, and in the center of the pre-planetary with a radius of more than 200 km, it can reach 2200 K [17]. This is quite enough to melt the Ca-Al material in the central part of the core, the melting point of which is 1830 K [1], and the iron-nickel mixture in its middle envelope. The outer envelope of the preplanetary, which transfers heat to the space, will remain solid. Further development of the process of formation of the planets is as follows. In accordance with the accumulation model of Safronov [21], the number of cores formed at the initial stage of the agglomeration process of condensation products is large, and they will often collide with each other. The collision of two pre-planetary with similar sizes, molten aluminosilicate core and middle envelope, folded with iron and a solid silicate outer envelope will lead to their destruction. Medium, molten envelopes in a collision will merge, forming a new pre-planet, the core of which consists of an iron-nickel alloy. The substance of the aluminosilicate core of the primary core will be extruded from their centers and thrown into the feeding zone for new core formed as a result of the collision. The outer hard envelope, the lower part of which could consist of a substance close to the composition of pallasites or ordinary chondrites, will be destroyed, and part of the fragments will also be thrown out of the feeding area of the growing planet. In this way, the metallic core of the Earth and the Moon is formed, and the separation of their chemical reservoirs of the core and the mantle takes place.

## 2. Mathematical modeling of the temperature distribution in the moon at the stage of its accumulation

The model [21] on the formation of planets and their satellites from a protoplanetary cloud is used as the initial one. The results of numerical modeling obtained by us [16, 17] showed that already at an early stage of the Earth's accumulation process, heat release during decay by short-lived naturally radioactive elements and above all 26Al is enough to exceed in the protoplanetary dimensions (50–100 km); a molten central region and a relatively thin, solid, predominantly

silicate upper envelope could be formed. This creates the conditions for the mechanism of differentiation of a substance into a chemical reservoir of the future predominantly iron core and a silicate reservoir of the Earth's mantle to be realized at small relative velocities of collisions of small bodies—pre-planets. The molten parts were combined, and the solid, mostly silicate, could not yet be held by the weak gravitational field of the planet and returned to the "feed zone." It is this material component that provided the chemical reservoir, from which the second largest body of the Earth-Moon system was formed. Fractionation went on in small bodies and was mostly completed in less than 10 million years, while the formation of the structure of the core and the mantle of the Earth continued for about 100 million years. The union of the liquid internal parts of the colliding bodies occurred as a result of inelastic collision; therefore, most of the potential gravitational energy was converted into heat through the kinetic energy of the collision. The process continued until the core reached a large part of the modern mass. At the final stage of the growth of the pre-planet, the mass of the Earth's pre-planet is already sufficient to hold an ever-increasing proportion of the silicate envelope of the falling out bodies. And the composition of the growing area is increasingly enriched with an admixture of silicates. The process of a completely inelastic collision of accumulated bodies with a high degree of potential energy conversion by gravitational interaction into heat, gradually turned into a mechanism of almost solid-state collisions, in which only a small part of the kinetic energy is converted into heat absorbed by the planet's pre-planet.

The carried out mathematical modeling of the thermal evolution of a growing planet implements the described process scheme. For the growth rate of the preplanet of the planet, the Safronov model is used in the variant [21]:

$$\frac{\partial m}{\partial t} = 2(\mathbf{1} + 2\,\theta)r^2 \,\alpha \left(\mathbf{1} - \frac{m}{M}\right)\sigma \tag{1}$$

<sup>ρ</sup><sup>T</sup> <sup>∂</sup><sup>S</sup> ∂ t þ V ! ∇ <sup>S</sup> 

where V !

transition, <sup>∂</sup> <sup>ψ</sup>

13

!

Initial Evolution of the Moon

DOI: http://dx.doi.org/10.5772/intechopen.84171

are the operators Nabla and Laplace.

∂ρ ∂ t

<sup>L</sup> <sup>∂</sup> <sup>ψ</sup> ! <sup>∂</sup> <sup>t</sup> <sup>¼</sup> <sup>q</sup> ! <sup>ξ</sup>þ<sup>0</sup> � <sup>q</sup> ! ξ�0

þ ∇ ρ V !

is centrifugal potential, ρ is density, η and ξ are coefficients of the first and second viscosity, λ is thermal conductivity coefficient, γ is gravitational constant, Q is total

heat flux density, respectively, before and after the phase boundary, and ∇ and Δ

The main difficulties are connected with the solution of the boundary value problem for the Navier-Stokes equation (Eq. 2). Even in the approximation with constant viscosity coefficients, as used in [16], finding a numerical solution in a 3D spherical layer is a significant problem. In addition, within the framework of equation (Eq. 2), it is difficult to describe the forced convective mixing of a substance near the surface of a growing body when individual bodies fall. The temperature distribution in the body of increasing radius is found from the numerical solution of the boundary value problem for the heat equation, taking into account the possibility of a melt appearing without explicitly highlighting the position of the crystallization front and

where cef , λef is the effective values of heat capacity and thermal conductivities,

Conditions are set on the surface of the pre-planet to ensure the balance of the incoming part of the potential energy of the gravitational interaction of bodies, the expenses of heat for heating the incoming substance and the heat flow re-emitted into the space taking into account the transparency of the external environment:

which take into account the heat of melting in Stefan's [24] problem and the presence of convective heat transfer, T is the sought temperature at a point at time t, and Q is the volume power of internal heat sources. The problem was solved by the finite difference method using an implicit, monotonic, conservative scheme. In Eqs. (1)–(8), the steps on the temporal and spatial grids are the same. The time grid step is variable and, with the density distribution chosen, as a function of depth, is calculated from Eqs. (1)–(2). Using these equations, at each time step, the mass of the growing planet and the distribution of lithostatic pressure in internal regions are calculated. For each value of the achieved size of a growing planet, the melting temperature distribution is calculated. In the core, the dependence of the melting point of mainly iron composition is calculated according to [25]. In the region of the predominantly forming silicate mantle, the dependence of the melting point on pressure is used [26]. The zone of complete and partial melting was determined for each time layer by comparing the calculated temperature distribution with the

the power of internal energy sources per unit volume, L is the heat of phase

<sup>∂</sup> <sup>t</sup> is the velocity of displacement of the interface, q

parametric accounting for convective heat transfer in the melt [23]:

cef ρ ∂T

distribution of the melting temperature at a given depth.

¼ λΔT þ Q (4)

¼ 0 (6)

! ξþ0

<sup>∂</sup> <sup>t</sup> <sup>¼</sup> <sup>∇</sup> <sup>λ</sup>ef <sup>∇</sup><sup>T</sup> <sup>þ</sup> <sup>Q</sup> (8)

and q! ξ�0 the

(7)

ΔW<sup>1</sup> ¼ �4πγ ρ W ¼ W<sup>1</sup> þ W<sup>2</sup> (5)

is fluid velocity, P is pressure, S is entropy, W<sup>1</sup> is gravity potential, W2

where ω is the angular velocity of the orbital motion, σ is the surface density of the substance in the "feeding" zone of the planet, M is the modern mass of the planet, r is the radius of the growing pre-planet, and θ is a statistical parameter taking into account the distribution of particles by mass and velocity in the "feeding" zone. Usually, the study of changes in PT conditions in growing satellites of the planets is not paid attention. In this work, on the contrary, we aim to study how strongly these conditions can depend on the rate of increase in body weight. Let us use the estimate for the mass of the largest-growing body m and the next largest body m1 obtained in the paper [12, 22] for the parameters of bodies in the Earth's group:

$$\frac{m\_1}{m} \approx \frac{1}{3} \left( 1 - 0.6 \left( \frac{m}{M} \right)^{0.3} \right) \tag{2}$$

The mathematical description of mass-energy transport in a growing selfgravitating body of variable radius consists in setting boundary value problems for a system of equations for the balance of momentum, energy, conservation of mass of matter, and the Stefan problem at the boundaries of regions with melt zones [16–18]:

$$
\rho \left[ \frac{\partial \vec{V}}{\partial t} + \left( \vec{V} \,\nabla \right) \vec{V} \right] = -\nabla P + \eta \Delta \,\vec{V} + \left( \frac{\eta}{3} + \xi \right) \nabla \left( \nabla \,\vec{V} \right) - \rho \nabla W \tag{3}
$$

Initial Evolution of the Moon DOI: http://dx.doi.org/10.5772/intechopen.84171

silicate upper envelope could be formed. This creates the conditions for the mechanism of differentiation of a substance into a chemical reservoir of the future predominantly iron core and a silicate reservoir of the Earth's mantle to be realized at small relative velocities of collisions of small bodies—pre-planets. The molten parts were combined, and the solid, mostly silicate, could not yet be held by the weak gravitational field of the planet and returned to the "feed zone." It is this material component that provided the chemical reservoir, from which the second largest body of the Earth-Moon system was formed. Fractionation went on in small bodies and was mostly completed in less than 10 million years, while the formation of the structure of the core and the mantle of the Earth continued for about 100 million years. The union of the liquid internal parts of the colliding bodies occurred as a result of inelastic collision; therefore, most of the potential gravitational energy was converted into heat through the kinetic energy of the collision. The process continued until the core reached a large part of the modern mass. At the final stage of the growth of the pre-planet, the mass of the Earth's pre-planet is already sufficient to hold an ever-increasing proportion of the silicate envelope of the falling out bodies. And the composition of the growing area is increasingly enriched with an admixture of silicates. The process of a completely inelastic collision of accumulated bodies with a high degree of potential energy conversion by gravitational interaction into heat, gradually turned into a mechanism of almost solid-state collisions, in which only a small part of the kinetic energy is converted into heat absorbed by the

The carried out mathematical modeling of the thermal evolution of a growing planet implements the described process scheme. For the growth rate of the pre-

where ω is the angular velocity of the orbital motion, σ is the surface density of the substance in the "feeding" zone of the planet, M is the modern mass of the planet, r is the radius of the growing pre-planet, and θ is a statistical parameter taking into account the distribution of particles by mass and velocity in the "feeding" zone. Usually, the study of changes in PT conditions in growing satellites of the planets is not paid attention. In this work, on the contrary, we aim to study how strongly these conditions can depend on the rate of increase in body weight. Let us use the estimate for the mass of the largest-growing body m and the next largest body m1 obtained in the paper [12, 22] for the parameters of bodies in the Earth's

<sup>2</sup> <sup>ω</sup> <sup>1</sup> � <sup>m</sup>

M � �

σ (1)

(2)

� ρ∇W (3)

planet of the planet, the Safronov model is used in the variant [21]:

<sup>∂</sup> <sup>t</sup> <sup>¼</sup> 2 1ð Þ <sup>þ</sup> <sup>2</sup><sup>θ</sup> <sup>r</sup>

<sup>1</sup> � <sup>0</sup>:<sup>6</sup> <sup>m</sup>

M � �<sup>0</sup>:<sup>3</sup> � �

¼ �∇P þ ηΔ V

The mathematical description of mass-energy transport in a growing selfgravitating body of variable radius consists in setting boundary value problems for a system of equations for the balance of momentum, energy, conservation of mass of matter, and the Stefan problem at the boundaries of regions with melt zones

> ! þ η <sup>3</sup> <sup>þ</sup> <sup>ξ</sup> � �

∇ ∇ V � � !

∂m

m<sup>1</sup> <sup>m</sup> <sup>≈</sup> <sup>1</sup> 3

þ V ! ∇ � �

" #

V !

planet's pre-planet.

Lunar Science

group:

[16–18]:

12

ρ ∂ V ! ∂ t

$$
\rho \, T \left[ \frac{\partial \mathcal{S}}{\partial t} + \left( \vec{\dot{V}} \, \nabla \right) \mathcal{S} \right] = \lambda \Delta \, T + Q \tag{4}
$$

$$
\Delta W\_1 = -4\pi\chi\rho \quad W = W\_1 + W\_2 \tag{5}
$$

$$\frac{\partial \rho}{\partial t} + \nabla \left(\rho \stackrel{\cdot}{V}\right) = 0 \tag{6}$$

$$L\frac{\partial\left.\overrightarrow{\boldsymbol{\Psi}}\right|}{\partial t} = \overrightarrow{\boldsymbol{q}}\Big|\_{\xi+\mathbf{0}} - \overrightarrow{\boldsymbol{q}}\Big|\_{\xi-\mathbf{0}} \tag{7}$$

where V ! is fluid velocity, P is pressure, S is entropy, W<sup>1</sup> is gravity potential, W2 is centrifugal potential, ρ is density, η and ξ are coefficients of the first and second viscosity, λ is thermal conductivity coefficient, γ is gravitational constant, Q is total the power of internal energy sources per unit volume, L is the heat of phase transition, <sup>∂</sup> <sup>ψ</sup> ! <sup>∂</sup> <sup>t</sup> is the velocity of displacement of the interface, q ! ξþ0 and q! ξ�0 the heat flux density, respectively, before and after the phase boundary, and ∇ and Δ are the operators Nabla and Laplace.

The main difficulties are connected with the solution of the boundary value problem for the Navier-Stokes equation (Eq. 2). Even in the approximation with constant viscosity coefficients, as used in [16], finding a numerical solution in a 3D spherical layer is a significant problem. In addition, within the framework of equation (Eq. 2), it is difficult to describe the forced convective mixing of a substance near the surface of a growing body when individual bodies fall. The temperature distribution in the body of increasing radius is found from the numerical solution of the boundary value problem for the heat equation, taking into account the possibility of a melt appearing without explicitly highlighting the position of the crystallization front and parametric accounting for convective heat transfer in the melt [23]:

$$
\omega\_{\text{cf}} \rho \frac{\partial T}{\partial t} = \nabla \left( \lambda\_{\text{f}} \nabla T \right) + Q \tag{8}
$$

where cef , λef is the effective values of heat capacity and thermal conductivities, which take into account the heat of melting in Stefan's [24] problem and the presence of convective heat transfer, T is the sought temperature at a point at time t, and Q is the volume power of internal heat sources. The problem was solved by the finite difference method using an implicit, monotonic, conservative scheme. In Eqs. (1)–(8), the steps on the temporal and spatial grids are the same. The time grid step is variable and, with the density distribution chosen, as a function of depth, is calculated from Eqs. (1)–(2). Using these equations, at each time step, the mass of the growing planet and the distribution of lithostatic pressure in internal regions are calculated. For each value of the achieved size of a growing planet, the melting temperature distribution is calculated. In the core, the dependence of the melting point of mainly iron composition is calculated according to [25]. In the region of the predominantly forming silicate mantle, the dependence of the melting point on pressure is used [26]. The zone of complete and partial melting was determined for each time layer by comparing the calculated temperature distribution with the distribution of the melting temperature at a given depth.

Conditions are set on the surface of the pre-planet to ensure the balance of the incoming part of the potential energy of the gravitational interaction of bodies, the expenses of heat for heating the incoming substance and the heat flow re-emitted into the space taking into account the transparency of the external environment:

One can only hope that further mineralogical and geophysical research results will reduce this uncertainty. One of the results obtained for the temperature distribution over the cross section of the globular sector of the growing Moon model is shown in Figure 2a. As can be seen from the above results, with the rapid growth of the Moon, the concentration of short-lived radioactive elements in the center of the growing body is significant, and their contribution to the energy balance can provide temperatures of 2500–3000 K in the region of R < 300 km. For large values of the lunar radius, numerous melt inclusions are recorded, which, with further evolution of the body, tend to unite. Thus, it becomes possible to trace the formation of the "ocean of magma" [27], the presence of which, according to modern concepts, is necessary for the formation of a powerful lunar crust of an anorthosite composition. Figure 2b presents the numerical results of the possible temperature distribution for a hypothetical body, the rate of increase in mass which is set significantly less compared to the rate of increase in the mass of the moon when it accumulates at (Eq. 1). This simulates the accumulation of the next mass in the Earth-Moon system. For this option, a slow increase in the volume of the nucleus leads to the fact that most of the decay energy of short-lived radioactive elements has time to dissipate into space and the temperature of the central region barely exceeds the melting point of pure iron. For large values of the radius in the variation of the rate of growth of body weight for the conditions of Figure 2b, the distribution of the current temperature values is lower than the melting temperature at a given lithostatic pressure. This eliminates the possibility of endogenous formation of a powerful anorthosite lunar crust and thereby imposes restrictions on the models of the formation of the moon. Common to the physical conditions considered in the variants of Figure 2a and b is that, due to the small magnitude of the gravitational acceleration, convective heat and mass transfer at the considered depths remains weak and for the analyzed period of time, the randomly distributed thermal heterogeneities caused by the falling of bodies still locally persist. Their presence pre-

Initial Evolution of the Moon

DOI: http://dx.doi.org/10.5772/intechopen.84171

determines the further dynamics of the bowels of the growing Moon.

