June, the 8th, 2002
FORWARD
In an unusual development, a small group of associates of American philosopher and economist Lyndon LaRouche demonstrated that what Gauss had called the "complex domain" is acting in decisive ways in our solar system, in ways never widely considered, since the time of Gauss, himself. They offered a devastating demonstration that the orbits and masses of the planets could not be arbitrary, but follow some as-yet undetermined precise principle. Thus, by the release of their findings, they have set off a "race against time," across the world, to see who will be the "next Kepler, " namely, who will complete this discovery, thus giving the world its first newly-discovered astronomical principles since the time of Kepler, 400 years ago. Will it be the adherents to the Keplerian-Leibnizian-LaRouchian scientific method who complete this discovery, or will it be the followers of Newton-Russell, and their school? Will it be LaRouche's close followers, including his scientific collaborators in such places as Russia and Italy, or will it be the stale and boring academic farts at such places as the physics departments of Princeton and Oxford? Such is the question on the table now, and one which shall soon be answered.
As the person who made the initial finds, laid forth below, which triggered this development, the responsibility falls upon me to demonstrate this in the clearest and briefest way possible, such that you, the reader of this memo, might decide, if you so choose, to take this matter up and run with it, as if your place in scientific history depended upon it. Thus, I call upon you to join us in this great endeavor, for very much is at stake in its resolution. You, yourself, could very well be the wise scientist and servant of mankind who solves this great riddle, thus changing the course of scientific history. Thus, I call upon you to redouble your efforts and wish you Godspeed, in studying our experiments, below.
PURPOSE AND EXPLANATION OF THE EXPERIMENTS:
Until now, it has been widely believed by the contemporary scientific world that the relative positions, masses and densities of the planets is not precisely determinable by any exact law. Although Kepler's three laws gave the nature of the individual orbits of the planets, and although his third law indicated an ordering principle acting upon the whole system of planets, yet, most still held that the exact location and mass of any particular planet is largely accidental to the particular conditions of its formation. Thus, there was no reason to believe that Saturn, for example, could not have any other mass or orbit than it does. If Saturn or any other planet had a slightly different orbit or mass, this would not fundamentally change the system, it was thought. Thus, our first step, in this regard, had to demonstrate that there was reason to say that this hypothesis was not correct, namely, that the masses were not arbitrary. It was some work by Russian investigator Alexandr Timofeev which suggested to us an inkling of such a solution, when we first viewed his work. But, he was looking at the system as a static one, comparing the individual masses of the planets to each other, without adequate regard to their motions. We suggested that the planets in the system were somehow "balancing out," into a stable system, and that if any one were moved or removed, the entire system would have to reorient itself, else collapse into chaos. Thus, we could not look at any individual planet or even a group of planets in solitude, but had to look for some over-riding order which was organizing the entire system, from the top down. Nature provides us with an example of such a function, in what is called the "Catenary," or "hanging chain." With the hanging chain, hung from two points at opposite ends, the entire length of the chain orients itself such that the tension between each link is equalized. Thus, the hanging chain forms itself into a stable system, which organizes each of the individual links. If any point along this hanging chain is changed, the entire length, and every link in it, must also change. It is not only that every link along the chain is acting upon all others, but that there is something above them all which is ordering the whole into a stable system. With planetary systems, of course, we are dealing with a far more complex function, involving differing periods, masses, and so forth. Yet, there is a clear similarity in the idea. LaRouche had repeatedly pointed to the catenary as being a superior type of geometry to those which were constructable by simple mathematical rules. The catenary is a complex physical function, with real physical causes, unlike a form which is constructable by simple naive geometrical rules.
Before laying forth the experiments, we demonstrate, below, some of the terms that will be used in them:
1) INCORPORATED ROTATIONS are periods during which two or more bodies complete one complete rotation of 360 degrees about the sun. For example, in the case of Mercury and Venus, we say that, where M is the period of Mercury and V the period of Venus, then 1 / [1/M + 1/V] is the period during which the sum of their motions complete one exact rotation about the sun. During this period, Mercury travels 0.718 of one rotation, while Venus travels 0.282 of one rotation. The sum of the two motions equals one complete rotation. Taking the period of a planet or body pu as unity, such that its period is one, incorporated rotations equal the inverse of the sum of all inverse periods of all bodies incorporated into the rotation. This takes the mathematical form:
R = 1 / [pu/p1 + pu/p2 + pu/p3 + ... pu/pn], where R is the incorporated period of rotation, pu is the period of the planet taken as unity, and p1, p2, p3 ...pn are successive periods of ascending planets, such that pn is the last of the series.
