(This article was published in the January 2012 issue of “Amateur Astronomer”; for the author, this is a wonderful New Year’s gift!)
In the seventh issue of Amateur Astronomer last year, we saw some symmetrical and beautiful periodic orbits presented by the N-body problem, which reflects the harmonious and orderly side of the N-body problem. However, this is only the tip of the iceberg. As the author has mentioned, the essence of the N-body problem is chaotic and disordered—in layman’s terms, it is extremely messy and cannot be accurately described by mathematical equations. This might seem like an imperfection. But imagine, if Galileo had seen a perfectly smooth lunar surface as imagined when he first pointed his telescope at the moon, would he still have marveled at the wonders of the universe?
In this article, let us take a deeper look into the research history of the N-body problem.
The Era of Observation and Fitting
Due to human self-importance and the experience of the sun, moon, and stars rising in the east and setting in the west, we long believed that the Earth was the center of the universe. The first person to systematically propose the geocentric model was the astronomer Eudoxus (who died around 347 BC), but his model was so crude that it could not explain many basic phenomena, such as accurately predicting solar eclipses or explaining planetary retrograde motion. However, Aristotle accepted the geocentric theory, and due to his political and scientific authority, the geocentric theory was saved from an early demise. Later, Ptolemy perfected the geocentric theory through his epicycles, allowing it to persist until the 16th century.
Many people have heard of Ptolemy’s epicycles but may not understand the underlying principle. In fact, Ptolemy’s theory was a very “compromising” and practical one. Since the traditional geocentric theory held that planets moved in circular orbits around the Earth, it could not explain retrograde motion. Ptolemy came up with a “clever solution”: he believed that a planet moved in uniform circular motion around a point (the epicycle), and it was this point that moved in a circle around the Earth (imagine the motion of the Moon as seen from the Sun). Through precise settings, planetary retrograde motion could be explained. Although it became much more accurate after improvements, there were still discrepancies with the data of the time, so Ptolemy added new, smaller epicycles onto the original ones. Ultimately, by adjusting more than 40 parameters, Ptolemy’s model finally matched the observational data of the era. It is important to note that Ptolemy was by no means mathematically sloppy; his model was quite precise. Later research discovered that if his epicycles were described in modern mathematical language, the result would be the Fourier series found in modern advanced mathematics!
As time passed and observational data became more precise, simply correcting the geocentric theory became increasingly difficult. In the 16th century, Copernicus proposed that the Sun was the center of the universe and that the Earth and the five known planets revolved around it. At that time, Copernicus’s theory seemed so simple because he only needed 15 parameters to explain historical astronomical phenomena. With Tycho Brahe, the “geocentric theory” made its final “struggle”; Tycho envisioned the Sun revolving around the Earth, while the five planets revolved around the Sun. This was essentially no different from the heliocentric theory, except that Tycho refused to give up the Earth’s unique position. Later, using Tycho’s observational data, Kepler summarized the three laws of planetary motion, which described the motion of planets in the solar system very accurately (at least for his time) using only 11 parameters.
Is the Solar System Stable?
Whether it was the geocentric theory, the heliocentric theory, or Kepler’s three laws, they all used mathematical formulas to fit observational data. This cannot be considered a rigorous analysis of celestial motion using mathematics combined with mechanics. The “first person” of celestial mechanics was Newton. Newton had two powerful tools: the law of universal gravitation and calculus. Using these, Newton rigorously derived Kepler’s three laws mathematically, thereby completely solving the two-body problem. However, Newton already realized that when N is greater than 2, the problem is exceptionally difficult, so much so that the three-body problem gave him headaches. Newton modestly stated: “...unless I am much mistaken, it exceeds the force of human wit to consider so many causes of motion at the same time, and to define the motions by exact laws which allow of an easy calculation.”
