TAB 4

4BP WITH A BINARY PLANET

It has been shown (1) that an ellipse of small eccentricity (i.e. .01 or less) can be constructed simply as:

Consider a 3BP set up so that the second primary mass (the smaller mass) is a binary planet - a small mass orbiting very close to a much larger mass. If the second mass makes one orbit per revolution (all three primary bodies are in the same orbital plane), this simulates the second term of the equation and the Center of Mass (CM) of the binary planet will move in an ellipse to O(e³). Or, if the larger binary mass is in an elliptical orbit, the action of the second mass of the binary makes the CM move in a circular orbit, to the same accuracy. These neglect the affect of the smaller planet on the barycenter, which parallels the second term of the equation ( 5 ); and the slight variation on mean motion, which is in any event a secular variation that averages to zero for each orbit.

Introducing a fourth mass orbiting the larger binary at 2x per revolution can make the fit to the CM orbit exact to O(e⁴) so that the entire system behaves as a large primary near barycenter and another CM in circular orbit around it. The same stable orbit could be simulated by a small mass orbiting the primary at barycenter 2x/rev, e.g. as an anomaly within the sun compensating for motion.

A more likely scenario would have the smaller planet(s) out of plane, all together simulating an orbit of CM in a plane at a small inclination to the original orbital plane, where the driving influence toward stability is a constant direction of the angular momentum vector, the CM remaining in a fixed orbital plane.

In so doing, placing the small fourth mass near the best orbit would drive it into that best orbit, making the fourth body exist in a stable Lagrange type orbit. Thus a new Lagrange point, or stable point, for the 4BP - actually two such points, as another is at barycenter.

Additional stable points happen at regular intervals as the nominal eccentricity of the large primary binary planet increases (ultimately building up a "system wave" ( 3 ) in a complex natural system). That is, the 4BP is first stabilized as indicated, upon a circular orbit (or consistent orbital plane, as the case may be); then a fifth body is added to compensate for a further increase in eccentricity/inclination - each stage with associated Lagrange points. This follows the method of consecutive CM calculations via a "chain of barycenters" technique, proven in the theory of Celestial Mechanics as a means of computing CM values for many body problems.

With each iteration the large primary mass is in an increasingly elliptical orbit (or out of plane orbit), whereas the small system remains in the same orbital plane, e.g. it's center of mass.

Eventually, as more terms are considered in the equation, the collection of masses becomes more like a single gaseous planetary entity where the gravitational field is comprised of several anomalies - e.g. Jupiter. This analysis has built up a Fourier Series representation of the orbit to match the Fourier Series way that a model of a planet's gravitational field is developed via satellite geodesy.

The solar system has an invariant plane, consisting of the angular momentum vector/plane. Let the primary body be Jupiter with the sun at barycenter. Now each planet is driven to a stable orbit, by mass distribution (including anomalies within the sun itself) to move in the plane perpendicular to the system angular momentum vector, i.e. the invariant plane.

This type of discrete mass distribution would presumably make it difficult for a large planet sized mass to reside at the sun-Jupiter equilibrium point. Hence, the asteroids that orbit the L4 and L5 points, but nothing exists at the Lagrange points themselves.

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© 2004 WH Clark