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About the connection of strong earthquakes with Earth crust’s tidal deformation.
The earthquake’s connection with the phases of Moon is more or less completely investigated by scientists in many countries.[1-15]. The discovered connections are usually of an ambiguous nature. They can either become apparent or not at the next shock, and be completely absent in other regions. In many author’s opinion a common mechanism has to exist, which controls the source’s discharge moment. Therefore, before talking about an earthquake’s prediction possibility, is necessary to find out and establish such a stationary factor, which appears as a regular connection in any geo-tectonic conditions.
This problem includes two basic objectives: the source’s physics and the “trigger mechanism’s” physics. Among other factors the lunar-solar deformations of the Earth crust can play this last role.
In order to uncover the role of tidal deformations in the “trigger mechanism” there were performed detailed investigations of many astronomic parameters at the earthquake time points in the Academy of Sciences of Russian Federation: in the Institute of Terrestrial Magnetism together with the Institute of Applied Astronomy. There were investigated Moon and Sun azimuth values, their first and second derivatives: the Laplace’s tidal potential’s vertical (Z) and horizontal (H) components. As source material has served the terrestrial globe 725 M>7 earthquake case’s catalogue, which were registered in 1904-1946. In 30% of cases from the general number there were intermediate depth earthquakes, and deep focused earthquakes in 70% of cases. There are shown Moon and Sun azimuth distribution functions in shock time points in fig. 1. Here we can see that the event case’s number is increasing at the 900 and 2700 azimuths, so, about 6 AM and 6 PM of the local time. In the upper and lower culmination zones the earthquake’s number is decreased noticeably.

,

fig. 1. Moon and Sun azimuth distribution function in local time,
obtained in real earthquakes’ time points.
   A similar result was obtained by American investigators for South California. The obtained distribution function could be interpreted as the Earth seismic activity daily cycle, then it could be asserted, that there is really a connection of the earthquakes with the Moon’s, Earth’s, and Sun’s positions. However, the time point’s and source coordinate’s random selection performed over the terrestrial globe within ± 700 latitudes gives a distribution function (fig. 2) with parameters identical to the one in fig. 1. Here and further the background in the pictures means the random selection. Superposing these function’s coordinate axes we can see, that the background curves become inscribed in real earthquake’s function’s confidence interval.

,

fig. 2. Moon and Sun azimuth distribution function in real
earthquake’s and randomly selected time points.

   Further forming the function of the difference’s distribution between the Moon and Sun azimuths of the real earthquakes and the random selection (fig. 3), they will be found identical too. We can see in the fig., that the background is within confidence interval limits.

,

fig.3. The functions of the differences’ distribution between Moon and Sun azimuths
of the real earthquakes’ time points and the random
selected time points and the epicenter’s geographic coordinates.

  The distributions of the local hour angle within earthquakes’ fixed time daily cycle and the random selection time are filling uniformly the local daily cycle. The azimuths’ calculation in this case gives practically indistinguishable from each other functions.

It follows that the discussed connection’s existence is not possible to demonstrate based on azimuths’ distribution within local daily cycles. Let’s observe Sun and Moon azimuths’ ratio:

(8)

then we have:

(9)

where tⁿ and t^M are Sun’s and Moon’s Greenwich hour angles, φ and λ are accordingly the epicenter coordinates.
The events number’s maximum falls in the zone, where Moon and Sun azimuth differences are close to zero. Supposing K=1 in (9) and resolving relatively tg we get:

(10)

According to (8)-(10) is possible to follow the random selection process and the “causation” of real earthquakes by Moon and Sun.
Let’s observe the case of well-known Chilean earthquake on May 22, 1960. The main shock was preceded by two foreshocks. The first one was located at a little more than 200 km from the main shock.

Date	Bodies’ altitude	Azimuth	Shift (in cm)
May 21, 1960 10h 02min	h_s = 210,5 h_m = 22.7	A_s = -80°,6 A_m = 65.8	H_s = 16,8 H_m = 35.9 Z = -14.8
May 22, 1960 10h 32min	h_s = 17.0 h_m = 15.9	A_s = -77.5 A_m = 65.6	H_s = -13.7 H_m = 25.9 Z = -19.4
May 22, 1960 19h 11min	h_s = 22.0 h_m = 14.0	A_s = -34.8 A_m = 65.7	H_s = 17.1 H_m = 22.6 Z = -19.4

