Abstract
An elegant sweep of solar system cosmology shows how planets were formed from the original cosmic gas cloud. Local eddies of gas coalesced then gradually became solid mass by increments represented by specific Fourier Series expansions. This exact mechanism is silhouetted in how Fourier Series expansions are used in satellite based geodesy to model planet gravity fields, like rings of a tree, and shows how geodetic summations correlate closely to specific aspects of the planet's orbit around the sun. The model posed here, which includes mechanistic explanations for major relativistic phenomena, leads to the dramatic conclusion that Fourier Series are in this instance more than a convenient mathematical tool but a physical representation of the allowable energy states for the major bodies in the solar system, analogous to allowable energy states in the atom. Similar correlations may exist in other Fourier Series models. The numbers in parenthesis refer to other papers with supporting material I have written. Please visit my website at http://www.celestialmechanics.org/stars.html to read these papers, and the rest of the theory. There are a total of 40 little papers in my PhD dissertation on the Three Body Problem, all on my website.

The Generalized 3BP
It is widely believed that the Three Body Problem (3BP) of Celestial Mechanics has no general solution. It will therefore come as a surprise to experts in mathematical physics that such a solution is presented here, in heuristic terms and without the use of any analytical tools. The alleged solution is shown to exist in nature, and this solution implies that the 3BP is a stable organization in nature - so much so that natural dynamical systems are driven into the circular coplanar configuration of the 3BP that mathematics has shown to be the most stable solution. In deference to extant analytical theory, the assumption is made herein that nature takes a rather devious approach to solving the generalized 3BP, in such a way that the overall system behaves like the most fundamental restricted 3BP to which a solution is known; in this case, the restricted circular coplanar problem.

This phenomena by which nature solves the 3BP, albeit subtly, takes place in the natural order of the solar system. The general 3BP occurs in a coordinate system local to the planet, whereas the circular coplanar restricted solution - presumably the most stable configuration over the long term - happens with respect to the coordinate system of the entire solar system. The latter inertial reference plane is the hyperplane ( 2 ) that is perpendicular to the angular momentum vector of the solar system, which is typically defined in two body motion as the orbital plane. Simply put, the generalized 3BP is shown to be a regularized solution in this dominant plane of the solar system, or mathematically a hyperplane in which upon which all motion is stable.

This transformation happens at discrete intervals or energy levels, so that each stage is translated to a stable hyperplane solution ( 2 ); then the next building upon that solution; and so forth, until many separate bodies are involved. ( 4 ) This process can theoretically continue indefinitely, until of course N bodies are done, at which time the whole configuration reduces down to the equations of motion for an incompressible fluid. This report shows how the summation happens for increments of inclination in the orbit, simultaneously as a similar process happens for increments of eccentricity of the orbit. The process is a little different in either case, but the result is always a circular coplanar restricted 3BP which has a known mathematical solution.

It will come as a surprise to experts in Celestial Mechanics, just how easily nature solves the generalized 3BP, and how elegantly the solution lends itself to the mathematics. This notion will immediately rankle skeptics, and it is in deference to their doubts that the solution will here be posed in a cosmological context. The explanation begins when the solar system was nothing more than a cloud of gas, and shows how planets were formed and how they came to be as they are today ( 1 ) - with particular reference to their orbital elements versus their gravitational properties; e.g. the distribution of mass, and its density, within individual planets.

It will be shown that the 3BP played an important role from the very beginning. The 3BP is more than an interesting mathematical problem, but the embodiment of a physical law of nature. The 3BP represents - at least in its stable and known solutions - a fundamental level of dynamical stability in nature, to the extent that natural motion is driven to be organized in such a scheme. The unsaid assumption in this hypothesis is that, given there are many versions of the 3PB with known solutions, each one of these represents a step in the evolution of a chaotic mass of random particles of matter into a consistent planet in a stable orbit around a central body. These problems, sequentially, form together an envelope curve ( 27 ) that is itself another solution to the 3BP.

This report makes the leap from chaotic gas cloud directly to the most stable 3PB of them all, the circular restricted coplanar 3BP, accepting the fact that there are many additional steps in between via quasi stable orbits, but here for purposes of brevity taking the overall perspective of the process. Experts in mathematical physics have already derived all the intermediate steps from one stable circular state to the next. Thus, far from posing a solution to the generalized 3BP on independent merits alone, this report shows that the general solution in fact involves the entire body of theory on the 3BP, with transitions between each quasi stable orbit happening perhaps by virtue of the various non-standard orbital elements. These details are beyond the scope of this report and, in any event, are moot if the basic process itself is not first established.

In the Beginning
For some reason each nascent planet was forced slightly out of the orbital plane of the sun-Jupiter system, and the planets went from gas to mass in order to maintain a dynamic equilibrium with the whole. These changes not only transformed gas to solid matter, but organized the matter by density in such a way that the gravitational field can be modeled by Fourier Series - as done in the science of satellite based geodesy.

In the next few pages you will see how the divergence of a planet's mass from the invariant plane transpired in discrete steps, each one a single term in a Fourier Series. The Fourier Series we now use to model a planet's gravitational field is a permanent record of these changes, like rings in a tree, solidified in the rock of the planet itself.

The important question now is, why did the small eddies of gas that were to become planets diverge from the invariant (i.e. angular momentum) plane at all? Relativity specifies a slight warping of space in the vicinity of a high gravitational field, such as the sun. In this case, however, the plane about which the mass was distributed could have had local discontinuities or bumps, projecting above or below the orbital plane formed by the summation of the whole solar system's cloud of gas, into a single common angular momentum vector. A facsimile of these local variations will be derived shortly.