Figure 2.

15

moon; (b) slow growth rate by the ratio (Eq. 2).

It is interesting to compare the resulting numerical modeling of the structure of the lunar crust, mantle, and core with the crust, mantle, and core at the stage of the Earth's accumulation. The distributions of hydrostatically varying pressures, as well as melting points in these structures, lead to their qualitative change. Thus, the relative

The dependence of temperature distribution in the three-dimensional sector of the growing moon on the growth rate of its mass by the time the radius reaches R = 1000 km. (a) a model with the physical parameters of the

Figure 1.

The distribution of the temperature in the model of pre-planetary body. Its radius is (1) 400 km, (2) 300 km, and (3) 250 km [16].

$$
\rho k \rho \frac{\gamma M}{r} \frac{dr}{dt} = \varepsilon \sigma \left[T^4 - T\_1^4\right] + \rho c\_P [T - T\_1] \frac{dr}{dt} \tag{9}
$$

where ρ is the density of matter, γ is the gravitational constant, M is the mass of the growing planet, and r is its radius. T and T1 are, respectively, the body temperature at the boundary, the external environment ε is the coefficient of transparency of the medium, cp is the specific heat, k is the fraction of the potential energy converted to heat, and σ is the Stefan-Boltzmann constant. In Eq. (9), just as in Eqs. (1)–(8), the steps on the temporal and spatial grids are used the same.

The qualitative difference between the obtained variants of the results of numerical simulation in a 3D model is that it was possible to trace the occurrence of thermal and density heterogeneities. The occurrence of these heterogeneities is due to the random distribution of bodies and particles by mass and velocity, which is described by parameter θ in Eq. (1) and is taken into account in Eq. (9) using a random number generator that determines M when calculating the left side of this equation inside the layer, on which is an increase in the radius of the body over time. On Figure 1 we show the results of calculating the temperature for a onedimensional spatial model of a growing body.

## 3. Results and discussion

The results of the numerical solution of the problem for the 3D model of the environment [16, 17] are obtained, which allow us to trace the formation and further dynamics of three-dimensional anomalous in temperature and composition areas resulting from the fall of bodies and particles on the surface of a growing planet when they are randomly distributed over masses. Quantitative estimates of the parameters that determine the solution of the problem are extremely difficult.

## Initial Evolution of the Moon DOI: http://dx.doi.org/10.5772/intechopen.84171

One can only hope that further mineralogical and geophysical research results will reduce this uncertainty. One of the results obtained for the temperature distribution over the cross section of the globular sector of the growing Moon model is shown in Figure 2a. As can be seen from the above results, with the rapid growth of the Moon, the concentration of short-lived radioactive elements in the center of the growing body is significant, and their contribution to the energy balance can provide temperatures of 2500–3000 K in the region of R < 300 km. For large values of the lunar radius, numerous melt inclusions are recorded, which, with further evolution of the body, tend to unite. Thus, it becomes possible to trace the formation of the "ocean of magma" [27], the presence of which, according to modern concepts, is necessary for the formation of a powerful lunar crust of an anorthosite composition. Figure 2b presents the numerical results of the possible temperature distribution for a hypothetical body, the rate of increase in mass which is set significantly less compared to the rate of increase in the mass of the moon when it accumulates at (Eq. 1). This simulates the accumulation of the next mass in the Earth-Moon system. For this option, a slow increase in the volume of the nucleus leads to the fact that most of the decay energy of short-lived radioactive elements has time to dissipate into space and the temperature of the central region barely exceeds the melting point of pure iron. For large values of the radius in the variation of the rate of growth of body weight for the conditions of Figure 2b, the distribution of the current temperature values is lower than the melting temperature at a given lithostatic pressure. This eliminates the possibility of endogenous formation of a powerful anorthosite lunar crust and thereby imposes restrictions on the models of the formation of the moon. Common to the physical conditions considered in the variants of Figure 2a and b is that, due to the small magnitude of the gravitational acceleration, convective heat and mass transfer at the considered depths remains weak and for the analyzed period of time, the randomly distributed thermal heterogeneities caused by the falling of bodies still locally persist. Their presence predetermines the further dynamics of the bowels of the growing Moon.

It is interesting to compare the resulting numerical modeling of the structure of the lunar crust, mantle, and core with the crust, mantle, and core at the stage of the Earth's accumulation. The distributions of hydrostatically varying pressures, as well as melting points in these structures, lead to their qualitative change. Thus, the relative

#### Figure 2.

The dependence of temperature distribution in the three-dimensional sector of the growing moon on the growth rate of its mass by the time the radius reaches R = 1000 km. (a) a model with the physical parameters of the moon; (b) slow growth rate by the ratio (Eq. 2).

kρ γ M r d r

Figure 1.

Lunar Science

and (3) 250 km [16].

dimensional spatial model of a growing body.

3. Results and discussion

14

d t <sup>¼</sup> εσ <sup>T</sup><sup>4</sup> � <sup>T</sup><sup>4</sup>

1 <sup>þ</sup> <sup>ρ</sup>cP½ � <sup>T</sup> � <sup>T</sup><sup>1</sup>

The distribution of the temperature in the model of pre-planetary body. Its radius is (1) 400 km, (2) 300 km,

where ρ is the density of matter, γ is the gravitational constant, M is the mass of the growing planet, and r is its radius. T and T1 are, respectively, the body temperature at the boundary, the external environment ε is the coefficient of transparency of the medium, cp is the specific heat, k is the fraction of the potential energy converted to heat, and σ is the Stefan-Boltzmann constant. In Eq. (9), just as in Eqs. (1)–(8), the steps on the temporal and spatial grids are used the same. The qualitative difference between the obtained variants of the results of numerical simulation in a 3D model is that it was possible to trace the occurrence of thermal and density heterogeneities. The occurrence of these heterogeneities is due to the random distribution of bodies and particles by mass and velocity, which is described by parameter θ in Eq. (1) and is taken into account in Eq. (9) using a random number generator that determines M when calculating the left side of this equation inside the layer, on which is an increase in the radius of the body over time. On Figure 1 we show the results of calculating the temperature for a one-

The results of the numerical solution of the problem for the 3D model of the environment [16, 17] are obtained, which allow us to trace the formation and further dynamics of three-dimensional anomalous in temperature and composition areas resulting from the fall of bodies and particles on the surface of a growing planet when they are randomly distributed over masses. Quantitative estimates of the parameters that determine the solution of the problem are extremely difficult.

d r

d t (9)

proportion of the lunar mantle, as can be seen from Figure 2a and b, is much larger than the fraction of the Earth's mantle (Figures 3 and 4) [18]. The emerging boundaries in the process of accumulation of the crust-mantle of the Moon and the crustmantle of the Earth are much more irregular, which is apparently due to significantly different pressure values at their boundaries. The same can be noted for the boundaries of the core-mantle of the Moon and the Earth. For the Earth, there is a vast solid core and an external molten core, whereas for the Moon, the inner core is either completely absent or does not appear significantly. In the future, already at the geological stage of development, this may lead to a significant difference in the mineral composition of the moon's crust from the crust of the earth. The lunar crust can be predominantly or even exclusively basalt composition.

4. Conclusions

Initial Evolution of the Moon

DOI: http://dx.doi.org/10.5772/intechopen.84171

the planet's dynamics.

data are required.

Author details

Khachay Yuriy

17

Acknowledgements

solutions to the problem.

A numerical solution of the problem is obtained for successively varying temperature and mass distributions over the cross sections of the three-dimensional spherical sector of the Moon model at the stage of its accumulation. It is shown that solutions can be obtained for temperature distribution, based on modern estimates of the physical parameters of the Moon, which provide the endogenous origin of a powerful anorthosite lunar crust. The random distribution of heterogeneities in the inner parts and on the surface of the Moon, caused by the fall of bodies on the growing surface during the accumulation process, controls the initial conditions of

The initial conditions for the Moon and the Earth are taken for a body with a radius of 10 km, the average composition of which corresponds to the matter of carbonaceous chondrites [17] from the common feeding zone of the Earth and the Moon. With their further growth in numerical simulation on the 3D surface of the sphere and at each step of the time grid, which means the radius value, the changing boundary conditions are calculated in accordance with Eq. (5) [18]. These conditions reflect the random distribution of the accumulated bodies in size, composition, and velocity of impact with both the growing Moon and the Earth, which leads to different composition and structure of their internal regions (Figures 2 and 3). In the results presented in Figure 2a and b, in the initial conditions, the concentration of Al2O3 is different: Figure 2a reflects the increased content of Al2O3, and Figure 2b shows the reduced content of Al2O3 [17]. To determine the preferred numerical simulation variant, additional geochemical and geophysical space–time

The author thanks his colleague Antipin A.N. for the participation in obtaining

Institute of Geophysics UB RAS, Yekaterinburg, Russian Federation

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: yu-khachay@yandex.ru

provided the original work is properly cited.

#### Figure 3.

The temperature distribution of the Earth's interior to the end of accumulation along the section of the 3D sector (without taking into account the heat dissipation energy of tidal friction) [18].

#### Figure 4.

An example of the temperature distribution and initial thermal heterogeneities in the protoplanet up to the end of its accumulation, without taking into account the heat of tidal friction [18].

## 4. Conclusions

proportion of the lunar mantle, as can be seen from Figure 2a and b, is much larger than the fraction of the Earth's mantle (Figures 3 and 4) [18]. The emerging boundaries in the process of accumulation of the crust-mantle of the Moon and the crustmantle of the Earth are much more irregular, which is apparently due to significantly different pressure values at their boundaries. The same can be noted for the boundaries of the core-mantle of the Moon and the Earth. For the Earth, there is a vast solid core and an external molten core, whereas for the Moon, the inner core is either completely absent or does not appear significantly. In the future, already at the geological stage of development, this may lead to a significant difference in the mineral composition of the moon's crust from the crust of the earth. The lunar crust

The temperature distribution of the Earth's interior to the end of accumulation along the section of the 3D sector

An example of the temperature distribution and initial thermal heterogeneities in the protoplanet up to the end

(without taking into account the heat dissipation energy of tidal friction) [18].

of its accumulation, without taking into account the heat of tidal friction [18].

can be predominantly or even exclusively basalt composition.

Figure 3.

Lunar Science

Figure 4.

16

A numerical solution of the problem is obtained for successively varying temperature and mass distributions over the cross sections of the three-dimensional spherical sector of the Moon model at the stage of its accumulation. It is shown that solutions can be obtained for temperature distribution, based on modern estimates of the physical parameters of the Moon, which provide the endogenous origin of a powerful anorthosite lunar crust. The random distribution of heterogeneities in the inner parts and on the surface of the Moon, caused by the fall of bodies on the growing surface during the accumulation process, controls the initial conditions of the planet's dynamics.

The initial conditions for the Moon and the Earth are taken for a body with a radius of 10 km, the average composition of which corresponds to the matter of carbonaceous chondrites [17] from the common feeding zone of the Earth and the Moon. With their further growth in numerical simulation on the 3D surface of the sphere and at each step of the time grid, which means the radius value, the changing boundary conditions are calculated in accordance with Eq. (5) [18]. These conditions reflect the random distribution of the accumulated bodies in size, composition, and velocity of impact with both the growing Moon and the Earth, which leads to different composition and structure of their internal regions (Figures 2 and 3).

In the results presented in Figure 2a and b, in the initial conditions, the concentration of Al2O3 is different: Figure 2a reflects the increased content of Al2O3, and Figure 2b shows the reduced content of Al2O3 [17]. To determine the preferred numerical simulation variant, additional geochemical and geophysical space–time data are required.

## Acknowledgements

The author thanks his colleague Antipin A.N. for the participation in obtaining solutions to the problem.

## Author details

Khachay Yuriy Institute of Geophysics UB RAS, Yekaterinburg, Russian Federation

\*Address all correspondence to: yu-khachay@yandex.ru

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Sagitov MG. Moon's gravimetry M. Science. 1979. 432 p

[2] Kovach RL,Watkins JS, Talwani P. Moon. The velocity structure of the lunar crust. Moon. 1973;7(1–2):63-75

[3] Latham C, Ewing M, Dorman J, Press F. All seismic data on the moon. Science. 1970;170(3958):620-626

[4] Nakamura Y, Lammlein D, Latham G. Science. 1973;181(4094):49-51

[5] Toksoz M, Press F, Dainty A, et al. Velocity structure and proportion of the lunar crust. Moon. 1972;4(3-4):490-504

[6] Frondel J. Lunar mineralogy. New York and London: J. Willey & Sons; 1975

[7] Galimov EM. Dynamic model of the moon formation. Geochemistry. 2005; (11):1139-1150

[8] Darwin GH. Satellite planet revolving around a tidally distributed planet. Philosophical Transactions. Royal Society of London. 1880;171: 713-891

[9] Ruskol E. Origin of the Moon. Moscow: Nauka; 1975. 188 p, (in Russian)

[10] Ringwood AE. Composition and origin of the moon. In: Hartman WK, Phillips RL, Taylor GJ, editors. Origin of the Moon. Houstan: Lunar Planet. Inst; 1986. pp. 673-698

[11] Schmidt OY. Four Lectures on the Theory of the Origin of the Earth. Publishing House of the USSR Academy of Sciences; 1957. Moscow, 144p

[12] Vityazev AV, Pechernikova GV, Safronov VS. Earth group planets. Origin and early evolution. Moscow: Science. 1990. 296 p

[13] Cameron AGV, Ward W. The Origin of the Moon. Houston: Proceedings of the Seventh Lunar Science Conference; 1976. pp. 120-122 Dynamics, Russian Academy of Sciences. M.: Academic Book; 2005.

DOI: http://dx.doi.org/10.5772/intechopen.84171

Initial Evolution of the Moon

[23] Tikhonov AN, Lubimova EA, Vlasov VK. On the evolution of the zones of melting in the thermal history of the earth. Dokl. of Akademie of Sciences.USSR. 1969;188(2):388-342

[24] Samarsky AA, Moiseenko BD. Computational Mathematics and Mathematical Physics. 1965;5(5):

[25] Stacy FD. Physics of the Earth.

[26] Kaula EM. Thermal evolution of the

[27] Shubert G,Turcotte DL, Olson P, Mantle convection in the earth and planets. London: Cambridge University

Moscow. Mir. 1972. 342 p

earth and moon growing by planetesimal impacts. Journal of Geophysical Research. 1979;84:

pp. 251-265

816-827

999-1008

19

Press 2001; p. 941

[14] Camp RM. Simulation of the late lunar-forming impact. Icarus. 2004;168: 433-456

[15] Taylor SR. The unique lunar composition and its bearing on the origin of the Moon. The Geochim. Et Cosmochim. Acta. 1987;51(5):1297-1310

[16] Anfilogov VN, Khachai YV. A possible variant of the differentiation of matter at the initial stage of the formation of the earth. DAN. 2005; 403(6):803-806

[17] Anfilogov V, Khachay Y. Some Aspects of the Solar System Formation. Briefs of the Earth Sciences. Cham, Heidelberg, New York, Dordrecht, London: Springer; 2015. 75p

[18] Khachay Yu, Hachay O, Antipin A. Dynamics of the earth's-moon system. Geophysics. Okiwelu A, editor. Intechopen. 2018:119-131

[19] Nichols RH Jr. Short lived radionuclides in meteorites: Constraints for space production. Space Science Reviews. 2000;(1-2):113-122

[20] Nyquist LE, Klein T, Shih C-Y, Reese YD. Distribution of short-lived radioisotopes and secondary mineralization. Geochimica et Cosmochimica Acta. 2009;73:5115-5136

[21] Safronov VS. Evolution of the Pre-Planetary Cloud and the Formation of the Earth and Planets. Moscow: Nauka; 1969. 244 p

[22] Pechernikova GV, Vityazev AV. Impacts and Evolution of the Early Earth/Catastrophic Impacts of Cosmic bodies. Institute of Geosphere

Initial Evolution of the Moon DOI: http://dx.doi.org/10.5772/intechopen.84171

Dynamics, Russian Academy of Sciences. M.: Academic Book; 2005. pp. 251-265

References

Lunar Science

Science. 1979. 432 p

1970;170(3958):620-626

(11):1139-1150

713-891

Russian)

1986. pp. 673-698

Science. 1990. 296 p

18

[1] Sagitov MG. Moon's gravimetry M.

