# The Moon turns out to be the perfect object to use the linear elasticity
theory

V.P. Pavlov  Steklov Mathematical Institute of RAS, 8, Gubkina Str, Moscow,
Russia<EMAIL_ADDRESS>
###### Abstract

The applicability of the linear theory of elasticity to the Moon has been
studied. As a criterion was taken the smallness of the strain tensor. The
elastic moduli are obtained from the data on the longitudinal and transverse
sound velocities in the Moon interior. The pressure was calculated in the
framework of the model of a homogeneous solid sphere under the action of its
own gravity. The strain tensor trace is of the order $0.02$, which indicates
the applicability.

The equilibrium condition in the body of the Moon is considered in the
reference system rigidly connected to the rotating Moon. Except the elastic
forces, all the mass forces in the body are potential. It means that
acceleration of each of them (gravity, the Earth’s tidal and solar,
centrifugal and inertia forces associated with the precession of the rotation
axis) is minus gradient of the corresponding potential.

It turns out that there is a hierarchy among these potentials. If you take the
order of gravity for 1, the relative order of the Earth’s tide is $\gamma\sim
10^{-5}$, the Sun’s one is 20 times smaller, and the rest - less than
$10^{-8}$. This allows you to keep in equilibrium condition only the Earth’s
tidal potential and apply the perturbation theory on $\gamma$.

The strain tensor in the body of the Moon is calculated in the first two (zero
and first) order in $\gamma$ (i.e. taking into account the Earth’s tidal
potential). Respectively the free energy density is calculated. Since the axis
of the Moon rotation has its own non-zero declination to the ecliptic plane,
the tidal potential variations take place during the rotation of the Moon
around the Earth.

The estimation of the corresponding free energy density variations are made.
Their dependence on the depth exhibit qualitative agreement with the data of
depth dependence in the energy of deep moonquakes obtained in the project
”Apollo.”

Integral estimations of variations in the free energy for the year shows that
it is many orders of magnitude greater than estimates of energy for the year
of deep moonquakes energy. Thus offering an answer to the fundamental
question: where is the source of energy released in the deep moonquakes.

###### keywords:

Linear elasticity, Moon, Tides, Strain tensor, Free energy density variations,
Seismicity

††journal: Earth and Planetary Science Letters

##  1 Introduction.

Any statements on the planets seismicity are based on the accumulated
information about their internal structure. To a large extent, this
information is obtained by processing of data on seismic events. First of all,
these are the data on the depth dependence of the longitudinal $c_{p}$ and
transverse $c_{s}$ speed of sound in a solid body of a planet. Here one should
note three circumstances. First, the speed values are reconstructed within
linear elasticity theory. Second, the conventional relation between elastic
moduli (bulk elastic modulus $K$ and shear modulus $\mu$) and speeds $c_{p}$
and $c_{s}$ and density $\varrho$

$\displaystyle\mu=\varrho c_{s}^{2},\quad
K=\varrho(c_{p}^{2}-\frac{4}{3}c_{s}^{2})$ (1)

holds only when the coefficients in the wave equation describing the
propagation of sound are constant, i.e. when all the above mentioned
parameters of the medium are constant. Third, there is a natural applicability
criterion of elasticity linear theory, the smallness of the strain tensor
$u_{ik}$. In this theory, the trace of the strain tensor $u_{ii}=p/K$, where
$p$ is pressure. Let us take as a criterion of applicability of the smallness
of the trace and see how it is met in geophysical models.

We note first of all that in those models (e.g., (Bullen and Haddon,, 1967;
Stacey and Davis,, 2008; Garcia at al.,, 2011)) is used for the pressure the
Bullen-Haddon hydrostatic approximation ($\mu=0$) in which the pressure at the
surface is zero. In general, this is not so: body forces create internal
stresses (including shear) even in the absence of external forces. In
particular, the model of a homogeneous gravitating sphere (exercise 3 of §7 in
(Landau and Lifshitz,, 2012)) gives

$\displaystyle
p(r)=p(0)\Bigl{(}1-\Bigl{(}1+\frac{8\mu}{15K}\Bigr{)}^{-1}\Bigl{(}\frac{r}{R}\Bigr{)}^{2}\Bigr{)}\Bigr{)}$
(2)

Accordingly, for estimation of the criterion we use the model (2) for the
pressure and the formulas (1) to the elastic moduli.

