We’re talking about islands of regularity

The three-body problem is one of the most popular in the field of astronomy, in part because it inspired one of the most important science fiction sagas of recent years. This problem has intrigued physicists around the world for centuries, and while it may not have a solution, the team has now found an important new clue.

Islands of stability. A team of physicists has discovered the existence of “islands of patterns” in the way groups of three astronomical objects interact gravitationally.

Three body problem. Let’s put it in context: the three-body problem refers to the difficulty encountered in calculating the orbital motions of three objects orbiting each other. The discovery of gravity and the mathematics behind it allowed us three centuries ago to mathematically describe the motion of two objects based on their gravitational interaction.

These movements are simple and allow for regular interactions, similar to the orbits of planets around a star. Thanks to this, we can, for example, calculate with great accuracy when a comet will visit us or what the distance between the Earth and the Sun will be in four and a half years.

The problem is when a third body comes into play. This simple addition could ruin any hope of calculating these long-term interactions. The reason is that the system becomes chaotic, that is, a system in which small changes in initial conditions lead to large differences in results, as happens, for example, with double pendulums.

From statistics to modeling. When there are three objects gravitationally interacting with each other, stable equilibrium is rarely achieved between these bodies, and usually one (or more) objects are thrown out of the system. Physicists often turn to statistics to avoid costly deterministic calculations when they need to know whether a system will eventually eject one of its elements to “solve” the three-body problem… by eliminating one of them.

One of these groups encountered a problem: when comparing statistical models and computer simulations, they noticed significant differences in results. In these experiments, the team started with a binary system, two bodies orbiting each other. The models simulated the approach of a third object from some random point in space and calculated the evolution of the system in time until one of the objects left the system.

Regular trajectories. In this way, they discovered a number of “regular trajectories”, that is, trajectories of these objects that were not chaotic. The regularity of these trajectories did not suggest stable equilibria, but rather non-chaotic, more easily predictable trajectories, which in this case lead to a faster and more immediate ejection of one of the objects.

If we create a graph based on the initial conditions of the system, we will find groups of similar conditions that lead to similar results. These will be the so-called “islands of regularity”. Details of the work were published in a journal article. Astronomy and astrophysics.

Back and forth. The result is progress in the sense that it allows us to learn more about this unwieldy problem. At the same time, this is a setback: it turns out that delving into the chaos of this problem is even more difficult than we thought.

The reason, as the team in charge explains, is that the coexistence of the chaotic and the predictable is even more difficult than the dominance of the chaotic. Thus, the calculations become even more complex.

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Image | Alessandro Alberto Trani