General Keplerian Dynamics (GKD). 2nd Edition

Abstract

Newton generalized Kepler's laws of planetary motion when he developed his laws of universal gravitation. Modifications to Newton's generalizations are submitted which offer a novel solution forthe galaxy rotation problem and a unified model ofthe Universe.Observable evidence and experimental predictions are also submitted which can prove the unified model.

Introduction

This paper will outline the basics of General Keplerian Dynamics (GKD) and discloses predictions which can prove the Lorentz-Mandelbrot Fractal Electrodynamic Astronomical Model (FLAME) illustrated below. To the author's knowlege, FLAME is the only model that can either be proven or disproven experimentally (unlike string theory, M-theory, etcetera). I leave it up to the experimental physicists to test the novel model.

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FIG. 1: The Lorentz-Mandelbrot Fractal Electrodynamic Astronomical Model (FLAME). The grey frame 1 zooms out to the black frame 2, which zooms out to the magenta frame 3, etcetera.

In honor of Einstein, let's assume there is an inertial reference frame RF(A) established for a train at rest and the position of a point P on the train is located at the origin of RF(A). While the position of P is static at the origin of RF(A), the velocity of P relative to a reference frame established for the Milky Way RF(B) is « 792,000 kph. The velocity of P is 0 when its position is certain in RF(A), dinate its exact position after an arbitrary amount of time in RF(B) to be determined with a wave function (analogous to Heisenberg's uncertainty principle). The angular momenta of celestial systems are therefore dependent upon the volume ofthe inertial coordinate system. An excellent resource for scale dependent celestial mechanics is Laurent Nottale's theory of Scale Relativity¹.

A temporal phenomenon also emerges from the scale dependency of angular momentum since, according to the Lorentz transformation equation:

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time is also dependent upon the volume ofthe inertial coordinate system. This would explain the incompatibility between general relativity and quanutm mechanics since angular momentum, time, and therefore gravity, are all scale dependent. This can be tested experimentally via the FLAME model in FIG. 1. According to special relativity, the velocity of light in a vaccum c is constant independent upon the scale of an inertial frame of reference. If time is scale dependent, and c is not scale dependent, then the effects of quantum entaglement will be be limited to c2 relative to our frame of reference (the fractal radiation of frame 1 in FIG. 1 would propagate at c2 relative to frame 6 and the radiation of frame 11 would be observed as cosmic microwave background radiation). Recursive fractal radiation can also explain zero point energy / dark energy and fractal matter can explain dark matter.

Of the four experimental predictions that will be disclosed in this paper, this is the most important, and fortunately we are on the brink oftesting quantum entanglement effects at scales comperable to prove or disprove FLAME².

A generalized 1st law of planetary motion (GKD1) will also be submitted which synthesizes the conic path of a secondary with its orbital inclination via a non-inertial toroidal reference frame embedded within an inertial Cartesian coordinate system.

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FIG. 2: Image is exaggerated. The center of mass (COM) is not positioned at one of two possible foci, but at the origin of the toroidal cavity (represented by the “x”). The torus can be constructed from an auxiliary circle with a radius equivalent to the secondary's apoapsis distance and a minor auxiliary circle with a radius equivalent to the distance of its periapsis (relative to the COM). The radius ofthe coplanar orbital axis (black) is equivalent to the secondary's average true anomaly distance (equivalent to the semi-minor axis of a Keplerian ellipse). The yellow and brown dots represent a primary and secondary respectively. The blue and green paths are Villarceau circles (example orbital paths ofa secondary).

GKD1 can be stated as: A secondary's geodesic is embedded within a degenerative fractal horn torus with the center of mass at the origin of the toroidal cavity.

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FIG. 3: An example half section ofa degenerative fractal horn torus. Circular, elliptical, parabolic, and hyperbolic trajectories are embedded within each quadrant of the torus along the z-axis when the coplanar orbital axis is oriented along the x and y plane.

It will be shown within the generalized 2nd law of planetary motion (GKD2) that the orbit of a secondary has three primary axes of rotation. I refer to the three axes as the major, minor, and torque axes.

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FIG. 4: Image is exaggerated. The major axis is represented by the blue vector, the minor axis is represented by the red vector, and the green pseudovector represents the torque axis.

Even though an ellipse and a Villarceau circle are geometrically equivalent relative to a two dimensional reference frame there is a subtle difference between the two in three dimensions. While an ellipse and a Villarceau circle both trace the oscillatory cycle of a secondary's apoapsis and periapsis relative to the COM, a Villarceau circle also traces an additional cycle which is perpendicular to the coplanar orbital axis. The additional cycle contains two reference points that I refer to as the crest and trough, in which the crest is perpendicular to the coplanar axis in the northernmost polar direction. The distance between the crest or trough from the coplanar axis will be referred to as the orbital amplitude.

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FIG 5: Image is exaggerated. An illustration of the orbital amplitude and the poloidal points of reference.

GKD3 will then show there are two periods that must be considered in Newton's version of Kepler's 3rd law instead of only one period. Villarceau trajectories result from a 1:1 ratio between a secondary's major and minor periods (which I refer to as its polarfrequency). The introduction of the polarfrequency term procures orbital paths that are not limited to conic sections. Geological evidence131 indicates a 4:1 polar frequency for our solar system's orbit.

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FIG. 6: Stellar orbital paths relative to galactic nuclei are not limited to conic sections according to the formalism of GKD3.

A Generalized 2nd Law of Planetary Motion (GKD2)

Kepler's second law of planetary motion can be stated as: A line joining a planet and the Sun sweeps out equal areas during equal intervals oftime.

GKD2(a) can be stated as: The radii of a secondary's major (r1), minor (r2), and torque (r3) axes, joining the major, minor, and torque points respectively, individually sweep out equal sectors during equal intervals oftime (GKD2(a) will be expanded upon).

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FIG. 7: Image is exaggerated. The radii ri and r2 are overlapping since the planet is at perihelion. The radius ofthe torque axis (r3) is miniscule. The planet orbits a perpendicularfractal torus along the minor coplanar axis which is observed as precession (represented by the torque axis pseudovector in FIG. 4).

GKD2(b) can be stated as: The radius of a secondary's major axis (r1) is equivalent to the semi-minor axis of its Keplerian ellipse when unperturbed by outside forces, or

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where b is the semi-minor axis, a is the semi-major axis, and e is the eccentricity of the ellipse. The radius (r1) is also equivalent to the geometric mean of the secondary's apoapsis

and periapsis distances relative to the major point (barycenter), or

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GKD2(c) can be stated as: The angle ofthe secondary's toroidal major axis, relative to any other plane of reference, can be determined from the angle of r1 relative to the plane of reference when the secondary is at crest or trough (when r2 is perpendicular to r1 and the hypotenuse is joined by the major and torque points).

GKD2(d) can be stated as: The relationship between the major axis revolution period, the minor axis revolution period, and the polarfrequency of a secondary's orbit is:

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