Conventional data (see, for example, (Stacey and Davis,, 2008)) on the
mechanical parameters of the Earth give $8\mu/15K\sim 0.27$ for the whole
mantle. For the pressure is reasonably to use normalization to the common
value of the pressure at the lower boundary of the mantle. Then there are the
following values $u_{ii}$ at depths of 60, 800 and 2878 $km$: 0.19, 0.15 and
0.21. It means that with such precision linear elasticity, in framework of
which the mechanical parameters and evaluated, is applicable. (Note that four
significant figures for them are issued everywhere in conventional tables!)

For the Moon ((Garcia at al.,, 2011)) $8\mu/15K\sim 0.3$, the value $p(0)$ is
calculated in (Pavlov,, 2015):

$\displaystyle
p(0)=\frac{2}{3}\pi\kappa\varrho^{2}R^{2}\Bigl{(}1+\frac{4\mu}{3K}\Bigr{)}^{-1}\Bigl{(}1+\frac{8\mu}{15K}\Bigr{)}\sim
29\times 10^{9}din\,cm^{-2}$ (3)

and estimates for the values of $u_{ii}$ at the depth of $60,270,400$ and
$800\,km$ are as follows: $0.008,0.011,0.012$ and $0.017$ respectively.
Estimation errors are relates only to errors in the measurement of sound
velocity. For $c_{p}$ they are within $3$% throughout the entire interior of
the Moon, and for $c_{s}$ they increase to $11$% only on the boundary of the
mantle and the liquid core. Moreover, with $7$% accuracy density and elastic
modules can be considered constant throughout the thickness of the mantle
since the depths of$28km$ to the border with the liquid core. Thus, unlike the
Earth, the Moon is a perfect object to use linear elasticity.

##  2 The Moon model.

Let us apply the linear elasticity theory to describe the variations of the
Moon interior stress state. We will work in a frame of reference rigidly
attached to the rotating Moon. In this system, all the mechanical forces
(except elastic) acting in the Moon interior (gravity, inertia forces and
tidal) are potential: the acceleration of each of them is the negative
gradient of the corresponding potential.

Since the polar flattening of the Moon is extremely small ($1.25\times
10^{-3}$), for the gravitational potential $\varphi_{0}$ it is enough to take
the approximation of a homogeneous sphere

$\displaystyle\varphi_{0}(r)=\frac{2}{3}\pi\kappa\varrho
R^{2}\Bigl{(}3-\Bigl{(}\frac{r}{R}\Bigr{)}^{2}\Bigr{)},$ (4)

where $\kappa=6.67\times 10^{-8}cm^{3}g^{-1}s^{-2}$ is the gravitational
constant, $\varrho=3.34g\,cm^{-3}$ the average density of the Moon,
$R=1.737\times 10^{8}cm$ is its radius, $r$ is the length of the radius vector
r of point inside the Moon. The order of $\varphi_{0}$ is determined by the
coefficients in front of the bracket (4) and is $1.4\times
10^{10}cm^{2}s^{-2}.$

Motion of the Moon in the solar system is rather complicated. Working in our
frame of reference, we must consider the potentials of all the inertial forces
associated with the acceleration of this movement. We are interested the
dependence on r of the forces generated by these potentials.

Centrifugal potential associated with the proper rotation of the Moon,
expressed by the formula

$\displaystyle\varphi_{cf}(\textbf{\emph{r}})=\frac{1}{2}(\omega_{j}^{2}r^{2}-(\omega_{j}r_{j})^{2}),$
(5)

where $\omega_{j}$ is angular velocity vector with the value of $1.75\times
10^{-7}sec^{-1}$ and an inclination relative to the ecliptic plane
$\epsilon=1.54^{\circ}$. The order of this potential we estimate as its
magnitude for $r=R$ and obtain $4.6\times 10^{2}cm^{2}s^{-2}$.

