Fluctuations of the planetary gravitational eld and nonlinear interactions with matter – triggering earthquakes.

The orbits of the big planets are very stable during millions of years. In addition to this, there is another important circumstance: the orbits of the planets lie almost on the same level. They represent natural oscillators on a big scale. Such a rhythm or such duration of vibration is determined by the time period from conjunction to conjunction of two planets. These are relatively stable frequencies. A non-linear correlation function forms a good way of describing these processes. It can be shown that this correlation function can also be interpreted as a non-linear interaction of the planetary fluctuations of the gravitational field with material structures. The vibrations of the planetary gravitational field lead to higher vibrations, to higher harmonics, in material structures. The problems of the correlation function are the coefficients a k and the meaning of H ij . In my researches I restricted myself to the polar qualities which are associated with the concepts of "stability" and "instability". The change from stable to unstable conditions and vice versa, can be observed in the evolution of many complex systems. Statistical researches will be presented and show this non-linear influence. Translated from German into English by DeepL.


Fluctuations of the planetary gravitational field
Galaxies in space, planetary systems, clouds, geological formations, plants and animals, human societies, our nervous system, quantum physical systems form simple and also complex structures on scales of different sizes. It is possible that the formation of such structures can be described from a model of more or less strongly coupled, oscillating subsystems. One such oscillating subsystem is the planetary system. The sun and moon are weakly coupled with the system of oceans, causing them to oscillate even in the ebb and flow of the tide. "Big earthquakes, such as the ones that devastated Chile in 2010 and Japan in 2011, are more likely to occur during full and new moons -the two times each month when tidal stresses are highest." [1] Cause and effect are related in a relatively simple and proportional way. But are there also non-linear relationships in which cause and effect are not directly proportional? The development of computer technology makes it increasingly possible to study complex systems with nonlinear dynamics in nature and society. One hypothesis underlying such investigations is the assumption that nature, but also society, can be modeled by nonlinear coupled oscillators on many scales. Starting with quantum fluctuations and ending with the "great cosmic rhythms of our solar system" [1], the complex human organism is influenced in its evolution but also in its individual development. The mathematical model for the influence of fluctuations of the gravitational field on complex systems in nature (triggering of earthquakes) and the human organism has emerged more or less by chance from different, originally separate investigations. The publication aims to draw attention to this oscillating subsystem (the solar system) and to stimulate further research. The computer program developed for this purpose is accessible for research projects. There are a number of indications that the relatively weak fluctuations of the planetary gravitational field influence structure formation processes in a nonlinear way. [2] [4] A correlation function that indicates stabilizing and destabilizing states with a certain probability is suitable for describing these processes. The underlying hypothesis is the oscillation between stable and unstable states throughout evolution. The pursuit of a stable state can only ever be a stage of evolution that maintains that state for more or less time. The structure and development of physical systems is determined by the interaction of different parts of the system with each other and between systems and the environment. Four groups of interactions are distinguished: strong, electromagnetic, weak and gravitational. These interactions are not equally effective on the different scales of nature, but they are also not completely decoupled in their effect.
The aim of these investigations here is to develop a model based on gravitational interaction that is suitable for demonstrating an influence of cosmic rhythms of the planetary system on structures and processes of varying complexity in nature.
The planetary system of the sun is on the one hand an object of research in astronomy, but on the other hand also a factor of influence on the evolution of the earth and its inhabitants. Thus, the Earth's moon not only acts in the formation of romantic and mystical ideas in human consciousness, but also through its stabilizing effect on the Earth's axis. Thus it guarantees the relative stability of climatic conditions necessary in biological evolution. Although Einstein's general relativistic theory of gravitation forms the basis for today's cosmology, Newton's theory of gravitation is sufficient for investigations on the scale of the solar system.

