Planetary Gravitational Resonances

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 complete 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.

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. 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 a very 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 then 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 showed us that there is Conservation of Energy, 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 = w * R, this is more conveniently described as 1/2 * I * w²) The (angular) momentum of a planet is described by (M * V) * R (Again, in circular coordinates, this is I * w). 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 w and the other on w squared 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 released, it necessarily ACCELERATES up to the final precessional rate, so what is the source of that energy that is used up in that acceleration?"

The gyroscope starts out with NO angular momentum around the precessional 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 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 plane to another does NOT conserve Angular Momentum in the process!

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.

The 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 come from a slight Z-axis (Solar-System-vertical) relative movement of the Sun/Earth and Moon/Earth, so Kinetic Energy would be conserved, even 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 * w². We know that the rotational inertia (I) of the earth is 8.07 * 10³⁷ kg-meters². We know that w 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!

Several years of research into the Physics of "forced vibrations" and resonance as related to the gravitational interactions between planets and satellites has suggested 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, primarily involving the planet Jupiter.

Some previously unconsidered conventional gravitational effects may have tended to "funnel" materials into an area near what is now the orbital path of the Earth. This would greatly have increased the local density of material that could eventually have accreted to become our Earth, creating essentially a ring-like structure around the Sun. This premise suggests that such a ring would then have been a precursor to 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 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.

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 long-term consequences of these effects has never been pursued. The research of the past several years has involved applying a standard Engineering/Physics concept of "forced vibration" to such gravitational systems. Traditionally, in Engineering, that field is applied to analyzing and eliminating destructive vibrations of machine elements or other feedback-type phenomena. It seems useful to apply it to astronomical gravitational interactions. A companion essay gives a more thorough discussion of the Physics and the mathematical analysis of forced vibration, at Gravitational Theory and Resonance. It involves the standard resonance "magnification factor" which greatly increases perturbative effects when the perturbing force has a frequency that is exactly fractionally related to the base frequency of the central phenomenon. In this case, when a perturbing planet's period is commensurately related (1/2, 1/3, 1/4, 2/3, 3/2, etc) to the orbital period of the planet being perturbed. This discussion will concentrate on several interesting resultant effects that have been observed in computer simulations:

• There is certainly a broad effect of very slowly altering the orbital elements of both objects in such an interaction, with the greater effects occurring to the orbit of the less massive object.
• One orbital element that is so altered is the orbital period, which has a slight second-order incentive to change TOWARD a simple fraction (or multiple) of the other object's orbital period.
• However, as an exactly commensurable relationship would eventually develop, the normal forced-vibration "magnification" effect (and a substantial phase shift) would develop, which acts to de-stabilize the orbital elements.
• The combination of these two effects is to create a meta-stability, where precise synchronicity is unstable, but NEARLY commensurate orbital times are a natural consequence, and are meta-stable.
• Another effect of the combination of the two effects is a general reduction of the orbital eccentricity (over extended time), but again with an instability for perfectly circular orbits.
• Yet another effect is a general reduction of the orbital inclination (over 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, and the others, 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 effect of forced vibration and gravitational resonance would gradually cause these various small particles to have orbits which had one of several specific orbital radii. They 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 would develop a substantial ring system, possibly all within the orbit of Jupiter. One favored radius might have been at a radius equivalent to an orbital period of 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.

This would then provide the environment where an Earth could then begin to gravitationally self-coalesce, from the appropriate (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. 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:

• The asteroids might then NOT be remnants of some previously destroyed larger object, but a future planet in the early process of forming, due to Jupiter's gravitational effect. This might even be confirmable, by collectively doing Fourier Analysis the orbital elements of many the asteroids over time, to determine whether they are becoming more random or more organized.
• All of the four inner planets might then still be in the process of "growing" as confirmed by the many tons of meteoritic dust that falls to earth every day. In the very distant past, those four planets might have been significantly smaller and less massive. Mercury would logically have been smaller, for having swept out a smaller volume of space anf for having been farthest from Jupiter's gravitational forced-vibration effects.
• The elemental composition of the four inner planets might then better be explainable, and even the variations between them might be a result of just what available particles and materials had been available at roughly those orbital distances from the Sun.
• This premise suggests the critical requirement for a Jupiter-like planet, to permit the efficient accumulation of materials in a restricted ring-like spaces. This might imply that virtually all stars that have a Jupiter-sized planet orbiting them (except double-stars), would eventually accumulate small inner planets, in commensurable orbits.
• This premise permits the original Solar System to have only contained "star-like" components and objects. No exotic explanation of initial small rocky planets would be necessary, thereby simplifying the theories of initial formation.
• It barely needs to be mentioned that this premise would necessarily cause all the inner planets to revolve about the Sun in orbits that are in roughly the same plane as Jupiter and in the same direction.

Rather than the Earth being formed as part of the original Solar System, this premise suggests that it developed as an "afterthought"! 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 could 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. 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.

Advanced Physics-related presentations in this Domain:

( http://mb-soft.com/public/othersci.html )

C Johnson, Physicist, Physics Degree from Univ of Chicago