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PROJECT INTRODUCTION AND SUMMARY

The gravity model for the Earth is solved by matrix and statistical methods. Being able to put the system in matrix form means it is a linear system, and the eigenvectors are linear independent and orthogonal - this is the nature of the Fourier Series used to create the model itself - and they are also subject to the superpostion of forces. The existence of eigenvectors and eigenvalues implies discrete allowable energy levels. All of this is in the time domain. The fluxions (phasors) developed later show it all in the frequency domain.

Scientists have iterated the equations of motion for the solar system forward in time millions of years and have found the whole system to be completely stable and predictable. Each planet and moon was modeled, in these simulations, as a point mass - e.g. each body acting as though it's entire mass were concentrated at a single point in space.

Contrast this notion to what we know about Earth from satellite geodesy. This science models the gravitational field of the Earth as the infinite sum of a Fourier Series. The matrix of these values has over 100,000 elements and requires the largest, most powerful computers in the world to solve.

It should be disconcerting, how these 100,000 parts of our crude simulation of the Earth's gravitational field - when you put them all together and sum their cumulative action upon other bodies in the solar system, it is as though all the mass were concentrated at a single point.

A similar phenomena is exhibited by the equations of motion of N bodies, which reduce down to the simple equations for incompressible flow in fluid dynamics. ( 11 )

The question arises, how was it that the solar system evolved from a huge cloud of cosmic dust into a star, nine planets, and many moons - all of which are quite physically complex, but infinitely simple at the same time. Just consider all the different minerals, elements, rocks, and disparate layers of Earth when summed together are so distributed as to have the sum total affect of being perfectly balanced versus a single point the size of the period at the end of this sentence.

Then consider that this same dynamic exists for the sun, the other eight planets, and all their respective moons, rings, and asteroids. More over, the N Body Problem says this also happens on a much larger scale - with vast collections of whole star systems.

The purpose of this investigation is to offer a theory about how this intricate, but completely balanced, organization of matter came to be as it is. This introductory paper is intended to discuss some rudimentary concepts needed to comprehend the research.

Gravity

An interesting aspect of gravity is that the internal behavior of a thin perfectly spherical shell of mass of constant density is the same as though all that mass were concentrated at one point, the center of the sphere. Thus, an infinite number of concentric shells, each of different mass, all assembled in a perfectly spherical planet or moon would behave like a point mass. The same applies for concentric elliptical homeoids, i.e. a thin shell between two similar ellipsoidal surfaces similarly placed.

Gravity of a thin sphere

The crust of the Earth, at least, has no such consistency. Moreover, the Earth is far from spherical, being more pear shaped and with a dramatic equatorial bulge. This makes it even more improbable that the whole planet Earth behaves just like a point mass.

This research will show how this whole intricate organization of different densities of mass in the planets happened, because of the way the original cloud of cosmic dust was organized, then compressed into matter of varying densities. Tesseral harmonics prove this phenomena of concentric spheres zeroing out, as shown later.

The transformation of mass from a swirling eddy of cosmic dust into spinning planets - first gas giants like Jupiter, then eventually solid planets like Earth or Mars - is a dynamic process. Any theory of cosmology that delves into the fine details must be derived from established dynamical theory.

Rigid Body Dynamics

The Three Body Problem (3BP) of Celestial Mechanics has the generality and versatility to encompass this cosmological process. The 3BP is a well known problem of mathematical physics, with many common, easily verifiable applications in solar system astronomy. In fact, the 3BP is so ubiquitous as to be much more than a mathematical curiosity but practically a physical law.

The difficulty in elevating the 3BP to the status of Law is that the general solution is not known in a closed form. The 3BP can be solved in specific situations such as circular coplanar orbits, but just not as the general 3D solution with random motion for all three bodies.

This research adopts the premise that there is in fact no exact mathematical solution to the general 3BP in closed form. However, the laws of nature act in such a way as to drive dynamical systems into known configurations of the 3BP - at which point the constituent bodies reach a dynamic equilibrium, and thus a stable long term periodicity of motion.

This is nothing more than a restatement of the well known scientific principle that natural systems seek dynamical equilibrium and stability.

The Three Body Problem

The 3BP of Celestial Mechanics is the study of three bodies in space under the influence of their respective gravitational forces. Typically, two of the bodies are much larger than the third, and are situated on an axis rotating at a constant angular rate such that the two large bodies remain on this axis.