Centrifugal potential $\varphi_{cf(1)}$ associated with the rotation of the
Moon around the Earth, expressed by the similar to (5) formula. It contains
instead of $\omega$ the angular velocity $\omega_{1}$ of the Moon orbital
motion around the Earth. The magnitude of $\omega_{1}$ is the same as of
$\omega$, the average inclination $i$ with respect to the ecliptic plane
equals $5.145^{\circ}$, and the order of the $\varphi_{cf(1)}$ is same that of
(5). A similar formula holds for the centrifugal potential $\varphi_{cf(2)}$,
associated with the rotation of the Earth-Moon pair around the Sun. The
angular velocity $\omega_{2}$ has the magnitude of $1.5\times 10^{-8}s^{-1}$,
$\omega_{2}$ is perpendicular to the plane of the ecliptic; the order of
$\varphi_{cf(2)}$ equals to $3.4cm^{2}s^{-2}$.

The last of the force of inertia potential $\varphi_{pr}$ is associated with
the precession of the lunar orbit axis with a period of $18.6\,years=7.8\times
10^{9}s$. Its angular velocity vector $\varphi_{pr}$ has the magnitude
$0.8\times 10^{-9}s^{-1}$ and deviates from the direction of $\omega_{1}$ at
an angle $i$. $\varphi_{pr}$ order is a hundred times less than that of
$\varphi_{cf(1)}$.

For the tidal potentials of the Earth and the Sun is sufficient to take the
Laplace approximation

$\displaystyle\varphi_{1}(\textbf{\emph{r}})=\frac{1}{2}\kappa\tilde{M}\tilde{R}^{-3}R^{2}\Bigl{(}\frac{r^{2}}{R^{2}}-3\frac{(\textbf{\emph{r}}\tilde{\textbf{\emph{R}}})^{2}}{R^{2}\tilde{R}^{2}}\Bigr{)},$
(6)

where $\tilde{M}$ and $\tilde{\textbf{\emph{R}}}$ are the mass of the tide
generating body and distance to it from the Moon center. Orders of these two
potentials are defined as for (4), by the coefficients before of the bracket;
for the Earth the order is $1.1\times 10^{5}cm^{2}s^{-2}$, and for the Sun it
is $6\times 10^{3}cm^{2}s^{-2}$.

We see that there is a kind of hierarchy for the potentials: if we take the
gravitational potential as 1, the Earth tidal potential has the order
$0.8\times 10^{-5}$, the Sun one has the order $4\times 10^{-7}$, the
centrifugal potentials $\varphi_{cf}$ and $\varphi_{cf1}$ have the order
$3\times 10^{-8}$, for $\varphi_{cf2}$ the order is $2.4\times 10^{-10}$. We
emphasize that the time dependence is only at the tidal potential with a
characteristic period of 1 month: in our frame of reference only $\tilde{R}$
depend on time. Therefore, only the variations of the tidal potential can
cause variations in the stress state in the the Moon interior. Since the own
oscillations are damped in the Moon interior in a few hours, to estimate the
tidal effects impact on the stress state of the Moon interior, we can use the
condition of equilibrium instead of the equations of motion.

##  3 The linear elasticity equations.

In the linear theory of elasticity, Hooke’s law is valid:

$\displaystyle\sigma_{ik}=K\delta_{ik}u_{jj}+2\mu(u_{ik}-3^{-1}\delta_{ik}u_{jj})=-p\delta_{ik}+2\mu
u^{\prime}_{ik},$ (7)

where $u_{ik}=2^{-1}(\partial_{i}u_{k}+\partial_{k}u_{i})$ is the strain
tensor, $u_{i}=u_{i}(\emph{{r}})$ is the displacement vector,
$\partial_{i}=\partial/\partial r_{i}$, $p$ is pressure and $u^{\prime}_{ik}$
defined by formula (7) is the deviator.

