**TITLE**
ASP Conference Series, Vol. **VOLUME**, **PUBLICATION YEAR**
**EDITORS**
Gravitational mechanisms of cyclic planet energization in
planetary systems
Yu.V. Barkin
Alicante University, Spain; Sternberg Astronomical Institute, Moscow,
Russia
J.M. Ferrandiz
Alicante University, Spain
Abstract. In the paper we discuss tidal and non-tidal mechanisms of
forced = endogenous activity of the given body PS (planet or satellite)
caused by the gravitational attraction of external celestial bodies (EB)
which are potential mechanisms determining induced rhythms o f variations of PS activity (seismic and volcano activity and others processes).
1.
Introduction
Every celestial body is subjected by a gravitational attraction of the external
celestial bodies which in definite measure influence on variations of planet endogenous activity in definite rhythms and provide an energization and cyclic
activation of the body. Here,in short form, we consider two possible mechanisms of the forced PS energization.
Tide mechanism. We suggest a new illustration of the mutual energetic
effect of superposition of tides produced by different external celestial bodies.
Every from external bodies of a planetary system (including central star) generates tides on the given planet (PS). In the classical approximation these tides
can be described by the linear theory of elasticity. A full effect of PS deformations is presented as the linear superposition of tides rised by the EB. The
tensional state of PS is characterized by the elastic energy stored in the superposition of tides. The new formula (1) for the elastic energy of a superposition
of tides was obtained on the base of classical solution of the problem of elasticity for a model of a planet with concentric mass distribution subjected to the
gravitational attraction of EB. We have shown that the elastic energy of tides
contain additional to classical crossed terms. They are caused by simultaneous
actions of each pair of planets (EB) and describe mutual effects of superposition
of tides generated by these EB.These additional terms for concreate sytems (for
example for Earth-Moon-Sun system) can be significant and lead to remarkable
periodic variations of elastic energy.
Shell mechanism. Gravitational interaction of EB of planetary system with
non-spherical, non-homogeneous shells of the PS body generates additional mechanical forces and moments between neighboring shells (core, mantle, mantle
shells). The acting of these forces and moments to shells generates cyclic per1
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Barkin & Ferrandiz
turbations (in the different time scales) of the tensional state of the shells, that
leads to deformations, small relative translational displacements and small relative rotational oscillations (Barkin, 2002). These additional forces and moments
being of the cyclic celestial-mechanical nature produce cyclic deformations of all
layers of the planet and influence on its inner and surface processes. Effects of
superposition of these forces are described in analytical form.
2.
Elastic energy of the planet tides
We will study a gravitational influence of the surrounding planets (including
central star) Pi (i = 1, 2, . . . , n) of a planetary system (considered as material
points with the masses mi (i = 1, 2, . . . , n)) on the given planet P0 (PS) considered as a finite deformable body with definite elastic properties. Let Oxyz be
an inertial Cartesian reference system with origin in the center of mass of the
PS and with axes saving directions in q
the space. Let xi , yi , zi be the Cartesian
coordinates of the planet Pi and ri = x2i + yi2 + zi2 the module of its position
vector in the mentioned reference system.
We will assume that the gravitational attraction of every planet generates
tides on the PS body which can be described by a well known solution of the
elasticity problem about tides of an elastic body with concentric distribution of
the density and elastic parameters, as in Getino and Ferrandiz, 1991 or Getino,
1992. In the fore mentioned classical approach, the elastic energy associated
to the tides rised by the gravitational attraction of the planet Pi accumulates
on the PS body as a simple sum. The analytical expression of each component
term of this energy is Ei = e2 m2i /ri6 , where e2 is a constant coefficient of second
order which depends the geometry and reology characteristics of PS.
It was shown that the full elastic energy produced by gravitational actions of the all EB on the
P considered body does not equal to the sum of those
previous energies, E 6= ni=1 Ei (Barkin and Ferrandiz, 2003). In addition to
them, new terms appear depending on the relative planet positions and having
a conditionally-periodic nature. Omitting analytical calculations, we give here
only the final full expression of the elastic energy for PS deformable body of the
considered system:
E=
N X
∞
X
i=n n=2
= en
m2i
2(n+1)
ri
+2
N X
∞
X
i,j=1
i>j
n=2
en
mi mj
n+1 n+1 Pn (cos Sij )
ri rj
(1)
Here en are new elastic parameters of arbitrary order n. Pn (cos Sij ) is Legendre
function of the order n. cos Sij = (xi xj + yi yj + zi zj )/ri rj is the cosine of the
angle between the position vectors of EB Pi and Pj .
3.
Induced interaction of the planet shells
Every planet renders a differential action on the non-spherical shells of the PS
and induces some additional interaction between them (Barkin, 2002). Here
we give a final expression for the module of the force of interaction between
APS Conf. Ser. Style
3
two shells of the PS (core and mantle) caused by the attraction of a system of
external bodies (including central star):
(
F
=D
=
N
´
X
m2i ³
2
4
1
−
2γ
+
5γ
i
i +
8
i=1
+2
N
X
i,j=1
i>j
=
ri
mi mj h
ri4 rj4
³
´
³
γi γj 4 − 5γ =2i −5γj2 + 1 − 5γi2
´³
´
1 − 5γj2 = cos Sij
(2)
Here D is a constant coefficient D = f m1 [(C2 − A2 )/m2 − (C1 − A1 )/m1 ].