The invariant plane as we now know it is dominated by Jupiter, which is by far the most massive planet. Before the formation of Jupiter from out of the gas cloud, all the mass in the solar system would have been roughly symmetrical to the invariant plane. Indeed, all mass in the solar system still moves perfectly symmetrical to such a "symmetric hyperplane," with the sun at (near) the barycenter. ( 2 )

Presumably this symmetric hyperplane is the same as the one that existed aeons ago when the solar system was all just gas. As for the force or phenomena which caused individual eddies of gas (future planets) to diverge from this symmetric plane, perhaps a pattern in the distribution of the planets in a "system wave" (or "system surface" as the case may be) in the current epoch offers an explanation. ( 3 )

The "system wave" is a study of the motion of the planets with respect to the ecliptic plane. The constraints for finding this helical 3D wave (longitudinal wave in 2D) are (i) a planet moves such that it is attached to the wave so that it's motion up/down in the course of a revolution is due to the inclination of its orbit; near/far on the wave is due to the eccentricity of its orbit; (ii) the planet's axis of rotation is perpendicular to the wave such that (iii) during a revolution this axis points always to the same place on the celestial sphere, like Earth and the North Pole. A helical wave was found that satisfies these criteria for all the planets such that each planet occupies a specific segment of the wave - a stable equilibrium position for the Ten Body Problem (10BP) - the point being that this wave pattern could have caused the small eddies of the primordial gas cloud to be pushed above/below the original plane of symmetry, e.g. the common orbital plane for the solar system gas cloud as a whole. Presumably this "system wave" is caused by galactic forces external to the solar system, as some aspect of the gravitational phenomena holding the entire galaxy in its specific configuration. This notion is discussed at length in other sections.

Mass from Gas
Now to consider the dynamic mechanism by which the planets became solid matter from out of a cloud of gas. Please note that this is not a trivial exercise in logic because, as you will see later, it silhouettes the opposite process - i.e. the conversion of solid matter to gas or plasma; a.k.a. atomic fission, a promising power source.

The following analysis is predicated on the assumption that the solar system, from beginning to end, is a single dynamic system and that each constituent part of the solar system is compelled to - within its physical limitations (which are nominal for a gas giant, but much more constrained for a solid planet) - seek a stable configuration. The strongest common force for the gas cloud solar system was that it was rotating in a single plane. Thus the organizing force or criteria was angular momentum alone and the constant direction (defining the plane and by this analysis the orbital inclination) and magnitude (defining the "strength" of the plane and by analogy the orbital eccentricity) of the angular momentum vector.

A simple study in geometry shows that an elliptical orbit in one plane can be modeled by a circular orbit in an adjacent plane at a small angle to the first one, by a Fourier Series type approximation. ( 4 )

Posing this as a 3BP, with the large second primary a gas giant of constant density, then these results are as stated and the analysis is valid in the present situation without loss of generality. The conclusion is that the whole body of the planetary gas cloud moves (or is compelled to move, if you accept the "system wave" hypothesis) away from the symmetric plane of the solar system. Once all the matter in the cloud is "solidified" the process stops, leaving the planet in just as perfect a dynamic equilibrium with the solar system as a whole, as it was in the beginning.

It is worth mentioning that the "Red Spot" on Jupiter may represent this ongoing process, and also the precession of Mercury's perihelion - Mercury being such a small planet, and unable to solidify since it is so close to the sun, the balancing mass has to be within the sun itself. ( 5 )

Most scientists believe the precession of Mercury's perihelion is "caused" by Relativity. However, there are still many who adhere to the equally plausible explanation of the precession being caused - in a process akin to that caused by Earth's equatorial bulge - by a specific density anomaly within the sun. The problem with the latter explanation is that it offers no explanation for the bending of light by the sun's intense gravitational field, but ( 5 ) may rectify this issue.

This raises the question, could gravity within a star or gas giant planet be proportional to distance or 1/r and not 1/r2 ? In such a force field, elliptical orbits are centered not at a focus but at the center of the ellipse, which would be a more logical place for it to be in some circumstance; e.g. motion of an anomaly within the interior of a star.

This would imply a specific transition between the original 1/r universe and the current 1/r2 (that's r squared) universe - and, given that Jupiter and perhaps the sun are remnants of the original 1/r universe - there must remain some boundary region between the two. This would fit the notion of weak and strong gravitational forces. This suggests that "black holes" might not be the drastic discontinuities specified by Relativity Theory, but simply transitions to the next level of reality, e.g. different powers of r than 1/r2.

The geometry of the 1/r and 1/r2 systems is that they are equivalent for circles. Ellipses are a combination of sinusoidal waves, circles being ellipses of e=0; and so waves emanating from the separate systems can coexist as in a 3D EM wave. That is, elliptical orbits exist in a 1/r system, only the central body is at the center of the ellipse and not at a focus, as in the 1/r2 system. Likewise, the intersection between two such systems would be a helix, the shape derived for the "system wave." Thus the body of theory as referenced in the footnotes () is geometrically consistent.

Conclusion
The analysis shows that the Fourier Series used to represent a system are more than a convenient mathematical tool of modeling, but a representation of actual physical processes - and perhaps even delineating the allowable discrete energy states for a system so represented. In the case of orbital perturbations about a unit circle, the Fourier Series are even more compelling as an analytical tool, being the equivalent of a Taylor Series expansion for linear systems, with the added bonus of representing actual dynamical processes integral to the solar system so represented. Given that neighboring optimal paths are simply trajectories differing by a Taylor Series from an existing local optimal path, the calculus of variations shows that the elliptical perturbation from a unit circle orbit is simply the local minimum energy trajectory for the system of particles in question.