Nonlinear interactions
The fundamental Newtonian equation of motion of N mass points has the form: (1) ri, rj = position vectors of planets i, j with masses Mi and Mj; This equation is the starting point for the derivation of the "Cosmic Fluctuations", but it is not yet in the form favorable for the present problem of the fluctuations. For this purpose, it becomes necessary to take into account first ordering points of view, which result from the structure and dynamics of the planetary system.
They are: A) The stability of the solar system. The present solar system is about 4.5 billion years old and consequently must have manifested itself as a quasi-stable structure during this time. Although Newton's equations of motion (1) are nonlinearly coupled, the structure of the planetary system is preserved over longer periods of time. The Lyapunov constant tL, which indicates the time at which the orbital shapes of the planets are entirely different, was determined by Laskar to be tL~ 5 million years. For the outer planets from Jupiter onwards, even larger Lyapunov periods were calculated. This gives fairly tight bounds on the orbital elements of the major planets over periods of time the size of the age of the solar system. B) Cosmic rhythms are considered over very long periods of evolution.
Therefore, above all the cosmic rhythms (frequencies), which are stable over longer periods, will be able to exert an influence. So it is not so much the absolute forces of the major planets, but rather their periodic changes that come into consideration. A stable alternating part is filtered out.
C) The planets of the solar system all move around the sun on circular orbits that are almost in one plane.
They represent natural oscillators whose couplings produce the superposition frequencies of the cosmic fluctuations. A cosmic cycle begins with the conjunction (seen from Earth) of two planets i, j and ends after the opposition with the next conjunction.
From the ordering points of view A, B and C a model of the cosmic fluctuation can be set up. Heliocentrically considered, circular frequencies i,j can be given for the cosmic cycles, which are relatively stable and change only little with time.
(2) Ti,j = time duration from conjunction to conjunction of planets i, j.
Without considering the direction of the resulting planetary forces (only direction-invariant processes are investigated), one can apply for the changes of the planetary forces (in a first approximation).

*Relation (3) follows from the vector addition of the forces Fi and Fj.
The quantities fi,j(t) and ki,j(t) contain the slowly and not very regularly changing components resulting from distance changes of the planets.
From a geocentric point of view, cosmic cycles are not quite so stable, so instead of i,j(t) it is easier to substitute the angle i, j at which planets i, j appear from Earth into (3). (4) For the further investigations only the faster and more "regular" changing cosine part in (4) is considered for the cosmic fluctuations. For a conjunction i,j = 0°) Fi,j is maximal and for the opposition i,j = 180°) minimal.
The weak gravitational field changes, in particular their cosine component, can be considered as a kind of excitation field strength on matter. (5) The quantities fi,j(t) and ki,j(t) are set approximately constant since they change weakly and less regularly with time.
The interactions of these "waves" (6) with matter and its different structures will be non-linear. It must be noted that these are not the gravitational waves derived from a linearization of Einstein's General Relativity. In analogy to other nonlinear interactions with matter (e.g. nonlinear optics), with (7) a general correlation function Hi,j for the influence of two planets i, j can be established.

(8)
Better suited is the transformation of (8) into a Fourier series.
(9) with a = a i,j The form (9) of the correlation function shows the emergence of "higher harmonics" in the interaction of cosmic fluctuations with matter.

The correlation function
The problem of the correlation function is the determination of the coefficients ak in (9) and the definition of the meaning of H. It is not intended to measure a force or a "deflection" with H. This would certainly cause insurmountable difficulties experimentally. This would certainly cause insurmountable difficulties experimentally, if one wanted to determine the influence of the fluctuations on test specimens with rotating lead balls (approximately according to Table 1). Moreover, evolution, which has extended over millions of years, is unlikely to be simulated experimentally. Since the fluctuations of the planetary gravitational field are very weak in their effect, only the following areas come into question for correlations: a) spatial structure-forming processes that are not or only very slightly determined by other effects. b) Formation of not fully determined biological patterns. c) Critical states in high-dimensional dissipative systems. d) Highly complex systems, far from thermal equilibrium and on the edge of chaos.
The coefficients ak will thus be determined from the study of interactions with regions a) to d). It is obvious to construct a correlation function H interacting with stable (harmonic) and unstable (disharmonic) states in regions a) to d). Determining the coefficients ak from statistical studies of unstable or chaotic processes, where small perturbations can have an effect, is very costly. Therefore, it seems reasonable to first obtain an approximation for the coefficients ak from theoretical considerations, which can then be adjusted by optimization procedures if necessary. Since we are dealing with cosmic cycles from conjunction to conjunction, one can take structural considerations about these oscillations as a starting point. If one takes the circle division as a basis (Fig 1), then the following structural points can be found: 2 points: polar structure; opposites that need balancing. Due to their tension and, if necessary, the impossibility of balancing them, they can nevertheless form a unity over a longer period of time. Score: strongly disharmonic 3 points: very stable structure; especially in engineering it is a prerequisite for stability in mechanical constructions. Score: very harmonious 4 points: unstable, dynamic structure; in engineering, this structure is often the basis for lever gears. Score: disharmonious 5 points: quasi-stable pentagram -structure; borderline between stability and instability. Complicated patterns and structures can be formed that do not repeat. Score: indifferent 6 points: Honeycomb -structure; close to the circle, relatively stable structure in the compound with good use of space.