[13] Cameron AGV, Ward W. The Origin of the Moon. Houston: Proceedings of the Seventh Lunar Science Conference; 1976. pp. 120-122

[14] Camp RM. Simulation of the late lunar-forming impact. Icarus. 2004;168:

[15] Taylor SR. The unique lunar composition and its bearing on the origin of the Moon. The Geochim. Et Cosmochim. Acta. 1987;51(5):1297-1310

[16] Anfilogov VN, Khachai YV. A possible variant of the differentiation of

[17] Anfilogov V, Khachay Y. Some Aspects of the Solar System Formation. Briefs of the Earth Sciences. Cham, Heidelberg, New York, Dordrecht, London: Springer; 2015. 75p

[18] Khachay Yu, Hachay O, Antipin A. Dynamics of the earth's-moon system. Geophysics. Okiwelu A, editor. Intechopen. 2018:119-131

radionuclides in meteorites: Constraints for space production. Space Science Reviews. 2000;(1-2):113-122

[20] Nyquist LE, Klein T, Shih C-Y, Reese YD. Distribution of short-lived

Cosmochimica Acta. 2009;73:5115-5136

[21] Safronov VS. Evolution of the Pre-Planetary Cloud and the Formation of the Earth and Planets. Moscow:

[22] Pechernikova GV, Vityazev AV. Impacts and Evolution of the Early Earth/Catastrophic Impacts of Cosmic

bodies. Institute of Geosphere

radioisotopes and secondary mineralization. Geochimica et

Nauka; 1969. 244 p

[19] Nichols RH Jr. Short lived

matter at the initial stage of the formation of the earth. DAN. 2005;

403(6):803-806

433-456

[2] Kovach RL,Watkins JS, Talwani P. Moon. The velocity structure of the lunar crust. Moon. 1973;7(1–2):63-75

[3] Latham C, Ewing M, Dorman J, Press F. All seismic data on the moon. Science.

[4] Nakamura Y, Lammlein D, Latham G. Science. 1973;181(4094):49-51

[5] Toksoz M, Press F, Dainty A, et al. Velocity structure and proportion of the lunar crust. Moon. 1972;4(3-4):490-504

[6] Frondel J. Lunar mineralogy. New York and London: J. Willey & Sons; 1975

[7] Galimov EM. Dynamic model of the moon formation. Geochemistry. 2005;

[8] Darwin GH. Satellite planet revolving around a tidally distributed planet. Philosophical Transactions. Royal Society of London. 1880;171:

[9] Ruskol E. Origin of the Moon. Moscow: Nauka; 1975. 188 p, (in

[10] Ringwood AE. Composition and origin of the moon. In: Hartman WK, Phillips RL, Taylor GJ, editors. Origin of the Moon. Houstan: Lunar Planet. Inst;

[11] Schmidt OY. Four Lectures on the Theory of the Origin of the Earth. Publishing House of the USSR Academy

of Sciences; 1957. Moscow, 144p

[12] Vityazev AV, Pechernikova GV, Safronov VS. Earth group planets. Origin and early evolution. Moscow: [23] Tikhonov AN, Lubimova EA, Vlasov VK. On the evolution of the zones of melting in the thermal history of the earth. Dokl. of Akademie of Sciences.USSR. 1969;188(2):388-342

[24] Samarsky AA, Moiseenko BD. Computational Mathematics and Mathematical Physics. 1965;5(5): 816-827

[25] Stacy FD. Physics of the Earth. Moscow. Mir. 1972. 342 p

[26] Kaula EM. Thermal evolution of the earth and moon growing by planetesimal impacts. Journal of Geophysical Research. 1979;84: 999-1008

[27] Shubert G,Turcotte DL, Olson P, Mantle convection in the earth and planets. London: Cambridge University Press 2001; p. 941

Chapter 3

Moon

Abstract

gravitation

21

1. Introduction

Boris P. Kondratyev

On the Deviation of the Lunar

Center of Mass to the East: Two

Possible Mechanisms Based on

Rounding Off the Shape of the

It is known that the Moon's center of mass (COM) does not coincide with the geometric center of figure (COF) and the line "COF/COM" is not directed to the center of the Earth, but deviates from it to the South-East. Here, we discuss two mechanisms to explain the deviation of the lunar COM to the East from the mean direction to Earth. The first mechanism considers the secular evolution of the Moon's orbit, using the effect of the preferred orientation of the satellite with synchronous rotation to the second (empty) orbital focus. It is established that only the scenario with an increase in the orbital eccentricity e leads to the required displacement of the lunar COM to the East. It is important that high-precision calculations confirm an increase e in our era. In order to fully explain the shift of the lunar COM to the East, a second mechanism was developed that takes into account the influence of tidal changes in the shape of the Moon at its gradual removal from the Earth. The second mechanism predicts that the elongation of the lunar figure in the early era was significant. As a result, it was found that the Moon could have been formed in the annular zone at a distance of 3–4 radii of the modern Earth.

Keywords: Moon, displacement of center of mass, formation and evolution,

optical libration of the Moon in longitude leads to a small (5<sup>0</sup>

At the dawn of modern astronomy, Hevelius and Galileo established that the

terrestrial observer) oscillations of the figure of our satellite in the East-West direction with a period in the anomalistic month. These oscillations disappear when the Moon is at perigee and apogee. Oscillations of a different kind—optical oscillations in latitude—occur with amplitude 6040<sup>0</sup> and a period of one draconic month with the disappearance of the deviation, when the Moon is at the nodes of the orbit.

–80

) seeming (for

Evolution of the Orbit and

## Chapter 3

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on Evolution of the Orbit and Rounding Off the Shape of the Moon

Boris P. Kondratyev

## Abstract

It is known that the Moon's center of mass (COM) does not coincide with the geometric center of figure (COF) and the line "COF/COM" is not directed to the center of the Earth, but deviates from it to the South-East. Here, we discuss two mechanisms to explain the deviation of the lunar COM to the East from the mean direction to Earth. The first mechanism considers the secular evolution of the Moon's orbit, using the effect of the preferred orientation of the satellite with synchronous rotation to the second (empty) orbital focus. It is established that only the scenario with an increase in the orbital eccentricity e leads to the required displacement of the lunar COM to the East. It is important that high-precision calculations confirm an increase e in our era. In order to fully explain the shift of the lunar COM to the East, a second mechanism was developed that takes into account the influence of tidal changes in the shape of the Moon at its gradual removal from the Earth. The second mechanism predicts that the elongation of the lunar figure in the early era was significant. As a result, it was found that the Moon could have been formed in the annular zone at a distance of 3–4 radii of the modern Earth.

Keywords: Moon, displacement of center of mass, formation and evolution, gravitation

## 1. Introduction

At the dawn of modern astronomy, Hevelius and Galileo established that the optical libration of the Moon in longitude leads to a small (5<sup>0</sup> –80 ) seeming (for terrestrial observer) oscillations of the figure of our satellite in the East-West direction with a period in the anomalistic month. These oscillations disappear when the Moon is at perigee and apogee. Oscillations of a different kind—optical oscillations in latitude—occur with amplitude 6040<sup>0</sup> and a period of one draconic month with the disappearance of the deviation, when the Moon is at the nodes of the orbit.

If the Moon was absolutely spherically symmetric, these optical librations would not have resulted in additional rotational oscillations of its body. But since due to the interaction with the Earth, the lunar body has tidal bulges, this leads to the appearance of moments of force from external celestial bodies. Newton [1] predicted that deviations of an elongated body of the Moon from the direction to the Earth must lead to real small rotational librations of the satellite relative to the inertial reference system. These small oscillations are called the physical libration of the Moon.

Besides, the shifts (1) and (2) of the center of the Moon's mass are global in nature, and, ultimately, they already include many different factors (see, e.g., [12]). Therefore, in particular, it is impossible to interpret the displacement of the center

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

Despite the seemingly geometric simplicity of the problem, the offset of the center of the Moon's mass remains an unexplored problem in the lunar science. The importance of this problem is that the Moon is close enough to the Earth and the accuracy of observations of its spin-orbital motion by the method LLR is now so much high that for correct interpretation of these movements it is necessary to take into account many celestial mechanical disturbances, including the indicated inter-

Here, we study the problem of the shift of the Moon's center of mass to the East. To do this, we consider two geometric mechanisms that allow us to explain this important feature of the internal structure of the Moon and shed light on some of the currently controversial features of its evolution and origin (see also [13–15]).

2. Optical libration of the Moon for the observer from the second focus

Instead of the term "the direction of the Moon's surface" often used in references, it is more accurate to speak of the direction of the main lunar axis of inertia, which only in two cases—at the position of the Moon at apogee and perigee—is directed to the center of mass of the Earth-Moon system. To do this, we first consider the optical libration of the Moon in longitude and place the observer in the

Recall that in the first approximation the Moon moves on ellipse (now the eccentricity of the orbit is e ¼ 0:0549 ), and this motion is synchronous, since there is the resonance 1:1 of periods of axial rotation and revolution of the Moon around the Earth. According to the Kepler's first law, the motion is described by the formula

, p ¼ a<sup>1</sup> 1 � e

Here, a<sup>1</sup> is the main semiaxis, and e is the eccentricity of an ellipse. The angle of

The time that has elapsed since the Moon was at perigee (E ¼ 0, υ ¼ 0), until the

cos <sup>υ</sup> <sup>¼</sup> cos <sup>E</sup> � <sup>e</sup>

<sup>t</sup> <sup>¼</sup> ð Þ <sup>E</sup> � <sup>e</sup>sin <sup>E</sup>

where T is the period of revolution on the ellipse. Since the lunar axial angular

<sup>2</sup> : (3)

<sup>1</sup> � <sup>e</sup> cos <sup>E</sup> : (4)

<sup>2</sup><sup>π</sup> T, (5)

<sup>2</sup><sup>π</sup> ð Þ¼ <sup>E</sup> � <sup>e</sup>sin <sup>E</sup> <sup>E</sup> � <sup>e</sup>sin <sup>E</sup>: (6)

<sup>T</sup> , the rotation angle δ of the major

of mass only as a displacement of the lunar core alone.

point of the second (empty) focus of the orbit [2].

<sup>r</sup> <sup>¼</sup> <sup>p</sup>

moment when the angles are equal ð Þ E; υ , is equal to

velocity <sup>Ω</sup> must be equal the mean motion <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

axis of inertia of the Moon (see Figure 1) in the time t will be

<sup>δ</sup> <sup>¼</sup> <sup>t</sup> � <sup>Ω</sup> <sup>¼</sup> <sup>t</sup> � <sup>n</sup> <sup>¼</sup> Tn

From the triangle f <sup>1</sup>MC (Figure 1) follows that

23

1 þ e cos υ

the true anomaly υ is associated with the angle of the eccentric anomaly E

nal asymmetry of the Moon's body.

DOI: http://dx.doi.org/10.5772/intechopen.84465

It is necessary to understand that when moving along the orbit of the Moon, its main axis is not directed at the center of mass of the Earth-Moon system and, on the average, at the second (empty) focus of the lunar orbit [2, 3]. The latter will play an important role in our theory.

Due to the proximity of the Moon in our time, the movement of our satellite is studied with such high accuracy that even a small asymmetry of its internal structure must be taken into account. This asymmetry is manifested in that the center of the Moon's mass COM is offset relative to the geometric center of the lunar figure COF.

This effect of shift is briefly mentioned in [4, 5]. Using astrometric data, an approximate numerical evaluation of the offset was given in [6] and in a more accurate version in [7]. A new approach based on the analysis of data obtained from the Lunar Laser Ranging experiment allowed in [8] clarifies the parameters of the shift of the Moon's center of mass.

Note that the definition of COF depends on the adopted model (sphere, ellipsoid, etc.), so that results of different researchers may be slightly different. However, according to many sources, it is reasonably safe to suggest that two points of the centers on the Moon really do not coincide.

To consider the internal asymmetry of the mass distribution in the lunar body, we introduce a coordinate system with the origin at the center of mass of the Moon, where the X-axis is directed (approximately) to the Earth, the Y-axis to the left (if viewed from the Earth), and the Z-axis—downward. Then, according to the United Lunar Control Network (ULCN), which takes into account the findings of many studies, including information from spacecraft [9], the displacement of the center of the figure relative to the center of mass "COM/COF" is equal to [10]

$$
\Delta \mathfrak{x} \approx -1.71 \text{ km}, \quad \Delta \mathfrak{y} \approx -0.73 \text{ km}, \quad \Delta \mathfrak{z} \approx -0.26 \text{ km}.\tag{1}
$$

Based on the results of a study of the topography of the lunar surface using laser altimetry from a satellite, the displacement of the "COM/COF" was determined more accurately [11]:

$$
\Delta \mathbf{x} \approx -1.7752 \text{ km}, \quad \Delta \mathbf{y} \approx -0.7311 \text{ km}, \quad \Delta \mathbf{z} \approx -0.2399 \text{ km}.\tag{2}
$$

As follows from the analysis of observational data (1) or (2), the effect of displacement of the center of the figure relative to the Moon's center of mass includes not only the shift of the center of mass toward the Earth 0:001 � R (R ¼ 1737:10 km� the average radius of the Moon) but also the spatial deviation of the line "COM/COF" to the North-West. Note that in the literature it often also speaks of the displacement of the center of mass of the Moon relative to the center of its figure; for the observer from the Earth, this shift of the center of mass occurs down (to the South) and to the left (to the East). Then, all the signs in (1) and (2) are reversed. According to (2), the total displacement of the lunar COM is equal to Δ ≈ 1:935 km:

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

Besides, the shifts (1) and (2) of the center of the Moon's mass are global in nature, and, ultimately, they already include many different factors (see, e.g., [12]). Therefore, in particular, it is impossible to interpret the displacement of the center of mass only as a displacement of the lunar core alone.

Despite the seemingly geometric simplicity of the problem, the offset of the center of the Moon's mass remains an unexplored problem in the lunar science. The importance of this problem is that the Moon is close enough to the Earth and the accuracy of observations of its spin-orbital motion by the method LLR is now so much high that for correct interpretation of these movements it is necessary to take into account many celestial mechanical disturbances, including the indicated internal asymmetry of the Moon's body.

Here, we study the problem of the shift of the Moon's center of mass to the East. To do this, we consider two geometric mechanisms that allow us to explain this important feature of the internal structure of the Moon and shed light on some of the currently controversial features of its evolution and origin (see also [13–15]).

## 2. Optical libration of the Moon for the observer from the second focus

Instead of the term "the direction of the Moon's surface" often used in references, it is more accurate to speak of the direction of the main lunar axis of inertia, which only in two cases—at the position of the Moon at apogee and perigee—is directed to the center of mass of the Earth-Moon system. To do this, we first consider the optical libration of the Moon in longitude and place the observer in the point of the second (empty) focus of the orbit [2].

