What really is the three-body problem?

Problem 3 body It is a new science fiction series from Netflix, one of the most anticipated series of the year, and is based on the novel of the same name by Chinese writer Liu Cixin. The title refers to one of the most well-known and discussed issues in celestial mechanics, the discipline that studies the motion of planets, satellites, asteroids, and other bodies using a mathematical physics approach. The actual “three-body problem” is central to the book's story and also has a certain significance in the TV series, which nonetheless took some license by partially distancing itself from the events imagined by Liu Cixin.

From here on out, a small spoiler is necessary to contextualize the explanation that follows, and does not reveal things that cannot be inferred from the title of the book or TV series itself: those who don't really want to know anything about it can skip the next two paragraphs.

In the novel
The book moves between the past, present, and future, and narrates humanity's contact with an alien civilization living in a solar system different from our own. One of the protagonists of the story discovers what appears to be a video game in which one must try to survive in a civilization living on a certain planet, where “eras of order” follow one another as day and night alternate and day and night alternate. The seasons are regular and the periods there are “chaos ages”, where the seasons and days are different and highly irregular periods. In “Ages of Order” civilization flourishes and develops, while in “Ages of Chaos” it is wiped out due to the unlivable conditions the planet ends up in.

Video game simulations continue throughout much of the book, while in the meantime other things happen at different narrative levels and eras that help us understand what this game is really about. It therefore becomes clear that the difficulty lies in the fact that the world of the video game is located in a system of three stars, which interact with each other gravitationally and in this way also affect the orbit of the planet, which alternates more stable phases with chaotic ones, with very long and cold winters. , days with moments of light that last a few moments or last a long time, mark the end (or rather the suspension) of a large part of life.

The three-body problem
The three-body problem in particular is that it is difficult to accurately predict the way the three celestial bodies move and the evolution of the system they form. It's a question that has fascinated mathematicians, astronomers and astrophysicists for centuries, and which, contrary to popular belief, is not always unsolvable. By deepening it, over the years it has become possible to better understand how the orbits of celestial bodies work and, above all, to understand how to exploit them, for example, managing interplanetary space missions using probes created to explore our solar system.

As we learned in school, the Sun is by far the largest body in our solar system, and due to its ability to attract other bodies (Gravity interactionIn the sense of classical physics, or simply what we call “gravitational force”) strongly depends on mass, the Sun was able to hold on to the planets (in cosmic terms) and make them revolve around itself. The planets also produce gravitational interactions with each other, but these interactions are all negligible because their mass is incomparable with the mass of the Sun (the latter's mass, for example, is more than 330,000 times that of the Earth).

At the end of the seventeenth century, the English mathematician and physicist Isaac Newton appeared He formalized it Equations of motion and gravity of celestial bodies, so that we can calculate in principle where the body orbiting another will be at a given time by knowing the starting position, speed, and some other variables. Newton's laws of motion were based on previous work and studies carried out by other important scientists, such as the German astronomer Kepler, who formulated the laws governing the movements of the planets.

Using Newton's laws and equations, the prescribed orbits of two bodies bound by gravitational interaction can be accurately calculated. The shapes of these orbits can be a circle, an ellipse, a parabola, or any other conic section (shapes that can be obtained by cutting a cone). The Earth, for example, describes an elliptical orbit around the Sun.

(Wikimedia Commons)

Newton's laws provide simple solutions when considering a scenario in which there are only two celestial bodies interacting with each other in the entire universe: the “two-body system.” Once you try to add a third object, things get very complicated as the orbits become irregular and often unpredictable in the long term. In this case, knowing the starting position and speed of the objects is no longer enough to make accurate predictions because everything becomes chaotic, and always after a certain period of time one of the three objects is thrown out of the system.

To be honest, any system with more than two bodies is a problem and that's why we generally talk about “Body problem n“, where “n” stands for any number. It is a question that has fascinated and occupied many mathematicians, physicists, and astronomers since the time of Newton, when people began to wonder how to predict the orbits of celestial bodies in more complex systems. And it was not just a theoretical question: predicting the motions of The moon and planets were essential for navigation at sea, for example, and more recently have become essential for organizing space missions.

