TAB 1

SMALL ECCENTRICITY ELLIPSE

ABSTRACT

It is desired to study of the Three Body Problem (3BP), in an effort to achieve a general solution in closed form. Clearly this is not possible in an unrestricted sense, so an attempt is made to form a general solution to a slight variation of the restricted circular 3BP. This is accomplished by assuming the circular orbit is a unit circle, as is customary in 3BP analyses, and that the orbit is an ellipse of small eccentricity. A small eccentricity ellipse is closely approximated by an off center circle. The purpose of this small paper is to derive an analytical solution for this type of ellipse. The solution is a Fourier Series in cosines that has a rapid convergence. Solutions of this form, in sines and cosines, are especially amenable to 3BP studies and this is shown in the general unrestricted 3BP.

The Unit Ellipse

In the late 1800’s it was popular to use Fourier Series to represent the orbits of the planets in the Solar System. The technique was suspended by Poincare et al because they believed Fourier Series solutions were unstable, and represented an unstable system. We know much more about the solar system now, enough to have confidence that a seemingly unstable system is likely due to aspects not explained by current theory.

Consider a unit circle inscribed within an ellipse of small eccentricity, with the center of the circle coinciding with a focus of the ellipse. It is easy to see that the variation between the radii varies systematically, if not exactly sinusoidally. In practice, an ellipse of eccentricity .01 varies from the unit circle by a cosine wave of magnitude .01

An ellipse of eccentricity .01 versus the unit circle

The maximum error can be modeled accurately by another cosine curve of half the wavelength, and a magnitude of .00005 And so forth.

An ellipse versus the first term approximation

This is the basis for Fourier Series: any curve can be reproduced exactly by the sum of sine and cosine waves. This is shown above for the case of a small eccentricity ellipse. A similar result can be achieved for a circle in an orbital plane at a small inclination to the original orbit. These two functions are orthogonal and, in fact, could be considered a 3D wave with sinusoidal variations in perpendicular planes similar to the behavior of electromagnetic waves. ( 29 ) Interference with another such wave can be made to vary the orientation of the ellipse and the direction of inclination of the orbital plane. All of these properties, which together comprise the orbital elements, arise from Fourier Series variations of a simple unit circle.

The formal equation is

Where the radius from the focus of the central body is accurate to eight significant digits with just two terms, for an ellipse of eccentricity 0.01 This size ellipse is assumed to be the highest eccentric ellipse for which this series representation is viable.

Mean Motion

There are two ways to consider the ellipse as the sinusoidal perturbation from the unit circle. Both schemes account for the orbital motion difference on an elliptical path versus the unit circle (e.g. the 3BP is in rotating coordinates).

In a system of nested increments of eccentricity, the radius vector is originated at the previous level; at the "empty focus" of the current level, so that the angle it sweeps out is the mean anomaly (accurate for small values of e). The overall path, like an envelope curve ( 27 ) is the sum of all the incremental e-steps.

In the mathematical treatment of an individual increment, the time differential for travel on an ellipse can be corrected by inclination of the orbit (adding sinusoidal wave(s) perpendicular to the orbital plane). Orbital inclination is irrelevant in the circular restricted coplanar 3BP, thus increments of i and e are added simultaneously, to keep the overall system in dynamic equilibrium. This notion is supported by a study of the motion of the planets in our solar system. ( 3 ) Both of these ideas will be developed further in subsequent reports.

Small Eccentricity Ellipse versus Motion of Barycenter

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© 2004 WH Clark