The Three-Body Problem, Revisited Statistically
Astrophysicists proposed a solution to the three-body problem inspired by statistical mechanics.
Many things go by a rule of three. A trio of words often creates a satisfying flow in prose or speech; a triad of notes helps construct the feel-good harmonies in a tune; a triple deity (such as the Holy Trinity) prevails throughout various cultures and religions. Three is also a magical number in physics. There are three spatial dimensions, three generations of particles, and three identical bosons that comprise the exotic “Efimov state”. Throw three massive objects together, and you now have the “three-body problem”.
The problem is the following: given three point masses under the influence of only their mutual gravitational attraction, how will their trajectories evolve if we know their initial positions and velocities? Writing in his 1687 masterpiece Principia, Isaac Newton formulated and solved the preceding two-body problem, where two masses are considered instead of three, with relative ease. The solutions are the familiar circular, elliptic, parabolic, or hyperbolic orbits that many physics students encounter in their first foray into celestial mechanics. Newton also took the first steps in tackling the three-body problem; little did he know, however, that it would remain unsolved today, over 300 years later.
To be precise, the question is unsolved in the sense that no general closed-form solution (i.e. exact formula for the orbits) is known to exist. Nevertheless, mathematicians and physicists have chipped away at the problem for centuries by providing explicit formulas for the orbits in special cases. The list of brave pioneers who have attacked the problem — Leonhard Euler, Joseph-Louis Lagrange, Henri Poincare, to name a few — hints at the problem’s difficulty. A natural question arises: how is it that when we add an additional mass to the two-body case, governed under the simplicity of Newton’s laws of motion, we face such notorious adversity in predicting the trajectories?
In the 1890s, Poincare discovered that the three-body problem exhibited chaotic dynamics. That is, the evolution of the system is random due to its extreme sensitivity to initial conditions. This leads to unpredictability in long-term, despite it being described by deterministic physical laws. Poincare thence showed that the general three-body problem is analytically unsolvable and provided an explanation for the difficulty even in finding exact solutions under restricted conditions.
Theorists have since branched into developing techniques to generate approximate solutions as ongoing efforts searched for explicit formulas under new families of conditions. Such approximate methods include perturbation theory, which produces solutions expressed in an infinite series; and numerical integration, whereby finite segments of orbits are calculated on a computer. A third approach — one inspired by statistical mechanics — has also been gaining traction.
In 2019, astrophysicists Nicholas Stone and Nathan Leigh from the Hebrew University of Jerusalem’s Racah Institute of Physics and Chile’s La Universidad de Concepción, respectively, derived a statistical solution to the chaotic, non-hierarchical three-body problem. While hierarchical regimes refer to situations where the masses or separations of the three bodies differ greatly, non-hierarchical regimes impose no such restrictions. This lack of a scale hierarchy has made analytical solutions describing a resulting orbit intractable in the latter regimes, compared to the former where analytic treatments exist. On the contrary, the statistical solution of Stone and Leigh takes the form of a probability distribution over a set of outcome trajectories.
The key to the duo’s approach is to leverage the chaotic nature of the trinary system instead of being intimidated by it. In particular, chaos allows one to reasonably invoke the ergodic hypothesis, which states that the system will uniformly explore the phase space volume accessible to it over a sufficiently long time. The assumption draws parallels with the microcanonical ensemble in statistical physics — the three bodies being isolated from their environment is analogous to a particle system not interacting with a heat bath. “In this way, we may turn the chaotic nature of the three-body problem — which has so far frustrated general, deterministic, analytic mappings from one set of initial conditions to one set of outcomes — into a tool that simplifies the mapping from distributions of initial conditions to distributions of outcomes, ” they explained in their paper, published in Nature.
Numerical integrations have shown that bound, non-hierarchical triple systems almost always disintegrate into a single escaping mass (the “escaper”) and a stable bound binary (the “surviving binary”). Considering this generic outcome, Stone and Leigh’s results built upon prior work involving similar statistical strategies dating back to 1976. Such analyses, however, were limited in several aspects: they yielded poor predictive power when verified against detailed numerical approaches, involved such mathematical difficulty that prevented the calculation of closed-form outcome distributions, or were only adequate for highly constrained configurations. The shortcomings have been attributed to several reasons. For one, early attempts failed to include the conservation of angular momentum in their derivations, which was assumed to be appropriate for low angular momentum systems. Even when such a kinetic constraint was accounted for, theorists were able to present a fully analytical formalism only for the special case of planar motion. Thirdly, and perhaps where Stone and Leigh made the most improvement, is that the interaction energy between the escaper and surviving binary was neglected.