Rating: harmonious
The addition of further points is possible, but the changes in the qualities become smaller as the structure becomes more similar to the circle. These qualitative statements are quantified step by step and plotted in a diagram (Fig. 2).

Fig. 2.
quantification of the circular pitch subdivided according to structural aspects. A symmetrical oscillation and decay process is assumed. The image is the basis for a Fourier transform for the 1st approximation of the coefficients ak.
Since this is a periodic cycle, a Fourier transform can be performed. The coefficients obtained are the first Fibonacci numbers (alternately mirrored, see 11.). The correlation function takes the following form: The first-order correlation function is shown in Fig 1, which is a first approximation for the study of the influence of cosmic fluctuations on the stable and unstable states of complex systems.
Consideration of higher orders may need to be made dependent on the problem under investigation. In general, it can be said that the higher orders will be more suitable for resonance and triggering. It must be said at this point that the hypothesis: "Stable and unstable processes of complex systems are reflected in the structures of the circle division" seems daring at first. Only practical investigations can bring confirmation that these assumptions are sufficient for a first approximation. For this purpose, it must be ensured that the correlation function (10) is not only suitable for describing one process, but also provides useful results for different processes and states. Expected values, at least in the tendency, must occur and no negative correlations may occur, in that, for example, the correlation function (10) indicates a higher probability for stability, but in reality there is a higher probability for an unstable state. Complex nonlinear processes are widespread in nature and society. High-dimensional complex systems are the rule. Far from thermodynamic equilibrium, these processes exhibit diverse spatiotemporal behavior. The fluctuations of the planetary gravitational field are, in absolute terms, certainly very weak. However, they act on a very large scale and on all material structures of the Earth. Crucial for the proof of the influence of these fluctuations is the emergence of the "higher harmonics" in the complex structures of matter. It is to be expected that the lower frequencies (1st order of correlation) will have a triggering or structuring effect on large-scale structures, the higher frequencies on small-scale areas.

An initial study of 41 of the strongest earthquakes
Are earthquakes triggered by the planetary gravitational field?
(For data see [2] The 41 strongest earthquakes) This is particularly interesting because when strong earthquakes occur in densely populated areas of the earth, there is usually a lot of damage to buildings and, above all, many lives are lost. Prior to an earthquake, stresses build up in the earth's crust, which then reach a critical state after a certain time. Generally starting with foreshocks, these tensions discharge in an earthquake, whereby a prediction of the strength of the earthquake is not possible. The investigations on the influence of planetary fluctuations on the triggering of earthquakes are based on the hypothesis that the reaching of a critical state of the stresses in the earth's crust happens within a certain time window. For this extremely unstable state, large-scale excitation field strengths of certain frequencies of the planetary fluctuations can then lead to the triggering of the earthquake and thus the relaxation of the earth's crust.
According to this hypothesis, the following results are expected: Only relatively stable excitation frequencies that are decoupled from the Sun will show correlations. Mercury and Venus always appear close to the Sun from Earth, they are not decoupled and are dominated by the Sun. Similarly, Mars' correlations are decoupled from the Sun, but because of the relatively large changes in distance from Earth, its excitation frequencies cannot be said to be stable. (Mars would first have to be removed from the statistical studies and examined separately). The correlation function (10) will take a negative value (instability) in the coherent superposition of all relevant planets, which is significantly far from the general expected value. Taking into account the foreshocks, the mean value of the first derivative will be positive. This means that the correlation function will be even more negative on average before the actual earthquake.
Only the frequencies of the fluctuations will show a correlation, which also gravitationally develop the largest forces. Pluto (and the planetoids) will therefore hardly show any correlation Can these expectations be confirmed?
These are the "strongest earthquakes" of the last century and the quakes with the highest loss of life, a total of 41 events [2] being studied.
To evaluate the influence of planetary fluctuations on "earthquake" events, the following calculations were performed:

1.
a) Superposition of the correlation function Hi,j (harmonic function) b) Superposition of the absolute amounts / Hi,j /("energy" function) c) Superposition of the 1st derivative according to the correlation function Di,j (time dynamics) d) Superposition of the absolute values of the 1st derivative according to the correlation function / Di,j /(time dynamics absolute) a) to d) Superposition of all 41 earthquake events related to sun, moon and selected planets.

