# Perturbation Theory

## A Mathematical and Physical Basis for Perturbation Theory

### A New Meta-Stable Three-Body Solution, of Two Small Nearly-Resonant Objects Gravitationally Revolving Around a Massive Primary Body

Isaac Newton developed a Theory of Gravitation that was entirely based on evidence and not theoretical logic. Einstein expanded the scope of Newton's equations by adding in relativistic factors that Newton could never have been aware of. But still, the understanding of gravitation is virtually entirely an empirical subject, with virtually no actual theoretical basis. Exactly WHY gravitation exists, or why it is an inverse-square function is yet unclear. However, Newton's laws, along with the relativistic improvements, have been extremely useful in making predictions of planets, moons, stars and artificial satellites. Without an actual theoretical basis, however, such (accurate) predictions are very complex.

Currently, accurate position and velocity values for an object are empirically measured. After a substantial body of such data has been accumulated, a mathematical process called a Fourier analysis is done to that data. Almost 200 years ago, Fourier developed and proved a mathematical process where any non-infinite mathematical function can be represented by a series of sine and cosine terms, each with specific amplitude and frequency. In general problems, these series have an infinite number of terms, however the amplitude of the terms is often extremely near zero for most terms. Therefore, in practical applications, there is nearly always a finite number of Fourier terms in a Fourier series.

This presentation was first placed on the Internet in December 2001.

In the case of the position data of a planet or any other astronomical object, if sufficient accurate data can be collected for an extended period of time, a Fourier Analysis can be done to the data. Since astronomers have been able to collect extremely precise data for at least a hundred years for most major Solar System objects, Fourier Analyses of such data generally results in many thousands of such terms. These terms are called "perturbations."

Patterns are easily seen in many of those terms, where many have periodicities (frequencies) that exactly match each other, and also match a known astronomic timing configuration.

For example, there are terms in the Fourier analysis of our Moon's movement that correspond to a precise period of about 782 days. This matches up with the synodic period of Mars (the time between Mars seeming to appear in the same location in the sky relative to the Earth-Moon and the Sun), and so those Fourier terms are seen to be very likely due to gravitational perturbations from Mars.

Separate Fourier Analyses are made on the data for Orbital radius, Orbital eccentricity, Orbital inclination and several other measurements that have long been measured. Each of these Fourier Analyses regarding the Moon's orbit include many terms that have that same periodicity of 782 days. Therefore, it is reasonably concluded that the presence of Mars has measurable effects on every aspect of the Moon's orbit. There are also Fourier terms of the Moon's motion that have timing that identifies Jupiter, Saturn, the Sun, and all the other major objects in the Solar System as being the source gravitational cause.

As an aside, the Fourier Analyses of such orbital parameters also include some terms that have large amplitude but extremely long periodicities. These are not understood to be directly caused by any specific planet, but they certainly exist! For example, a parameter that defines the direction of the long axis of the Earth's orbit, called the line of apsides, is known due to Fourier Analysis to currently be very slowly revolving eastward at a rate that would take it about 108,000 years to entirely circle the orbit. However, there is an even slower Fourier term that shows that the speed of this shift will not stay constant! In the Fourier Analysis of the eccentricity of the Earth's orbit, terms exist that indicate that the current eccentricity of 0.016 is currently diminishing, and will do so for around 24,000 more years, at which time the eccentricity will be around 0.003 (extremely circular). After that, the eccentricity will increase for around 40,000 years, after which it will be around 0.070 (a very eccentric orbit, nearly as eccentric as the orbit of Mars is now). All the other parameters of the Earth's orbit are known to have similar long-term (called secular) perturbations, and all are exclusively known because of terms in the Fourier Series.

As mentioned above, there are actually an infinite number of terms in a Fourier Series, but in such an orbital data derived series, the terms eventually become exceedingly small, and further terms are ignored. As long as sufficient terms are included in calculations, reasonable future position predictions can be accurate. But since many of those tiny terms are excluded, no prediction from this (current) method can be EXACT. The current handling and calculating of perturbations is always done in this way, through Fourier Analysis. That analysis is exceedingly powerful, but it has the limitation of being inaccurate for long-term predictions, because of those many tiny terms being ignored.

Part of the point of this current premise is to suggest that a differential equation might be defined for that effect of Mars on the Moon, and for each of the other perturbations of planets and satellites on each other. The existence of such an equation would represent a fairly simple way of representing the whole (infinite) series of Fourier terms (due to Mars, in this case) as a single exact quantity. If perturbations were mathematically represented by differential equations rather than truncated infinite series, extremely accurate long-term position predictions should be possible.

The other shortcoming from the current procedures at calculating predicted positions and elements of planets and moons is that a numerical integration is necessary, with a specified time interval. By calculating all those terms repetitively and sequentially, after each time interval, very small initial errors eventually grow into substantial errors due to the well-known geometrical damage done in such extrapolation. And truncating the infinite series necessarily creates small initial errors. This has the effect of limiting the future time range of accurate position calculations for all Solar System bodies. Should a differential equation be usable for such calculations, then "true" Integration could be done, and those extrapolative errors avoided.

