Planetary Gravitational Resonances in the Solar System

A new approach suggests an extremely logical explanation for how and why the Earth first formed, and even how and why the Moon formed.   A number of additional issues are also described in logical terms, such as why the inner planets have much higher densities than the outer ones.

There is an assumption that has always been made which had appeared to eliminate this possibility, but which is now seen to be incorrect.   There are six "orbital elements" which completely define a planet's (or other revolving object) orbit.   They are (1) average orbital radius or orbital semi-diameter; (2) orbital eccentricity; (3) inclination of the orbital plane; (4) direction if the perihelion or line of apsides; (5) direction of the tilt of the orbital plane or the longitude of the ascending node; and (6) the location of the planet or object along that orbit at some instant of time.

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In the study of Perturbations of planets on each other's orbits, it is fully accepted that all of these orbital elements except the first can be affected and therefore changed.   Even Laplace and LaGrange and all the early giants in astronomical calculations accepted that as absolutely true.   Therefore, Perturbation Theory attempts to track the slow changes in the eccentricity of the orbit, and all the other parameters that can have such secular variations.   For example, it is well established that the Earth is presently advancing toward a much more circular orbit. Currently, our eccentricity is around 0.016 (which means that we vary from a circular orbit by about one part in 60 in distance from the Sun, meaning that our average 93 million mile distance varies from 91.5 million (in early January) to 94.5 million (in early July).   It is well established that in around 24,000 years, our eccentricity will have greatly reduced, to 0.003, which is extremely circular.   After that, the eccentricity will increase for around 40,000 years until we are in an extremely eccentric orbit, (0.070 eccentricity).   We can know these things due to a mathematical analysis of careful observations which is called a Fourier Analysis.   The Earth's eccentricity is therefore able to vary in the range of 0.003 to 0.070 but that it will always stay within that range.   The other orbital elements can equally be secularly perturbed in such ranges.

The single exception has always been the orbital semi-diameter.   This orbital element has always been denied any possible variation!   The reason seems to be obvious!   Newton and others showed us that there is Conservationof Energy, which is also called the First Law of Thermodynamics, where energy can neither be created nor destroyed.   He also showed us that there is Conservation of Angular Momentum, where that too can not be created nor destroyed (except due to the effect of an external action).   When Laplace, Lagrange and others first carefully studied the motions of planets two hundred years ago, they applied these two Conservation laws, and that forced the assumption that the orbital semi-diameter cannot be perturbed.

The energy of an orbiting planet is in kinetic energy, which is described by 1/2 * M * V2. (In circular coordinates, since I = M * R2 and V = ω * R, this is more conveniently described as 1/2 * I * ω2)   The (angular) momentum of a planet is described by (M * V) * R (Again, in circular coordinates, this is I * ω).   Kepler (and then Newton) determined that the orbital semi-diameter is directly related to the orbital period, which means that the (average) velocity in the orbit is uniquely specified when the orbital semi-diameter is known.

The fact that one of these quantities depends on ω and the other on ω2 means that it is not possible to conserve both in any possible Perturbation transfer between two planets or other objects.   As one of the objects would gain potential energy and the other would lose the exact same amount, there would necessarily be a violation regarding the changes of the angular momentums, therefore Laplace, Lagrange and all others have always concluded that no such perturbation of the orbital semi-diameter has been said possible.

Those statements are true, if only one plane of motion is considered. However, Euler expanded Newton's equations of motion for three dimensions, and a new possibility then arises.

The most obvious example of this is a high-quality child's gyroscope.   Consider one where the support bearings are perfect, that is, there is no friction whatever, and it is operated in a total vacuum, such that the gyroscope rotor will spin forever and never slow down.   Placed on the usual pedestal in a axle-horizontal position, we all know that the gyroscope will then precess around the pedestal.   The question is "when the gyroscope is first released, it necessarily accelerates up to the final precessional rate, so what is the source of that energy and angular momentum that appears during that acceleration?"

The gyroscope starts out with no angular momentum around the precessional (Z) axis, but quickly develops the angular momentum due to the precession.   According to the conventional description, this is a violation of the Conservation of Angular Momentum!   The rotor did not slow down, so that was not the source of any angular momentum.

The answer is in the (third) Euler equation which describes the dynamics around the precessional axis.   There is no external moment applied, so M = 0. The result is therefore that the angular acceleration of the precessional motion is due to (vertical) motion in a different plane!   The support angle of the gyro body is very slightly lowered, which gives up some gravitational potential energy, which is then converted into the kinetic energy of the precessional motion.   Conservation of Energy is maintained.   The significant fact is that this transfer of Energy from one form (potential) to another (precessional kinetic) does not conserve Angular Momentum in the process!

