Eclipses and The N-Body Problem: Determining Coherence from Chaos
Eclipses and The N-Body Problem: Determining Coherence from Chaos
Stephan Arndt
Thomas Jefferson High School for Science and Technology
This article was originally included in the 2020 print publication of the Teknos Science Journal.
What determines the motion of the skies? How do eclipses behave in general planetary systems? Both of these questions, fundamental to astronomy and pondered on for thousands of years, intrigued me. Although eclipses on Earth happen with regularity, in most planetary systems, they occur inconsistently and are difficult to forecast. In the Astronomy and Astrophysics Research Lab, I have been working with these planetary systems to tackle this issue, which is key to understanding tidal forces, planetary alignment, and much more. However, the hardest part of eclipse forecasting does not actually concern eclipses, but is about determining the evolution of the systems themselves. Understanding their motion is not just about astronomy‒ it is about understanding the nature of chaos itself.
For thousands of years, ancient and modern astronomers alike have worked to understand planetary motion, coming up with various models and equations to describe it: Aristotle’s geocentric model, Copernicus’ heliocentric model, Galileo’s telescopes, and Kepler’s laws of planetary motion. Newton continued Kepler’s advancements with the creation of calculus and his understanding of gravity, which birthed the famous n-body problem: how can we describe the motion of a system with n number of planets, where the motion is determined by gravity [1]? This question has captivated researchers for hundreds of years, many of whom dedicate their life’s work to its advancement. Understanding the n-body problem is also essential to predicting eclipses, which are reliant on the motions of planetary bodies. Richard Montgomery, a mathematics professor at University of California, Santa Cruz, has worked for over 20 years on a specific sub-question of the n-body problem, dealing with eclipse sequences in periodic three-body systems [3].
The n-body problem is not just a contrived physics question. Over hundreds of years, its study has generated great progress in the field of chaos theory, which aims to study the evolution and nature of chaos. The “butterfly effect” is one commonly known phenomenon in chaos theory: if a butterfly flaps its wings, could the tiny air disturbance eventually form a hurricane? In the particular case of the butterfly and the hurricane, the answer is actually no. The wind generated by the flap would dissipate too quickly, unable to grow large enough to cause a hurricane. Thus, when it comes to the weather, the idea of the “butterfly effect” is more of a thought experiment, rather than a theory based in science. However, the three-body problem is so complex and intricate that a small perturbation in the system, such as a flap of a butterfly’s wings, would actually have a dramatic effect. In August 2018, Zwart & Boekholt [6] found that all it takes is a perturbation on the order of 1 in 1 quintillion to drastically change a planetary system’s motion. To put how small this is into perspective, there are about 1 quintillion grains of sand on Earth. If the Earth were only made up of all its grains of sand, it would only take the shift of one of those grains for the entire Earth to change course. This makes numerical simulations of these systems incredibly difficult, as even the best approximation techniques fail relatively quickly.
Despite the difficulty of the three-body problem, researchers have made significant progress. For example, most three-body systems, over time, will eject one of the bodies and leave the remaining two orbiting each other. In other words, most chaotic systems eventually achieve regularity. The motion of the bodies while the system is intact is still incredibly difficult to describe, but a breakthrough in December 2019 helped answer this elusive question. Instead of trying to work around the chaotic nature of three-body systems, Nicholas C. Stone and his team took advantage of them by using a concept also serving as the basis of weather prediction [2]. A 70% chance of rain on the weather forecast means that meteorologists looked at thousands of past days with similar conditions and found 70% of them had rainfall. In a similar way, these researchers looked at sets of similar three-body systems and generated a probability distribution of their outcomes. In other words, instead of determining exact solutions, they found the probabilities of each possible solution occurring, just as meteorologists predict weather [5]. Although this research has only been done on three-body systems, Nathan Leigh described recent work on four-body systems and hopes to be able to generalize the results to any n-body system. Leigh (personal communication, Jan. 11, 2020) wrote, “There is something of an ansatz approach we started developing in that paper, which we have continued with and it works out quite well for most of the parameter space so far!” An ansatz is an educated guess in mathematics or physics, and the parameter space is the set of all possibilities for the system to look like. Although Leigh has not rigorously proven anything for four-body systems, his educated guesses appear to have worked well for simulating their motion.
Although the three-body problem is strongly related to more earth-based questions, its primary applications are astronomical. For example, Samsing & Ilan [4] detailed the use of the three-body problem in predicting the motion of interacting black holes. Their results were consistent with the chaotic and unpredictable nature of the three-body problem, but they noted that some initial conditions held more influence over the outcome than others. When considering the motion of black holes, knowledge of these more important conditions can be sufficient in solving the system’s motion. Researchers can then work backwards to understand the system’s past motion, helping to describe the system’s influence on the stars, planets, and asteroids around it. In other words, a clear understanding of these complex systems allows us to not only predict the motion of the skies, but also to reverse time and understand how things fell into place.
Over its centuries-old history, the n-body problem has sparked numerous developments in mathematics, statistics, physics, and astronomy. Its applications extend beyond Earth into space, reaching forwards and backwards in time. Though recent technologies and breakthroughs have allowed us to make progress on specific sub-problems of the n-body problem, truly understanding planetary motion still remains out of our grasp. Enduring through mankind’s timeless fascination with the skies, the n-body problem remains one of the most famous unanswered questions in history.
References
[1] Benavides, J. C. (2010). Trajectory Design Using Approximate Solutions of the N-Body Problem (Doctoral dissertation, The Pennsylvania State University, State College, PA). Retrieved from https://etda.libraries.psu.edu/files/final_submissions/2573
[2] Hebrew University of Jerusalem. (2019, December 18). Researchers crack Newton's elusive three-body problem. Retrieved January 11, 2020, from https://phys.org/news/2019-12-newton-elusive-three-body-problem.html
[3] Montgomery, R. (2019, August 1). The Three-Body Problem. Retrieved from https://www.scientificamerican.com/article/the-three-body-problem/
[4] Samsing, J., & Ilan, T. (2018). Topology of black hole binary-single interactions. Monthly Notices of the Royal Astronomical Society, 476(2), 1548-1560. https://doi.org/10.1093/mnras/sty197
[5] Stone, N. C., & Leigh, N. W. C. (2019). A statistical solution to the chaotic, non-hierarchical three-body problem. Nature, 576, 406-410. https://doi.org/10.1038/s41586-019-1833-8
[6] Zwart, S. F. P., & Boekholt, T. C. N. (2018). Numerical verification of the microscopic time reversibility of Newton's equations of motion: Fighting exponential divergence. Communications in Nonlinear Science and Numerical Simulation, 61, 160-166. https://doi.org/10.1016/j.cnsns.2018.02.002