Recall that in the first approximation the Moon moves on ellipse (now the eccentricity of the orbit is e ¼ 0:0549 ), and this motion is synchronous, since there is the resonance 1:1 of periods of axial rotation and revolution of the Moon around the Earth. According to the Kepler's first law, the motion is described by the formula

$$r = \frac{p}{1 + e \cos \nu}, \quad p = a\_1 (1 - e^2). \tag{3}$$

Here, a<sup>1</sup> is the main semiaxis, and e is the eccentricity of an ellipse. The angle of the true anomaly υ is associated with the angle of the eccentric anomaly E

$$\cos \upsilon = \frac{\cos E - e}{1 - e \cos E}. \tag{4}$$

The time that has elapsed since the Moon was at perigee (E ¼ 0, υ ¼ 0), until the moment when the angles are equal ð Þ E; υ , is equal to

$$t = \frac{(E - e\sin E)}{2\pi}T,\tag{5}$$

where T is the period of revolution on the ellipse. Since the lunar axial angular velocity <sup>Ω</sup> must be equal the mean motion <sup>n</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>T</sup> , the rotation angle δ of the major axis of inertia of the Moon (see Figure 1) in the time t will be

$$
\delta = t \cdot \Omega = t \cdot n = \frac{T n}{2\pi} (E - e \sin E) = E - e \sin E. \tag{6}
$$

From the triangle f <sup>1</sup>MC (Figure 1) follows that

If the Moon was absolutely spherically symmetric, these optical librations would not have resulted in additional rotational oscillations of its body. But since due to the interaction with the Earth, the lunar body has tidal bulges, this leads to the appearance of moments of force from external celestial bodies. Newton [1] predicted that deviations of an elongated body of the Moon from the direction to the Earth must lead to real small rotational librations of the satellite relative to the inertial reference system. These small oscillations are called the physical libration of

It is necessary to understand that when moving along the orbit of the Moon, its main axis is not directed at the center of mass of the Earth-Moon system and, on the average, at the second (empty) focus of the lunar orbit [2, 3]. The latter will play an

Due to the proximity of the Moon in our time, the movement of our satellite is studied with such high accuracy that even a small asymmetry of its internal structure must be taken into account. This asymmetry is manifested in that the center of the Moon's mass COM is offset relative to the geometric center of the lunar

This effect of shift is briefly mentioned in [4, 5]. Using astrometric data, an approximate numerical evaluation of the offset was given in [6] and in a more accurate version in [7]. A new approach based on the analysis of data obtained from the Lunar Laser Ranging experiment allowed in [8] clarifies the parameters of the

Note that the definition of COF depends on the adopted model (sphere, ellipsoid, etc.), so that results of different researchers may be slightly different. However, according to many sources, it is reasonably safe to suggest that two points of

To consider the internal asymmetry of the mass distribution in the lunar body, we introduce a coordinate system with the origin at the center of mass of the Moon, where the X-axis is directed (approximately) to the Earth, the Y-axis to the left (if viewed from the Earth), and the Z-axis—downward. Then, according to the United Lunar Control Network (ULCN), which takes into account the findings of many studies, including information from spacecraft [9], the displacement of the center

Δx≈ � 1:71 km, Δy≈ � 0:73 km, Δz≈ � 0:26 km: (1)

Δx≈ � 1:7752 km, Δy≈ � 0:7311 km, Δz≈ � 0:2399 km: (2)

Based on the results of a study of the topography of the lunar surface using laser altimetry from a satellite, the displacement of the "COM/COF" was determined

As follows from the analysis of observational data (1) or (2), the effect of displacement of the center of the figure relative to the Moon's center of mass includes not only the shift of the center of mass toward the Earth 0:001 � R

(R ¼ 1737:10 km� the average radius of the Moon) but also the spatial deviation of the line "COM/COF" to the North-West. Note that in the literature it often also speaks of the displacement of the center of mass of the Moon relative to the center of its figure; for the observer from the Earth, this shift of the center of mass occurs down (to the South) and to the left (to the East). Then, all the signs in (1) and (2) are reversed. According to (2), the total displacement of the lunar COM is equal to

of the figure relative to the center of mass "COM/COF" is equal to [10]

the Moon.

Lunar Science

figure COF.

important role in our theory.

shift of the Moon's center of mass.

more accurately [11]:

Δ ≈ 1:935 km:

22

the centers on the Moon really do not coincide.

#### Figure 1.

The large ellipse is the orbit of the Moon M (for clarity, the ellipticity is exaggerated), and P and A are the points of perigee and apogee. The point of active focus f <sup>1</sup> is the center of mass of the Earth-Moon system, and f <sup>2</sup> is the point of the second (passive) focus.

$$\frac{d\_{Cf\_1}}{\sin \chi} = \frac{r}{\sin \left(\nu - \chi\right)}, \quad \delta + \chi = \nu,\tag{7}$$

It is important to emphasize that, according to formula (11), the effect of the

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

Thus, when the Moon is moving on the ellipse around the Earth, the end of the major axis of inertia will be approximately directed to the point of the second focus. Strictly speaking, this end of the axis will perform (without taking into account the very small physical libration of the Moon in longitude) oscillatory motions in the

> Δ a1

The results of calculations (12) and (13) show a small asymmetry oscillations (� 11%) relative to the right and left sides of the point f2. Emphasize that the physical libration of the Moon in longitude has a very small amplitude and with a

3. Resolution alternatives to choose between two options for the lunar

Since Darwin [16], many efforts were made to examine the secular evolution of

but so far it has not been established whether the orbit of the Moon in the past more or less oblate than now. In the literature, this issue is still under discussion. In this regard, the study of the shift of the Moon's center of mass to the East may shed

Many researchers agree that gravitational differentiation of the Moon occurred in the early era (see, e.g., [17]), with the result that the Moon's center of mass is slightly ð Þ � 0:001 � R shifted toward the Earth. We shall not discuss here the question of the gravitational differentiation of the Moon and just to note that one of the reasons for the displacement of the Moon's center of mass to the Earth can be some asymmetry of tidal forces from the Earth into two hemispheres of the Moon

<sup>≤</sup>1:<sup>4275</sup> � <sup>10</sup>�<sup>3</sup>

�612km ≤ Δ ≤548km: (13)

: (12)

�1:<sup>5933</sup> � <sup>10</sup>�<sup>3</sup> <sup>≤</sup>

In our era, in a linear measure, this is approximately

<sup>a</sup><sup>1</sup> as a function of the true anomaly υ.

<sup>a</sup><sup>1</sup> is already in the first approximation proportional to the square of the

deviation <sup>Δ</sup>

Graph of deviation <sup>Δ</sup>

Figure 2.

eccentricity of the Moon's orbit.

DOI: http://dx.doi.org/10.5772/intechopen.84465

large reserve of fits in the interval (13).

some light on this important issue.

vicinity f <sup>2</sup> in the interval

orbit evolution

the Moon's orbit,

25

so

$$d\_{Cf\_1} = r \cdot \frac{\sin \chi}{\sin \left(\nu - \chi\right)} = r \frac{\sin \chi}{\sin \delta}.$$

Then, the distance Δ ¼ dC f <sup>2</sup> ¼ 2a1e � dC f <sup>1</sup> is

$$\frac{\Delta}{a\_1} = 2e - \frac{1 - e^2}{1 + e \cos \nu} \frac{\sin \chi}{\sin \delta} = 2e - \frac{1 - e^2}{1 + e \cos \nu} \{ \sin \nu \text{ ctg } \delta - \cos \nu \}. \tag{8}$$

Here, ctg δ is the function of the angle E (or true anomaly υ)

$$\text{ctg } \delta = \text{ctg}\left(E - e\sin E\right) = \frac{\mathbf{1} + \frac{\sqrt{1 - e^2}\sin\nu}{\epsilon + \cos\nu} \cdot \text{tg}\left[\frac{\epsilon\sqrt{1 - \epsilon^2}\sin\nu}{1 + \epsilon\cos\nu}\right]}{\frac{\sqrt{1 - \epsilon^2}\sin\nu}{\epsilon + \cos\nu} - \text{tg}\left[\frac{\epsilon\sqrt{1 - \epsilon^2}\sin\nu}{1 + \epsilon\cos\nu}\right]} . \tag{9}$$

Therefore, the required distance <sup>Δ</sup> <sup>a</sup><sup>1</sup> from the point f <sup>2</sup>, which is a continuation of the lunar major inertia axis that crosses the apsidal line, is not, generally speaking, zero and equal to

$$\frac{\Delta}{a\_1} = e + \cos E - \text{ ctg}\,\delta\sqrt{1 - e^2}\sin E \tag{10}$$

Expanding in powers of a small eccentricity gives

$$\begin{aligned} \frac{\Delta}{a\_1} &= -\frac{\cos\nu}{2}e^2 - \frac{1}{3} \left( 1 + \frac{\cos^2\nu}{2} \right) e^3 - \frac{\cos\nu}{8} \left( 7 - 4\cos^2\nu \right) e^4 + \dots; \quad \Delta \le 0;\\ \frac{\Delta}{a\_1} &= -\frac{\cos E}{2}e^2 + \frac{1}{3} \left( \frac{1}{2} - 2\cos^2 E \right) e^3 - \frac{\cos E}{24} \left( 1 + 8\cos^2 E \right) e^4 + \dots; \quad \Delta \ge 0. \end{aligned} \tag{11}$$

The results of calculations using formula (10) are shown in Figure 2.

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

#### Figure 2.

dC f <sup>1</sup>

Then, the distance Δ ¼ dC f <sup>2</sup> ¼ 2a1e � dC f <sup>1</sup> is

sin χ

ctg δ ¼ ctg ð Þ¼ E � esin E

Δ a1

Expanding in powers of a small eccentricity gives

<sup>2</sup> � 2 cos <sup>2</sup>

� �

cos <sup>2</sup>υ 2 � �

e

E

e

The results of calculations using formula (10) are shown in Figure 2.

so

Figure 1.

Lunar Science

Δ a1

zero and equal to

¼ � cos <sup>υ</sup> <sup>2</sup> <sup>e</sup>

¼ � cos <sup>E</sup> <sup>2</sup> <sup>e</sup> 2 þ 1 3 1

Δ a1

Δ a1

24

<sup>¼</sup> <sup>2</sup><sup>e</sup> � <sup>1</sup> � <sup>e</sup><sup>2</sup>

the point of the second (passive) focus.

1 þ e cos υ

Therefore, the required distance <sup>Δ</sup>

<sup>2</sup> � <sup>1</sup> 3 1 þ

sin <sup>χ</sup> <sup>¼</sup> <sup>r</sup>

dC f <sup>1</sup> <sup>¼</sup> <sup>r</sup> � sin <sup>χ</sup>

sin <sup>δ</sup> <sup>¼</sup> <sup>2</sup><sup>e</sup> � <sup>1</sup> � <sup>e</sup><sup>2</sup>

Here, ctg δ is the function of the angle E (or true anomaly υ)

sin ð Þ <sup>υ</sup> � <sup>χ</sup> <sup>¼</sup> <sup>r</sup>

The large ellipse is the orbit of the Moon M (for clarity, the ellipticity is exaggerated), and P and A are the points of perigee and apogee. The point of active focus f <sup>1</sup> is the center of mass of the Earth-Moon system, and f <sup>2</sup> is

1 þ e cos υ

<sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffi <sup>1</sup>�e<sup>2</sup> <sup>p</sup> sin <sup>υ</sup>

> ffiffiffiffiffiffiffi <sup>1</sup>�e<sup>2</sup> <sup>p</sup> sin <sup>υ</sup>

the lunar major inertia axis that crosses the apsidal line, is not, generally speaking,

<sup>¼</sup> <sup>e</sup> <sup>þ</sup> cos <sup>E</sup> � ctg <sup>δ</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>3</sup> � cos <sup>υ</sup>

<sup>3</sup> � cos <sup>E</sup> 24

<sup>8</sup> <sup>7</sup> � 4 cos <sup>2</sup> υ � �e

> <sup>1</sup> <sup>þ</sup> 8 cos <sup>2</sup> E � �e

sin ð Þ <sup>υ</sup> � <sup>χ</sup> , <sup>δ</sup> <sup>þ</sup> <sup>χ</sup> <sup>¼</sup> <sup>υ</sup>, (7)

f g sin υ ctg δ � cos υ : (8)

h i : (9)

<sup>1</sup>�e<sup>2</sup> <sup>p</sup> sin <sup>υ</sup> 1þe cos υ h i

<sup>1</sup> � <sup>e</sup><sup>2</sup> <sup>p</sup> sin <sup>E</sup> (10)

<sup>4</sup> <sup>þ</sup> …; <sup>Δ</sup> <sup>≤</sup>0;

<sup>4</sup> <sup>þ</sup> …; <sup>Δ</sup> <sup>≥</sup>0:

(11)

<sup>1</sup>�e<sup>2</sup> <sup>p</sup> sin <sup>υ</sup> 1þe cos υ

<sup>a</sup><sup>1</sup> from the point f <sup>2</sup>, which is a continuation of

sin χ sin δ :

<sup>e</sup><sup>þ</sup> cos <sup>υ</sup> � tg <sup>e</sup> ffiffiffiffiffiffiffi

<sup>e</sup><sup>þ</sup> cos <sup>υ</sup> � tg <sup>e</sup> ffiffiffiffiffiffiffi

Graph of deviation <sup>Δ</sup> <sup>a</sup><sup>1</sup> as a function of the true anomaly υ.

It is important to emphasize that, according to formula (11), the effect of the deviation <sup>Δ</sup> <sup>a</sup><sup>1</sup> is already in the first approximation proportional to the square of the eccentricity of the Moon's orbit.

Thus, when the Moon is moving on the ellipse around the Earth, the end of the major axis of inertia will be approximately directed to the point of the second focus. Strictly speaking, this end of the axis will perform (without taking into account the very small physical libration of the Moon in longitude) oscillatory motions in the vicinity f <sup>2</sup> in the interval

$$-1.5933 \cdot 10^{-3} \le \frac{\Delta}{a\_1} \le 1.4275 \cdot 10^{-3}.\tag{12}$$

In our era, in a linear measure, this is approximately

$$-612\,\mathrm{km} \le \Delta \le 548\,\mathrm{km}.\tag{13}$$

The results of calculations (12) and (13) show a small asymmetry oscillations (� 11%) relative to the right and left sides of the point f2. Emphasize that the physical libration of the Moon in longitude has a very small amplitude and with a large reserve of fits in the interval (13).

## 3. Resolution alternatives to choose between two options for the lunar orbit evolution

Since Darwin [16], many efforts were made to examine the secular evolution of the Moon's orbit,

but so far it has not been established whether the orbit of the Moon in the past more or less oblate than now. In the literature, this issue is still under discussion. In this regard, the study of the shift of the Moon's center of mass to the East may shed some light on this important issue.

Many researchers agree that gravitational differentiation of the Moon occurred in the early era (see, e.g., [17]), with the result that the Moon's center of mass is slightly ð Þ � 0:001 � R shifted toward the Earth. We shall not discuss here the question of the gravitational differentiation of the Moon and just to note that one of the reasons for the displacement of the Moon's center of mass to the Earth can be some asymmetry of tidal forces from the Earth into two hemispheres of the Moon

where R<sup>0</sup> is the distance between the centers of the Earth-Moon and R is the distance from the center of the Moon to the near (far) points of its surface. The

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

<sup>Δ</sup>F<sup>⊕</sup> <sup>¼</sup> <sup>F</sup><sup>1</sup> � <sup>F</sup><sup>2</sup> <sup>≈</sup> <sup>6</sup>GM<sup>⊕</sup>

3.2 The first version: the evolution of the lunar orbit with increase in its

First, suppose that in the early epoch the orbit of the Moon was more circular than in our epoch. Consequently, during the secular evolution, the Moon's orbit became more and more eccentric, up to its modern value of eccentricity e ¼ 0:0549: Recall now that the Moon's COM, already shifted toward the Earth, after the solidification of the lunar body will be fixed relative to its main axes of inertia. Since in the early epoch the orbit of the Moon was almost circular, the line connecting the geometrical center of the figure of the Moon and its center of mass was directed

However, since in this version of the secular evolution the orbit of the Moon becomes more eccentric, two foci appear (Figure 3b). In accordance with the laws

From Figure 3b, it can be seen that, for the observer from the Earth (point f1), the center of mass S will now be located on the left (to the East) from the direction to the center of the Moon (see also Figure 5). Thus, in the first variant of the evolution of the Moon's orbit, the modern Earth's observer, in accordance with Figure 3b, will see the Moon's center of mass displaced to the left (to the East) from the direction to the center of the figure. It is this location of the center of mass of the Moon relative to the center of its

The contribution of this mechanism to the displacement of the Moon's center of

3.3 The second version of the evolution: from more flattened to less flattened

If we assume that the orbit of the young Moon was more eccentric in the early

Thus, Figure 4 shows that in the second version of the evolution of the orbit a modern observer from the Earth would see that the center of mass of the Moon is

era than it is now, that is, during the secular evolution, the Moon's orbit was rounded; then in our era, when the orbital eccentricity decreased to the current value e ¼ 0:0549, instead of Figure 3b, we will see the location of the center of

In the era of its formation, the Moon could be much closer to Earth than in our era (see, e.g., [16, 18, 19]). Due to the proximity to the Earth of the young Moon, the difference in tidal forces (15) in both lunar hemispheres was much more in the early era than it is now. In the era of the differentiation of the Moon, it was this difference in tidal forces (15) that caused the displacement of the center of mass of the Moon toward the Earth. Based on these provisions, we note that the very solution to the question of the displacement of the Moon's COM to the East is closely related to the further secular evolution of its form and orbit. In particular, to find out how the lunar COM would be located relative to the Earth's observer in the modern era, when its orbit evolved and eccentricity acquired modern significance, consider two possible options with the initial eccentricity of the young Moon orbit.

