TAB 14

THE SZEBEHELY EQUATION

The following study suggests that the solar system behaves just like three bodies - i.e. the equation developed for the "system wave" solves the equation for a regularized 3BP. The three bodies presumably are the sun, the inner planets, and the outer planets. You may recall that all the solar systems discovered around other stars have a massive sun plus just one or two gas giants, bigger than Jupiter.

Consider the equation for the system wave, where in the case of fitting the position of the planets to some common waveform, the value of a in the helix equation was close to zero. From the analytic geometry, when a=0 the solution is the unit circle. Consequently, the planets in this new scheme follow orbits of small eccentricity and it takes their combined influence to collectively avoid the singularity of a unit circle. The mathematics of the situation follows.

Which means motion is restricted about the unit circle, and although x and y are not identically equal to the sine and cosine terms respectively, their derivatives can be accurately represented as

Now consider the Jacobi constant for this rotating system,

from which because is constant for the circular coplanar 3BP. Notice that =1 is motion on the unit circle but < 0 for motion to exist - i.e. rotation - so motion on the unit circle is not possible; it's a singularity. If you evaluate the problem in the complex plane, then

Stability Points in the Ten Body Problem

Now consider the Szebehely Equation for the restricted 3BP

Substituting for ,

To put this into context, consider the earlier statement that the fact that the system wave "solves" the Szebehely Equation means the planets of the solar system are in some context organized as a 3BP. The system wave places the Earth near an equilibrium stable point and the moon in a stable orbit around this point. ( 3 ) The graphical similarity implies, as expected, that each planet orbits a stability point on the system wave, like a halo orbit in the 3BP. (It is likely these stability points also exist in the 3BP.) Notice that the 10BP rotates into the plane of the page, while the 3BP rotates in the plane of the page. Thus lines in the 10BP will be points in the 3BP, and the equilibrium stability points in the 10BP will be orbits in the 3BP - theoretically.

Projecting the Equilibrium Points into Three Dimensions

 

It was suggested before that the Szebehely Equation, or the system wave, is the transition between fractal levels, which apparently alternate coordinate systems which are perpendicular to each other. The only configuration of stability points - if they are to be considered an orthogonal/independent and mutually exclusive set, level to level - is for them to be the corner of a box in a 3D transition space. Thus the Lagrange Points in the transition coordinate system are at an angle of pi/4 as derived above.

Analysis

How to analyze the Szebehely Equation as a dynamic system when the analysis versus the system wave has been of a virtually static system? Even the projection of the whole body of work into three dimensions (refer to the next figure) is static, if complicated. It is important to establish motion because this shows how the system changes over time, evolving from a cosmic cloud into a stable system of planets, as it is now. Not even Relativity has a mechanism for change, having instead a fixed cosmic constant.

The Three Body Problem in the Context of Relativity Theory

(find the box of the previous figure)

 

It is helpful to return to the motor/generator analysis, which served so well ( 3 ) to conceptualize the complex interaction of gravitational and electromagnetic fields. Consider now a six pole stator, but now with windings on the rotor as well. That is, the analysis so far has been static - focusing upon the stator, which is unchanging - so now that a dynamical element to the system is sought, it is logical to look at the role of the rotor in the electromechanical analogy. In this case, the rotor moves but so slowly in terms of Earth orbits or years, as to be practically imperceptible. Rather this rotor turns infinitely slow so that only over the course of the evolution of the solar system does the system behave like a motor/generator set.

With the rotor, having six windings in the stator, the whole system all together - ten bodies, perhaps even N bodies - behaves as an elaborate 3BP, just as the Szebehely Equation supposes. The figure suggests a rather uncomfortable presentation of a single planet system, with the central body and moons/rings. This corresponds, however, with the projection of the essentially 2D planar theory so far, into a complicated but not abstract three dimensions. This process will continue more or less until the end of the text, as the theory of Three Bodies is applied to even more systematized 3D systems such as atomic orbitals, which not only have barbell and donut shapes; but have concentric layers of the shapes, all of which implies a powerful central force capable of exacting complex force patterns at a distance.

The Rotor versus the Symmetric Plane

 

The use of complex numbers previously in this study is natural in the study of rotating dynamical systems, as is a natural rotating frequency and the phasors it defines in the frequency domain establish the axis of the planet's rotation, in the complex domain. As illustrated above, the individual planets acts as a salient pole rotor. Presumably this applies mostly to the gas giant planets, which are large enough to have a strong localized gravitational field that can establish an order between rings especially, that is relatively independent of any external influences. Hence, the invariant and symmetric planes establish the field strength of the stator, and a complimentary system of forces exists in the stator for individually independent planet/moon/ring systems as shown here.