These two effects are the motivation for considering this premise, because it would permit exact calculations rather than incremental extrapolations based on incomplete infinite series.

## Current Unsolved Gravitation Puzzles

Along the history of the growth of research in astronomy, there have been a number of "peculiar" discoveries associated with gravitation. A hundred years ago, astronomers were aware that the planet Mercury's orbit was consistently drifting in position in a way than Newton's equations could not explain. Einstein used that discrepancy as a stimulus toward developing part of his theories of relativity, and his prediction of the relativistic adjustment that should apply to Mercury's motion was impressively confirmed.

However, several other "puzzles" have remained. They include

• The collection of many thousand asteroids in our Solar System, has a number of obvious "empty areas" where no asteroids seem to be. As early as the 1860s, a scientist named Kirkwood recognized these gaps, and it was soon realized that the gaps occurred at distances from the Sun that matched up with objects that would revolve around the Sun at simple fractions of the huge planet Jupiter, like 1/2, 1/3, 1/4, 1/5, 2/5, 2/7, 3/7, 4/7, 2/3, etc. Physicists have always considered this to be too sharply identified to just be a random situation. Statistically, such patterns can be shown to be unsupportable from a random distribution, but no strong theory has been presented to describe or explain the Kirkwood Gaps.

• Saturn's rings have many 'gaps' in them, and those gaps are nearly always at distances from Saturn which would represent orbital periods of simple fractions of one of the several large moons that also circle Saturn. In addition, these 'gaps' are not always circular and perfect, but sometimes have wavy edges and are slightly off center!

• For hundreds of years, astronomers have noted a "long inequality" of Jupiter and Saturn, where five orbital periods of Jupiter is nearly (but not exactly) equal to two orbital periods of Saturn.

• Of the comets with known orbits, a large number have orbits where their aphelion distances are approximately the radii of the four large planets. Statistically, this could not happen, so there must be a cause for the pattern.

• The four very large moons of Jupiter, which Galileo first discovered in 1610, have periods of rotation around Jupiter that are very close but not precisely in the ratio of 1:2:4:8. Again, this was recognized as probably not being random, but no adequate theory has been presented regarding that pattern. Similar patterns exist among the moons systems of Saturn, Uranus and Mars.

• For more than 100 years, astronomers have noticed another very odd thing about those same four moons. They have NEVER (and apparently can never) all be on the same side of Jupiter at the same time! Often, three of them are on one side, but the fourth is always on the other side.

• An assortment of unusual aspects associated with gravitation apply to our Milky Way Galaxy. The most seemingly irrational of these is that the Spiral Arms that we know our galaxy (and millions of other galaxies) have, should quickly wind themselves up around a faster rotating Core, but they don't. On this issue, I wrote a separate essay a number of years back, which presents a completely logical and rational explanation for the persistence of Galaxy Spiral Arms, without having to resort to "dark matter", centrally located "black holes" or any of the other speculative theories presented to try to explain that stability. That essay is at Galaxy Spiral Arms Stability and Dynamics http://mb-soft.com/public/galaxy.html

• Near our Galaxy, there are around a hundred Globular Clusters which are self-contained collections of hundreds of thousands of stars, in apparently beautifully symmetric patterns, in remarkably close proximity to each other. If such groupings had been discovered to be very young, maybe there would be some explanation regarding a brief grouping like that, because there just seem to be WAY too many stars that are very close to each other to remain that way very long. Logic would suggest that they would constantly be perturbing each other and therefore regularly colliding with each other, and certainly continually be altering the courses of each other due to the gravitational effects of close passes. However, these Globulars are believed to be among the oldest objects in the Universe! There seems to be no logic that could support those findings.

• The Globulars have another interesting aspect. They are NOT so far away that they hover around the perimeter of the Galaxy, but they seem to be in two groups relatively near the Core of our Galaxy and above and below the Plane of the Galaxy. Our understanding of gravitation insists that they cannot stay on one side of the Galaxy. They must therefore pass through the Plane fairly regularly in orbiting the Core in accordance with our understanding of gravitation. A number of years back, I wrote an essay on that issue, and it contains the reasoning that concludes that those Globulars must have passed through the Plane of the Galaxy many hundreds of time during their existence. The process of passing through that Plane, where there are countless thousands of Galaxy stars, would necessarily produce enormous tidal stretching of the Cluster, as well as statistically allowing many opportunities of individual stars from the Globular to be re-directed by a near pass of a Galaxy star, and thereby be perturbed completely free of the Globular Cluster. And yet, the Globulars are impressively symmetric! Globulars have many great mysteries! My essay on them is at: Globular Clusters Must Regularly Pass Through the Galaxy Plane http://mb-soft.com/public/globular.html

## Recent Research

I believe that my recent research has developed something that can contribute toward a theoretical understanding of the first five of these areas listed above. It would seem that the traditional Physics field of resonance and vibration might offer a nice and fairly simple explanation and description of those behaviors.