γi (i = 1, 2, . . . , n) are cosines of angles between the radius-vector of a EB Pi
(w.r.t. a reference system Oxyz ) and the axis of rotation of the PS. f is a
gravitational constant, m1 and m2 are the mass of core and mantle of the PS;
C1 , A1 and C2 , A2 are polar and equatorial moments of inertia of the core and
mantle.
Function (1) and (2) have a conditionally-periodic structure for a given
planetary system. They are changed following the corresponding frequencies or
rhythms, which should agree with the rhythms of the PS endogenous energy
variations due to the forces addressed in this paper.
4.
About correlation of earthquakes and moonquakes with variations of elastic energy
The relative orbital motions of the Moon and the Earth are identical. Periodicities in their orbital motions in same style influence on the ti dal processes on
both celestial bodies and, consequently, rhythms at identical periods can be expected in seismic processes for the Moon and for the Earth. First confirmation
of mentioned correlations has been obtained for shallow earthquakes (with magnitude ¿7.3) and shallow moonquakes in period 1971-1976 (Schirley, 1985/1986).
Tidal nature of moonquakes has been discussed by an interpretation of results
of their spectral analysis (Lammlein, 1977; Oleinic et.al, 20 00). In mentioned
papers the periods of the Moon orbital perturbations in 27.4 d, 13.6 d, 206 d and
some others have been determined. Our spectral-temporal analysis of the full
series of moonquakes from catalogue which has been kindly presented us by Y.
Nakamura (12558 events) let us to confirm mentioned periods and to establish
some fine structure of tidal periodicities (M. Garcia). The main cyclicities of the
Moon seismic process is characterized by the half of draconic period (13.62 d),
the half of the synodic period (14.77 d) and the draconic period (27.20 d). Also
were determined variations of Moon seismicity with another periods multiple to
orbital draconic, anomalistic and synodic periods (in days): 5.50; 6. 75; 9.15;
9.80; 13.15; 22.8; 32.8 ant others. Obtained results were analyze d and confirmed in comparison with similar results obtained for a random distribution of
quakes. So, a celestial mechanical nature of seismic rhythms on the Moon looks
sufficiently clear. Although, the spectral-temporal analysis has revealed some
temporal instability of the rhythms observed (Oleinic et.al, 2000). As known
the many from the Earth processes are characterized by a similar behavior. It
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Barkin & Ferrandiz
points that some contribution to observed picture of seismicity can be caused
by another (non-tidal) mechanism.
Basing on these results we have studied theoretical curves of elastic energy
change of lunar-solar deformations of the Earth mantle and terrestrial-solar deformations on the Moon in period 1971-1976 years. Preliminary analysis has
been shown that the moments of quake events as usually are situated closely to
moments of extreme values of elastic energy. Moments of big earthquakes (with
magnitude 7.3 and more) and moonquakes in mentioned period of time are correlated with moments of elastic energy extremes. In 1975 year, for example, 12
from 13 quakes are situated closely to extremes of elastic energy curve (with a
deviation in 1-1.5 days).
Assuming that mentioned particularity is general for the longer periods of
time we fulfilled statistical analysis of differences of dates of big earthquakes (in
the last 30 years with a magnitude ¿7) and close dates of extremes of elastic
energy (seismosurfing.html). Obtained results in general confirm the discussed
tendency and a new phenomenon of displacement of dates of the big quakes on
1.5-2.0 days with respect to the dates of extremes of elastic energy has been
observed. However, the temporal distribution of earthquakes is characterized
more complex structure comparatively with moonquakes. To explain observed
data we plan to study in future a possible role of the plate motion and a role
of mechanical interaction between non-spherical mantle and core of the Earth
induced by gravitational action of the Moon and the Sun.
Barkin’s work was supported by grant SAB2000-0235 of Ministry of Education of Spain and by RFBR grant 02-05-64176.
5.
References
Barkin, Yu.V. 2002, Izvestia sekzii nauk o Zemle RANS,
N9(Moscow, VINITI), 45.
Barkin, Yu.V., & Ferrandiz, J.M. 2003, in Geophys. Res. Abstr., Vol. 5,
03227, 2003 =A9 European Geophysical Society 2003. EGS-AGU-EUG Joint Assembly (Ni
Getino, J. 1992, Celes. Mech. and
Dyn. Astron., 53, 11.
Getino, J., & Ferandiz, J.M. 1991, Celes. Mech., 51, 17.
Lammlein, D.R. 1977. Phys. Earth. and planet. Inter., 14, 224-273.
Oleinik, O.V., Galkin, I.N., Gamburtsev, A.G.2000. LPS XXXI.
Seismosurfing the Internet for Earthquakes Data: http://www.geophys.washington.ed
Shirley, J.H. 1985/1986, Earth and Planetary Science Letters, 76,
241.