TAB 15

TRANSITIONING BETWEEN COORDINATE PLANES

It has been established that orbits are made up of summed sinusoidal waves. Now there is an hypothesis that two levels of orbits summed this way intersect at an angle of ninety degrees, orthogonal. The problem then becomes how to coordinate these two systems - i.e. the focus of orbits for physical bodies in a dynamically stable system must coincide.

This section has some sketches drawing attention to the fact that the mean anomaly is equal to the angle from the empty focus for small eccentricity orbits. Transitioning from one sinusoidal term to the next for high eccentricity orbits, the actual focus is the empty focus of the previous term, in between there being an increment of inclination added to make this geometrically correct. This inclination term is the primary step function in the symmetric plane, where ellipses remain centered at the center of the ellipse via 1/r forces.

An Eccentric Anomaly for the Four Body Problem

This kind of system is amenable to the use of quaternions, and in some cases octurnions.

Now look a little more closely at the figure, to see how the Earth to Mars trajectory can be modeled as a 3BP. The figure below shows how versus the offset angle it is possible to approximate the third body forces on a spacecraft on the heliocentric ellipse to a single quantity, the center of mass of the Earth-Mars bodies. So whenever the position of the spacecraft is known, you also know the forces from Earth and Mars acting upon the spacecraft. Notice that the center of mass is on an elliptical orbit with center at the center of the ellipse, indicative of a 1/r force system.

 

Dynamical Relationships of a Four Body Problem