It should be noted that this premise does not involve any new "mechanisms" such as gravity waves or electromagnetic phenomena. Rather, this premise is simply the long term result of repetitive patterns of the standard mutual gravitational attraction of two or more objects orbiting the same primary object, and the resonance effects that would therefore exist. Nothing more than standard Newtonian gravitation is involved, F = GMm/r2. The Engineering field of Forced Vibration becomes valuable in this exploration.

This is not to imply that any attack is intended regarding current methods. Actually, they would become even more important, in determining certain constants that would be in the differential equation terms. The premise here is that, as an addition to current methods, adding in some aspects of the Engineering fields of Resonance and Forced Vibration would both improve long-term position predictions of astronomical objects and also hopefully assist in generating differential equations for the component gravitational effects.

No set pre-conditions seem necessary regarding the orbits. Rather than such obvious resonance patterns developing immediately, this premise suggests that standard application of forced vibration analysis shows that almost any initial orbits would gradually evolve into the currently observed patterns over very long time intervals. There are two specific consequences of a standard forced vibration analysis:

• (1) the resultant differential gravitational (amplitude) effects can become "magnified" under certain specific conditions;
• (2) under those same conditions, the resultant differential gravitational effect has a significant "phase angle" which means that the effect does NOT act along the line between the two bodies, as would normally seem obvious with gravitational effects.

These characteristics result in a number of surprising consequences. Two are of specific interest here:

• The magnification effect of the forced vibration equations would make any orbital period that was an EXACT simple fraction of the orbital period of a massive outer planet to be extremely unstable. This results, for example, in asteroids that might momentarily have such simple-fractional orbits of Jupiter to be strongly perturbed into some other radius orbit. The net result appears as though there is a forbidden orbit.
• The combination of the magnification effect and the changing phase-angle create a subtle general tendency toward that unstable exact fractional period. This pair of consequences creates a pair of meta-stable orbits that are nearly a simple fraction, on either side of the commensurate relationship.

Preliminary computer simulations have seemed to support this premise, although more extensive and even longer-term simulations are needed in order to confirm some of the near-synchronous meta-stabilities. Precise synchronicities calculate to be extremely unstable situations according to the preliminary results, due to the denominator of the amplitude of the forced vibration effect approaching zero, which therefore results in extremely large amplitudes of perturbations. This extreme instability describes the situation that is seen in Nature.

## The General Logic

The logic is essentially the same for each of the five subject areas mentioned above. We will consider the largest asteroid, Ceres. Jupiter takes around 12 of our years to circle the Sun, while Ceres takes around 5 years. If we were on Ceres, we would see that, as we would pass Jupiter on an inner path, Jupiter would seem to gradually move across the stars 'backwards' (retrograde), and it would take a 'synodic period' for Jupiter to go from one 'opposition' to the next, of around 8 years.

Starting halfway through that period, when Ceres and Jupiter are on opposite sides of the Sun, let's consider the gravitational effect Jupiter would have on Ceres. At that first moment, it is VERY far away, and the force, being inverse-square, is rather weak. However, continually for the following 4 years, every second, the direct line force vector of the attraction of Jupiter on Ceres will have a component FORWARD along Ceres orbit around the Sun. Until Jupiter gets to the point of Quadrature, the other component acts to accelerate Ceres inward toward the Sun. After Jupiter passes Quadrature, the forward component continues, and keeps growing in magnitude due to the lessening distance between Jupiter and Ceres, but now the radial component now acts to accelerate Ceres outward away from the Sun.

One result of this is therefore a slight warp or distortion of the otherwise elliptical orbit of Ceres, first slightly inward followed by a slight outward distortion until opposition occurs.

These resultant accelerations of Ceres are all very small, always being under one one-millionth of a meter per second per second. However, since the forward-acting component of that acceleration continually acts to 'speed up' Ceres for that 4 year period, by the time of opposition, Ceres has an additional 'differential' orbital velocity at that point of over 8.6 meters per second. That's well under one one-thousandth of Ceres' orbital velocity, so it might not seem to be significant. However, it does advance Ceres forward along its orbit, while some of the other orbital elements are also altered.

As soon as opposition is passed, Jupiter's gravitation will act identically as before, except Jupiter is now 'behind' Ceres, so that acceleration will have a component along Ceres' orbit to decelerate it. The cumulative effect over the following 4 years will be the same as before, and at the very end of that four-year period, the deceleration would have exactly reduced (the first-order component of) that 'differential' velocity to zero. (This description is of the simplest possible situation, where both Jupiter and Ceres follow perfectly circular orbits in the same plane. The actual situation is similar but more complex.)

This might make it seem as though there was no net resultant effect. However, the first 4 years of accelerating force from Jupiter acted to build up forward orbital velocity that would not have existed if Jupiter had not been present. The remaining 4 years of decelerating force acted to REDUCE that forward orbital velocity, but it was still a forward orbital velocity through the entire period. Even though that differential orbital velocity began at zero and ended at zero and was never greater than 9 meters/second, a total of eight years of applying that differential forward velocity can really add up! In fact, Ceres would arrive back on the opposite side of the Sun, around 226,000 km ahead of where it otherwise would have been, if Jupiter had not existed! That means that it would arrive at the opposition point (conjunction) more than 1,000 seconds (or 16 minutes) earlier than it would have without Jupiter existing!