R2 0 x2

: (15)

difference of these forces will be

DOI: http://dx.doi.org/10.5772/intechopen.84465

eccentricity

exactly to the Earth (Figure 3a).

of celestial mechanics, as we know

figure that we observe in our era.

lunar orbit

27

mass to the East will be made in Section 4.

mass of the Moon, as shown in Figure 4.

#### Figure 3.

(a) Orientation of the displaced center of mass S of the young Moon after the differentiation of the substance of its body. A large circle is the orbit of the Moon in the early epoch, and a small ellipse is the cross section of the Moon. Since the orbit is circular, the focuses f<sup>1</sup> and f<sup>2</sup> coincide with the center O. Relative sizes are not respected. The line O<sup>0</sup> S is directed straight to the Earth; therefore, the Earth's observer would see both points coinciding with each other from Section 2, the motion on the ellipse the line passing through the center of the Moon's figure and its center of mass be directed to the second (empty) focus of the orbit. Therefore, in our era, when the eccentricity of the lunar orbit has increased to its current value e ¼ 0:0549, we will observe the picture as in b. (b) The orientation of the lunar center of mass S in our era in the first version of the evolution of the Moon's orbit. The large ellipse is the orbit of the Moon, and the small ellipse is the cross section of the Moon. The Earth is in the first focus f<sup>1</sup> of the lunar orbit. The angle E characterizes the orientation of the Moon COM S relative to the direction to Earth.

(Sect. 4.1). One of the manifestations of the offset center of mass can be a different thickness of crust in the near side and the far side of the Moon [18].

Thus, the core of the Moon was formed during the gravitational differentiation, and then under the influence of a small asymmetry of tidal forces, the process of displacement of the lunar center of mass toward the Earth began to occur. This offset COM for the Earth observer can be characterized by the orientation angle E between the line }COF=COM} and the direction to the center of the Earth (Figure 3b).

## 3.1 On the difference on tidal forces from the Earth in near and far lunar hemispheres

Assuming that the differentiation of the Moon occurred (according to cosmogonic times) rather quickly, it is necessary to require that the shift of the lunar center of mass toward the Earth occurred even before the Moon hardened.

The real cause of the displacement of the Moon's center of mass to the Earth could be some asymmetry of tidal forces. Let us perform the required calculations. After the capture of the Moon in resonance 1:1, it was possible to talk about near and far of its hemispheres. It is clear that the forces in the nearest and farthest points are, respectively, equal to

$$\begin{aligned} F\_1 &\approx \frac{2GM\_\oplus}{R\_0^2} \varkappa \left( 1 + \frac{3}{2}\varkappa \right), \\ F\_2 &\approx \frac{2GM\_\oplus}{R\_0^2} \varkappa \left( 1 - \frac{3}{2}\varkappa \right), \quad \varkappa = \frac{R}{R\_0}, \end{aligned} \tag{14}$$

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

where R<sup>0</sup> is the distance between the centers of the Earth-Moon and R is the distance from the center of the Moon to the near (far) points of its surface. The difference of these forces will be

$$
\Delta F\_{\oplus} = F\_1 - F\_2 \approx \frac{\xi G M\_{\oplus}}{R\_0^2} \varkappa^2. \tag{15}
$$

In the era of its formation, the Moon could be much closer to Earth than in our era (see, e.g., [16, 18, 19]). Due to the proximity to the Earth of the young Moon, the difference in tidal forces (15) in both lunar hemispheres was much more in the early era than it is now. In the era of the differentiation of the Moon, it was this difference in tidal forces (15) that caused the displacement of the center of mass of the Moon toward the Earth. Based on these provisions, we note that the very solution to the question of the displacement of the Moon's COM to the East is closely related to the further secular evolution of its form and orbit. In particular, to find out how the lunar COM would be located relative to the Earth's observer in the modern era, when its orbit evolved and eccentricity acquired modern significance, consider two possible options with the initial eccentricity of the young Moon orbit.

## 3.2 The first version: the evolution of the lunar orbit with increase in its eccentricity

First, suppose that in the early epoch the orbit of the Moon was more circular than in our epoch. Consequently, during the secular evolution, the Moon's orbit became more and more eccentric, up to its modern value of eccentricity e ¼ 0:0549:

Recall now that the Moon's COM, already shifted toward the Earth, after the solidification of the lunar body will be fixed relative to its main axes of inertia. Since in the early epoch the orbit of the Moon was almost circular, the line connecting the geometrical center of the figure of the Moon and its center of mass was directed exactly to the Earth (Figure 3a).

However, since in this version of the secular evolution the orbit of the Moon becomes more eccentric, two foci appear (Figure 3b). In accordance with the laws of celestial mechanics, as we know

From Figure 3b, it can be seen that, for the observer from the Earth (point f1), the center of mass S will now be located on the left (to the East) from the direction to the center of the Moon (see also Figure 5). Thus, in the first variant of the evolution of the Moon's orbit, the modern Earth's observer, in accordance with Figure 3b, will see the Moon's center of mass displaced to the left (to the East) from the direction to the center of the figure. It is this location of the center of mass of the Moon relative to the center of its figure that we observe in our era.

The contribution of this mechanism to the displacement of the Moon's center of mass to the East will be made in Section 4.

## 3.3 The second version of the evolution: from more flattened to less flattened lunar orbit

If we assume that the orbit of the young Moon was more eccentric in the early era than it is now, that is, during the secular evolution, the Moon's orbit was rounded; then in our era, when the orbital eccentricity decreased to the current value e ¼ 0:0549, instead of Figure 3b, we will see the location of the center of mass of the Moon, as shown in Figure 4.

Thus, Figure 4 shows that in the second version of the evolution of the orbit a modern observer from the Earth would see that the center of mass of the Moon is

(Sect. 4.1). One of the manifestations of the offset center of mass can be a different

(a) Orientation of the displaced center of mass S of the young Moon after the differentiation of the substance of its body. A large circle is the orbit of the Moon in the early epoch, and a small ellipse is the cross section of the Moon. Since the orbit is circular, the focuses f<sup>1</sup> and f<sup>2</sup> coincide with the center O. Relative sizes are not respected.

S is directed straight to the Earth; therefore, the Earth's observer would see both points coinciding with each other from Section 2, the motion on the ellipse the line passing through the center of the Moon's figure and its center of mass be directed to the second (empty) focus of the orbit. Therefore, in our era, when the eccentricity of the lunar orbit has increased to its current value e ¼ 0:0549, we will observe the picture as in b. (b) The orientation of the lunar center of mass S in our era in the first version of the evolution of the Moon's orbit. The large ellipse is the orbit of the Moon, and the small ellipse is the cross section of the Moon. The Earth is in the first focus f<sup>1</sup> of the lunar orbit. The angle E characterizes the orientation of the Moon COM S relative to

3.1 On the difference on tidal forces from the Earth in near and far lunar

Assuming that the differentiation of the Moon occurred (according to cosmogonic times) rather quickly, it is necessary to require that the shift of the lunar center of mass toward the Earth occurred even before the Moon hardened.

The real cause of the displacement of the Moon's center of mass to the Earth could be some asymmetry of tidal forces. Let us perform the required calculations. After the capture of the Moon in resonance 1:1, it was possible to talk about near and far of its hemispheres. It is clear that the forces in the nearest and farthest points

> x 1 þ 3 2 x

> <sup>x</sup> <sup>1</sup> � <sup>3</sup> 2 x

,

, x <sup>¼</sup> <sup>R</sup> R0 , (14)

<sup>F</sup><sup>1</sup> <sup>≈</sup> <sup>2</sup>GM<sup>⊕</sup> R2 0

<sup>F</sup><sup>2</sup> <sup>≈</sup> <sup>2</sup>GM<sup>⊕</sup> R2 0

Thus, the core of the Moon was formed during the gravitational differentiation, and then under the influence of a small asymmetry of tidal forces, the process of displacement of the lunar center of mass toward the Earth began to occur. This offset COM for the Earth observer can be characterized by the orientation angle E between the line }COF=COM} and the direction to the center of the Earth (Figure 3b).

thickness of crust in the near side and the far side of the Moon [18].

hemispheres

Figure 3.

Lunar Science

The line O<sup>0</sup>

the direction to Earth.

are, respectively, equal to

26

sin <sup>E</sup> <sup>¼</sup> <sup>2</sup>esin <sup>υ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>e</sup> cos <sup>υ</sup>

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

In particular, for the Moon's orbit, the current value of eccentricity is equal

Taking into account (18), in the framework of the first variant of the evolution mechanism of the lunar orbit from the circle to the ellipse with the modern value of eccentricity, we find that the ratio of the average angle h i <sup>E</sup> to the angle arctg <sup>Δ</sup><sup>y</sup>

Therefore, the first orbital evolution mechanism helps to explain approximately 18% of the observed current Moon's offset COM to the East. In the linear measure,

We emphasize that the conclusion of the theory that evolution of the orbit of the Moon occurred with increasing eccentricity is consistent with the fact that at the present time the eccentricity of the orbit of the Moon is really growing and, there-

Besides, the following should be noted. As is well known, due to perturbations, all elements of the lunar orbit are subject to periodic perturbations [20, 26]. Thus, for several thousand years, the eccentricity of the Moon's orbit changes due to solar perturbations in the range from 0.0255 to 0.0775. However, here we do not consider the periodic perturbations: throughout in this chapter, we are talking about tidal secular change in the average eccentricity of the Moon's orbit, which is now equal

5. Second mechanism of displacement of the Moon's center of mass

Because of proximity of the Moon to Earth during an early era, which is offered by many researchers, the main factor of formation for the Moon is a tidal force from our planet. In the tidal field of the Earth, the figure of the early Moon stretched out, which was also facilitated by its capture in spin-orbit resonance 1:1. Therefore, for our approximate calculations, we can simulate the figure of the Moon using the elongated (toward the Earth) spheroid with the semiaxes a<sup>1</sup> . a<sup>2</sup> ¼ a3. The equation

arcsin <sup>2</sup>esin <sup>υ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>e</sup> cos <sup>υ</sup> 1 þ e<sup>2</sup> þ 2e cos υ

Then, the average angle E is given by the integral

h i <sup>E</sup> <sup>¼</sup> <sup>1</sup> π ð π

e≈0:0549, and formula (17) gives

DOI: http://dx.doi.org/10.5772/intechopen.84465

will be

it is

e≈ 0:0549:

29

to the East

0

<sup>κ</sup> <sup>¼</sup> h i <sup>E</sup>

fore, in the past it was less than today [20, 21] (see also [22–25]).

of the surface of this spheroid in Cartesian coordinates Ox1x2x<sup>3</sup> is

x2 1 a2 1 þ x2 <sup>2</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 3 a2 3

<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>e</sup> cos <sup>υ</sup> : (16)

� �: (17)

h i E ≈ 0:0700: (18)

arctg 0ð Þ :7311=1:<sup>7752</sup> <sup>≈</sup> <sup>0</sup>:18: (19)

j j Δy ≈ 0:132 km: (20)

¼ 1: (21)

Δx

#### Figure 4.

The final configuration of the location of the lunar COMS in the second version of the evolution of its orbit. The line O<sup>0</sup> f <sup>1</sup> sets the direction (for the Earth observer) to the center of mass of the Moon in the early era, and the line O0 f 0 <sup>1</sup> sets the direction to the center of the figure of the Moon for the observer from the Earth in our time. In this version, the observer would see that the center of mass of the Moon S is shifted to the right (to the West, as indicated by the arrow) from the average direction to the center O<sup>0</sup> of the figure of the Moon.

#### Figure 5.

Elongated figure of the Moon in the early era (its cross section of the plane x<sup>2</sup> ¼ 0 is shown by the ellipse with the semiaxes a<sup>1</sup> . a3). The arrows represent the directions from the center O<sup>0</sup> of the Moon to both foci f<sup>1</sup> and f<sup>2</sup> of its orbit around the Earth (the Earth in focus f1), as well as to the center of the mass S of the Moon. The angle α between the line on S and the line to f<sup>1</sup> measures orientation of the Moon's center of mass, and the angle E between the directions to the first focus f<sup>1</sup> and second focus f<sup>2</sup> measures the deflection of an ellipse from a circle.

shifted to the right (to the West) from the direction to the center of the figure. However, this is contrary to observations, so the second version of the evolution must be discarded.

## 4. Correction factor to mechanism of orbit evolution

Let us consider again (Figure 1) the motion of a satellite in an elliptical orbit around a body of greater mass. The equation of an ellipse is given by formula (3). From the triangle O<sup>0</sup> f <sup>1</sup> f <sup>2</sup> by the sine theorem, we find the relation

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

$$\sin E = \frac{2e\sin\nu(\mathbf{1} + e\cos\nu)}{\mathbf{1} + e^2 + 2e\cos\nu}. \tag{16}$$

Then, the average angle E is given by the integral

$$\langle E \rangle = \frac{1}{\pi} \int\_0^{\pi} \arcsin \left[ \frac{2e \sin \nu (1 + e \cos \nu)}{1 + e^2 + 2e \cos \nu} \right]. \tag{17}$$

In particular, for the Moon's orbit, the current value of eccentricity is equal e≈0:0549, and formula (17) gives

$$
\langle E \rangle \approx \mathbf{0}.0700. \tag{18}
$$

Taking into account (18), in the framework of the first variant of the evolution mechanism of the lunar orbit from the circle to the ellipse with the modern value of eccentricity, we find that the ratio of the average angle h i <sup>E</sup> to the angle arctg <sup>Δ</sup><sup>y</sup> Δx will be

$$\kappa = \frac{\langle E \rangle}{\text{arctg}(0.7311/1.7752)} \approx 0.18. \tag{19}$$

Therefore, the first orbital evolution mechanism helps to explain approximately 18% of the observed current Moon's offset COM to the East. In the linear measure, it is

$$|\Delta y| \approx 0.132 \text{ } km.\tag{20}$$

We emphasize that the conclusion of the theory that evolution of the orbit of the Moon occurred with increasing eccentricity is consistent with the fact that at the present time the eccentricity of the orbit of the Moon is really growing and, therefore, in the past it was less than today [20, 21] (see also [22–25]).

Besides, the following should be noted. As is well known, due to perturbations, all elements of the lunar orbit are subject to periodic perturbations [20, 26]. Thus, for several thousand years, the eccentricity of the Moon's orbit changes due to solar perturbations in the range from 0.0255 to 0.0775. However, here we do not consider the periodic perturbations: throughout in this chapter, we are talking about tidal secular change in the average eccentricity of the Moon's orbit, which is now equal e≈ 0:0549:

## 5. Second mechanism of displacement of the Moon's center of mass to the East

Because of proximity of the Moon to Earth during an early era, which is offered by many researchers, the main factor of formation for the Moon is a tidal force from our planet. In the tidal field of the Earth, the figure of the early Moon stretched out, which was also facilitated by its capture in spin-orbit resonance 1:1. Therefore, for our approximate calculations, we can simulate the figure of the Moon using the elongated (toward the Earth) spheroid with the semiaxes a<sup>1</sup> . a<sup>2</sup> ¼ a3. The equation of the surface of this spheroid in Cartesian coordinates Ox1x2x<sup>3</sup> is

$$\frac{\mathbf{x}\_1^2}{a\_1^2} + \frac{\mathbf{x}\_2^2 + \mathbf{x}\_3^2}{a\_3^2} = \mathbf{1}.\tag{21}$$

shifted to the right (to the West) from the direction to the center of the figure. However, this is contrary to observations, so the second version of the evolution must be

Elongated figure of the Moon in the early era (its cross section of the plane x<sup>2</sup> ¼ 0 is shown by the ellipse with the

orbit around the Earth (the Earth in focus f1), as well as to the center of the mass S of the Moon. The angle α between the line on S and the line to f<sup>1</sup> measures orientation of the Moon's center of mass, and the angle E between the directions to the first focus f<sup>1</sup> and second focus f<sup>2</sup> measures the deflection of an ellipse from a circle.

of the Moon to both foci f<sup>1</sup> and f<sup>2</sup> of its

The final configuration of the location of the lunar COMS in the second version of the evolution of its orbit. The

indicated by the arrow) from the average direction to the center O<sup>0</sup> of the figure of the Moon.

f <sup>1</sup> sets the direction (for the Earth observer) to the center of mass of the Moon in the early era, and the line

<sup>1</sup> sets the direction to the center of the figure of the Moon for the observer from the Earth in our time. In this version, the observer would see that the center of mass of the Moon S is shifted to the right (to the West, as

Let us consider again (Figure 1) the motion of a satellite in an elliptical orbit around a body of greater mass. The equation of an ellipse is given by formula (3).

f <sup>1</sup> f <sup>2</sup> by the sine theorem, we find the relation

4. Correction factor to mechanism of orbit evolution

semiaxes a<sup>1</sup> . a3). The arrows represent the directions from the center O<sup>0</sup>

discarded.