Although the N-body problem is difficult, our solar system is a real multi-body system. To accurately describe the trajectories of planets and satellites, considering only the two-body problem is clearly insufficient. Since we have no way to solve the N-body problem yet, we can take a “compromising approach” by considering corrections to the orbits of the two-body problem. Because 99.8% of the mass of our solar system is concentrated in the Sun, the Sun is our primary source of gravity. Other large bodies, such as Jupiter and Saturn, although they exert some influence, are not that large and can be regarded as “perturbations.” Thus, the problem becomes studying how a small body moving around a massive body is “perturbed” by the gravity of a third body and deviates from its conic section orbit. This developed into “perturbation theory.” This method is not only applied in celestial mechanics but has also permeated many disciplines, such as being called “perturbation theory” in certain electronic fields.
Before the 1880s, perturbation theory was one of the cores of celestial mechanics. Many famous mathematicians and mechanicians made outstanding contributions to it, such as Euler, Lagrange, Jacobi, Laplace, Poisson, and Haret. During this period, one of the most famous questions was: is our solar system a stable system? Of course, the stability discussed here is not about whether the Sun will explode, but rather treating the celestial bodies in the solar system as point masses and discussing whether they will collide or drift to infinite distances under mutual gravitational influence. It helps us answer questions like “Will Mars or Venus collide with the Earth?” If such an event were truly to happen, we could predict it in advance.
In 1773, the mathematician Laplace made the first prediction regarding the stability of the solar system: in the first-order power function approximation of eccentricity, the semi-major axes of the planets do not contain secular terms. What do these technical terms mean? Simply put, the higher the order of the power function approximation, the more accurate the description of the orbit (the two-body problem can be seen as 0-th order), and the appearance of secular terms means that the solar system might be unstable. Clearly, Laplace’s result gave us a “reassurance”: at least in his prediction, the solar system is stable. From 1774 to 1776, Lagrange generalized this result, and his findings also indicated that the solar system is stable. In 1809, the French mathematician Poisson went a step further, proving that in the second-order approximation, the semi-major axes of the planets also do not contain secular terms. It seemed we could rest assured that the solar system was very stable.
However, do not forget that these were all results of “perturbation theory,” which is to say, approximate answers. In 1877, the Romanian mathematician Spiru Haret proved that in the third-order approximation, secular terms appeared in the semi-major axes of planetary orbits! This clearly led to the opposite conclusion: the solar system is not necessarily stable! At this point, the essence of the N-body problem began to reveal itself. Next, we will see the epoch-making contribution made by a great French mathematician.
The Birth of Chaos
To attract more people to participate in solving the N-body problem, in 1885, an announcement appeared in the seventh volume of the Swedish mathematical journal Acta Mathematica: to celebrate the 60th birthday of King Oscar II of Sweden and Norway in 1889, Acta Mathematica would hold a mathematical competition with a prize of 2,500 kronor and a gold medal. There were four problems, the first of which was to find all solutions to the N-body problem. This competition caused a sensation at the time. Although the prize money was not high, such high honor was rare; one must remember that the more famous Nobel Prize was not established until 1896. However, due to the difficulty of the problem, most mathematicians who were initially eager eventually withdrew. In the end, only four or five mathematicians actually submitted their papers. The winner was not hard to choose; although no one had completely solved any of the problems, all the judges agreed that one submission made a critical contribution to the solution of the N-body problem and that the prize should be awarded to this mathematician. This winner was the French mathematician and physicist Henri Poincaré.
Someone once said:
“Relativity eliminated the illusion of absolute space and time; quantum mechanics eliminated the Newtonian dream of a controllable measurement process; and chaos eliminated the Laplacian illusion of deterministic predictability.”
Undoubtedly, this is a very high evaluation of Poincaré. So, what outstanding contribution did Poincaré make to deserve such praise? It was because he discovered the phenomenon of chaos in differential equations and found that the solutions to almost all differential equations are chaotic. What does this mean? He actually told us: our world is almost entirely chaotic!
The word “chaos” has been repeated many times here. The butterfly effect we are familiar with also tells us something about chaos. If any readers are loyal fans of Liu Cixin’s The Three-Body Problem, I think your impression of chaos will be even deeper. Now let us understand chaos a bit more deeply.