This VERY significant advancement along the orbit effectively alters the orbital period of Ceres by around that 10 minutes. That's not a lot in a five-year orbital period, but it turns out to be significant. Essentially, it is a second-order effect of the gravitational attraction of Jupiter. Other orbital elements of Ceres are similarly affected. These various second- and higher-order effects have long-term implications. Some of the more prominent effects occur for elliptical orbits, but the math becomes far more complex.

### Forced Vibration

In Physics, a circular or elliptical motion can be treated as a vector sum of two sinusoidal 'vibrations' at right angles to each other and 90 degrees out of phase with each other. This is described as being a motion with two degrees of freedom (two perpendicular axes). The periods of those two vibrations must be identical, and if the axes are chosen carefully, their amplitudes determine the major and minor axes of the orbit. Another basic fact of Physics is that any 'real' object that is vibrating has certain RESONANT frequencies. There is a NATURAL resonant frequency that nearly always exists. The Earth can be considered to be "vibrating" about the Sun with a period of exactly one year. In these cases, that principal frequency is the inverse of the orbital period, so these frequencies are all extremely low.

Yet another basic Physics fact is that such real objects generally also can have HARMONIC resonances, at double, triple, quadruple, etc, of the natural resonant frequency. Fourier Analysis results show that such resonant considerations apply to the orbital motion of planets and moons and ring particles. The natural orbital period can be considered to represent a "free" vibration. The action of a third gravitationally attractive body can be treated as a harmonic disturbing force (using the inverse of the synodic period as the disturbing frequency), which can then be treated with relatively standard logic as a "forced vibration". First-order effects tend to cancel out but second-order effects remain. Some of the calculations of a harmonic forced vibration are below.

In addition to Jupiter's gravitational effect ALONG the orbit of Ceres, there are radial effects that we briefly mentioned. Before Quadrature (for most of the orbit, then) this component acts to accelerate Ceres toward the Sun. However, after Quadrature, the component acts to accelerate Ceres radially outward. Even though this effect occurs for a shorter period, the fact that Jupiter and Ceres are closest allows the gravitational effect, and therefore the resultant acceleration, to be much higher. By the time of Quadrature, the cumulative inward acceleration would give Ceres about 2.4 m/s differential inward velocity, but that is soon overcome and the net effect would be an outward differential velocity of around 3.8 m/s at opposition. That value continues to increase to around 9.6 m/s a few months later, but then the later inward component reduces the total average effect for the orbit to be about 7.5 m/s outward.

The fact that these values change from inward to outward, and have various effective values due to the cumulative effect of gravitational acceleration, causes asymmetric distortions in the orbital path of Ceres, which can be described as a Fourier series of perturbation terms. Each of these Fourier terms has a period which is the synodic period of Jupiter for Ceres. In this particular case, the ellipticity of Ceres' orbit is significantly affected, in a long-term relatively consistent (second-order) manner.

It is suggested that this premise offers a theoretical basis for the many terms used to empirically describe the motion of the Moon and planets and the Galilean moons of Jupiter. In addition, since this analysis provides differential equations which can be integrated, exact solutions would be possible, rather than the approximations which are now necessary.

Consider a reference frame that uses a Sun-Jupiter radius vector as its x-axis, which would therefore rotate at the rate of Jupiter's period or just under once each twelve years. In this reference frame, Ceres would appear to have two nearly harmonic motions, one along that axis and the other lateral to it. That would essentially be the equivalent to describing a "natural harmonic vibration" of Ceres.

Consider the situation if there was exactly a simple fraction orbital time ratio of Jupiter and Ceres. Each time around, the same disturbing, distorting forces/accelerations would occur at the same locations along the orbit of Ceres. The (second-order) perturbative effect of Jupiter on the motion (and orbit) of Ceres would then be somewhat cumulative, as in forced vibration at a resonant (or harmonic) frequency. This is the situation where the denominator of the amplitude of the forced vibration equations (see below) approaches zero, allowing an extreme increase in the amplitude of the cumulative perturbation, generally called the "magnification factor" to apply to the vibration (perturbative) amplitude.

As in any other mechanical situation where that applies, an unrestrained magnification factor soon results in mechanical instabilities and the system and the vibration soon becomes destructive and breaks itself down. Orbital stability would therefore not be possible in an exact synchronous or harmonic orbital relationship with another planet or moon.

This represents a logical description of why two bodies orbiting a much more massive third body can never maintain a stable simple fractional orbital period relationship. Each of the smaller objects are always subjected to harmonic repetitive perturbations due to the gravitational attraction of the other. If their periods happened to be some exact simple fractional relationship, then the magnification factor effect of the forced vibration would soon alter their orbits into a non-commensurable relationship. If their periods initially happened to be far from a simple fractional relationship, the combined effects of the amplitude and phase-angle changes alters their orbits toward a commensurable simple fraction relationship. The result of these two effects is an eventual near-synchronicity or near-harmonicity as a meta-stable arrangement. When the orbits are elliptical, these effects are enhanced, as the repetitive perturbations would occur at the same location along the ellipse, which generally creates greater orbital distortions.