28

Figure 5.

Figure 4.

Lunar Science

line O<sup>0</sup>

O0 f 0

From the triangle O<sup>0</sup>

The main symmetry semiaxis a<sup>1</sup> of this spheroid was initially directed exactly to the Earth.

Let us consider Figure 5. Due to the small orbit eccentricity, the angle E between the main axis of the Moon's figure and the direction to f<sup>1</sup> was also initially small. However, in the evolution of the Moon's orbit from the less eccentric to the more eccentric, as was shown in the first mechanism, the angle E will increase monotonically. This factor changes the orientation of the figure of the Moon relative to the observer on the Earth, and the angle α will also increase. From a geometrical point of view, during the evolution of the lunar orbit, the angle E can change only in the interval of values 0 ≤E ≤2e≈ 0:11: Moreover, taking into account the averaging performed above (see form. (18)), the right part of the interval will be adjusted

$$0 \le E \le 0.070. \tag{22}$$

In addition, although the angle α can vary from zero (in the early era of lunar evolution) up to the current value <sup>α</sup><sup>0</sup> <sup>¼</sup> arctan <sup>0</sup>:<sup>7311</sup> <sup>1</sup>:<sup>7752</sup> <sup>≈</sup> <sup>0</sup>:39, but also taking into account the action of the first mechanism, the interval will be changed:

$$0 \le a \le a\_0. \tag{23}$$

two components (first and third) of the velocity field in (25) taking into account a

Streamlines at deformation of the Moon's shape (the section is shown by ellipse). Arrows depict the direction of

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

2

In the plane Ox1x3, the condition x<sup>2</sup> ¼ 0 is satisfied, and expressions for angles E

, E � α ¼ �arctg

� arctg

are the coordinates of the points of intersection of

x3 x1

� <sup>x</sup>1x\_ <sup>3</sup> � <sup>x</sup>3x\_ <sup>1</sup> x2 <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 3

S, respectively. Therefore,

<sup>4</sup> <sup>γ</sup>½ � sin 2<sup>E</sup> � sin 2ð Þ <sup>E</sup> � <sup>α</sup> : (32)

γ sin α � cos 2ð Þ E � α : (33)

<sup>γ</sup>x3; <sup>γ</sup> <sup>¼</sup> <sup>1</sup>

a1 da<sup>1</sup>

> x0 3 x0 1

dt : (27)

: (28)

: (29)

; (30)

<sup>4</sup> <sup>γ</sup> sin 2ð Þ <sup>E</sup> � <sup>α</sup> : (31)

condition of incompressibility (26) will take the form

deformation at the stage of rounding the figure in the early era of evolution.

E ¼ �arctg

<sup>α</sup>\_ <sup>¼</sup> <sup>x</sup><sup>0</sup> 1x\_ 0 <sup>3</sup> � x<sup>0</sup> 3x\_ 0 1

Thus, the derivative of the angle α will be equal to

<sup>α</sup>\_ ¼ � <sup>3</sup>

dα dt ¼ � <sup>3</sup> 2

<sup>E</sup>\_ ¼ � <sup>3</sup>

<sup>1</sup>; x<sup>0</sup> 3

and α, (see Figure 5) will be equal:

DOI: http://dx.doi.org/10.5772/intechopen.84465

Here, ð Þ x1; x<sup>3</sup> and x<sup>0</sup>

Figure 6.

31

the Moon's surface by the rays O<sup>0</sup>

<sup>u</sup><sup>1</sup> <sup>¼</sup> <sup>γ</sup>x1, u<sup>3</sup> ¼ � <sup>1</sup>

x3 x1

f <sup>1</sup> and O<sup>0</sup>

x0 3 x0 1

By substituting in (30) the components of the velocity field (27), we obtain

More convenient than (32), below will be the next form of differential equation:

<sup>4</sup> <sup>γ</sup> sin 2E; <sup>E</sup>\_ � <sup>α</sup>\_ ¼ � <sup>3</sup>

α ¼ arctg

Differentiating expression (29) with respect to time t, we find

x02 <sup>1</sup> <sup>þ</sup> <sup>x</sup>0<sup>2</sup> 3

where

$$a\_0 = \arctan\left(\frac{0.7311 - 0.1243}{1.7752}\right) \approx 0.329. \tag{24}$$

We emphasize that because of inequalities (22) and (23), the center of mass of the Moon will have that arrangement which is shown in Figures 3b and 5.

The problem consists in studying dependence between the angle α and the changing form of the Moon during the secular evolution in the gravitational tidal field of the Earth.

## 6. Differential equation for evolution of the angle α

As you know (see, e.g., [27]), a change in the shape of an ellipsoidal body can be described by a linear velocity field. In particular, the evolution of the prolate spheroid (21) in the moving frame of reference, whose axes coincide with the main axes of this body at any time, can be represented by the velocity field:

$$
\mu\_1 = \frac{\dot{a}\_1}{a\_1} \varkappa\_1, \quad \mu\_2 = \frac{\dot{a}\_2}{a\_2} \varkappa\_2, \quad \mu\_3 = \frac{\dot{a}\_3}{a\_3} \varkappa\_3. \tag{25}
$$

Here, the point above denotes the time derivative <sup>d</sup> dt. Since for incompressible figures the condition of volume preservation should be fulfilled (in this case—for the volume of the prolate spheroid (21)), we have the additional ratio

$$\mathbf{div \ v} \cdot \mathbf{u} = \frac{\dot{a}\_1}{a\_1} + 2\frac{\dot{a}\_3}{a\_3} = \mathbf{0}. \tag{26}$$

In the velocity field (25), the Moon's shape will always remain a second-order surface, and the streamlines will be represented by pieces of hyperboles (Figure 6). Owing to symmetry, the elongation of the spheroid (21) is described by the only

polar oblateness <sup>ε</sup> <sup>¼</sup> <sup>1</sup> � <sup>a</sup><sup>3</sup> a1 . Consider changing ε for the Moon's shape. In this case On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

Figure 6.

The main symmetry semiaxis a<sup>1</sup> of this spheroid was initially directed exactly to

Let us consider Figure 5. Due to the small orbit eccentricity, the angle E between the main axis of the Moon's figure and the direction to f<sup>1</sup> was also initially small. However, in the evolution of the Moon's orbit from the less eccentric to the more eccentric, as was shown in the first mechanism, the angle E will increase monotonically. This factor changes the orientation of the figure of the Moon relative to the observer on the Earth, and the angle α will also increase. From a geometrical point of view, during the evolution of the lunar orbit, the angle E can change only in the interval of values 0 ≤E ≤2e≈ 0:11: Moreover, taking into account the averaging performed above (see form. (18)), the right part of the interval will be adjusted

In addition, although the angle α can vary from zero (in the early era of lunar

0:7311 � 0:1243 1:7752 

We emphasize that because of inequalities (22) and (23), the center of mass of

As you know (see, e.g., [27]), a change in the shape of an ellipsoidal body can be

a2

figures the condition of volume preservation should be fulfilled (in this case—for

a1 þ 2 a\_ 3 a3

In the velocity field (25), the Moon's shape will always remain a second-order surface, and the streamlines will be represented by pieces of hyperboles (Figure 6). Owing to symmetry, the elongation of the spheroid (21) is described by the only

<sup>x</sup>2, u<sup>3</sup> <sup>¼</sup> <sup>a</sup>\_ <sup>3</sup>

a3

. Consider changing ε for the Moon's shape. In this case

described by a linear velocity field. In particular, the evolution of the prolate spheroid (21) in the moving frame of reference, whose axes coincide with the main

<sup>x</sup>1, u<sup>2</sup> <sup>¼</sup> <sup>a</sup>\_ <sup>2</sup>

axes of this body at any time, can be represented by the velocity field:

the volume of the prolate spheroid (21)), we have the additional ratio

div <sup>u</sup> <sup>¼</sup> <sup>a</sup>\_ <sup>1</sup>

the Moon will have that arrangement which is shown in Figures 3b and 5. The problem consists in studying dependence between the angle α and the changing form of the Moon during the secular evolution in the gravitational tidal

account the action of the first mechanism, the interval will be changed:

1:7752

evolution) up to the current value <sup>α</sup><sup>0</sup> <sup>¼</sup> arctan <sup>0</sup>:<sup>7311</sup>

α<sup>0</sup> ¼ arctan

6. Differential equation for evolution of the angle α

<sup>u</sup><sup>1</sup> <sup>¼</sup> <sup>a</sup>\_ <sup>1</sup> a1

Here, the point above denotes the time derivative <sup>d</sup>

a1

0≤E≤ 0:070: (22)

0≤α≤ α0, (23)

≈ 0:39, but also taking into

≈ 0:329: (24)

x3: (25)

dt. Since for incompressible

¼ 0: (26)

the Earth.

Lunar Science

where

field of the Earth.

polar oblateness <sup>ε</sup> <sup>¼</sup> <sup>1</sup> � <sup>a</sup><sup>3</sup>

30

Streamlines at deformation of the Moon's shape (the section is shown by ellipse). Arrows depict the direction of deformation at the stage of rounding the figure in the early era of evolution.

two components (first and third) of the velocity field in (25) taking into account a condition of incompressibility (26) will take the form

$$u\_1 = \gamma \mathbf{x}\_1, \quad u\_3 = -\frac{1}{2} \gamma \mathbf{x}\_3; \quad \gamma = \frac{1}{a\_1} \frac{da\_1}{dt} \,. \tag{27}$$

In the plane Ox1x3, the condition x<sup>2</sup> ¼ 0 is satisfied, and expressions for angles E and α, (see Figure 5) will be equal:

$$E = -\text{arctg}\frac{\varkappa\_3}{\varkappa\_1}, \quad E - a = -\text{arctg}\frac{\varkappa\_3'}{\varkappa\_1'}.\tag{28}$$

Here, ð Þ x1; x<sup>3</sup> and x<sup>0</sup> <sup>1</sup>; x<sup>0</sup> 3 are the coordinates of the points of intersection of the Moon's surface by the rays O<sup>0</sup> f <sup>1</sup> and O<sup>0</sup> S, respectively. Therefore,

$$a = \operatorname{arctg} \frac{\varkappa\_3'}{\varkappa\_1'} - \operatorname{arctg} \frac{\varkappa\_3}{\varkappa\_1}. \tag{29}$$

Differentiating expression (29) with respect to time t, we find

$$\dot{a} = \frac{\mathbf{x}\_1^{\prime}\dot{\mathbf{x}}\_3^{\prime} - \mathbf{x}\_3^{\prime}\dot{\mathbf{x}}\_1^{\prime}}{\mathbf{x}\_1^{\prime 2} + \mathbf{x}\_3^{\prime 2}} - \frac{\mathbf{x}\_1\dot{\mathbf{x}}\_3 - \mathbf{x}\_3\dot{\mathbf{x}}\_1}{\mathbf{x}\_1^2 + \mathbf{x}\_3^2};\tag{30}$$

By substituting in (30) the components of the velocity field (27), we obtain

$$
\dot{E} = -\frac{3}{4}\gamma\sin 2E; \quad \dot{E} - \dot{a} = -\frac{3}{4}\gamma\sin 2(E - a). \tag{31}
$$

Thus, the derivative of the angle α will be equal to

$$\dot{a} = -\frac{3}{4}\gamma[\sin 2E - \sin 2(E - a)].\tag{32}$$

More convenient than (32), below will be the next form of differential equation:

$$\frac{da}{dt} = -\frac{3}{2}\gamma \sin a \cdot \cos\left(2E - a\right). \tag{33}$$

Lunar Science

## 7. Solution of Eq. (33)

Let us turn to the analysis of the differential equation (33) and transform the derivative <sup>d</sup><sup>α</sup> dt:

$$\frac{da}{dt} = \frac{da}{d\varepsilon} \frac{d\varepsilon}{dt}.\tag{34}$$

ε ≈ 0:0125, E ¼ 0:07, α ¼ arctan

DOI: http://dx.doi.org/10.5772/intechopen.84465

Thus, the solution of equation (41) will get in the form

ε αð Þ¼ ; E 1 � 0:713 � exp

then the formula (41) gives

limits given in (23).

Figure 7.

value

Figure 7.

33

0:7311 � 0:1243

sin α cos 2ð Þ E � α � � <sup>1</sup> cos 2<sup>E</sup> !

Formula (44) represents the solution of the problem: it describes the change in the Moon's oblateness ε during the tidal evolution and establishes the dependence between ε and the angle α. Recall that α is the angle between the directions (from the center of the Moon) to the first focus of the orbit and the Moon's COM. As we already know, in the course of evolution, the angle α varied (in radians) within the

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

The graphic image of the function of two variables from (44) is shown in

<sup>ε</sup> <sup>¼</sup> <sup>1</sup> � <sup>a</sup><sup>3</sup> a1

Graphs for the two extreme values of the angle E are shown in Figure 8. As seen in Figure 8, the oblateness ε of the figure is very little depending on the angle E. Moreover, in the initial era, ε for all E has the same value and could not exceed the

3D image of the function ε αð Þ ; E . The angle α is set in radians. The oblateness of shape of the Moon ε αð Þ ; E in the early era very little depends on the value of the parameter E, and its value does not exceed ε≈0:285.