Readers who have studied calculus know that through Newton’s calculus, given the initial velocity and position of a particle, we can describe its motion at any time (differential equation theory); this is the so-called “determinism.” The problem is that measurement has precision limits. Beyond that precision, we do not know the specific details. If the precision is 0.01 m/s, we can control the initial velocity at v=6.82 m/s, but in reality, the initial velocity might be 6.821 m/s or 6.822 m/s. How much impact will this 0.001 m/s difference have on the future motion of the object? If the impact is small, we say the motion is predictable; if this tiny difference can cause a “world of difference” in the future, then the motion is unpredictable, which is chaos. This is like the ancient saying, “A miss is as good as a mile.” At the same time, we cannot repeat this motion because repeating it requires absolutely precise correspondence of initial velocity and position, which we cannot achieve!
Of course, this does not mean Newton’s theory is wrong. Given the initial velocity and position of a particle, we indeed can know its motion at any time; it is just that we can never precisely know the initial conditions. Through measurement, we might find two velocities are the same, but in fact, there are tiny unmeasurable differences that lead to future unpredictability. And as it happens, the N-body problem is such a system; when N > 2, chaos appears.
A folk song vividly illustrates this theory:
For want of a nail, the shoe was lost;
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, the battle was lost;
For want of a battle, the kingdom was lost.
It turns out the fall of an empire was all because of a missing nail!
Our World is So Messy
Mathematicians and astronomers before Poincaré used quantitative methods to study the motion of objects—that is, to determine the motion of an object after a certain period, you had to calculate the motion from now until that moment. Poincaré’s discovery of chaos was key because he opened a path completely different from his predecessors: using “qualitative” methods to study motion. This means I only need to know the “trend” of the object’s future motion, rather than exactly which point it moves to. For example, if I want to know whether the product of two integers is odd or even, I don’t need to calculate the result; I only need to look at the last digits.
Poincaré also found that the solutions to almost all differential equations are chaotic, which means our entire world is in chaos. The most obvious chaos is the movement of the Earth’s atmosphere. We know that accurately predicting the weather is extremely difficult because the unpredictability of atmospheric motion makes it hard to know for sure whether it will be rainy or sunny in a few days. Similarly, earthquakes, volcanoes, biological evolution and reproduction, and human thought patterns all carry great uncertainty; these are all examples of chaos.
But we do live in a multi-body system. Why don’t we feel the chaos of the N-body problem? Why do the planets in the solar system still move in orderly, nearly elliptical orbits around the Sun? Computer simulations show that the early solar system was very chaotic. After long-term evolution, those bodies with too high eccentricity either flew out of the solar system or collided with other bodies, eventually leaving only those in nearly circular orbits. Another reason is that the stability of the N-body problem is built on “differences in magnitude.” Taking the three-body problem as an example, there are only two ways for a three-body system to exist stably: one is for the mass of the third body to be many orders of magnitude smaller than the first two, and the other is for the distance of the third body to be many orders of magnitude larger than the first two, so that its gravitational pull is so small it can almost be ignored. Such problems are called “restricted three-body problems,” which Poincaré was the first to study. The mass of the solar system is concentrated in the Sun, which basically satisfies the magnitude difference in mass, so it is not surprising that it can exist stably for a long time.
Now back to that interesting question: is the solar system stable? We now have a definitive probabilistic answer (99%) to this centuries-old question. In the next 50 million years, no planet will go out of control. When the Sun faces its end 6 billion years from now, the planets will indeed show unusual trajectories. The remaining 1% probability refers to the possibility that Mercury’s orbit will go out of control, which could truly lead to a devastating direct collision between the Earth and Mars. But that is something hundreds of millions of years away.
Conclusion
The story of the N-body problem ends here for now. With the advent of the computer age, the work of celestial mechanics has gradually faded from the view of amateur astronomers and been handed over to computers. Therefore, revisiting that history of celestial mechanics not only allows us to understand the development process but, more importantly, to grasp the changes in thinking patterns. Astronomy has never been just a science; it is also a philosophy. Poincaré’s landmark results heralded a new era, bringing not only progress in mathematics and physics but also a brand-new way of thinking about the world. Just like the N-body problem, the world is chaotic and unpredictable—is our life not the same?
References
A History of the Circle: Mathematical Reasoning and Physical Reasoning
Celestial Encounters: The Origins of Chaos and Stability
Science Squirrel Garden: Is the Solar System Stable?
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