We write the equilibrium condition as a differential equation for the
displacement vector:

$\displaystyle\varrho\partial_{i}\varphi=\partial_{k}\sigma_{ik}=((K+3^{-1}\mu)\partial_{i}\partial_{k}+\mu\delta_{ik}\triangle)u_{k}\equiv
L_{ik}u_{k},$ (8)

where $\varrho$ is density, $\triangle=\partial_{j}^{2}$ is Laplace operator
and $\varphi=\Sigma_{a}\varphi_{a}$ is the potentials sum of mass forces
(gravity, centrifugal and tidal). $L_{ik}$ for our Moon model is the second
order differential operator with the constant coefficients. (Everywhere in
previous formulas and later the sum on repeated indices is implied:
$a_{k}b_{k}\equiv\Sigma_{k}a_{k}b_{k}$.)

Boundary condition for a differential equation (8) for $u_{i}$ is set on the
Moon surface $\Sigma$ defined as $\varphi=const$:

$\displaystyle\sigma_{ik}\partial_{k}\varphi|_{{}_{\Sigma}}=0.$ (9)

The solution of equilibrium condition for $u_{i}$ is the sum of a particular
solution of the inhomogeneous equation
$L_{ik}u_{k}=\varrho\partial_{i}\varphi$, represented as a simple gradient,
$u_{i}=\partial_{i}u(\emph{{r}})$, and a general solution of the homogeneous
equation $L_{ik}\tilde{u}_{k}=0$, represented as the sum
$\tilde{u}_{k}=\partial_{i}\tilde{u}(\emph{{r}})+w_{i}(\emph{{r}})$ of the
gradient and the vector $w_{i}$ divergence of which is zero:
$\partial_{i}w_{i}=0.$ The scalar functions $u$ and $\tilde{u}$ naturally
called deformation potentials.

The inhomogeneous equation takes the form of equation for $p=-K\triangle u$ –
the contribution to the pressure from the inhomogeneous equation solution:

$\displaystyle\partial_{i}p=-A\varrho\partial_{i}\varphi,\quad A=(1+4K/3\mu).$
(10)

The solution of equation (10) can be obtained by the following argument
((Pavlov,, 2015)). The variables $p,\varrho$ and $\varphi$ are the scalars
relative to rotation group $O(3)$ and must depend on scalar combinations of
their argument r only. (Here we may temporarily consider $\varrho$ as
changeable.) We may take the potentials $\varphi_{0}$ and two $\varphi_{1}$
for such scalar combinations denoting them as $\varphi_{a},a=0,1,2$. Then the
equations (10) take the form

$\displaystyle(\partial_{a}p=A\varrho)\partial_{i}\varrho_{a},\partial_{a}=\partial/\partial\varphi_{a}$
(11)

for each $a=0,1,2$. But the gradients $\partial\varphi_{a}$ are linearly
independent and therefore $\partial_{a}p-A\varrho=0$. It means that $p$
depends on whole sum $\varphi$ only and it is valid the formula

$\displaystyle p(\emph{{r}})=-A\int\varrho
d\varphi=-A\varrho\varphi(\emph{{r}})+p_{0}.$ (12)

(In (12) we come back to the constant $\varrho$ of our Moon model.) Note that
formula (12) gives the generalization of the above relation (2) and allows for
corrections by other than the gravitational potential. Integration constant
$p_{0}$ is to be found from the boundary conditions (9).

The homogeneous equation has the form

$\displaystyle\partial_{i}\tilde{p}-A\mu\triangle
w_{i}=0,\quad\tilde{p}=-K\triangle\tilde{u}.$ (13)

Take its divergence. We obtain the Laplace equation for $\tilde{p}$. The
boundary condition for it can be taken zero as the pressures $p$ and
$\tilde{p}$ are included in boundary condition (9) as summands. But harmonic
function with zero values at the boundary is equal to $0$ everywhere.
Therefore $\tilde{u}$ and $w_{i}$ are harmonic functions.