<sup>1</sup>:<sup>7752</sup> <sup>≈</sup> <sup>0</sup>:32937, (42)

: (44)

C≈ 0:713: (43)

≈ 0:285: (45)

As

$$\frac{d\varepsilon}{dt} = \frac{d}{dt}\left(1 - \frac{a\_3}{a\_1}\right) = \frac{a\_3\dot{a}\_1 - a\_1\dot{a}\_3}{a\_1^2},\tag{35}$$

therefore, in agreement with (34),

$$\frac{d\varepsilon}{dt} = \frac{3}{2}\gamma(\mathbf{1} - \varepsilon). \tag{36}$$

Substituting (36) in (34) and then the result in (33), we have

$$
\dot{a} = \frac{3}{2}\gamma(1-\varepsilon)\frac{da}{d\varepsilon} = -\frac{3}{2}\gamma\sin a \cos\left(2E-a\right).
\tag{37}
$$

As a result, the differential equation for the angle α takes the form

$$\frac{da}{d\varepsilon} = -\frac{\sin a \cos \left(2E - a\right)}{1 - \varepsilon}.\tag{38}$$

Separating the variables in (38) and integrating and taking into account the auxiliary formula

$$\int \frac{da}{\sin a \cos \left(2E - a\right)} = \frac{1}{\cos 2E} \ln \frac{\sin a}{\cos \left(2E - a\right)},\tag{39}$$

we obtain a solution for equation (38) in the form

$$\frac{1}{\cos 2E} \ln \frac{\sin a}{\cos \left(2E - a\right)} = C + \ln \left(1 - \varepsilon\right),\tag{40}$$

where C is the integration constant. Potentiating expression (40), we find the solution in the form

$$\varepsilon(a,E) = \mathbf{1} - \mathbf{C} \cdot \exp\left\{ \left[ \frac{\sin a}{\cos \left(2E - a\right)} \right]^{\frac{1}{\cos 2E}} \right\}.\tag{41}$$

## 8. Analysis of the solution (41) and estimation of the elongation of the lunar figure in early era

In formula (41), the constant integration C is defined by the known observational data. As in the modern epoch of tidal evolution of the Moon the supplemented relations

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

$$a \approx 0.0125, \ E = 0.07, \quad a = \arctan \frac{0.7311 - 0.1243}{1.7752} \approx 0.32937,\tag{42}$$

then the formula (41) gives

7. Solution of Eq. (33)

dt:

derivative <sup>d</sup><sup>α</sup>

Lunar Science

As

auxiliary formula

solution in the form

lunar figure in early era

supplemented relations

32

Let us turn to the analysis of the differential equation (33) and transform the

<sup>¼</sup> <sup>a</sup>3a\_ <sup>1</sup> � <sup>a</sup>1a\_ <sup>3</sup> a2 1

dt : (34)

γð Þ 1 � ε : (36)

γ sin α cos 2ð Þ E � α : (37)

<sup>1</sup> � <sup>ε</sup> : (38)

cos 2ð Þ <sup>E</sup> � <sup>α</sup> , (39)

: (41)

, (35)

dα dt <sup>¼</sup> <sup>d</sup><sup>α</sup> dε dε

dt <sup>1</sup> � <sup>a</sup><sup>3</sup> a1 � �

> dε dt <sup>¼</sup> <sup>3</sup> 2

Substituting (36) in (34) and then the result in (33), we have

<sup>d</sup><sup>ε</sup> ¼ � <sup>3</sup> 2

<sup>d</sup><sup>ε</sup> ¼ � sin <sup>α</sup> cos 2ð Þ <sup>E</sup> � <sup>α</sup>

Separating the variables in (38) and integrating and taking into account the

where C is the integration constant. Potentiating expression (40), we find the

8. Analysis of the solution (41) and estimation of the elongation of the

In formula (41), the constant integration C is defined by the known observa-

tional data. As in the modern epoch of tidal evolution of the Moon the

cos 2<sup>E</sup> ln sin <sup>α</sup>

sin α cos 2ð Þ E � α � � <sup>1</sup> cos 2<sup>E</sup> ( )

cos 2ð Þ <sup>E</sup> � <sup>α</sup> <sup>¼</sup> <sup>C</sup> <sup>þ</sup> ln 1ð Þ � <sup>ε</sup> , (40)

As a result, the differential equation for the angle α takes the form

<sup>γ</sup>ð Þ <sup>1</sup> � <sup>ε</sup> <sup>d</sup><sup>α</sup>

dα

sin <sup>α</sup> cos 2ð Þ <sup>E</sup> � <sup>α</sup> <sup>¼</sup> <sup>1</sup>

ln sin <sup>α</sup>

dε dt <sup>¼</sup> <sup>d</sup>

therefore, in agreement with (34),

<sup>α</sup>\_ <sup>¼</sup> <sup>3</sup> 2

ð dα

1 cos 2E

we obtain a solution for equation (38) in the form

ε αð Þ¼ ; E 1 � C � exp

$$
\mathbf{C} \approx \mathbf{0}.713.\tag{43}
$$

Thus, the solution of equation (41) will get in the form

$$\varepsilon(a,E) = \mathbf{1} - \mathbf{0}.7\mathbf{1} \mathbf{3} \cdot \exp\left( \left[ \frac{\sin a}{\cos \left(2E - a\right)} \right]^{\frac{1}{\cos 2E}} \right). \tag{44}$$

Formula (44) represents the solution of the problem: it describes the change in the Moon's oblateness ε during the tidal evolution and establishes the dependence between ε and the angle α. Recall that α is the angle between the directions (from the center of the Moon) to the first focus of the orbit and the Moon's COM. As we already know, in the course of evolution, the angle α varied (in radians) within the limits given in (23).

The graphic image of the function of two variables from (44) is shown in Figure 7.

Graphs for the two extreme values of the angle E are shown in Figure 8. As seen in Figure 8, the oblateness ε of the figure is very little depending on the angle E. Moreover, in the initial era, ε for all E has the same value and could not exceed the value

$$
\varepsilon = 1 - \frac{a\_3}{a\_1} \approx 0.285.\tag{45}
$$

#### Figure 7.

3D image of the function ε αð Þ ; E . The angle α is set in radians. The oblateness of shape of the Moon ε αð Þ ; E in the early era very little depends on the value of the parameter E, and its value does not exceed ε≈0:285.

Figure 8.

The dependence of the oblateness ε of the Moon shape from the angle α between the line }COM=COF} of the Moon and the mean direction to the Earth. The graph shows the change ε during the tidal evolution. Two extreme angle values E ¼ 0:0 (upper curve) and E ¼ 0:07 are taken for comparison. The beginning and the end of the evolutionary process correspond to the values α ≈0 and α ≈0:329.

Thus, the second mechanism explains both the displacements of the center of mass of the Moon to the East and predicts that the oblateness of the Moon in the early era could not exceed the value ε ≈ 0:285:

## 9. Some consequences: how close to the earth could the Moon be formed

Above we established that on the known shift of the Moon's center of mass to the East, we can find the oblateness (45), which the Moon could have in the epoch of its formation. The corresponding spheroid eccentricity will be equal to

$$
\epsilon \approx 0.70.\tag{46}
$$

Here, p is the pressure, ρ is the density, φ is the quadratic internal gravitational potential of the satellite, Ω is the angular velocity rotation of the satellite, and R<sup>⊕</sup> is the distance between the centers of the Earth and the Moon. For satellite with the

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

<sup>1</sup> � <sup>A</sup><sup>3</sup> <sup>x</sup><sup>2</sup>

;

� 2

<sup>2</sup>e<sup>3</sup> ln <sup>1</sup> <sup>þ</sup> <sup>e</sup> 1 � e :

The internal pressure of the equilibrium figure should also be a quadratic func-

1 a2 1 � x2 <sup>2</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 3 a2 3

2πρR<sup>3</sup> ⊕ ¼ κ

; <sup>κ</sup> <sup>¼</sup> <sup>2</sup> 3 ρ⊕

<sup>x</sup><sup>3</sup> <sup>¼</sup> <sup>A</sup><sup>1</sup> � <sup>1</sup> � <sup>e</sup><sup>2</sup> ð ÞA<sup>3</sup>

Substituting the value e from (46) into the right-hand side (53), we obtain the

Thus, the Moon with oblateness (45) could form at a very close distance from the Earth: at a distance of only three and a quarter of the mean radii of the modern

Note that the prolate spheroid with meridional eccentricity (45) is a stable figure

of equilibrium. In fact, the instability of this type of figure occurs only when

<sup>A</sup><sup>1</sup> � <sup>1</sup> � <sup>e</sup><sup>2</sup> ð ÞA<sup>3</sup>

From the first integral (47) is possible to find a square of angular velocity

<sup>2</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 3

: (49)

<sup>4</sup> � <sup>e</sup><sup>2</sup> : (50)

<sup>x</sup><sup>3</sup> , (51)

<sup>ρ</sup> <sup>≈</sup> <sup>1</sup>:09875, (52)

<sup>4</sup> � <sup>e</sup><sup>2</sup> : (53)

<sup>x</sup><sup>3</sup> <sup>¼</sup> <sup>0</sup>:0324, (54)

x≈ 3:24: (55)

(48)

<sup>1</sup> � <sup>e</sup><sup>2</sup> <sup>e</sup><sup>2</sup> ;

form of the prolate spheroid (21), we have [27]

DOI: http://dx.doi.org/10.5772/intechopen.84465

tion from the coordinates

rotation of satellite

Since

cubic equation

e≥0:883 (see, e.g., [28]).

35

<sup>φ</sup> <sup>¼</sup> <sup>π</sup>G<sup>ρ</sup> <sup>I</sup> � <sup>A</sup>1x<sup>2</sup>

<sup>e</sup><sup>2</sup> � <sup>1</sup> � <sup>e</sup><sup>2</sup>

<sup>p</sup> <sup>¼</sup> <sup>p</sup><sup>0</sup> <sup>1</sup> � <sup>x</sup><sup>2</sup>

Ω2 <sup>π</sup>G<sup>ρ</sup> <sup>¼</sup> <sup>2</sup>

> Ω2 <sup>2</sup>πG<sup>ρ</sup> <sup>¼</sup> <sup>M</sup><sup>⊕</sup>

where we have identified the following characters

the ratio (51) can be represented as

from which we find the required distance

<sup>x</sup> <sup>¼</sup> <sup>R</sup><sup>⊕</sup> R<sup>⊕</sup>

κ

κ

Earth. This result slightly corrects the one we received earlier [15].

<sup>e</sup><sup>3</sup> ln <sup>1</sup> <sup>þ</sup> <sup>e</sup> 1 � e

<sup>A</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> � <sup>e</sup><sup>2</sup>

<sup>A</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>

Proceeding from (46) and using the theory of tidal equilibrium figures, it is possible to estimate how close to each other might be the Earth and the Moon in the early era. For this purpose, without loss of generality, we assume that the satellite is uniform (at the Moon, as we know, and now concentration of substance very small), and its mass in comparison with the mass of the Earth can be neglected. Then, in the tidal approach for the potential of the Earth, the equation of hydrostatic equilibrium of the satellite with synchronous rotation has the first integral [28]:

$$\frac{p}{\rho} + \text{const} = \rho + \frac{1}{2}\Omega^2 \left(3\mathbf{x}\_1^2 - \mathbf{x}\_3^2\right); \qquad \Omega^2 = \frac{GM\_{\oplus}}{R\_{\oplus}^3}. \tag{47}$$

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

Here, p is the pressure, ρ is the density, φ is the quadratic internal gravitational potential of the satellite, Ω is the angular velocity rotation of the satellite, and R<sup>⊕</sup> is the distance between the centers of the Earth and the Moon. For satellite with the form of the prolate spheroid (21), we have [27]

$$\begin{aligned} \varphi &= \pi G \rho \left[ I - A\_1 x\_1^2 - A\_3 \left( x\_2^2 + x\_3^2 \right) \right]; \\ A\_1 &= \frac{1 - e^2}{e^3} \ln \frac{1 + e}{1 - e} - 2 \frac{1 - e^2}{e^2}; \\ A\_3 &= \frac{1}{e^2} - \frac{1 - e^2}{2e^3} \ln \frac{1 + e}{1 - e}. \end{aligned} \tag{48}$$

The internal pressure of the equilibrium figure should also be a quadratic function from the coordinates

$$p = p\_0 \left( 1 - \frac{\varkappa\_1^2}{a\_1^2} - \frac{\varkappa\_2^2 + \varkappa\_3^2}{a\_3^2} \right). \tag{49}$$

From the first integral (47) is possible to find a square of angular velocity rotation of satellite

$$\frac{\Omega^2}{4\pi G\rho} = 2\frac{A\_1 - (1 - e^2)A\_3}{4 - e^2}.\tag{50}$$

Since

Thus, the second mechanism explains both the displacements of the center of mass of the Moon to the East and predicts that the oblateness of the Moon in the early era could

The dependence of the oblateness ε of the Moon shape from the angle α between the line }COM=COF} of the Moon and the mean direction to the Earth. The graph shows the change ε during the tidal evolution. Two extreme angle values E ¼ 0:0 (upper curve) and E ¼ 0:07 are taken for comparison. The beginning and the

9. Some consequences: how close to the earth could the Moon be formed

Proceeding from (46) and using the theory of tidal equilibrium figures, it is possible to estimate how close to each other might be the Earth and the Moon in the early era. For this purpose, without loss of generality, we assume that the satellite is uniform (at the Moon, as we know, and now concentration of

substance very small), and its mass in comparison with the mass of the Earth can be neglected. Then, in the tidal approach for the potential of the Earth, the equation of hydrostatic equilibrium of the satellite with synchronous rotation has the first

formation. The corresponding spheroid eccentricity will be equal to

end of the evolutionary process correspond to the values α ≈0 and α ≈0:329.

1 2

Ω<sup>2</sup> 3x<sup>2</sup>

<sup>1</sup> � <sup>x</sup><sup>2</sup> 3 ; <sup>Ω</sup><sup>2</sup> <sup>¼</sup> GM<sup>⊕</sup>

Above we established that on the known shift of the Moon's center of mass to the East, we can find the oblateness (45), which the Moon could have in the epoch of its

e≈ 0:70: (46)

R3 ⊕ : (47)

not exceed the value ε ≈ 0:285:

Figure 8.

Lunar Science

integral [28]:

34

p ρ

þ const ¼ φ þ

$$\frac{\Omega^2}{2\pi G\rho} = \frac{M\_{\oplus}}{2\pi\rho R\_{\oplus}^3} = \frac{\kappa}{\varkappa^3},\tag{51}$$

where we have identified the following characters

$$
\kappa = \frac{R\_{\oplus}}{R\_{\oplus}}; \quad \kappa = \frac{2}{3} \frac{\rho\_{\oplus}}{\rho} \approx 1.09875,\tag{52}
$$

the ratio (51) can be represented as

$$\frac{\kappa}{\kappa^3} = \frac{A\_1 - (\mathbf{1} - \mathbf{e}^2)A\_3}{4 - \mathbf{e}^2}. \tag{53}$$

Substituting the value e from (46) into the right-hand side (53), we obtain the cubic equation

$$\frac{\kappa}{x^3} = 0.0324,\tag{54}$$

from which we find the required distance

$$x \approx \text{3.24.}\tag{55}$$

Thus, the Moon with oblateness (45) could form at a very close distance from the Earth: at a distance of only three and a quarter of the mean radii of the modern Earth. This result slightly corrects the one we received earlier [15].

Note that the prolate spheroid with meridional eccentricity (45) is a stable figure of equilibrium. In fact, the instability of this type of figure occurs only when e≥0:883 (see, e.g., [28]).

## 10. Discussion and conclusions

Here, it is necessary to add the following. As is well known, in the problem of secular perturbations, the perturbation function is replaced by its secular part. The influence of the Sun leads only to periodic perturbations of the eccentricity of the lunar orbit, which we do not take into account here. In this chapter, we ignore periodic oscillations and consider only tidal secular changes in the average eccentricity of the lunar orbit.

shown that from the hidden fact that in our era there is a slightly shift of the center of the Moon's mass to the East, and not to the West, you can get valuable information about the evolution of the orbit of the Moon and its shape. This finding supports the scenario [29] that the Moon could be formed about 4.5 billion in the surrounding "donut" from the hot gas that appeared after the collision of Theia with

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

1 Sternberg Astronomical Institute, M.V. Lomonosov Moscow State University,

2 Faculty of Physics of the M.V. Lomonosov Moscow State University, Russia

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

3 Сentral Astronomical Observatory at Pulkovo, Russia

provided the original work is properly cited.

\*Address all correspondence to: work@boris-kondratyev.ru

proto-Earth.

DOI: http://dx.doi.org/10.5772/intechopen.84465

Author details

Russia

37

Boris P. Kondratyev1,2,3

As for the tidal influence of the Sun on the figure of the Moon, it turns out to be insignificant compared to the influence of the Earth. Indeed, the ratio of force ΔF<sup>⊙</sup> to force ΔF<sup>⊕</sup> from (15) is equal to

$$\frac{\Delta F\_{\odot}}{\Delta F\_{\oplus}} = \frac{M\_{\odot}}{M\_{\oplus}} \left(\frac{R\_{\oplus}}{R\_{\odot}}\right)^{4} \approx 10^{-5} \text{ .}$$

Therefore, to solve the posed problem within the framework of our model, the influence of the Sun can be neglected.

In the theory of the tidal evolution of the Moon's orbit and its form, we encounter problems that are difficult to give exact answers. Above, we examined some of the conclusions from those observational facts that the center of mass of the Moon is slightly shifted to the East. Two geometrical mechanisms have been developed to explain this shift.

The first mechanism considers the secular evolution of the Moon's orbit, using the effect of the preferred orientation of the satellite with synchronous rotation to the second orbital focus. According to this mechanism, only the scenario of secular evolution of the orbit with the increase of eccentricity leads to the desired offset of the center of the Moon's mass to the East. It is important to note that this conclusion that the evolution of the Moon's orbit occurred with an increase e is consistent with the fact that at present the eccentricity of the lunar orbit is indeed increasingly, and therefore in the past, it was less than today [20, 21] (see also [22–25]).

To fully explain the displacement of the center of the Moon's mass to the East, a second mechanism was developed, which takes into account the influence of tidal changes in the shape of the Moon as it gradually moves away from the Earth. The essence of the second mechanism is fully consistent with the fact that the distance between Earth and Moon is now really increasing and the Earth's spin is slowing in reaction.