Dynamic Integration between Planetary Systems

The above illustration addresses the existence of dozens of complex resonances between the gas giant planets. These resonances, or comensurabilities, show that the whole body of outer planets has a very intricate sub-organization that is quite beyond the capability of any known theory to explain even superficially. This dialog does not do much better, except that to suggest as the second half of the text proceeds into the complex 3D mechanics of atomic orbitals which are well known; that because the solar system is a planar representation of these same forces, the better known atomic orbital scheme has its analogy in the comensurabilities between the gas giants.

Typically, each gas giant system is one level of the atomic orbital scheme - e.g. the 1s, 2s, 3s, 4s, 5s… - and the resonances between the gas giant planets are structural bonds between these schemes on the orbital level. These exist because of the fractal nature of the forces that are at play in both systems. The method is to apply the 3BP to the better known system, then to overlay this upon the lesser known system and perhaps to use what is well known about the latter to develop insights into the former, and so forth. The more commonalities that can be shown to exist between the disparate realms, the stronger the proof of fractal theory correlations; and the more robust the resulting model.

The best known, but least explained, phenomena in all of this is the fine structure of the ring systems, especially of Saturn. The recent flyby shows ice collecting in some rings; debris in others. This implies a dynamical system that is many orders of magnitude more complex than any known body of theory, which justifies the complexity of this analysis.

Fractals are not necessarily nested

The above illustration of a special periodic orbit in the Earth-moon system found by Davidson shows similar orbits around Earth and the moon linked by a small loop. This supports the notion that fractal patterns may be interlocked in subatomic or quantum systems, but exist in the same plane in systems on the scale of planets. This will henceforth be a recurring idea in the text, and is a useful way to understand the forces at play.

Chaotic Attractors

It is helpful now to study in greater detail the concept of a Lagrange Point being itself a local center or origin of force, as suggested before in this dialogue. Mathematically, regions traversed in phase space are strictly bounded when there is an attractor. In chaotic motion, nearby trajectories in phase space are continually diverging from one another but must eventually return to the attractor. These abstract things are illustrated in the next figure in the general case.

The attractors in such chaotic systems are called strange or chaotic attractors. They are bounded in phase space, as the attractors must fold back into the nearby regions of phase space. They are a kind of transition between fractal levels.

Strange attractors create intricate patterns because the folding and stretching of the trajectories must occur so that no trajectory in phase space intersects any other. Numerical studies of the 3BP show many such patterns, as will be presented in context in later chapters. Mathematically, these periodic orbits are caused by strange attractors.

Chaotic Attractors as Fractals

Returning to the more comfortable realm of Three Body motion, consider elliptical motion in a halo orbit around say the L4 Lagrange Point. This would look like the middle figure because the motion is stable (an orbit around one of the colinear Lagrange Points can also be a stable elliptical orbit, but the slightest perturbation or external force applied will cause the orbiting body to quickly spiral away, never to return). In dynamical systems analysis, such points where motion converges for dissipative systems are called attractors. Where beyond the closed ellipses there is what dynamical systems analysts call a limit cycle, but which motion slowly escapes.

The separtrix ( 9 ) is another example of this kind of motion, in which two equal masses are the primary bodies (instead of one central body as above). You might say that the figure-8 shaped free return orbit is a stable orbit in the two body system which dissipates into either a higher orbit around both bodies or an orbit around just one of the primaries, when perturbed. This is a more complex type of stability, which implies that the L2 Lagrange Point is actually a stable equilibrium point when the masses are equal, and not an unstable equilibrium point. This principle is important when considering a system of two masses, which over time evolve so that eventually both masses become equal; at which time a new dynamical equilibrium happens between them and a whole new families of stable and quasi stable orbits are possible, is in the nested systems of atomic orbitals. The following figure attempts to show this idea as it applies to a gas giant with a stable "dark spot" and several systems of complex ring systems. It builds upon the 3D box figure of the previous illustration.

Saturn's Ring Systems

These multiple dimensions are all associated with the same central body - e.g. the main 3BP is the sun-Saturn system, and the orthogonal iterations go sequentially as planets, ring bands (A,B,C,D,E, and F ring systems), and then the fine structure within each ring system. These are consecutive fractal levels that all exist around a single central body. This is quite a complicated force structure to emanate from a single body, which supports the idea of an equally complicated inner structure to the gas giant - here shown with a single core mass and one other mass at the surface manifesting itself as a "dark spot." (Well, Jupiter and Uranus have such spots; this model for Saturn assumes there is a spot, but it is below a surface cloud layer.)

This is just the arrangement of the three types of spherical harmonics associated with the Legendre ploynomials and are actually an amplification thereof - whose study will show ( 31 ) exactly how the system is structured, and maybe even how it evolves through time.

________________________________
your banner could be here

© 2004 WH Clark