The chronology and circumstances of source’s “causation” or “trigger mechanism’s” action were as follows: in the first two cases Moon azimuth was the same, and the shocks arisen at that apparently could cause internal strains’ redistribution in the source. At the main discharge Moon azimuth has changed its sign. It is possible to suppose, that the tidal shift is probably acting to compensate internal strains in the source zone.
In fig.fig. 4 and 5 are shown dA/dT distribution functions: the azimuth changes in time units. The calculations were made according the formula (11).
,

(11)

where h is the celestial body’s altitude in shock’s time point.

fig. 4. Sun’s dA/dt distribution functions (the background is a random selection in the time and epicenter coordinates).,

fig. 5. Sun’s dA/dt distribution functions (the background is a random selection in the time and epicenter coordinates).,
fig. 6. Moon’s dA/dt distribution functions (the background is a random selection only in epicenter coordinates).,

   In order to determine the reason of symmetric maximums’ appearance on the background curve, the shock time was shifted selecting randomly in the real source. Such time shifts (fig.fig. 5 and 6) are not changing noticeably the dA/dt distribution function of main shocks (fig. 7).

fig. 7. Sun’s distribution function dA/dt (the background is random selection only in coordinates).,

   The investigations show, that the maximums on background curve appear on latitude ±45°, so, a latitudinal effect arises. On latitude φ = 45° we have the main shocks’ distribution function:

,

(12)

   The expression in parentheses in the second member in most cases is close to one, so, it ensures the maximum. The latitudinal effect is peculiar not only to ±450 latitudes, but also to others. Differentiating (12) one more time, we get:
,

(13)
fig. 8. Moon’s distribution function dA2/dt2 (the background is random selection in the time and epicenter coordinates).,

fig. 9. Sun’s distribution function dA2/dt2 (the background is random selection in the time points and epicenter coordinates).,

   From (13) we can see, that the acceleration can achieve high values at high altitudes of celestial bodies. High altitudes are observed near sub solar and sub lunar points: at the celestial bodies’ culmination on the local meridian, in the general case (fig. 8).

   The background’s and real shocks’ d²A/dt² distribution functions can be considered identical, and, refusing the background curve’s observation, we can conclude, that the azimuth alteration’s speed doesn’t determine any connection with the concerned events.

   It was supposed, that any clarifications in observed connection is possible to get from the analysis of dh/dt, the celestial bodies’ altitudes’ alterations in earthquakes’ time points. The calculations were made according to the formula:

dh/dt = 15’ cosφ sinA.

(14)
   In fig. 10 are presented dh/dt changes 12 hours before and after the shock for all 725 earthquake cases superposed in one epicenter. There are shown dh/dt changes on a surface, tangential to the epicenter, taking into account the dh/dt projection on the surface z. There are almost not observed any differences in the behavior pattern of the mathematical expectation M and dispersion D in the “Sun” case. Moreover, M and D are insignificantly changed about 4 hours before the shock. This can point to the approaching of the observed process to a stationary one, probably peculiar to any shock.

fig. 10. The mathematical expectation M and dispersion D of dh/dt 12 hours before and after the shock.,

   In “Moon” case the D curve is more broken compared with “Sun” case, though the M behavior is generally similar to “Sun” case. While taking into account the dh/dt projection on the Z-axis, M and D are getting abrupt variations. This can be explained by Moon’s higher mobility.
The curves’ analysis results in the conclusion, that the dh/dt projection on the axis Z has to play significant role in the earthquakes’ “trigger mechanism”. It is known, on the other hand, that the tangential strains are concentrated in Earth’s crust and can reach high values.
   It is reasonable to suppose, that the tidal deformations, superposed on already existing ones in the source area, can result in its discharge. The tide’s horizontal H and vertical Z components were obtained according to formule:

,

(15)

,

(16)

Here Z is the celestial body’s zenithal distance, a is Earth’s radius.
   Carrying in a diagram the calculated H and Z components, we get the picture, shown in fig. 11.

,
fig. 11. The tidal shifts’ distribution in earthquakes’ time points.