It soon became clear that the n-body problem, at least for our cosmic neighbours, was not insoluble, as long as we were satisfied with some degree of approximation. In fact, a system containing multiple celestial bodies sufficiently distant from each other can be visualized as a series of two-body systems. In the case of the solar system this work is facilitated by the presence of one star, the Sun, which, as we have seen, has a strong influence on everything else because of its enormous mass. So we can roughly think of two-body systems that always include the Sun and one of the other planets, neglecting everything else.

In this way we obtain a solar system in which each planet completes an elliptical orbit around the Sun as predicted by Kepler's laws. It is a more than acceptable approximation of the time horizon of our existence and that of the entire human race, considering that any changes in orbits due to minimal gravitational interactions become noticeable over millions if not billions of years.

Non-scale representation of the solar system (NASA)

These less-than-ideal solutions fall into what is called “Three bound object problem“, where it is acceptable to obtain the best possible approximation of the motion of celestial bodies. In the case of the solar system, this is done by assuming, for example, that they have a negligible mass compared to the Sun and also compared to Jupiter, which is the largest planet in the family of eight planets, including That land clearly.

For those who study its orbits, our solar system also offers the advantage of having only one star, unlike many other systems that have two stars (“binary systems”). The latter's suns usually form stable orbits around a certain point that corresponds to their center of mass. The planets of binary systems usually have stable orbits themselves, due to their distance and lower mass compared to stars, and thus their movements over time can be predicted to a good approximation.

However, there are ways to work on the three-body problem and get results that are as close to reality as possible. Only a very small portion of the path can be calculated accurately, so if you divide the entire problem into small sections, you can calculate the motions of all objects in the system in a short period of time and update them as you calculate the sections. (Mathematically speaking, numerical integration is said to be performed to solve differential equations.)

By applying this technique we obtain an n-body simulation, that is, a simulation to predict the movements of a certain number of planets even in the very distant future. Simulations can be used to try to predict the formation or evolution of entire galaxies, that is, a large collection of solar systems, stars, dust, and interstellar gas. Today these calculations are performed using very powerful computers, whereas previously they had to be done manually with a lot of patience and the risk of making more errors.

Approximation of the trajectories of three identical bodies whose center of mass remains constant (Wikimedia)

Many mathematicians and astronomers have approached the three-body problem to find specific cases in which a complete functional solution can be found without approximations (i.e. an analytic solution). In the 18th century, the Swiss mathematician Euler found A A series of solutions Three hypothetical celestial bodies always remain aligned along an imaginary straight line passing through their center of mass. The Italian Joseph Louis Lagrange found a more elegant solution to the three-body problem, account Points in space where two objects of very large mass allow a third object (of much less mass) to maintain a fixed position relative to the other two objects. These points are known today as “Lagrange points” and are necessary to place probes and telescopes in space at great distances from Earth for research purposes.

Lagrange points (NASA)

The advent of computers and the possibility of increasingly powerful computing systems have provided new opportunities to address the n-body problem in recent years. In 2008, a research team conducted simulation Which sparked a debate among experts and orbiting enthusiasts: He collected as much information as possible about our solar system and tried to predict what would happen to each planet that makes it up in the next 5 billion years.

The research group conducted more than two thousand simulations, always starting from the same conditions except for one case: the distance between the Sun and Mercury, the planet closest to our star. In each simulation, the distance was changed by a few millimeters to see if this difference could have any very long-term consequences for the rest of the planets (between the Sun and Mercury there are on average 58 million km). In 1% of the simulations, Mercury's orbit changed over time to the point that the planet could collide with the Sun or its closest neighbour, Venus. In one simulation, an initial change in Mercury's orbit was enough to destabilize the entire inner solar system, which in addition to Mercury and Venus includes Earth, the Moon, Mars and its two moons.

The paper is a good example of the difficulties posed by the n-body problem. The system itself is in fact not completely random: if we always start many simulations from the same starting conditions, we will always get the same result, but if we modify a variable even by just a small value in the long run the result changes: it is the definition of a chaotic system. The amount of variables complicates the calculations to such an extent that it is impossible to arrive at a general solution to the problem, which applies to all possible cases.

As we have seen, the problem can be controlled in some way, but it is still not a trivial issue, and where limits of this kind are encountered, philosophical considerations often take over. But again, as Liu Cixin wrote in his novel: “Should philosophy lead experiments, or should experiments lead philosophy?”

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