3.
Monte Carlo simulation was used to calculate the density function, since an exact calculation for 41 events leads to unacceptable computing times. As a control, the exact density function was calculated numerically for up to 6 events. 10 000 groups of 41 events each were randomly selected in the period from 1900 to the end of 2000.

4.
To test the hypothesis: "The correlation function of the 41 earthquakes is significantly discordant", a one-sided significance test is performed. The percentage of randomly selected event groups that have equal or smaller values for the superimposed correlation function Hi,j is calculated. This percentage value represents the probability of error of the hypothesis.
If one first looks at the density distribution of Hi,j (Fig 3) for the Sun, Moon and all planets and compares both with the mean value (expected values), the sum of all 41 earthquakes Hi,j is definitely still within the range of the expected values. The correlations of sun, moon and all planets are below the expected value and also the "energy" is below the expected value but all in all there is no significant influence of the planetary fluctuations. This changes immediately when the influences of the Sun, Moon, Jupiter, Uranus and Neptune, which are to be expected according to the hypothesis, are considered separately (Fig 4.). The harmonic function Hi,j is now highly significant far below the expected value (0.03% probability of error for the hypothesis). If Saturn, whose frequencies do not play a major role here, is added, the result is still highly significant 0.85% (99.15% of the 10 000 control groups are more harmonic) Here is the computer printout for all major planets (red and blue indicate significance):  It was not the aim of this first investigation to derive concrete probabilities for the triggering of earthquakes. First of all, it is important to prove the effectiveness of planetary fluctuations of the gravitational field on highly complex processes on Earth, as represented by earthquake dynamics. This has been confirmed with the above investigations with an error probability of less than 1%. On the other hand, the correlation function derived from structural considerations of stability and instability is to be tested for its ability to describe the probability of stability and instability of complex processes and structure formation processes. It was therefore logical to apply this function also and perhaps primarily to a process which makes an influence of gravitational fluctuations on complex physical systems seem plausible from the outset. Two investigations are still connected here. 1. Are the higher orders (harmonics) better at indicating triggering of earthquakes? 2. s the period before and after the earthquake more meaningful? 3. Which frequencies could be relevant for triggering?
The following table shows the probabilities for orders 1 to 12 of the correlation function for Sun, Moon, Jupiter, Saturn, Uranus, Neptune.   Table 3. time shift to 5 days before and after the event for the 7th order.
The energy is relatively low for the entire period. A trend cannot be identified with certainty. While the 1st order correlates more strongly with the quality of time (stability-instability), the triggering effect of the higher frequencies of the 7th order is remarkable for the energy. Generally, it is expected that the energy for triggering could be high. Moreover, the high frequencies of the Sun and Moon should be particularly suitable. The correlation function for the 12th order does not indicate this. The low energy for the Order of the correlation 98.14% (of the 10 000 control groups have a higher energy) at the time of the earthquake seems strange. It is reasonable to assume that before the time of the event the energy is higher. An investigation can confirm this assumption for the sun and moon:  Table 4. time shift for correlation of sun and moon. Accordingly, 10 hours before an earthquake, the correlation is very discordant, with simultaneous increases in energy first in the dynamics and then in the correlation function. Are these random oscillations? Can this be generalized? Does this only apply to these very large earthquakes?