## Brief Mathematical Discussion

A more technical presentation of this is that in standard engineering, the equation of motion of a forced vibration can be represented by a differential equation (in this example, a weight suspended by a spring in a gravitational field): Where W represents the weight or inertia of the object, g is the acceleration due to gravity, the x terms represent the first and second derivatives of position (essentially velocity and acceleration), P represents the amplitude of the harmonic disturbing force, t is time and ω is the angular frequency of the disturbing force, the other items being constants.

In any system, this initially results in a vibration made up of two parts, a free damped vibration at the natural frequency (ωn) of the object (due to the interplay of the inertia of the object's mass and the restoring force of the spring), which virtually always soon gets damped out in engineering applications, and a forced vibration which will continue as long as the disturbing harmonic force is applied.

For that relatively simple case, as the free vibration is quickly damped out, the vibration is then just the forced vibration component. The expression for this is: Where X (amplitude) and φ (phase) are given by:  Zeta is the 'damping factor' that acts to dissipate the vibration, and the ωs are again the 'forced' frequency and the 'natural' frequency.

In the case of gravitational applications, P would represent the distance-variable gravitational attractive force of the perturbing body (nearly sinusoidal), ω would be the synodic frequency, ωn would be the orbital frequency of the object being perturbed, and the left side terms in the equation of motion would represent the original unperturbed orbit characteristics. There are some aspects of gravitational systems that are sufficiently different from standard engineering forced vibration, as to require modifying these factors and equations. Specifically, the force of gravitation is an inverse-square dependence rather than the simple linear dependence of a spring. For co-planar orbits, the vector distance between the two objects varies during the synodic period as the square root of (R12 + R22 - 2 * R1 * R2 * cos(θ)), where θ is the instantaneous primary (solar) angle between the two orbiting objects. The cumulative effect of the perturbative effect would involve the Integral of the inverse square of this quantity. The point here is primarily to present the basic concept.

It is easy to see from the resultant amplitude equation above, if the forced frequency is exactly the same as the natural frequency, the denominator of the X term drops to its lowest value, which causes X, the amplitude, to reach its greatest value, the so-called 'magnification factor'. The limit on the magnification factor is therefore based on the amount of damping present. It would seem that in gravitational problems little damping would be present, and the amplitude of the perturbation could become extreme. In the case of an orbit, such a very large perturbation amplitude could cause the existing orbit to rapidly become unstable, shifting the orbital elements to a different orbit that was stable or at least meta-stable.

Notice that the phase angle changes very rapidly if the forced frequency is very close to the natural frequency. In the case of the orbital situation, if the forcing frequency is far lower than the natural orbital frequency, the phase angle is near zero and the effect is a normally expected distortion of the orbit. If the forcing frequency is far higher than the natural orbital frequency, the phase angle is near 180 degrees, and again the effect is a fairly reasonable distortion of the orbit. In both of those cases, we get an expected effect along the line between the two bodies, which enables current perturbation calculations to achieve good accuracy.

But for a forcing frequency exactly at the natural orbit frequency, the phase angle is 90 degrees, meaning that the object would be affected in a totally different direction, not at all along the line between the two objects. Combining the effect of the magnification factor and the phase angle shifts seems to give two results. There is a general "preference" toward commensurate periods, but also a very sharp aversion to an exactly commensurate relationship. That seems to imply that, on BOTH sides of the commensurate frequency, there are 'preferred states' where the orbit is much more stable (actually meta-stable).

Also, note that the amplitude and the phase angle do NOT vary identically for forcing frequencies near the natural orbital frequency, although for more different frequencies they are much more similar. Again, this situation results in an instability of a commensurate orbital time period, and preference for a slightly non-commensurate relationship.

There are some apparent variations in applying forced vibration analysis to astronomic situations. Two are related to the fact that there is not an obvious source for any damping action. This certainly allows for the persistence of the natural 'free' vibrations (of the initial unperturbed orbit). The extremely low (or zero) value of a damping factor implies that the magnification factor would become extremely large, with extremely disruptive results for the orbit for exactly commensurate relationships. An interesting speculation is that the collective gravitational effect of all the other objects in a system (or external to it) might act in a damping manner. If this should turn out not to be true, this logic still all applies with zero damping factor. I have not yet investigated this area.

Also, it has certainly been noted that the resonant frequencies of the perturbing action and the body being perturbed are NOT generally at the same frequency but in a harmonic relationship, such as an octave (1:2 relationship) apart. The reasoning still applies, although the math is somewhat more complicated.

Finally, orbital paths are three dimensional oscillations, where we have been discussing the simpler one dimensional case. Again, the reasoning still applies, and the math is more complex.

These calculations suggest that there is great value regarding stability, if the two orbital periods are NEARLY commensurable. Given the significant effects noted, advancement along the orbital path, distortions lateral to that path, effects on elliptic orbits, and the effects of the phase angle variations, an ALMOST commensurable relationship between the two orbital periods permits and encourages a "meta-stable" amount of "forced vibration magnification". The magnification effect, with both its amplitude effects and its phase angle effects, thus creates a meta-stability for a near-synchronization pattern of the two objects' orbital periods. It appears that the unstable and destructive nature of nearly unlimited magnification, combined with the 90° phase angle, keeps the synchronization from ever being able to too closely approach a precise synchronization. This therefore results in a meta-stability rather than a true stability.