2) SYNODIC PERIODS are periods involving two orbiting bodies during which the lower body "catches up" with the higher body, such that it lies exactly between the higher body and the sun. Mathematically, this takes the form:
S= 1 / [ 1/p1 - 1/p2], where p1 and p2 are the respective orbital periods of the two bodies involved. At first glance, we note that there are two basic types of synodic period:
a) In cases where the higher planet has a period more than twice that of the lower, the synodic period is greater than the period of the lower planet and is lesser than that of the higher. Let us call this an "internal synodic," since the period is internal to the interval of the two periods that result from it. If we call an ascending adjacent synodic the synodic period of a planetary body or asteroid belt with that body or belt directly above it, then we note that five out of nine known possible adjacent ascending synodics in our system are of such a type.
b) We also have synodic periods, like that of Earth with Venus, where the period of the higher planet is less than twice the period of the lower. In such cases, the synodic period is greater than the periods of both planets which result from it. Let us call this an "external synodic," since it lies outside the interval that it produces. Four out of nine known adjacent ascending synodics in our system, namely, those of Venus, Earth, Uranus and Neptune, are of this type.
3) PREDECESSION OF A SYNODIC PERIOD: We also must consider the rate at which the position of a synodic, itself revolves about the sun. For example, during Mercury's ascending synodic, Mercury travels about its orbit 1.64337 times, while Venus travels about its orbit 0.64337 times. The decimal 0.64337, for both planets, is that portion of the total revolution about the sun that the synodic itself completes during the period. Thus, the inverse of this, or 1.5543 is the number of times this synodic period will have to complete itself to fill in one complete rotation about the sun. Multiply this by the period of the synodic (1.5543 x 0.39578, in this case) to get the period during which the synodic "orbits" the sun. This turns out to be the period of the higher planet. Indeed, for all internal synodics, the period of revolution of the synodic period, itself, will always be the period of the higher planet. For external synodics, it is a bit more complex.
4) INCORPORATED SYNODIC PERIODS we define as the inverse of the subtraction of all successive inverse periods incorporated into the incorporated synodic, or:
1 / [1/p1 - 1/p2 - 1/p3 - ... 1/pn] , where p1, p2. . . pn are successive periods, such that pn is the last one to be incorporated. This is not the period wherein a group of planets all line up, but is a type of inversion of an incorporated rotation. As for the predecession of an incorporated synodic, there seems to be more than one possibility, the best of which I am not certain.
5) FULFILLMENT OF MASS is the key conception in the organization of the experiments, and has much similarity to Kepler's use of "area swept out" to determine the order of individual orbits. We take the mass of a planet, such as the Earth, and say that, during Earth's orbital period about the sun, it "fulfills" this mass. Thus, we take the total mass, not as an object, or "point mass" but as a reflection of some process going on in a higher domain. Take the case of Venus, which has an orbital period of 0.61518 that of Earth, and a mass 0.815 times that of Earth. During the period of Earth, what mass does Venus fulfill? We take the period of Earth, and divide it by that of Venus, to determine how many times Venus completes its own orbit during the period of Earth (1.6255 times), and multiply Venus' mass (0.815) times this:
[Period of Earth / Period of Venus ] x [Mass of Venus]. Thus, during the period of Earth, Venus "fulfills" 1.3247 times the mass of the Earth. (For all of our experiments, below, we are taking Earth's mass as one, or "unity.")
EXPERIMENT SERIES ONE: Since we are dealing with groups of planets, let us begin with the most basic interval, the relationship of the first planet, Mercury, to Venus. There are two basic incorporated periods for a pair of planets, namely their synodic period and their incorporated rotation.
a) To begin, what mass do they fulfill during the period of Venus? If we divide the period of Venus by that of Mercury, we get the number of rotations Mercury fulfills during the period of Venus (2.5543). We multiply this by Mercury's mass, to get 0.14118, then, add Venus's mass of 0.815 which it fulfills once during its own period, for a final sum of 0.95618. This is the mass the two planets fulfill during Venus' period.
b) In the case of their synodic period, which is 0.39578, what mass do the two planets fulfill in unison? Mercury fulfills 0.090835, while Venus fulfills 0.52434. The sum of the two masses is 0.61518, which is exactly the period of Venus! Were either of the two masses or periods different than they are, this would not be.
c) Next, take the incorporated rotation of the two (0.17308) and the sum of mass fulfilled by the two: (0.039722 + 0.22929 = 0.269012). We will make use of this later.
d) And, since, as we showed above, the predecession of an internal synodic, is the same as the period of the higher planet, we now ask what mass the two fulfill during the period of 0.61518, which is both the period of Venus and the period of the predecession of the synodic. This is the same as in experiment 1-a, of course. Thus, we have three different masses fulfilled, namely, those fulfilled during the synodic period, the incorporated rotational period, and the predecession of the synodic period.
e) Let us now compare these three masses. Divide that mass fulfilled by this latter period by that fulfilled during the incorporated rotation and synodic respectively. This gives us 3.5544 and 1.5543, respectively. Thus, these are in some sort of synodic relation, since they contain the same decimal.
f) Let us also ask, during which period do the two planets fulfill the sum of their own masses? If the sum of their mass is 0.870274, then the period they need to fulfill this mass is 0.5599221. If we divide this by the period during which Mercury fulfils its own mass (Since there are no planets beneath Mercury, this is a series.), we obtain 2.3248, which is exactly the mass fulfilled by Venus and Earth during the period of Earth!