Here is the (third) Euler equation.


M3 is the net external force (moment) acting around the Z-axis, which is zero.   The first term on the right includes the acceleration of the precessional rotation rate and the other term involves the constant angular rotation of the spinning rotor and the angular rotation rate of falling (around axis 2).   If we integrate this over time, we get:


We can see that we can now solve this for the relationship between ω3 (the final precessional angular velocity) and the total change in ω2 (which is the vertical angle through which the mass of the gyro has fallen, or θ2.   The vertical angle of drop of that mass is therefore directly related to the increase in precessional angular velocity.

Further analysis of the energy content of the two show that Energy is Conserved, as they are the same quantity.

This suggests that the long-held assumption that Angular Momentum is always conserved is not really necessarily true when more than one plane of motion is considered, that gyroscopic precession certainly shows that flaw of reasoning.

Solar System objects move in various planes.   This fact results in effects that are similar to the non-Conservation of Angular Momentum of the toy gyroscope.   For example, the earth has an equatorial bulge that is rotating in a plane where the Sun and Moon nearly always act to gravitationally try to tilt that plane, which causes the Precession that the Earth experiences.   Consider a "new earth" exactly like ours but not precessing.   It would start to precess, in other words, the Precessional motion of the earth would accelerate up to the rate it is now at.   The energy that would supply that motion would seem to come from a slight Z-axis (Solar-System-vertical) relative movement of the Sun/Earth and Moon/Earth, (the earth would move through the potential energy field of the Sun) so Energy would be conserved with the "precessional acceleration up to the new precession rate".   However, Angular Momentum in the Plane of the Ecliptic would not be conserved!   New Angular Momentum would arise in that Plane.

The effect described here is very small, and very slow.   In all practical situations, Conservation of Angular Momentum will be seen to appear true.   It is only where Euler equation transfers from one plane to another can occur that any variances with that Conservation can occur.   Conservation of Energy appears to still always be true.

For that "new Earth" that is initially not precessing, we can easily calculate how much kinetic energy there is in our precession.   It is 1/2 * I * ω2.   We know that the rotational inertia (I) of the earth is 8.07 * 1037 kg-meters2.   We know that ω is one precessional revolution in 25,800 years or one radian in 1.296 * 1011 seconds.   Therefore, the kinetic energy the Earth has in precessing is around 2.4 * 1015 joules.   In planetary dynamics, that is not very much, but it still is kinetic energy that did not used to exist, and must have been converted from some other previous form of energy!

Several years of research into the Physics and Engineering of "forced vibrations" and resonance as related to the gravitational interactions between planets and satellites has suggested a number of potential consequences of this phenomenon that seem never to have been previously discussed.

Asteroids in mutual perturbation with the planet Jupiter would have several slow, long-term effects.   The semi-major axis dimension of individual asteroids would be able to change slowly, to come into commensurable, or actually near commensurable, orbits with Jupiter.   This could explain the presence and the origin of the Kirkwood gaps in the asteroid belt.   The same effect would also apply to the four Galilean satellites of Jupiter, to cause the very distinctive near commensurable relationships to have developed over long periods of time (less than a million years in their case).   The extreme thin shape and the gaps in the rings of Saturn, and the complex interrelationships between those ring particles and the various satellites of Saturn might also show the same reasoning.   There are many other similar relationships.   Interestingly, this seems to suggest that our common reasoning might have been backwards regarding the rings of Saturn;   that instead of being the remains of previous satellites ripped apart by going within the Roche limit, they may instead represent particles that are gradually collecting to form future satellites.

That reasoning is this.   By a particle being able to have its semi-major axis altered by the perturbative effect of an existing satellite, the particles might tend to be "channeled" into preferred orbital distances.   There appear to be similar effects that might reduce eccentricity and orbital inclination.   Among other things, this might explain the extremely thin thickness of the rings of Saturn and the other planets.   This reasoning would suggest that a long-term study of Saturn's rings might show certain dynamic changes where some areas might be becoming more dense, while other areas of the rings might tend to clear out.   This seems to show that the orbital inclination and eccentricity of the individual particle orbits have already been greatly affected but that there might eventually be generated a rather narrow and fairly circular well-defined, higher density ring of material circling the planet.   The reasoning would continue in suggesting that certain locations along that ring might have incrementally greater density, and therefore greater gravitational attraction, suggesting that the ring might gradually "clump up" into significant chunks, and that eventually, those co-revolving chunks would collect into a planet or moon.