In addition, the second mechanism predicts that the Moon's figure flattening in the early era was very significant and reached the value of ε≈ 0:285: In turn, based on the theory of tidal equilibrium figures, it allowed us to estimate how close to Earth could the Moon be formed as an astronomical body. According to formula (55), the Moon was formed in the ring zone at a distance of 3–4 medium radii of the present Earth. This result seems to be consistent with the modern view that the Moon was formed as a result of a gigantic impact in the immediate vicinity of the proto-Earth.

Since the formation of the Moon as a celestial body and so far the Earth-Moon system has been and remains a binary planet, the physical laws of its development have always been the same. In the early era, however, the tidal forces between the Earth and the Moon were much more important. Indeed, now the tidal force has very little effect on the Moon, because of which it is removed from the Earth for only 3.8 cm per year. However, studying the evolution of the moon still requires a great effort of researchers.

In summary, we can say that the method presented here really allows to take into account additional observational facts in the structure of the Moon. We have On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

shown that from the hidden fact that in our era there is a slightly shift of the center of the Moon's mass to the East, and not to the West, you can get valuable information about the evolution of the orbit of the Moon and its shape. This finding supports the scenario [29] that the Moon could be formed about 4.5 billion in the surrounding "donut" from the hot gas that appeared after the collision of Theia with proto-Earth.

## Author details

10. Discussion and conclusions

tricity of the lunar orbit.

Lunar Science

explain this shift.

reaction.

proto-Earth.

36

great effort of researchers.

to force ΔF<sup>⊕</sup> from (15) is equal to

influence of the Sun can be neglected.

ΔF<sup>⊙</sup> ΔF<sup>⊕</sup> <sup>¼</sup> <sup>M</sup><sup>⊙</sup> M<sup>⊕</sup>

therefore in the past, it was less than today [20, 21] (see also [22–25]).

Here, it is necessary to add the following. As is well known, in the problem of secular perturbations, the perturbation function is replaced by its secular part. The influence of the Sun leads only to periodic perturbations of the eccentricity of the lunar orbit, which we do not take into account here. In this chapter, we ignore periodic oscillations and consider only tidal secular changes in the average eccen-

As for the tidal influence of the Sun on the figure of the Moon, it turns out to be insignificant compared to the influence of the Earth. Indeed, the ratio of force ΔF<sup>⊙</sup>

> R<sup>⊕</sup> R<sup>⊙</sup> <sup>4</sup>

Therefore, to solve the posed problem within the framework of our model, the

In the theory of the tidal evolution of the Moon's orbit and its form, we encounter problems that are difficult to give exact answers. Above, we examined some of the conclusions from those observational facts that the center of mass of the Moon is slightly shifted to the East. Two geometrical mechanisms have been developed to

The first mechanism considers the secular evolution of the Moon's orbit, using the effect of the preferred orientation of the satellite with synchronous rotation to the second orbital focus. According to this mechanism, only the scenario of secular evolution of the orbit with the increase of eccentricity leads to the desired offset of the center of the Moon's mass to the East. It is important to note that this conclusion that the evolution of the Moon's orbit occurred with an increase e is consistent with the fact that at present the eccentricity of the lunar orbit is indeed increasingly, and

To fully explain the displacement of the center of the Moon's mass to the East, a second mechanism was developed, which takes into account the influence of tidal changes in the shape of the Moon as it gradually moves away from the Earth. The essence of the second mechanism is fully consistent with the fact that the distance between Earth and Moon is now really increasing and the Earth's spin is slowing in

In addition, the second mechanism predicts that the Moon's figure flattening in the early era was very significant and reached the value of ε≈ 0:285: In turn, based on the theory of tidal equilibrium figures, it allowed us to estimate how close to Earth could the Moon be formed as an astronomical body. According to formula (55), the Moon was formed in the ring zone at a distance of 3–4 medium radii of the present Earth. This result seems to be consistent with the modern view that the Moon was formed as a result of a gigantic impact in the immediate vicinity of the

Since the formation of the Moon as a celestial body and so far the Earth-Moon system has been and remains a binary planet, the physical laws of its development have always been the same. In the early era, however, the tidal forces between the Earth and the Moon were much more important. Indeed, now the tidal force has very little effect on the Moon, because of which it is removed from the Earth for only 3.8 cm per year. However, studying the evolution of the moon still requires a

In summary, we can say that the method presented here really allows to take into account additional observational facts in the structure of the Moon. We have

≈ 10�<sup>5</sup> :

Boris P. Kondratyev1,2,3

1 Sternberg Astronomical Institute, M.V. Lomonosov Moscow State University, Russia

2 Faculty of Physics of the M.V. Lomonosov Moscow State University, Russia

3 Сentral Astronomical Observatory at Pulkovo, Russia

\*Address all correspondence to: work@boris-kondratyev.ru

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

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[2] Kondratyev BP. On one inaccuracy of Isaac Newton. Kvant. 2009;5:38

[3] Murray K, Dermott S. Dynamics of the Solar System. M.: FIZMATLIT, 2010; p. 588 (trans. from English).

[4] Yakovkin AA. The radius and shape of the moon. Bulletin AOE. 1934;4

[5] Bohme S. Bearbeitung der Aufnahmen von F. Hayn zur Ortsbestimmung des Mondes. Astronomische Nachrichten. 1953;256: 356

[6] Shakirov KS. The influence of the internal structure of the moon on its rotation. Izvestia AOE. 1963;34

[7] Lipsky YN, Nikonov VA. The position of the center of the figure of the moon. Astronomicheskii Zhurnal. 1971; 48:445

[8] Calame O. Free librations of the Moon determined by an analysis of laser range measurements. Moon. 1976;15:343

[9] Archinal BA, Rosiek MR, Kirk RL, Redding BL. Completion of the Unified Lunar Control Network 2005 and Topographic Model. Virginia: US Geological Survey. 2006

[10] Iz H, Ding XL, Dai CL, Shum CK. Polyaxial figures of the Moon. Journal of Geodesy. 2011;1(4):348

[11] Barker MK, Mazarico E, Neumann GA, Zuber MT, Kharuyama J, Smith DE. A new lunar digital elevation model from the Lunar Orbiter Laser Altimeter and SELENE Terrain Camera. Icarus. 2016;273:346

[12] Lemoine FG, Goossens S, Sabaka TJ, Nicholas JB, Mazarico E, Rowlands DD, et al. GRGM900C: A degree-900 lunar gravity model from GRAIL primary and extended mission data. GeoRL. 2014;41: 3382. DOI: 10.1002/2014GL060027

[21] Goldreich P. History of the lunar orbit. Reviews of Geophysics and Space

DOI: http://dx.doi.org/10.5772/intechopen.84465

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on…

[22] Simon JL, Bretagnon P, Chapront J,

Chapront-Touze M, Francou G, Laskar J. Numerical expressions for precession formulae and mean elements for the moon and planets. A&A. 1994;

[23] Chapront J, Chapront-Touzé M, Francou G. A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements. A&A. 2002;387:700

[24] Laskar J, Fienga A, Gastineau M, Manche H. La2010: A new orbital solution for the long-termmotion of the

[25] Folkner WM, Williams JG, Boggs DH, Park RS, Kuchynka P. The Planetary and Lunar Ephemerides DE430 and DE431. The Interplanetary Network Progress Report. 42-196. 2014.

[26] Deprit A. The movement of the moon in space. In: Kopal Z, editor. Physics and Astronomy of the Moon. New York and London: Academic Press;

[27] Kondratyev BP. Dinamika ellipsoidal'nykh gravitiruiushchikh figure. Moscow: Nauka; 1989

[28] Chandrasekhar S. Ellipsoidal Equilibrium Figures. New Haven and London: Yale University Press; 1969

10.1002/2017JE005333

[29] Simon SJ, Stewart ST, Petaev MI, Leinhardt ZM, Mace MT, Jacobsen SB. The Origin of the Moon Within a Terrestrial Synestia. JGR. 2018. DOI:

Earth. A&A. 2011;532(A89):15

Physics. 1966;4:411

282:663

pp. 1-81

1971

39

[13] Kondratyev BP. The deviation of the lunar center of mass to the east of the direction toward the earth. A Mechanism Based on Orbital Evolution. Astronomy Reports. 2018;62(8):542. DOI: 10.1134/ S106377291808005X

[14] Kondratyev BP. The deviation of the lunar center of mass to the east of the direction toward the earth. A Mechanism Based on Rounding of the Figure of the Moon. Astronomy Reports. 2018; 62(10):705. DOI: 10.1134/S106377 2918100062

[15] Kondratyev BP. On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon. Astrophysics and Space Science. 2018;186(186)

[16] Darwin GH. Tidal Friction in Cosmogony, Scientific Papers 2. Cambridge University Press; 1908

[17] Urey HC. Chemical evidence relative to the origin of the solar system. MNRAS. 1966;131:212

[18] Wieczorek MA, Neumann GA, Nimmo F, Kiefer WS, Taylor GJ, Melosh HJ, et al. The crust of the Moon as seen by GRAIL. Science. 2013;339(6120):671. DOI: 10.1126/science.1231530

[19] Zhong S. Origin and Evolution of the Moon. 2014. 2014IAUS, 298, 457Z. DOI: 10.1017/S1743921313007229

[20] Macdonald GJF. Tidal friction. Reviews of Geophysics. 1964;2:467. DOI: 10.1029/RG002i003p00467

On the Deviation of the Lunar Center of Mass to the East: Two Possible Mechanisms Based on… DOI: http://dx.doi.org/10.5772/intechopen.84465

[21] Goldreich P. History of the lunar orbit. Reviews of Geophysics and Space Physics. 1966;4:411

References

Lunar Science

[1] Newton I. Mathematical principles of natural philosophy. In: Bernard Cohen I, [12] Lemoine FG, Goossens S, Sabaka TJ, Nicholas JB, Mazarico E, Rowlands DD, et al. GRGM900C: A degree-900 lunar gravity model from GRAIL primary and extended mission data. GeoRL. 2014;41: 3382. DOI: 10.1002/2014GL060027

[13] Kondratyev BP. The deviation of the lunar center of mass to the east of the direction toward the earth. A Mechanism Based on Orbital Evolution. Astronomy Reports. 2018;62(8):542. DOI: 10.1134/

[14] Kondratyev BP. The deviation of the lunar center of mass to the east of the direction toward the earth. A Mechanism Based on Rounding of the Figure of the Moon. Astronomy Reports. 2018; 62(10):705. DOI: 10.1134/S106377

[15] Kondratyev BP. On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon. Astrophysics and Space

S106377291808005X

2918100062

Science. 2018;186(186)

MNRAS. 1966;131:212

[16] Darwin GH. Tidal Friction in Cosmogony, Scientific Papers 2. Cambridge University Press; 1908

[17] Urey HC. Chemical evidence relative to the origin of the solar system.

[18] Wieczorek MA, Neumann GA, Nimmo F, Kiefer WS, Taylor GJ, Melosh HJ, et al. The crust of the Moon as seen by GRAIL. Science. 2013;339(6120):671.

[19] Zhong S. Origin and Evolution of the Moon. 2014. 2014IAUS, 298, 457Z. DOI: 10.1017/S1743921313007229

[20] Macdonald GJF. Tidal friction. Reviews of Geophysics. 1964;2:467. DOI: 10.1029/RG002i003p00467

DOI: 10.1126/science.1231530

[2] Kondratyev BP. On one inaccuracy of

[3] Murray K, Dermott S. Dynamics of the Solar System. M.: FIZMATLIT, 2010; p. 588 (trans. from English).

[4] Yakovkin AA. The radius and shape of the moon. Bulletin AOE. 1934;4

Astronomische Nachrichten. 1953;256:

[6] Shakirov KS. The influence of the internal structure of the moon on its rotation. Izvestia AOE. 1963;34

[7] Lipsky YN, Nikonov VA. The

[8] Calame O. Free librations of the Moon determined by an analysis of laser range measurements. Moon. 1976;15:343

[9] Archinal BA, Rosiek MR, Kirk RL, Redding BL. Completion of the Unified Lunar Control Network 2005 and Topographic Model. Virginia: US

[10] Iz H, Ding XL, Dai CL, Shum CK. Polyaxial figures of the Moon. Journal of

[11] Barker MK, Mazarico E, Neumann GA, Zuber MT, Kharuyama J, Smith DE. A new lunar digital elevation model from the Lunar Orbiter Laser Altimeter and SELENE Terrain Camera. Icarus.

Geological Survey. 2006

Geodesy. 2011;1(4):348

2016;273:346

38

position of the center of the figure of the moon. Astronomicheskii Zhurnal. 1971;

Whitman A, editors. A Guide to Newton's Principia. University of

Isaac Newton. Kvant. 2009;5:38

[5] Bohme S. Bearbeitung der Aufnahmen von F. Hayn zur Ortsbestimmung des Mondes.

356

48:445

California Press; 1999

[22] Simon JL, Bretagnon P, Chapront J, Chapront-Touze M, Francou G, Laskar J. Numerical expressions for precession formulae and mean elements for the moon and planets. A&A. 1994; 282:663

[23] Chapront J, Chapront-Touzé M, Francou G. A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements. A&A. 2002;387:700

[24] Laskar J, Fienga A, Gastineau M, Manche H. La2010: A new orbital solution for the long-termmotion of the Earth. A&A. 2011;532(A89):15

[25] Folkner WM, Williams JG, Boggs DH, Park RS, Kuchynka P. The Planetary and Lunar Ephemerides DE430 and DE431. The Interplanetary Network Progress Report. 42-196. 2014. pp. 1-81

[26] Deprit A. The movement of the moon in space. In: Kopal Z, editor. Physics and Astronomy of the Moon. New York and London: Academic Press; 1971

[27] Kondratyev BP. Dinamika ellipsoidal'nykh gravitiruiushchikh figure. Moscow: Nauka; 1989

[28] Chandrasekhar S. Ellipsoidal Equilibrium Figures. New Haven and London: Yale University Press; 1969

[29] Simon SJ, Stewart ST, Petaev MI, Leinhardt ZM, Mace MT, Jacobsen SB. The Origin of the Moon Within a Terrestrial Synestia. JGR. 2018. DOI: 10.1002/2017JE005333

**41**

**Chapter 4**

**Abstract**

Rock Massive

new principles of monitoring

**1. Introduction**

*Olga Hachay and Oleg Khachay*

New Principles of Monitoring

Seismological and Deformation

Processes Occurring in the Moon

Currently, the interest in studying the processes occurring in other planets surrounding the Earth is becoming increasingly important. The Moon-satellite planet is the closest to the planet Earth, and therefore, it makes sense to organize a system for studying it first and foremost, incorporating the most advanced ideas about the physics of processes in rock massive, which are also used in terrestrial conditions. In this paper, new ideas on the organization of seismological and deformation monitoring are set out, based on the results obtained for the rock massive of the Earth and the theoretical ideas presented in the works of I. Prigogine and S. Hawking.

In recent decades, a new science was born—the physics of non-equilibrium processes associated with such concepts as irreversibility, self-organization, and dissipative structures [1]. It is known that irreversibility leads to many new phenomena, such as the formation of vortices, vibration chemical reactions, or laser radiation. Irreversibility plays a significant constructive role. It is impossible to imagine life in a world devoid of interrelations created by irreversible processes. The prototype of the universal law of nature is Newton's law, which can be briefly formulated as follows: acceleration is proportional to force. This law has two fundamental features. It is deterministic: since the initial conditions are known, we can predict movement. And, it is reversible in time: there is no difference between predicting the future and restoring the past; movement to a future state and reverse movement from the current state to the initial state are equivalent. Newton's law is the basis of classical mechanics, the science of the motion of matter, of trajectories. Since the beginning of the twentieth century, the boundaries of physics have expanded significantly. Now, we have quantum mechanics and the theory of relativity. But, as we shall see from the sequel, the basic characteristics of Newton's law—determinism and reversibility in time—are preserved. Is it possible to modify the very concept of physical laws so as to include in our fundamental description of the nature of irreversibility, events and the arrow of time? The adoption of such a program entails a thorough

**Keywords:** Moon rock massive, seismological and deformation processes,

## **Chapter 4**
