If we are not to take any single planet as unity, then, we must find some period which is not arbitrary, but which also incorporates every other period into one. In this exercise, the search is for a possible fundamental period of the system as a whole. To begin, let us look at synodic periods, the periods wherein two bodies in different orbits, "catch up" to each other and line up such that the lower planet of the two lies exactly between the higher planet and the sun. A synodic is already a superior order than a simple sidereal period, since it incorporates two such periods. Indeed, we have reason to suspect that it is the synodics, and possibly still-higher periods which produce the ordinary sidereal periods, and not the other way around.
At first glance, we note that there are two basic types of synodic period:
1) 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.
2) 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.
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. Ha! This is 0.61518, 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. As we demonstrate below, this means that the synodics are a lot more than meets the eye at first glance.
A long-term study at the Jet Propulsion Laboratory recently stated that: "as the sun rotates, the same magnetic structures have tended to face the earth every 27.03 days (27 days and 43 minutes) over the past 38 years." I take this to mean that this period corresponds to the earth's solar position synodic, the period during which the earth comes to lie over the same point on the solar "surface." This would give an actual rotation of the sun of 0.06891 earth years. Thus, based on this, we can easily see, based upon the forgoing, that all of the periods of rotation of the solar position synodics, for all of the planets, will be equal to the periods of the planets themselves. For example, the solar position synodic of Earth, is 0.074 Earth years, as above. Since 0.074 is less than half of the period of Earth's year, we are dealing with an internal synodic. As we noted above, the period of revolution of an internal synodic is always the period of the higher body. Thus, the revolution of the Earth's solar position synodic is the same as the period of the earth. Similarly, such is the case for all other planetary bodies in our system. I will show why this may be relevant shortly.
The fundamental period of any individual planet, it seems to me, would be that period during which it went from maximum to minimum gravitational "attraction". In first approximation, of course, this is its orbital period, since, during this period, it goes from being closest to being furthest from the sun. Yet, I think we must also incorporate some other elements. For example, what if we were to take the synodic period for each planet of a) the revolution of its solar position synodic and b) the revolution of its Jovian synodic, clearly the two dominating periods of the planet?
Thus, from the forgoing, were we to consider the fundamental period of a planet to be that synodic which determines its two dominant periods, namely, the revolutions of its solar position synodic and Jovian synodic, we would immediately note that this fundamental period is the same as its Jovian synodic, for all planets up to Jupiter. In the case of Jupiter itself, the resulting period would be the same as its period of revolution about the sun. Thus, we would get:
Mercury----0.24583
Venus------0.64882
Earth-------1.0920641
Mars-------2.2352323
Asteroids-7.5137978
Jupiter-----11.862
These periods are very different than the sidereal periods of revolution, and, yet, would seem to be more fundamental in terms of the actual gravitational "experience" of each individual planet itself. But, look at what happens, beyond Jupiter: For the case of any planet above Jupiter, the revolution of ifs Jovian synodic is it's own period, since it is the higher planet of the two. Thus, were we to follow the same rule, we would get as the corresponding periods for the bodies above Jupiter the following:
Saturn-----0
Uranus----0
Neptune--0
Kuiper Belt-0
Either we must find a different way to incorporate the two periods for the case of planets above Jupiter, or we must accept that there is a flaw in the approach we are taking here.
As an experiment, take this one step further: Take the Saturnian synodics of each planet, since Saturn is, by far, the third dominating influence on each planet after Jupiter and the Sun. Next, we will take that synodic, for each planet, which produces both its Jovian and Saturnian synodics, the synodic of its Jovian and Saturnian synodics. (I'll let people do this experiment themselves. It is a true delight, and teaches us something else about the nature of synodics.)
PROBABLE ORIGIN OF THE SOLAR CYCLE, REVEALED:
If my reading of the JPL study is correct--and I see no other possibility--then we can take the period of rotation of the solar magnetic field to be 0.06891 Earth years. This, then, will be the basis of the next construction. Think back to our Catenary as Metaphor memo. We are going to ask the question: "During which period does the entire system complete one rotation?" To begin, we determine what angular proportion of its complete rotation about the solar axis each body completes during a given arbitrary period of time. Let us take, again, the period of one Earth year, which is easiest. (I actually did this with the sun as unity, but this makes it simpler.) We add the inverse of each body in the system's period with that of all others, which will give us the total angular rotation the system performs in the given period. But, since we are seeking the period during which the whole system performs one complete rotation, we now take the inverse of this sum, to obtain our final figure, which is 0.045088 Earth years.
During this period, the Sun completes 0.6543 of one rotation, while Mercury completes 0.18721 of its orbit, Venus completes 0.07329 of one complete rotation, and so forth. If we add all of the angular motions which every body in the system performs during this period, from the Sun to Pluto, we get one exact rotation of 360 degrees. Let us call this the "Period of Rotation of the System." This is a real period, with real physical existence, not an imaginary one.
For those who don't know this, this is a very remarkable thing, indeed, for, every approximately 11 years, the sun goes through what is called "the solar cycle," where the magnetic polarity of the solar field reverses itself. This period also corresponds to the period during which solar activity and sunspot activity increases to its maximum and, then, decreases to its minimum. Astronomers think not of the 11 year cycle, but of the 22 year period wherein the Sun reverts to its original magnetic polarity. This period varies, and has not yet been nailed down to a precise average. However, let me venture to say that, perhaps, we have found it, for, the inverse of our period of 0.045088 is 22.17885 Earth years! Thus, we might fairly suspect that, "The solar period is the inverse of the period of rotation of the system."
Furthermore, assuming this is true, then the following universal principle would follow (, reasonably assuming that this also applies in other stellar systems): "The Solar Period would be the total number of angular rotations performed by all of the main bodies of the system, during the period of revolution of any of the main bodies, divided by the period of that body in question." Put in mathematical language (which is sometimes a crude language for grunting gorillas and apes), this would be:
P = [pu/S + pu/p1 + pu/p2 + pu/p3 + . . .pu/pn] / [pu] ,
Where P is the solar period, S is the solar rotation period, P1, P2, P3...Pn are successive orbital periods of planetary bodies or asteroid belts of the system, and Pu is the period of the planet we choose as "unity", or the measure of the calculation.
Put differently, and perhaps more lawfully, we could also describe the solar period as, "That period during which the body whose period is taken as unity fulfills all of the complete angular rotations fulfilled by all other bodies during its own period." A subsidiary reflection of this would be that, "The square of the solar period is the number of angular rotations which must be completed by the Sun to fulfill the period." (P^2=P/S).
It would seem that the "balance" of the planets, when more force is located above or below the solar equator, based upon the distribution of the planets during the course of the cycle, causes the solar magnetic polarity to flip over every half-cycle, and return to its original polarity after the completion of the cycle. The reason that the length of the cycle varies, thus, would be that this "balance" never follows a simple pattern, but changes depending on the particular orientations of the planets at any particular time. However, over long periods of time, assuming this hypothesis to be correct, the cycle would average out to be precisely the value which we have found here. (Moreover, it would be possible to predict the length of any particular cycle by taking the particular configuration of the system over a particular interval.)
I believe that we now have some useful "tools" for our next steps ahead.