To What Do the Planets Dance?

The motion of the newly-discovered asteroids, Ceres and Pallas, presented to Gauss a renewed opportunity to advance the Platonic method of astrophysical investigation that Kepler had presented to the world nearly 200 years earlier. In the intervening 200 years, the oligarchy, through their Leporellos, Euler and Newton, had perpetrated a hoax, to undo Kepler's accomplishment, not by locking up what Kepler wrote, but by locking up the minds of those who would read it, and not wish to be on the wrong side of popular opinion. Gauss, with his determination of the orbit of Ceres, wielded the power of the universe against those oligarchical mind games, and, like a good commanding general, pursued the enemy, with the investigation into the tiny perturbations of the Ceres and Pallas orbits.

This oligarchical mind control exists to the present day. Last month, Roger Ham forwarded a copy of the Summer Fidelio to one Brian Marsden who specializes in the determination of orbits at the Smithsonian Astrophysical Observatory of Harvard University. Within 3 days time, the professor replied, "I fully agree that Gauss' method of orbit determination has great elegance and power, and we certainly make use of it in our own work, including what we did on 1997 XF11. Nevertheless, that is not the whole story, and it was quite improper of the Fidelio article to criticize Newton and Euler in the way it did...."

Another professor, MIT's Laurence Taff, displays a similar ignorance in his 1985, text "Celestial Mechanics--A Computational Guide for the Practitioner." In the section on the determination of planetary orbits, Taff quotes extensively from Gauss' Theoria Motus. Then, after a brief description, albeit in formal terms, of Gauss measurement of non-linear curvature in the small, Taff states, "Although this picture of the Gaussian process is interesting, it is also irrelevant." Under the heading, "The Historical Myth--Ceres," Taff says, "We do not know how Gauss actually computed the orbit of Ceres or any of the other big four minor planets. We do know that however he did it."

Both men trumpet the superiority of the statistical methods, rejected by Gauss, because those methods are more practical, especially with the use of high speed computers. Their obsession, however, is not driven by the quality of their results, but by the emptiness of their minds. To them, there are those methods of elegance and power, and then there are the not "improper" practical ones.

In this light, let's continue to look at Gauss' investigation of the perturbations of the orbit of Ceres and Pallas, from Gauss' standpoint.

As Kepler, following Plato and Cusa, demonstrated, the solar system is a harmonically-organized process. For the mind to grasp that harmony, one must be able to "hear" the polyphony of interactions among all the heavenly bodies. Kepler presented that polyphony to us, in the form of his three principles of planetary motion, and the musical relationships of the planet's extreme angular velocities.

If, however, when you think of Kepler's solar system, you think of each planet, separately moving about the Sun, you are already on the slippery slope into the Sarpi/Newton abyss. It is but a short step, in this mode of thinking, to translating Kepler's principles, by algebraic manipulation, into a seemingly more practical system of formal equations such as Newton's inverse square law. Will you fall into the trap, that Kepler's principles are of "elegance and power," but Newton's laws say the same thing in a way more suitable for calculation as, Cal Tech's Richard Feynman, who, in his famous "lost lecture," committed the sophistry of pretending to prove "geometrically" that the planets obey Kepler's principles as a consequence of the inverse square law?

Nothing strikes terror into the mind of the formalist, more than the elegance and power of the Socratic method, just as Gauss and Kepler to this day strike terror in the minds of Marsden, Taff, and Feynman. That terror grows stronger when confronted with the physical evidence that the planets don't obey the formalist's rules. If Kepler's and Gauss' work on the larger motions of the planetary orbits were not enough, the whole Newtonian edifice, and the mental states of its adherents is completely thrown into disarray, when the tiny perturbations of the planetary orbits are considered. As Taff admits in the above mentioned text, "In my view, the problem is that when you really need perturbation theory (in celestial mechanics) it does not work very well (as applied in the past). The existing applications of it do not explain the Kirkwood gaps in the asteroid belt. They do not explain the fascinating discoveries in the Saturnian rings/moon system. They do not predict physically correct rates of change of the orbital elements of a satellite revolving about an oblate primary. Some of these failures may be attributable to faulty mathematics, some to the difference between a trajectory and a field theory. The reader is advised that the above is a minority opinion (and taken slightly out of context)."

Taff's uneasiness, once again, comes from his obsessive devotion that has lead him, like the Harvard professor, to turn away from methods of power and elegance, for the more practical ones.

The problem is straight forward. All the heavenly bodies that move about the Sun are moving in Keplerian orbits, and, these orbits, as Kepler demonstrated in the Harmonies, are interacting with each other. The effect of these inter-orbital interactions, is measurable in terms of small variations in the orbital elements that characterize the size, shape, inclination, and position, of the orbits. In other words, the planetary orbits are not fixed tracks in the sky, but regions of motion, in which the planets move, similar to the regions of ambiguity that characterize the tones of the well-tempered system of polyphony. The curves traced out by these motions, therefore, are not the simple curves you draw on a piece of paper, but much more complicated. (The reader can make a representation of this, by drawing a circle with a compass and slowly making the radius slightly larger and smaller as you rotate the compass around the center.) This representation is merely the footprint of the planet's motion. Now think of how this type of motion would be reflected in changes in the orbital elements.

The measurable variations of these orbits occur in very long cycles, but, these cycles are very significant. For example, the precession of the equinoxes is one effect of these perturbations. A longer 100,000 year cycle in the eccentricity of the Earth's orbit, resulting from these orbital interactions, accounts for the cycle of ice ages.

The scientific question posed, is, what is the harmonic relationships that are reflected in these inter-orbital interactions? In the larger planets, as Gauss says, these cycles are so small and slow as to be nearly insensible, over short intervals of time. But, in the orbits of the smaller planets like Ceres and Pallas, these cycles are measurable during shorter time intervals, and in them, the underlying harmonic ordering principle of all the inter-orbital interactions are potentially discoverable.

How do we measure the interactions between orbits?

In an 1819 paper, "The Determination of the Attraction of an Elliptical Ring," Gauss discusses the following principle of physical astronomy:

Changes in the elements of a planet's orbit, that are due to the interaction with another planet, are independent of the position of the disturbing planet. That is, the long range changes in the size, shape, inclination and position of a planet's orbit, that arise from the interaction with another planet, arise from the interaction of the two orbits. Furthermore, Gauss says, that if the total time it takes for both planets to make one complete revolution around the Sun, are incommensurable, then the effect of the disturbing planet on the elements of the disturbed planet, can be calculated using the following metaphor:

Think of the mass of the disturbing planet as spread out in a ring of uniform, but infinitely small thickness, such that the mass is distributed in this ring, proportional to the speed of the planet. That is, since the planet spends more time in the parts of the orbit farther from the Sun, and less time in the parts of the orbit closer to the Sun, and, according to Kepler's area principle, equal times sweep out equal areas, any piece of the elliptical ring, in which the planet spends equal times, will have equal amounts of mass. Or, in other words, the mass of the planet should be thought of spread out non-uniformly, in proportion to, the non-uniform motion of the planet.

The problem Gauss then solved, is how to determine the effect of this elliptical ring, on any other position in space. This lead Gauss to a surprising application of the arithmetic- geometric mean and a deeper insight into modular functions.

For now, however, it were useful to turn again to Friedrich Schiller, whose poem, "The Dance" presents to us the appropriate conceptions for the task.

See how with hovering steps the couple in wavelike motion

Rotates, the foot as with wings hardly is touching the floor.

See I shadows in flight, set free from the weight of the body?

Elves in the moonlight there weaving their vapor-like dance?

As by zephyr 'twere rocked, the nimble smoke in the air flows,

As so gently the skiff pitches on silvery tide,

Hops the intelligent foot to melodic wave of the measure,

Sweet sighing tone of the strings lifts the ethereal limbs.

Now, as would they with might traverse through the chain of the dances,

Swings there a valorous pair right through the thickest of ranks.

Quickly before them rises the path, which vanishes after,

As if a magical hand opens and closes the way.

See! Now vanished from view, in turbulent whirl of confusion

Plunge the elegant form of this permutable world.

No, it hovers rejoicing above, the knot disentangles,

Only with e'er-changing charm rule does establish itself.

Ever destroyed, creation rotating begets itself ever,

And an unspoken law guides the transformative play.

Say, how's it done, that restless renews the swaying formations

And that calmness endures even in moveable form?

Is each a ruler, free, to his inner heart only responding

And in hastening course finds his own singular path?

Wish you to know it? It is the mighty Godhead euphonic

Who into sociable dance settles the frolicking leap,

Who, like Nemesis fair, on the golden rein of the rhythm

Guides the raging desire and the uncivilized tames.

And do the cosmos' harmonies rustle you to no purpose,

Are you not touched by the stream of this exalted refrain,

Not by the spirited pulse, that beats to you from all existence,

Not by the whirl of the dance, which through eternal expanse

Swings illustrious suns in boldly spiraling pathways?

That which you honor in play--measure--in business you flee.

Translation by Marianna Wertz