EXPERIMENT SERIES TWO: This brings us to the next planet up, namely, the Earth. Now we have three bodies to deal with, and a far more complex picture. To begin, let us note that, since the synodic of Venus and Earth is an external one, the predecession of its synodic will not be the period of Earth. The synodic is 1.59869, while its predecession is 2.6703. However, we have an additional element, namely the incorporated synodic, (which incorporates Mercury, Venus and Earth), which is 0.65507. We also have the incorporated rotation of Earth with Venus (0.38088) and with Venus and Mercury (0.14754).
a) Let us begin by establishing how much mass all three planets fulfill during Earth's own year: This is 2.5543. This is exactly the ratio of the period Venus divided by that of Mercury!
c) And, if the mass fulfilled by Mercury and Venus, during their synodic, is 0.61518, during which period does the group of Mercury, Venus and Earth fulfill this same mass? We take this mass of 0.61518 and divide it by the mass fulfilled by the three bodies per Earth year (2.5543). The result is 0.24084, which is exactly the period of Mercury! Thus, The period during which the first three planets fulfil that mass which is fulfilled by the first two planets during their synodic, is exactly equal to the period of the first planet! . Thus, the three planets are organizing themselves into an organized system. Were any of the values for the masses or the periods for any of the three planets to be even slightly different than they are, then this would not be.
d) Let us take the sum of masses fulfilled by Earth and Venus during their synodic: (2.1179 + 1.59869 = 3.7166) and compare it to the sum of masses fulfilled by Venus and Mercury during their incorporated rotation (0.039722 + 0.22929 = 0.26902). Indeed, the two sums are almost exactly inverses of each other, such that their product is the mass of the Earth. (Note that since we are using the value of 1.0000174 for Earth's period, we must take this discrepancy into account.)
CONCLUSION
There is much more to say than we have laid forth here, including many things which are as important as anything we have so far said. For example, when we bring the matter further, to the orbit of Mars and Ceres, we find other such things. I suspect there may be a fulcrum effect at Jupiter. However, since we have already made the point sufficiently, let us conclude with a quote by our teacher. In a recent paper, The Rules by Which Games are Played [EIR, May 17, 2002], Lyndon LaRouche stated the matter thusly:
"Kepler's revolution in science, based explicitly on the foundations provided Plato, Cusa, Pacioli, and Leonardo da Vinci, typifies the great achievements of modern European civilizations' physical science and technology. Unfortunately, that great revolution has never been completed. Venice-centered forces struck back, beginning their role in the Fall of Constantinople, as their Forth Crusade earlier, drowning Europe in Venice-orchestrated religious warfare, during the interval 1511-1648, launching a Romantic revival of the twin, rival forms of Aristoteleanism and the empiricism of Paulo Sarpi and Sarpi's lackey Galileo...."
But, now, it is we who are going to strike back. With our determined minds fixed upon the eternity, let us set forth to complete this great revolution, for the sake of mankind, and for the glory of Him who is our Maker.
APPENDIX: TABLE OF DATA
The table below shows, for each planet: a) orbital period, b) mass, where Earth's mass is one, c) mass fullfilled per Earth year, d) axis, where Earth's is one, e) area swept out per Earth year, in square AU's, and f) ellipsoidal volume of the whole orbit swept out per earth year. That is, we take the ellipsoid (solid) corresponding to the ellipse (plane) of the planet, calculate its volume, and multiply the inverse of the planet's period in Earth years, with the total volume of its ellipsoid. The last two values, may be slightly off, since I have calculated them according to less precise data. These two values will, in most cases, approximate the values of circles or spheres with the same radius as the semi-major axis of the ellipse. The other values are taken from NASA. For the two asteroid belts, we take Ceres and Pluto as being typical of their belts as a whole. Also note that I am in need of more precise values for Ceres. The reason that the mass for the two asteroid belts is left blank is that we can only guess it, until such time that the discovery of the principle allows us to project it precisely. Nonetheless, in each case, this mass is so small as to be nearly negligible.