Lagrange Points for the Three Body Problem

The center of mass or barycenter for the two large bodies is at the origin of the coordinate system. A common study of the 3BP is the circular coplanar problem, in which m2 is in a circular orbit around m1 (e.g. the sun and Earth), in which case the two bodies in the rotating coordinate system are in circular orbits around barycenter.

The 3BP itself is a study of motion of the third small body. In the simplest case, for the circular coplanar (all motion is in a single plane) 3BP, there are several important features called Lagrange Points, labeled L1 through L5.

The points L1, L2, and L3 are along the m1-m2 axis and they are all unstable equilibrium points. If the third small body is placed at one of these colinear points it remains there, although the slightest perturbation or external force will cause the body to quickly move away, never to return. However, if the small third body is put in a small orbit around one of these points (called a halo orbit) perpendicular to the m1-m2 axis and to the plane of the page, it will remain there indefinitely in a stable orbit.

The L2 Lagrange point has another interesting feature. There is a figure 8 shaped orbit with L2 at the cross point between the two bodies. Note that all of these phenomena exist in the rotating coordinate system - called a "free return" orbit. This closed "free return" loop was used by the Apollo moon missions (e.g. an equilibrium solution for the Earth-moon system) so that if anything happened in transit, the spacecraft would always return to Earth. Apollo 13 lost power on the dark side of the moon, and fortunately the space capsule was in such a "free return" orbit that got them back home - or at least close to Earth - without any further thrusts. Notice that this kind of orbit is the only orbit (other than an orbit very far away from the masses) that is inclusive of both large masses, that is stable.

The Sun-Jupiter-Asteroids Three Body System

Two more Lagrange points, L4 and L5, are at the apex of an equilateral triangle with the two large masses as one side. These are stable equilibrium points - i.e. a small body located at the L4 or L5 point will remain there, even in the absence of perturbations. Moreover, small orbits around these two equilibrium points are also stable. In the sun-Jupiter system, the L4 and L5 points are occupied by small clusters of orbiting asteroids, the Trojan and Apollo Asteroid groups.

The mathematics proving these concepts is straightforward. However, the general 3BP itself has no solution. There are solutions to many other restricted cases - e.g. elliptical coplanar motion - but no solution has yet been found for the 3BP with no restrictions on the motion of the three bodies.

Phasors in a Rotating Coordinate System

Dynamically, particles near L4 and L5 are free to move anywhere in the xy plane. Consequently, the forces must be strong indeed for bounded motion at the equilateral points to exist (i.e. gravity is balanced by centrifugal forces near L4 and L5 so that the sum of forces in the rotating system is zero).

The notion of a rotating coordinate system is not the customary way of looking at things. However, nature is in constant motion and actually it is more irrational to assume that natural laws all operate in a fixed, static reference frame. Rotating coordinates are much more natural. That being the case, then centrifugal forces at L4 and L5 are irrelevant and there must be some real forces acting there to keep objects in orbit there such that L4 and L5 are the foci of elliptical halo orbits (actually, not at the foci, but at the center of the ellipse).

Frequency Domain ~ A Rotating Coordinate System

A rotating coordinate system often used in dynamical system analysis is called the frequency domain. In the complex plane, is a point on the unit circle rotating at a constant rate (=1 for the typical restricted Three Body Problem). The starting angle for motion is called the phase angle (notice the project of the rotating phasor into the frequency domain is a sinusoidal wave). The two equilateral points in the phase domain are represented by phasors. Likewise with the colinear points.

In complex analysis, points to the left of the imaginary axis are stable, those on the axis itself are quasi stable, and those to the right of the imaginary axis are unstable. A more consistent representation is shown below, which uses the reflection principle of the Laplace equation and/or the complex domain. The plot is similar to the three branches or phasors in three phase power systems, with the characteristic out of phase sine waves.

As posed here, the L1, L2, and L3 points would be on the real axis and L2 and L1 would be unstable. The only configuration that satisfies this is the free return orbit that goes through L2 and close to L1 and L3. Notice that a halo orbit at L4 or L5 forms a carrier wave to the respective phasors.

 

Lagrange Points of the Three Body Problem

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© 2004 WH Clark