A study of 588 earthquakes
For data see [7] The study of the strongest earthquakes of a century has shown that a correlation with the harmonics of the planetary gravitational field can be demonstrated. This could be proven with an error probability of less than one percent. Nevertheless, it cannot be ruled out that it is an artifact. Therefore, further groups of earthquakes in smaller time periods were examined. The addition in magnitude of smaller earthquakes could cause a stronger noise, so that no significant correlations are detectable.
The following investigations refer to earthquakes in the years 1996 to 2002, in total earthquakes with a magnitude of m = 6.5 and greater or which caused severe damage.
The following questions were examined: Which order of correlation best describes a possible triggering of the earthquakes. Are there special frequencies that are suitable for triggering?
The results are shown in the following  For this list of earthquakes only the energy of sun and moon is significant and highly significant. This is also true for a larger time period (1900 to 2100) of the comparative calculations according to the Monte Carlo simulation. The 4th order shows for the matrix of correlation (harmony and dysharmony) the largest values for disharmony. With 85% the control groups are more harmonious than the earthquake group. A look at the matrix shows that strongly differentiated behavior of the individual correlations: strongly disharmonious are Sun-Venus, Moon-Mars, Venus-Saturn, Saturn-Uranus, Moon-Neptune, Venus-Pluto, Mars-Pluto, Venus-IC (Imum Coeli, represents the center of the Earth), Saturn-IC. Looking at the row sums of the correlation matrix, Venus and the IC are significantly disharmonic. There does not seem to be an explanation for this based on the effect of gravity. 9 out of 55 elements of the matrix are significant at <=5% p<=0.05 Probability of error: 0.0015 2 of 11 elements are significant with p<=2% p=0.025 Probability of error: 0.0296 It can be seen from Table 5 that the energy is significant over all orders for the Sun and Moon and even highly significant from order 7 on wards. The density function for the energy can be seen in  This result suggests that for this group of earthquakes the energy could be a trigger. 588 earthquakes distributed over 7 years corresponds to an average of 7 earthquakes per month. It is understandable that in such short periods the major planets show only small changes in their correlation function. The sun and moon are better for that. To illustrate this, December 2000 is examined in more detail here. During this period 8 earthquakes took place.  The high significances for the high orders are remarkable. Can these results be used for earthquake forecasting? According to this, only 13 events would be due to a triggering of the sun and moon, which is 2.2%. This is too low for forecasting, but it clearly shows that there is also a certain increase in probability from the many other influences that can trigger an earthquake. This probability can be increased somewhat by adding other frequencies (those of Jupiter, Saturn and the IC) and the 1st derivative of the correlation function. These initial investigations are only intended to show that further investigations appear to be useful. As can be seen in Fig. 8, in such a small period of time only high frequencies, as they are given by the sun and the moon, are suitable for a possible triggering of earthquakes. At the time of the full moon no earthquake took place. However about 24 hours later. Further investigations would have to show whether this is significant. Figure 9 shows the correlation function for the 1st order for comparison. It does not seem to be suitable for triggering.  In Table 8, we can at least see that at the time of the event, the energy in the correlation function was very high, as was the energy in the dynamics. A low energy (-11h) is driven to a high energy by a high dynamic (1st derivative), likewise the energy of the dynamic increases until the event. Can this scenario also be stated for the much larger period from 1900 to 2100. The results are shown in Table 9.  The expected value for high energy is 203 earthquakes. 222 have a higher energy in the correlation function. According to this, 19 earthquakes could be triggered by the sun, moon, and IC, which is 3.23 percent. That's a 1% increase. The IC, as expected, brings an increase in the probability of triggering because the local energy maxima indicated by the IC with the Sun and Moon occur at different times than those of the Sun and Moon. Certainly the major planets Jupiter and Saturn (lower frequencies) in interaction with the high frequency of the earth's rotation are also of influence. This is shown in Table 11:

Summary
According to the calculations, it seems possible that about 6% of the 588 earthquakes are triggered by the Sun, Moon, IC, Jupiter and Saturn. This figure of 6% can certainly be increased if the energy level is optimized and other elements of the correlation function are added. For further investigation, it can be hypothesized that a trigger or threshold energy exists that is constantly decreasing. Before this threshold energy becomes zero, small external disturbances (e.g., weather events) may be triggering. But this can also be the fluctuations of the planetary gravitational field in the higher frequencies. Earthquakes occur at all times. When the threshold energy drops, they can also be triggered by harmonics of the gravitational field. This seems to be a characteristic of highly complex, nonlinear systems, that small external energies can trigger large changes .   Fig 10. model of the triggering of earthquakes.
Our planetary system is highly complex. The nonlinear dynamics of this system also has an influence on the triggering of earthquakes. This now seems to be a fact and opens the door for further investigations. From [6]