This does NOT mean that absolute synchronization is denied, but merely that it is an extremely unstable arrangement, which would soon act to alter (perturb) one or both of the orbits to become the much more stable near-resonant situation. The destructive aspects of this are actually a combination of the effect on the orbital velocity and that of the orbit's semi-major axis and eccentricity, in large enough amounts to deny exact-synchronous stability. In short-term situations, perfect commensurability could exist, but, as the planets and asteroids and moons and rings have apparently existed for very long times, the forced vibration and near-resonance arguments would have gradually created the near-synchronicity we now see in many examples, as indicated in the first five entries in the initial list above. All five of those phenomena would thus seem to be explained by this premise.

This premise seems to imply a number of predictions. First, that there are two equally meta-stable states which are on opposite sides of an unstable synchronous or commensurate state. Second, that the specific masses and orbits of any pair of orbiting objects should be able to be mathematically solvable to accurately determine just how much of a difference there should be between the commensurate ratio and the meta-stable ratio. Such calculations should therefore predict the accurate specific relationships between the four major moons of Jupiter, for example, possibly giving the number of seconds different from commensurability for pairs of those moons. Such a calculation would seem to represent a good proof of this premise.

The basic equations of forced vibrations and of resonance considerations are long established in Physics and Engineering. Any Physicist should be able to confirm the mathematics behind the above assertions. If the field is unfamiliar, "Theory of Vibration" by William Thomson (1973) is a good starting point.

There IS an aspect of this that I am still investigating. It would appear that relatively small massed particles, such as ring particles and asteroids, tend to be displaced in orbital period and radius in more broad of a range from a precisely synchronized situation. However, larger bodies, such as the Galilean moons of Jupiter and the planets, seem to have preferred patterns much closer to resonance, and usually slightly lower than resonant resultant orbits. (The long-inequality of Jupiter and Saturn would be an exception, being higher). If this represents a real phenomenon, it seems probable that the mathematical support for it is somewhere in the time period involved. The planets and their larger satellites have had time to make many millions of orbits with relatively consistent conditions, so these small second-order resonance effects would have had plenty of time to establish meta-stable arrangements. For the particles in the rings of Saturn and for the asteroids, sufficient time may not have yet passed for final meta-stability to be achieved, and they may be subject to regular collisions or other perturbations that might disturb the slow and methodical procedure described here. If this is true, then ultimately the particles and asteroids might eventually bunch together into much narrower bands. However, considering all the other gravitational effects perturbing each of them, they might just as likely stay spread out as they are. The difference may therefore just be an effect of "maturity" of the meta-stability.

In the example of the three-body problem of Jupiter, Io and Europa, Io's rotational period is about 561 seconds less than half that of Europa. During each synodic period (of around 3.53 earth days), both Io and Europa are affected as described above. As to the along-orbit components, Io is moved forward in its orbit by around 74.6 km (which effectively reduces its orbital period by a little over 2 seconds) and Europa is moved backwards along its orbit by around 76.4 km (which effectively increases its orbital period by around 5.5 seconds). Both of these effects are further modified by the existence of Ganymede and Callisto, but the point here is to suggest that merely the along-orbit relative movement is NOT sufficient to account for the 561 second variance from synchronicity. This adds to the evidence for a NON-synchronous meta-stable solution, rather than a synchronous solution that was "artificially" distorted.

The deviation from pure synchronicity seems to vary for different situations. For Jupiter-Saturn, it is around 1/155. For Io-Europa, it is around 1/275. For Europa-Ganymede, it is around 1/135. For Ganymede-Callisto, it is much poorer, around 1/7. In each of the planetary moon systems, inner moons tend to be very close to synchronicity, while outer moons tend to have poorer ratios. This seems to suggest that the long-term effects of the forced vibration and resonance would be much more pronounced where the ambient gravitation field of the central body are stronger, a logical situation. It also represents a situation where a larger number of these forced vibration interactions have occurred because of the shorter orbital periods of near satellites. This might also reflect on the broader range of small objects like ring particles or asteroids, but that situation might be different because the very small particles have so little inertia of their own, a factor in the equations.

This near-resonance argument might also offer a theoretical basis for the historically intriguing Titius-Bode's Law. The sequence of numbers in Bode's Law, where multiples of '3' are involved, could actually be seen as harmonic resonant relationships. The need to add '4' to each term would represent the "NEAR" aspect of the forced vibration resonance as suggested here.

A full analysis of possible application to theoretically explaining Titius-Bode's Law would involve extensive research on its own, separate from the premise of this essay.

In any case, the long-recognized near-resonances of planets might therefore find an explanation. Jupiter-Saturn is nearly 2:5 in orbital periods. Saturn-Uranus is nearly 1:3. Uranus-Neptune is nearly 1:2. Even Neptune-Pluto is nearly 2:3. Randomly spaced planets would not have so many near commensurabilities. I suspect that mathematical analysis of the combinations of perturbations and resonance, for an initially random spacing of a few planets, will (with the help of long-term computer simulations) eventually show such evolvement of near-resonant relationships. This premise might therefore imply that all mature planetary systems around other stars might have orbital ratios resembling Bode's Law.