To use the boundary condition, we need to find the deformation potentials $u$
and $\tilde{u}$ and harmonic vector-function $w_{i}$.

Having formula (12) for pressure and the relation $p=-K\Delta u$ between
pressure and the trace of the strain tensor, we can recover the deformation
potential $u$ and then the stress and strain tensors, as well as satisfy the
boundary condition (9).

According to the arguments presented above, the scalar function $u$ depends
only on $\varphi_{0}$ and $\varphi_{1}$. In what follows, as the scalar
arguments of the function $u$ it is convenient to take the variable part of
the gravity potential $x=\alpha_{0}r^{2}/2$ and the dimensionless ratio
$z=\varphi_{1}/x$ while distinguishing the small parameter
$\gamma=\alpha_{1}/\alpha_{0}\sim 10^{-5}$ in it:
$z=\gamma\zeta,\,\zeta=-r^{-2}\tau_{ik}r_{i}r_{k}.$ (Here we denote by
$\alpha_{0}$ and $\alpha_{1}$ the coefficients before $r^{2}/2$ in expressions
(4) and (6).) In these terms, the equation for recovering $u$ has the form

$\displaystyle\Delta u(x,z)=B(x(1+\gamma\zeta)-c),\quad B=AK^{-1}\varrho,\quad
c=\frac{3}{2}\alpha_{0}R^{2}+(A\varrho)^{-1}p_{0}.$ (14)

It is shown in (Pavlov,, 2015) that a solution to equation (14) is given by

$\displaystyle
u=\tilde{B}\Bigl{(}x^{2}\Bigl{(}\frac{1}{5}-\frac{2}{7}\gamma\zeta\Bigr{)}-\frac{2}{3}cx\Bigr{)},\quad\tilde{B}=(2\alpha_{0})^{-1}B.$
(15)

It gives us a particular solution of the inhomogeneous equation
$L_{ik}u_{k}=\varrho\partial_{i}\varphi$ for the deformation vector
$u_{k}=\partial_{k}u.$

The general solutions for harmonic functions $\tilde{u}$ and $w_{i}$ are
constructed within the framework of perturbation theory in the small parameter
$\gamma.$ The Laplace equation for $\tilde{u}(x,z)$ is given by

$\displaystyle\Delta\tilde{u}(x,z)=2\alpha_{0}x^{-1}\Bigl{(}\frac{3}{2}xu_{x}+x^{2}u_{xx}-\frac{3}{2}\gamma\zeta
u_{z}+\gamma^{2}(\zeta^{2}-\zeta+2)u_{zz}\Bigr{)}=0.$ (16)

In the first two orders in $\gamma$ the general solution for $\tilde{u}(x,z)$
with the natural condition $\tilde{u}(0,z)<\infty$ has the form

$\displaystyle\tilde{u}=c_{1}xz,$ (17)

where $c_{1}$ is an arbitrary constant. It is shown in (Pavlov,, 2015), that
the vector function $w_{i}$ vanishes in the first two orders in $\gamma$.

Thus, in the first two orders in $\gamma$, the general solution of the
equilibrium conditions for the displacement vector $u_{i}(\textbf{r})$ is
given by the sum of the gradient of the deformation potential $u$ defined by
formula (15) and the gradient of the deformation potential $\tilde{u}$ defined
by (17). It is shown in (Pavlov,, 2015), that the terms of the zeroth order in
$\gamma$ of the boundary condition (9) fix the constant $c$:

$\displaystyle c=\frac{\alpha_{0}}{2}R^{2}\Bigl{(}1+\frac{8\mu}{15K}\Bigr{)},$
(18)

and the terms of the first order do it for the constant $c_{1}$:

$\displaystyle c_{1}=-\frac{\alpha_{0}}{2}R^{2}\frac{2}{35}.$ (19)

As a result, formula (15) and (17) – (19) allow to calculate the strain tensor
and stress tensor up to first order in $\gamma$. In particular, we get for the
pressure

$\displaystyle
p(\textbf{r})=\frac{1}{2}\alpha_{0}\varrho\Bigl{(}1+\frac{4\mu}{3K}\Bigr{)}^{-1}\Bigl{(}\Bigl{(}1+\frac{8\mu}{15K}\Bigr{)}R^{2}-r^{2}\Bigl{(}1-\gamma\tau_{ik}\frac{r_{i}r_{k}}{r^{2}}\Bigr{)}\Bigr{)}.$
(20)

Obviously when $\gamma=0$ we obtain (2) for uniform solid sphere, and when
$r=0$, – the value $p(0)$ of formula (3). It should be noted that the members
of the first order in $\gamma$ in (20) do not contribute to the value $p(0).$

##  4 Variations of the free energy density

In linear elasticity theory, the volume density $f$ of the free energy of
elastic stresses is expressed in terms of the stress tensor:

$\displaystyle f=\frac{K}{2}u_{jj}^{2}+\mu(u^{\prime}_{ik})^{2}.$ (21)

In Section 3, we have calculated the trace of the strain tensor $u_{jj}=\Delta
u$ (formula (14)). The results of the same section allows to calculate the
deviator $u^{\prime}_{ik}=u_{ik}-3^{-1}\delta_{ik}u_{jj}.$ In both formulas,
only the dimensionless function $\zeta=1-3\cos\vartheta$ depends on time,
where $\vartheta$ is the angle between the radius vector r of a point in the
Moon’s body and the radius vector $\tilde{\textbf{\emph{R}}}$ of the Earth.

The function $\zeta$ depends periodically (with a period of $T$ = 1 month) on
time and experiences variations $\delta\zeta$. Define $\delta\zeta$ as the
difference of absolute values of $\zeta$ for two extreme positions of
$\tilde{\textbf{\emph{R}}}$ in our coordinate system. In the spherical
coordinates, $\emph{{r}}=r(\cos\theta\cos\phi,\cos\theta\sin\phi,\sin\theta)$,
where $\theta$ and $\phi$ are the latitude and longitude of the radius vector
r in the Moon’s body (the axis $z$ is directed along the self-rotation axis of
the Moon). Since the self-rotation of the Moon and its motion along the orbit
are synchronized and the direction of the angular velocity vector $\omega_{j}$
is constant, from the standpoint of the Moon the extreme positions of the
Earth correspond to
$\tilde{\textbf{\emph{R}}}=R(\cos\varepsilon,0,\pm\sin\varepsilon)$, where
$\varepsilon$ is the angle of inclination of $\omega_{j}$ to the ecliptic
plane. Corrections related to the ellipticity of the Moon’s orbit for the
r-dependent part of $\zeta$ are on the order of $0.05,$ and we neglect them.
As a result, we obtain the following formula for the Earth tides:

$\displaystyle\delta\zeta=|\sin 2\varepsilon|\cdot|\sin
2\theta|\cdot|\cos\phi|.$ (22)

The variation of the function $\zeta$ corresponds to the variation of the free
energy density (21):

$\delta f=Ku_{0jj}\delta u_{1jj}+2\mu u^{\prime}_{0ik}\delta
u^{\prime}_{1ik},$ (23)

where $u_{0jj}=B(x-c)$ is the trace of the strain tensor in the zeroth order
in $\gamma$, $\delta u_{1jj}=B\gamma x\delta\zeta$ is the variation of the
trace in the first order in $\gamma$, and the convolution
$u^{\prime}_{0ik}\delta u^{\prime}_{1ik}$ is calculated in paper (Pavlov,,
2015):

$u^{\prime}_{0ik}\delta
u^{\prime}_{1ik}=B^{2}\gamma\frac{22}{105}x\Bigl{(}x+\frac{21}{22}c_{1}\Bigr{)}\delta\zeta.$
(24)

As a result, we obtain the following formula for the variations of the free
energy density:

$\delta f=B^{2}\gamma\delta\zeta
x\Bigl{(}K(x-c)+\frac{44}{105}\mu\Bigl{(}x+\frac{21}{22}c_{1}\Bigr{)}\Bigr{)}.$
(25)

Here it is useful to introduce a new variable $y=r/R$ and substitute the mean
values of elastic moduli, $\mu/K\sim 0.57.$ Then (25) reduces to

$\delta f=-0.75\gamma\delta\zeta K^{-1}p^{2}(0)y^{2}(y^{2}-1.04^{2}).$ (26)

The graph of the variation of the free energy density versus the dimensionless
radius $y=r/R$ has a zero at the center, a maximum at $y=0.52$, and one-third
of the maximum value on the surface (in the hydrostatic approximation it would
be zero).

