Thoughts about Orbital Mechanics & Trading Markets

When I was about 15 I developed an interest in mechanics, particularly orbital mechanics. It seems pretty simple, if potentially elaborate: smaller ball orbits bigger ball, perhaps while aforementioned bigger ball orbits still bigger ball. But I was curious about what determines which ball orbits which. We're accustomed to suns thousands of times more massive than their planets, but what about celestial objects of the same order of magnitude? Or very different spatial distribution (e.g., a massive but dense object, versus a partner that happens to be a gas cloud)?

The moment I considered there might exist a tipping point, the answer came to me. Objects don't really orbit their partner—they orbit a common center of mass. If you had two objects of identical mass, the they would be compelled to accelerate around a common point in order to maintain an orbit around each other. This common center of gravity is the barycenter.

The moon has a mass 1/81 that of earth; the barycenter is thus 1/82 x 384,405 Km from the earth's midpoint, or 4,641 km from the earth's midpoint—still inside the earth, as it happens. As a result, the earth wobbles on its axis every 27.3 days (a lunar month is longer because of the earth's own sidereal rotation every 24 hours). If the moon were more massive, or farther away, the wobble would be bigger.

At the time, this insight was not terribly interesting, but decades later I learned about the Three-body problem. The problem is that there's no way of establishing universally what the three orbits (or wobbles) and their corresponding periods are. One can only approach the solution through successive approximations. The reason is that the orbital paths are affected by the angle between any two partners of an orbiting body, like the angle of the earth to the sun (as observed on the moon). If one increases the number of planets to eight, plus asteroids and moons, then the math is massively more demanding. The orbits are slightly different for each and every orbital period. For instance, the barycenter of the solar system moves about over time; it's presently outside of the sun itself.

What makes this interesting to me is the fact that the solar system will never return to its exact configuration, ever. One might imagine that, for any alignment of planets that could exist, the system will eventually return to that alignment: for instance, where all 8 planets line up¹ in an identical quandant of the solar system. But this is not the case: the prior position of the planets slightly alters their orbits for all future time, and the exact position will never recur.

Now, there's another concept known as ergodicity. Consider a conventional mechanical clock. The clock's movement is a trivial example of an ergodic process, because there are (60 x 12=720) possible states for the clock's hands, and each one occurs exactly one 720th of any really long time interval one chooses (i.e., any whole number multiple of 12 hours). One can imagine a magic clock in which the customary cyclical sequence of minutes is rearranged in totally random order, but with the same 1-minute duration for each position every 12 hours. The next day, another totally random, sequence in which 3:45 is followed by 11:12, then 5:22, and so on (but with one minute allocated to each possible configuration).

So far, so boring. Let's have another look at that clock, though. You might think of the clock as having two hands to indicate two distinct values, the hour and the minute; but to be honest, the hands only express one value (but use two hands for easy reading). The mechanical clock has two hands, each capable of taking one of 60 positions on the clock face, but there are not 3600 possible positions—there are are only 720, whether in random order or not. If we were to include impossible positions, like the big hand on 10 and the little hand on 3 and one-fifth, then there would be. But further: supposing the hands moved randomly, independent of each other, and there were literally hundreds of hands?

Supposing you could read all the hands at a glance; reporting the “time” would be difficult all the same. 12:59 AM (the time at which I am typing this) merely means it is the 59th minute of a 1440-minute day; likewise, 9:16 AM = (9 x 60) + 16 = minute 556, and if history had turned out differently it's not difficult to imagine people telling the time like that instead. But if 9:16:32:23:51. . . . meant several hundred different values of “time,” none of which occurred in linear sequence, then you might feel the need to express time as the sum of all those number: 9 + 16 + 32 . . . (formally, as n1:n2:n3...n200)

So let's imagine all those numbers separated by colons can be between 0 and 59, and there are in fact 200 of them. The sum of these numbers could be as small as 0 and as large as 11,800. There are 4.26825 x 10355 possible combinations, but obviously, a lot fewer potential values for the sum. In fact, we will notice that, while we treat each combination as equal in likelihood, not all sums of values are equal. The most likely sums will be the average of 0→59 (viz., 29.5) times 200, or 5900. Values diverging far from that in either direction will occur less frequently, and in fact, be distributed like a Gaussian (normal) distribution.

That's why there are so many uses for Gaussian distributions. It's not just that natural phenonena may sometimes behave that way; it's that the likelihood of a statistical error (by, say, accidentally sampling an unrepresentative population) is distributed normally as well, so clinical trials of medicines, for instance, also assume the familiar "bell curve." In economics, the significance is more important yet. In economics, there's the concept of market equilibrium, in a surprisingly strong form.² When a shock causes the supply of a commodity to dry up, the price is expected to rise sharply, exceeding the equilibrium price, then fall to somewhat less, with each pass getting closer. This process is known as tâtonnement, or “groping.” Each successive pass gets closer to a moving target because high guesses and low guesses represent the long tails of the distribution, and as new information reaches the market, then the market's consensus of the true price of the commodity converges.

Convergence, here, is the outcome of an ergodic process. We can imagine that, as time passes, information about the true market conditions at some specific moment in the future is constantly improving, in the sense that estimates by a diverse group of well-informed traders keep getting closer and closer to some particular value. By way of comparison, estimates of the age of the planet earth have naturally tended to converge as more sophisticated earth sciences emerge. Estimates of what the market-clearing price for benchmark grades of crude oil will be on a particular day, for instance, include a range of plausible guesses made at different times in the past.³ Therefore, any spot price is partly the result of a random distribution of guesses, with quantifiable results than can be “summed.” Ergodicity would lead one to expect stable prices, since extreme variations around a mean would reflect costly outliers; catastrophic events like the outbreak of war between Iraq and the West would usually be predictible well in advance, with pricing effects that would diminish with time.⁴

That's not remotely how things work: not only are commodity prices volatile beyond any industrial explanation, the prices do not tend to regress to any mean: tâtonement is a mythical process. Prices jump around, feeding on the reaction of traders to each other. Even setting aside the psychology of the trading floor, there are arithmetic limits to the reserves of market makers. And this process duplicates itself outside of formal trading venues like the commodities exchanges or forex.

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NOTES

1. On their respective planes of the ecliptic, of course. Most sites make this a huge sticking point, but full radial alignment would theoretically recur every 1.1218 trillion trillion years. The concept I was going for here was the solar system as an enormous clock with eight hands.


2. Hypotheses sometimes have different degrees of “strength,” or predictive force. For example, there are three official versions of the Efficient Markets Hypothesis—strong, medium, and weak—that vary in how fundamentally efficient they claim markets to be. The strongest version of any hypothesis is the hardest to prove and easiest to refute. Neoclassical economics insists on an extremely precise and fast price equilibrium, including in the markets for labor, capital, and natural resources.


3. These include futures contracts that can be for several years duration, as with Southwest Airlines. Many of the forward positions taken are by direct participants in the industry or else major consumers, and reflect complex derivative strategies. Naturally, a refinery or aircraft fleet represents decades of stranded costs, which reflect an inherent assumption of the future price of petroleum made long ago.


4. As for the long-run effect of wholly unexpected political turmoil in certain Middle Eastern countries, the logical impact on price seems to be small: many countries, such as the USA, have immense stockpiles, and the effect of any such crisis could be expected to have a minor effect on world prices after a couple of years. The markets, of course, see things differently.