This line of reasoning seems to suggest an entirely new premise regarding the origin of the Earth!   It has always been assumed that the Earth formed, in its present place, pretty much due exclusively to its own gravitational actions among its component parts.   This must certainly have been true, of course, but a critically important external effect may also have been necessary, or at least important, primarily involving the planet Jupiter.

The previously unconsidered conjectured gravitational effect discussed here may have tended to "funnel" random materials of the Solar System into an area near what is now the orbital path of the Earth.   We referred to that above as a narrow and compact ring.   This would greatly have increased the local density of material that could eventually have accreted to become our Earth, temporarily creating essentially a wide and then very narrow ring-like structure around the Sun.   This premise suggests that such a ring would then have been a precursor to relatively conventional view of accreting planetary formation, and not the result of the destruction of an earlier planet.

It should be noted that this premise does not involve any "mechanisms" such as gravity waves or electromagnetic phenomena. Rather, this premise is simply the long term result of repetitive patterns of the positions of objects orbiting the Sun, and the perturbations and resonance effects that would therefore exist. Nothing more than standard Newtonian gravitation is involved. The Engineering field of Forced Vibration becomes valuable in this exploration.

The full reasoning of the Forced Vibration aspects shows that it would be extremely unstable of two planets were precisely commensurate, as Laplace and LaGrange and the others regularly noted. Forced vibration analysis shows that the amplitude of perturbations would be extreme for exact commensurable orbiting objects.

In the field of Engineering, the mechanically destructive effects of resonant vibrations due to forced vibrations are analyzed. The differential equations of motion of an object having a natural frequency of ωn while being forced by an exterior force acting at a frequency defined by ω, can be written in the form of:




The solution to these differential equations can be written in the form of:




where e represents a variable generally called eccentricity (but which has a different meaning than the astronomical meaning of the term).

One can see that if the forcing frequency were exactly the same as the natural frequency, the denominator goes to zero and the amplitude of the oscillatory motion therefore goes to infinity. In mechanical systems, this is akin to the situation when a device disintegrates due to unexpected vibrations. These equations are for the situation for a system which has no damping factor, which cannot actually occur in any real mechanical device, but which is essentially true for systems of planets orbiting the Sun.

The claim here is that this Engineering approach of forced vibration can be applied for the situation of one planet perturbing another. The equations are more complex than these simple ones because the strength of the perturbing force constantly varies with the distance between the two planets per the inverse square rule, and they are each travelling in elliptic orbits. So the actual mathematics of this is more complex, but the reasoning is as indicated here.

Therefore, this shows that if the perturbing (forcing) frequency were commensurate with the natural frequency (inverse of period) of a planet, the perturbing effect would be extremely unstable and essentially catastrophic. This shows the well known effect of this fact that exact commensurability cannot exist among planets or satellites or asteroids with Jupiter.

There are therefore two adjacent orbital radii of near commensurability, one on each side of the exact commensurable orbital radius, where a meta-stable relationship can and will occur. By inserting the actual parameters for any specific two planets in the differential equations above, it should be possible to solve them to calculate the actual meta-stable solutions. Such solutions contain the above and also factors that depend on the cosine or sine of the motion of the perturbed planet, so the perturbative effects vary by both orbital motions. It is my belief that this should then indicate/calculate the slight differences of the Galilean satellites from exact commensurability with each other, and the other slight differences which exist in such relationships.

The Earth happens to be near the outer of the two which would represent the perturbations of Jupiter (at a harmonic factor of 12:1). Roughly 1.35 million miles inward of where we now are, there should be another meta-stable location, where debris should collect, as a response to Jupiter's perturbations. It seems reasonable to speculate that if a smaller amount of material initially went into that other ring, and it also coalesced into an object, we might have a very practical explanation for the Moon. It can even include why the Moon is made of different material than the Earth. The two might have been able to revolve in extremely similar orbits for a long time, but the great mass of both, and the very near passes they would make every century or so, and the rather small differential velocity that would exist between them, might easily explain how they later captured each other and became our double planet of the Earth and Moon.

For several centuries, it has been noted that there are an assortment of "interesting patterns" involving some Solar System objects. Titius-Bode's Law presented a surprisingly accurate simple formula regarding the relative distances of the planets from the Sun. No one has ever found a theoretical explanation for it. The four large Galilean Moons of Jupiter have orbits that are fairly close (but not exactly!) to having periods in the ratio of 1:2:4:8. Jupiter and Saturn have orbital periods around the Sun that are very close (but not exactly!) in the ratio of 2:5. Among the many thousands of asteroids circling the Sun, none seem to have orbital periods that are exactly simple fractions of the period of Jupiter (Kirkwood gaps), but many are fairly close to one or another of such fractions.