   The central ellipse belongs to the shifts, caused by Sun, the external one is caused by Moon. The formulae (15) and (16) are working so, that an m mass particle is forced to move continuously on the elliptic trajectory’s sector, describing some lunar-solar ellipse. We can see in fig., that the radial Z component tide is non-symmetric. It is in accord with (16).
   Moon ellipse’s dispersed outline is explained by Moon’s geocentric distance’s alterations. The same effect, noticed in Moon’s and Sun’s azimuth distribution functions is also here observed, the earthquakes are mainly grouped in morning and evening hours of the local time. Their number is sharply reduced in the noon.
   Based on this we can conclude, that the negative Z component is more active, than the positive one. Projecting Sun’s horizontal component on the lunar ellipse’s corresponding axis and denoted the angle between Moon’s and Sun’s tidal vectors as Y, this angle’s values can be obtained from the following formula:
,

(17)

where i_M and i_S correspondingly are Moon’s and Sun’s tidal vectors’ dip angles, A is the difference between Sun’s and Moon’s azimuths. At A = 0°, 90° and 180° we successively find:

,

(18)

,

(19)

,

(20)
   From these formule we can see, that the source’s actuation moment depends only on Moon’s and Sun’s tidal vectors’ dip angles. At A=90° are excluded shifts caused by Sun. This points to the fact, that the lunar tide possibly can independently cause a shock. The tangential strains, existing in the source area, in Earth’s crust, apparently can reach noticeable higher values, than in the area surrounding the source. If so, then horizontal tidal strains can either intensify or attenuate them, and, at certain favorable conditions, lead the source to discharge.

fig. 12

fig. 13

   In fig. 12 is shown horizontal tidal shift amplitudes’ distribution in real earthquakes’ time points. The epicenters coincide with the coordinate origin. There are marked horizontal shifts in the source area, caused by Sun (x-axis) and by Moon (y-axis) in the shock time points.
    In fig. 13 is shown horizontal tidal shift amplitudes’ distribution in randomly selected earthquake time points and epicenter geographic coordinates. This effect is expressed slightly weaker in fig. 12, the points are more evenly distributed on the surface. The hydro-cycloid outlined central picture areas include essentially different earthquake cases number. We can clearly see here, that earthquakes don’t exist at zero shifts. Moreover, here is revealed the coordinated working of lunar-solar tidal deformations, except the cases, when maximal lunar tides cause the shock independently, without Sun’s participation.
fig. 14. The “trigger mechanism’s" model.

   Holding the standpoint, that the key role in the “trigger mechanism” belongs only to horizontal shifts and observing a source model, which works in conditions of the following equation’s existence

H_Scosx = cos[180°-(ΔA + x)],

(21)

or
,

(22)

where ΔA is the azimuth difference (ΔA =0° and 180°), then it is possible to obtain the supposed strain axes direction’s azimuth in the source. The source’s or, more precisely, “trigger mechanism’s” schematic model is shown in fig. 14.

    In this model Z vertical component is compensated by N reaction. The arrows are showing the strain directions, acquired by the source in its development process. The break’s one board’s directions will always coincide with lunar-solar horizontal shift directions. From the other hand X in (21) is the angle between the normal to the break and the maximal horizontal shift direction, caused by the celestial body.

   So, H_S and H_M projections on this normal can attenuate the break’s compression to a condition, ensuring the possibility to slide its one board relatively to the other one.

   The seismo-active breaks’ forming during Earth’s geologic development history happened not only under the influence of internal dynamic factors, but also external ones. Among such external factors have to be considered first of all lunar-solar tidal deformations of not only the proper Earth’s crust, but also the entire terrestrial body.

   This is why the breaks forming and their spatial orientation have to reflect definite way the tidal influences. The tidal deformations, as we can see from the formulae (15) and (16), don’t reach zero simultaneously, therefore Earth’s crust is continuously in deformation condition.
   The source’s discharge happens at unknown circumstances of its internal condition. This circumstance is the proper obstacle, behind which lies the earthquakes prediction. We can see in fig. 11, that while the tide amplitude increase the earthquakes number is increased. Less frequently the sources’ actuation is observed also at slight shifts.

   So, obtained results’ analysis leads to following conclusions:
    - The investigations of the planetary seismic activity’s connections with such astronomic parameters, as Moon and Sun altitudes, their first and second derivatives, Moon’s age, has shown, that such searches of the correlative connections without taking into account the random selection comparisons are methodically unacceptable and lead to ungrounded results.
   - The earthquakes sources’ stimulation is ensured by Earth’s crust’s tidal deformations in favorable ratio time points of the strain axes in the source area to the tidal shifts horizontal and vertical components’ axes.
   Further investigations have to be directed to the study of seismic activity’s connection with Earth’s crust’s tidal deformations in different regions together with the observation of more wide magnitude range earthquakes.

Translated from Russian by George Vardanyan.

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