##  5 Comparison with seismic data

The data on moonquakes obtained in $1969-1977$ by the expeditions of the
Apollo project ((Nakamura at al.,, 1973; Lammlein at al.,, 1974)) have been
discussed in the literature for more than four decades. Today, there is a
catalogue ((Nakamura,, 2003, 2005)) of more than 12 500 seismic events. Most
events (more than 7000) are identified as deep moonquakes whose sources are
concentrated at about 300 ”nests” at a depth of between 700 and 1200 $km$.

According to the modern views on the mechanics of strength and destructions
(see, for example, (Volkov and Korotkikh,, 2008)), periodic variations of
pressure make the main contribution to the variations of the free energy
density and lead (in addition to ordinary dissipation into heat) to the
accumulation of defects in a solid medium (dislocations, cracks, etc.). Such a
process is accompanied by the concentration of energy in these structures.
When the concentration reaches a certain critical limit, the medium is
destroyed with a release of accumulated energy.

We draw attention to the qualitative similarity between two patterns: depth
distributions for deep focus moonquakes and variations of the free energy
density due to tides. This similarity suggests that it is the energy
concentrated at defects that is released in deep focus moonquakes.

This suggestion is corroborated by the integral (over the depths between 700
and 1200 $km$, which correspond to the interval $[0.3R,0.7R]$) estimate for
the variations of the free energy, which is possible due to formula (26):

$\delta F=\int\delta f(r)r^{2}drd\cos\vartheta d\varphi$ (27)

over a half-period $T/2$. For the Earth tides, this estimate yields $0.9\times
10^{29}erg$ during a period of $T/2=1.8\times 10^{7}s$, or $2\times
10^{30}erg$ per year. This is mainly the energy of oscillations. However, in a
nonideal continuum, the dissipation of the energy of oscillations occurs both
due to viscosity and due to the accumulation of defects.

Available estimates for the fraction of dissipating energy are very rough,
depend on how the nonideality of the medium is modeled, and are on the order
of $10^{-2}$. Apparently, the overwhelming part of this energy is converted
into heat and is spent on heating the body of the Moon. If all the heat goes
outside, then the power of its flux amounts to $1.4\times
10^{2}ergs^{-1}cm^{-2}=14\times 10^{-6}Wcm^{-2}$. The available experimental
estimates ((Langseth at al.,, 1976)) yield a value of $2\times
10^{-6}Wcm^{-2}$. Hence, theoretically, all the heat is spent on heating the
body of the Moon.

The fraction of dissipating energy that is spent on the accumulation of
defects is also estimated roughly ((Volkov and Korotkikh,, 2008)). The data of
laboratory experiments are formulated in terms of the number of cycles of
periodic loading that lead to the destruction of a sample. Translation of
these data into the language of energy stored in defects depends on the model
of the destruction process and is estimated as $10^{-5}$ of the oscillation
energy. In any case, this estimate is many orders of magnitude greater than
the estimate of energy release in deep focus moonquakes (see Goins at al.,
(1981)), which amounts to $8\times 10^{13}erg$ per year. We conclude that the
energy of tidal oscillations is more than enough to explain where the energy
released in deep focus moonquakes comes from.

##  6 Acknowledgements

I am grateful to N.A. Slavnov and to the participants of the seminar led by
A.G. Kulikovskii for the discussion of the results and valuable remarks.

This work is supported by the Russian Science Foundation under grant
14-50-00005.

References

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