Fourier Series - in Celestial Mechanics

Gravity is a wave; that much is accepted by all scientists. This short paper shows how gravity, as a wave, acts as a perturbing force to cause the elliptical shape of the orbits of planets around our sun. That is, as the constructive interference between sinusoidal waves. Historically, the body of theory using this kind of method to approximating gravity - using Fourier Series - was common in the late 1800's. Then Poincare "proved" that Fourier Series solutions were unstable, and the whole body of theory has since literally vanished from the scientific corpus.

It has previously been shown that an Invariant Plane can be found for the solar system by which the motion of the nine planets is completely symmetrical. That is, a body in an elliptical orbit exhibits sinusoidal behavior in two respects: (1) in the plane of the orbit (this causes the ellipticity), and (2) perpendicular to the plane of the orbit (this causes the inclination of the orbit with respect to the Invariant Plane). This is simply an application of the well known principle of analytical geometry by which it can be shown that an ellipse is the combination of two sinusoidal wave forms.

The exact position of a planet with respect to this Invariant Plane (henceforth the analysis will pertain to a single satellite orbiting a central body) can therefore be represented by the coordinates (in a rotating coordinate system):

x = a + b*sin(phi + c)
z = d*sin(phi + e)


Where phi is an angle from some reference point, x is the radial distance from the central body, z is the perpendicular displacement relative to the Invariant Plane, c and e are phase angles. This is simply a mathematical representation of the principles stated above. The first equation for x is the radius vector, and this equations corresponds to the well known expression

r = a - ae*cosE

where E is the eccentric anomaly (one could just as easily use cos in the expressions for x and z; the phase angles c and e make it possible to use either cos or sin). The eccentric anomaly is very nearly the same as the angle phi for orbits of small eccentricity. (Observe that this solution attempts to use a Fourier Series solution to the orbit - a solution practiced 100+ years ago by Poincare' et al but discarded because it implied an unstable dynamic system - thus the term we have here is the first term in a Fourier Series, and need only be approximate because other terms will be added for the exact solution. This is not hard to conceptualize; all orbital mechanics assumes gravity of bodies acts as a single point mass. All this series solution implies is that the body is not of uniform density, and that different aspects of density act individually upon an object in orbit; summing to the actual orbit.)

The correspondence to the normal orbital elements is easily visualizeded in this coordinate system.

(1) eccentricity is the ratio of b/a
(2) inclination is the ratio of d/a
(3) argument of periapse is the angle c
(4) longitude of ascending node is the angle e
(5) eccentric anomaly corresponds to phi
(6) semi major axis corresponds to a

Variations in the orbital elements is an important part of orbital mechanics. Unlike in the conventional coordinate system, these are not abstract concepts. They are rather the result of combining sinusoidal wave forms, as in the interference patterns between gravitational waves. Any shape or orientation of an orbit can be made by the combination of sinusoidal waves. That is the basis of Fourier Analysis. It is how the gravitational field of the Earth is modeled, in fact, as terms in a Fourier Series; making it a straightforward process to correlate aspects of the gravitational model to their precise affect upon the orbit of a satellite.

(1) a sine wave in phase with x changes eccentricity
(2) a sine wave in phase with z changes inclination
(3) a sine wave in phase with c causes precession of the periapse
(4) a sine wave in phase with e causes the ascending node to precess

The coordinate system thus rationalizes the actual force of gravity, and its affect upon physical bodies by virtue of a construtive interference between waves.

Recall that in the graphical derivation of the Invariant Plane in a previous paper, the phase angles for many of the planets were the same. This reduces the number of unknowns in the analysis of a many body problem. The results of that paper - showing all the planets with prograde rotation in phase; all the planets with retrograde rotation in phase; and the two sets of planets pi radians out of phase - also is a proof of this hypothesis, as it shows that all nine planets can be represented by the set of equations stated.

Be that as it may, eliminating unknowns is very important in celestial mechanics. The derivation of the Jacobi Integral in the early days of celestial mechanics resulted in literally tens of thousands of new papers in a field that had been in the doldrums. The above coordinate transformation makes it possible to represent a body in orbit using just five variables; whereas all existing systems must use six. This is a significant development. (The transformation of coordinates into the Invariant Plane eliminates the phase angle for x from both equations if all planets are in prograde rotation or all are in retrograde rotation.)

A common way to reduce the number of variables in problems involving three or more bodies is to assume all the bodies move in the same plane. When studying such a problem in the new coordinate system, two variables are eliminated (d and e); this means that a coplanar 4BP in this coordinate system has only 12 unknowns - the same number of unknowns as the coplanar 3BP in conventional coordinates. This means it is now possible to extend our understanding of the 4BP from a handful of papers into many thousands of papers with relative ease. Also, notice that this so called "Clark Transformation" creates a system of variables which are a function of sines and cosines only. Thus, all higher order derivatives exist when they are used in the differential equations of motion.

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© 2002 bill h. clark ii