A surprising preliminary finding in computer simulations is that eccentric or inclined orbits seem to tend to gradually reduce in both eccentricity and inclination. This is interesting in that it might be possible to consider an initial collection of planets with somewhat random orbital radii, eccentricities and inclinations, and given enough time, a system such as we now see might develop.

Extending this premise slightly farther, a collection of initially randomly moving particles around the Sun (or any star), with some existing aggregation of mass already orbiting, would then actively create planets in pre-arranged orbits! The individual randomly moving particles would have their orbital inclinations gradually come to be nearly in the plane of the orbit of the aggregation. Simultaneously, the individual particles would also have their orbits modified to become less eccentric, more circular. Finally, the particles would be perturbed into meta-stable orbits that were near commensurable fractions of the period of the aggregation. These three effects would "collect" very large numbers of particles into all having the same orbital radius, the same orbital plane, and zero eccentricity. That describes a natural collection of random particles into a very narrow ring structure (actually a close pair of rings). Once in a meta-stable ring, the individual particles could logically start to clump together along their one remaining degree of freedom, along the orbit, over time, slightly shifting forward or back in the orbit to join with other particles.

This scenario might represent a possible explanation for the formation of the Earth and other planets. If this is true, then EVERY star (that is not multiple) must similarly accrete planets, and they must eventually be arranged in a pattern that resembles our planets.

This approach describes the creation of a set of planets that does not necessarily have to be in the plane of the rotation of the Sun, which is a serious problem for many planetary genesis hypotheses. The Sun's rotation axis is tilted about seven degrees from the orbital planes of the planets, a situation that otherwise produces enormous problems in accounting for conserving the angular momentum of everything. This premise suggests that all the planets would have a natural preference for an orbital plane that matches the orbital plane of Jupiter, and which could be unrelated to the spin axis of the Sun. This is observed to be the case for the Solar System.

In the rings of Saturn and the other planets, close examination has established that they certainly have many gaps, and the gaps generally correspond to orbital radii where a particle would have a simple fractional relationship with one or more large moons of that planet. (Saturn's innermost four large moons are VERY well "near-synchronized". This agrees with the general premise offered here.

In addition, the presence of "spokes" in Saturn's rings might then be described as gravitational short-term-metastable resonance conditions also associated with those moons. The fact that a seemingly non-Keplerian spoke could rotate as an entity at various radii from Saturn, is a situation that seems to violate basic understandings of Physics. This observation might also be explainable by this premise if those various low-mass component particles were briefly experiencing a meta-stable resonance with a particular moon. Particles at various radii, being of very low mass, could, due to forced vibration, be quickly accelerated or decelerated into a synchronicity or near-synchronicity with that moon. This explanation would necessarily insist that all such spokes fade out at both ends, being most sharply defined nearer the middle of a spoke. This agrees with observations. Such spokes would also necessarily have rather short existence, which is not yet known.

Another seemingly illogical discovery was that the ring edges do not appear to always be circular, or even concentric with the center of Saturn. Some waviness of the edge shapes has been noted. On a related subject, some very small "shepherd" moons have been discovered that share orbits just exterior to the rings, and even they have irregularities in their motions.

These various peculiarities might also be explainable by standard applications of Physics forced vibration theory and resonance theory. They would thus only occur as a direct result of the existence of Mimas and the other large inner moons of Saturn, and the various phenomena would necessarily have periodicities that were NEARLY commensurate simple fractions of the orbital periods of those moons.

The "shepherd" moons that are commonly credited with sharply defining the edges of the rings, may thus instead just be another example of this near-resonance premise. In that case, the ring particles would be interior to an exact resonance radius, while the "shepherd" moons would be just outside of it. The empty space in between would just be the unstable position of a commensurate orbit to one of the larger moons. This implies that the small moons are not CAUSING the sharp edge but rather are just a different manifestation of the same near-resonance stable situation.

These matters regarding the Saturnian system are other areas that I have not yet had a chance to investigate thoroughly. It seems like a good reason to re-examine the data collected by our various fly-by spacecraft, to look for accurate periodicity information on the various phenomena recorded.

Another interesting possibility is regarding the eccentricity of orbits. The origin and the variation of the eccentricities of revolving bodies (within families of such bodies) might be explained as a first-order consequence of such gravitational forced vibration effects. Even more interesting is the possibility that the unusually large eccentricity of Mercury and Mars might be an expected result! Mercury is a relatively small body, with all of the forced perturbations arising from much more massive bodies farther out than it is. Mars is in a similar situation, with the massive Outer Planets being next farther out. Much more investigation is needed in this area, but some computer simulation frequency configurations would seem to act to increase the orbital eccentricity for such situations. (The above mentioned apparent gradual action toward reducing ellipticity could still exist. It is another of the meta-stable effects, where most situations result in gradually reducing eccentricity, while certain situations result in great instability that would act to do just the opposite.) There might thus be a theoretical reason for why the orbits of Mercury and Mars have such high eccentricity.