I am suggesting that there is a possibility that the violation of Conservation of Angular Momentum due to the precession-gyroscope effect might enable slow alteration of the semi-major axes of celestial objects, and the effects of Engineering's forced vibration analysis might clarify why exact commensurability cannot occur, and that this combination might provide some answers to some previously unanswered questions.

Jupiter is, by far, the most massive planet in our Solar System, and so it is universally accepted that it gravitationally affects all of the other objects, but deep analysis of the very-long-term consequences of these effects seems never to have been pursued. My research of the past several years has involved attempting to apply commonly accepted standard Engineering/Physics concepts regarding gyroscopes and "forced vibration" to such gravitational systems.

Another effect of the combination of these effects seems to be a general reduction of the orbital eccentricity (over very extended time), but with an instability for perfectly circular orbits.

Yet another effect seems to be a general reduction of the orbital inclination (again over very extended time), but again with an instability for perfectly co-planar orbits.

Planetary Formation

Over extremely long periods of time, stray atoms, molecules and gas pockets and dust grains, would have been affected by Jupiter's gravity, either to be eventually pulled into Jupiter itself or to have orbital elements (around the Sun) altered as mentioned above. These materials might have been left-over components from the original formation of the Solar System OR they may have been deep-space materials that just happened to be in the path of the Sun and its family on its way through the Galaxy. The objects would (virtually) all be very small and easily perturbed in the ways described.

Over an extended period of time, a substantial amount of this material would initially have orbits that were relatively random in orbital radius, in orbital eccentricity, and in orbital inclination. The effects discussed here would gradually cause these various small particles to have orbits which tended toward having one of several specific preferred orbital radii and shapes and locations. Many would gradually tend to all have relatively circular orbits around the Sun, in a plane that is relatively the same as that of Jupiter's orbit.

Essentially, the Sun (with an axis of rotation completely independent of this, around 7° different in inclination, would develop a substantial ring system, possibly all within the orbit of Jupiter. One (pair of) favored radii might have been at a radius equivalent to an orbital period of around 1/12 that of Jupiter. This would then have created a relatively circular, relatively narrow ring around the Sun, at the orbital radius of where the Earth is now, or a pair of such rings around 1.35 million miles different in semi-major axis.

This would then provide the environment where the conventional view of an Earth then beginning to gravitationally self-coalesce, from the appropriate ring or pair of rings, with the same effect happening to create Venus, Mercury and Mars. The present asteroid belt may not then have existed, or it might have been an unsuccessful accretion into a planet for some reason, or it may still be an ongoing attempt at creating another planet. The former seems like a more likely scenario, suggesting that the objects in the present asteroid belt may be extremely old but the collecting them into a "belt" may have been a more recent process, which is still proceeding.

This premise suggests several interesting consequences:

Rather than the Earth being formed as part of the original Solar System, this premise suggests that it developed as an "afterthought"! The initial solar system seems logically to have had a preferred coordinate frame of the Sun's axis, given that it represents virtually all of the mass. Yet, all of the planets tend to lie in a rather different plane, relatively close to the Invariable Plane. Assuming that there was a good deal of stray material left over from the initial formation of the Solar System, the rate of meteoritic influx would have been much higher than today. As to how high, it may never be possible to know. But with the combination of Jupiter "aligning" space debris in and near the orbital path of the Earth, and the Earth's self-accretion, the Earth might have formed in a relatively reasonable period of time.

Note: A careful study of particle densities in the Solar System could constitute a confirmation of this premise. If this is valid, then there should be a number of exceedingly faint rings around the Sun, in orbits that are approximately commensurate with Jupiter, such as at radii that would have slightly different than 3 year orbital periods, or 4 year periods, or 6 year periods, or other simple fractions of Jupiter's orbital period. Many of these are known to exist, as they are within the asteroid belt. The exactly commensurate periods, being unstable, do not contain asteroids, and are known as Kirkwood Gaps in the asteroid belt. But there are many asteroids that have orbital periods that are quite close, the meta-stable situation described above.

Note: Another possible confirmation of this premise would be a thorough analysis of Saturn's ring system, as to orbital elements of the ring particles, particularly the dynamical changes that occur over years. If this premise is valid, statistically those ring particles should be trending toward more organized patterns, possibly in physically smaller volumes. This would be occurring because of the presence of the several massive satellites of Saturn, and the resultant forced-vibration resonant effects.

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C Johnson, Physicist, Physics Degree from Univ of Chicago