A step forward
The researchers began with a general geometry and built into their formalism an interaction energy between the escaper and surviving binary, in addition to the conservation of angular momentum. Not shy to the rule of three themselves, they made three significant assumptions: (i) the ergodic hypothesis; (ii) that the disintegration in the triple system is instantaneous; and (iii) that the escaper sees the receding binary as a point particle. As explained earlier, assumption (i) underpins Stone and Leigh’s results. Assumption (ii) is vital in calculating the probability distributions from the accessible phase space. Assumption (iii) factors into the interaction between the escaper and binary. Overall, their derivation bore out of fewer premises compared to their predecessors.
Working through pages of complex integrals and grueling algebra, the duo succeeded in producing a mathematically well-defined (that is, non-divergent) estimate of the phase space volume and ultimately a closed-form expression for the distribution of outcomes. Their distributions are also qualitatively different compared to those from past analyses.
Comparing against numerical experiments
Thus, having established appropriate presumptions and powered through the math, here comes the third act — testing the results. The pair executed this stage by conducting a series of numerical experiments and comparing the outcomes with their probability distributions. They numerically integrated the equations of motion starting from initial conditions that described several ensembles of non-hierarchical three-body systems. This generated ensembles of orbits from which an empirical distribution was constructed. It is here that they ran into a worrying observation.
“Many of our [experiments] do not form resonant three-body systems, but instead resolve abruptly in a prompt exchange, where it is unlikely that the ergodic hypothesis can be applied,” they noted in their paper. A resonant three-body system describes triples that exert regular, periodic gravitational forces on each other over a time interval, which can place the system in a metastable state. For the ergodic hypothesis to be valid, interactions will have to unfold over several dynamical times before the system disintegrates, allowing for chaotic evolution to arise.
But there must be a resolution to the conundrum. Indeed, Stone and Leigh found a key dynamical mechanism that helped them identify subsets of experiments exhibiting a high degree of ergodicity — the “scramble”. As frisky as its name is, a scramble is defined as a period of time when no pairwise binaries exist in the trinary system. This occurs when all three bodies interact so strongly with each other that they enter a phase of intense chaos. The researchers kept track of the number of scrambles throughout each integration in each ensemble and used it to filter trajectories that are suitably ‘ergodicized’. With this extra tool in hand, they returned to comparing their derived distributions against the numerical experiments.
“We…find good agreement, so long as we restrict ourselves to ‘resonant’ encounters,” Stone and Leigh concluded in their paper. These “‘resonant’ encounters” are precisely the trajectories that include multiple scrambles and undergo chaotic evolution. Of course, as with any research endeavor, there are still aspects to improve upon. “In most cases we see data that match analytic predictions to leading order but also exhibit some level of higher-order structure, ” they cautioned, “The nature of these superimposed, second-order structures is not altogether clear.” The investigation of these features has been deferred to future work.
The duo’s findings are not only interesting from a mathematical point of view but also have substantial applications in the study of astrophysical processes. Three-body processes are ubiquitous in the universe and are responsible for phenomena, including planetary evolution in the solar system, binary–single star scattering in dense star clusters, and the formation of binary black holes. In particular, Stone and Leigh cite the study of leftover binaries produced in binary–single scattering events as a possible application of their formalism.
In the arena of celestial mechanics, it seems that one’s lonely, two’s company, and three’s a chaotic crowd. As we’ve seen, chaos doesn’t just spell trouble, though — it allowed one statistical inspiration to lead two to tackle the three-body problem, bringing to the fore intriguing new results. Moreover, one can’t deny the beautiful connection the problem has threaded between brilliant thinkers through the fabric of time. Stone and Leigh now add their names to the ends of that thread, having helped chip away at the centuries-old problem’s defenses.