## Conclusion

By applying standard forced vibration theory and resonance theory reasoning and equations to the explanation of the perturbations of these several situations, it seems that a logical theoretical basis may exist for explaining the phenomena. An empirical basis for the long-term stability of near-commensurability of orbital periods, and also the apparent lack of long-term exact commensurability seems valid. That would seem to support these theoretical premises that give those mathematical results.

I am not familiar with previous researchers who have attempted to apply such resonance arguments to reasonably explain both the gaps (in rings and asteroids) and the long-recognized patterns of mutual orbiting objects.

In one sense, this argument, based on theory, seems to offer an additional family of solutions to the restricted family of stable solutions to the three-body problem. The two stable and three unstable Lagrangian points, and the recently discovered Lissajous choreography solution, might need to be joined by this meta-stable pair of three-body solutions.

I believe that, given the equation of standard Newtonian gravitation and the ones for forced vibration, and applying them to the group of the four large moons of Jupiter, a long-term computer simulation, beginning with four random distance moons, would eventually result in, and therefore "predict" the currently seen meta-stable periods for each of them. Such an experiment would seem to offer a solid test of this premise.

In another sense, this near-resonance forced-vibration analysis might offer a mathematical and theoretical basis for the many hundreds of empirical Fourier terms used to calculate the orbits of objects like the Galilean moons or our Moon. Better still, if there is validity here, a differential equation might be able to replace a whole (infinite) series of Fourier terms, potentially permitting both true Integration and exact solutions.

In the event that useful validity is found in this premise, there might be many other applications. For example, the closely packed but apparently very stable stars in Globular Clusters might have one or more unrecognized dominant resonances (or near-resonances) that somehow enable long-term existence of such an apparently unstable configuration.

In addition, the recently found "figure-8" stable solution to the three-body problem, might be theoretically explained as such a resonance situation where a meta-stability could exist.

Yet another possible application would be in mathematically explaining the persistence of atmospheric features such as the Great Red Spot on Jupiter. Frictional energy losses to surrounding atmosphere would seem to quickly degrade such a Spot by natural damping, but possibly a resonant forced vibration effect could be 'magnifying' that Spot to an extent that overcomes the substantial frictional losses and allows persistence. In that situation, there would essentially be a NEGATIVE net damping factor, but the same equations should hold. That same logic might also be applicable to the Physics of earth's cyclones and anti-cyclones. I have done some preliminary work on analyzing these same forced vibration harmonic effects on Hurricanes, the Physics and Analysis A Credible Approach to Hurricane Reduction regarding both the genesis of them and the potential for artificially developing destructive resonances inside them to degrade them.

### NOTE

If this approach of "near" commensurability being based in forced vibration harmonic effects is valid, another intriguing possibility seems to loom. In my college education in Quantum Mechanics, the permissible energy levels in electron shells was always described as being in absolute and commensurable terms. If this current premise regarding near commensurability is valid for celestial mechanics, it seems possible to also be true regarding mutual perturbations of electrons in atoms, and essentially the arguments of forced vibrations and essentially the same equations and the same reasoning should apply. This is NOT to suggest that there is any great similarity in astronomic gravitational and nuclear electrical situations, except that they are each inverse-square dependencies. It is ONLY meant to suggest that curious patterns exist among the planets and moons and so similar patterns might exist in atomic electron structure, and that an investigation into seeing if forced vibration and nearly-commensurate relationships might be worthwhile in that venue.

This could therefore suggest that the atomic energy levels generally described might actually be slightly incorrect, as being the commensurate energy level (which would actually be unstable) between two very closely spaced meta-stable energy levels. Given that electrons make unimaginable numbers of orbits in a second, the evolution of a system into a "mature" meta-stable configuration would be almost instantaneous, and those countless orbits would likely be evenly split in the 'under' and 'over' meta-stable radii, giving any long term average as seeming to be a value exactly AT the commensurate value. Very accurate measurement would then suggest finding twinned preferred energy levels, extremely close to each other. Some recent research seems to suggest some level of fine detail in those matters. In such a case, some basic understandings of quantum theory may need to be re-examined.

During my education in Physics, and in my researches afterward, I do not recall ever hearing a theoretical basis for the specific positions of the various energy levels within the atom. Empirically developed rules like the Pauli Exclusion Principle describe disallowed orbitals for electrons, but not why that might be the case. This premise might offer a start at building a theoretical foundation for atomic electron shell structures. It might not be that there are actually any disallowed orbits after all, but merely that electrons become perturbed into the configurations always seen in far under one one-billionth of a second, and we have just never observed any immature states.

Link to an older essay of mine regarding Io, on Geysers on Io - Source of Driving Energy, and a suggested telescopic experiment to confirm that those volcanoes are sulphur.

Link to a newer essay of mine regarding the planetary formation implications mentioned above is at Origin of the Earth New Arguments Regarding the Formation of the Earth.

This concept was developed by August 2001, and first published on the Internet on December 10, 2001.

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Carl Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago