Three-Body Problem
The Three-Body Problem refers to the challenge of predicting the motion of three celestial bodies interacting with each other through gravity. Originating from the work of Sir Isaac Newton in the late 17th century, it was later expanded upon by mathematicians like Jules Henri Poincaré, who described the system as chaotic and deterministic, indicating that small changes in initial conditions could lead to unpredictable outcomes. A central concept in understanding the Three-Body Problem is the common center of mass, or barycenter, around which the bodies orbit.
While solutions for specific configurations exist, such as those identified by Joseph-Louis Lagrange and Leonhard Euler, a general solution remains elusive. Various families of solutions have been discovered, including the figure-eight orbits and more complex arrangements identified in recent research. Notably, a 2021 study introduced a probabilistic approach to understanding the interactions of the three bodies, akin to predicting weather patterns. This ongoing area of research highlights the intricate dynamics of celestial mechanics and the complexities involved in multi-body gravitational interactions.
Three-Body Problem
FIELDS OF STUDY: Astrometry; Orbital Mechanics; Theoretical Astrophysics
ABSTRACT: The three-body problem is a concept that involves the orbital motions of celestial bodies in a three-body system. The problem arises when attempting to predict how three given bodies will orbit one another around the common center of mass. Sir Isaac Newton addressed orbital motions in the 1680s, but could not predict the way three bodies would orbit each other. Jules Henri Poincaré developed the three-body problem in the 1880s. Other scientists, including Joseph-Louis Lagrange and Leonhard Euler, Roger Broucke and Michel Hénon, and Milovan Šuvakov and Veljko Dmitrašinović, discovered solutions to the three-body problem. Each solution is generally grouped into one of sixteen families.
Brief History
The three-body problem has roots in the work of English physicist Sir Isaac Newton (1643–1727) in the 1680s. Newton’s law of gravity proved that the orbits of two celestial bodies could always be accurately predicted, and that they almost always orbit each other in an elliptical path. Newton, however, was unable to predict the way in which three celestial bodies in a three-body system would repeatedly orbit one another. Scientists had struggled with this problem for about two centuries by the time Jules Henri Poincaré (1854–1912) developed his work on it in the late 1880s.
Poincaré claimed that a three-body system is a chaotic deterministic system. The term "deterministic" suggests that the positions and speeds of the three bodies are fixed, but Poincaré believed that the system is also chaotic. This means that a disturbance in the original state of the bodies could cause differences in a later state, and the final state would not be able to be predicted. An example of such a disturbance is a small change in the initial position of one of the bodies. Poincaré’s work on the three-body problem contained the first description of chaotic behavior in a dynamic system. His paper on the subject won an international competition sponsored by King Oscar II of Sweden and Norway in 1889 and was published in Acta Mathematica in 1890.
Basics of the Three-Body Problem
The three-body problem involves predicting the way the three objects will repeatedly orbit one another in a three-body system. To grasp the concept, an understanding of the common center of mass is generally required. Every celestial body has a center of mass, or balance point. This balance point may be located in the middle of the body or somewhere else. For example, if a celestial body was thought of as a ruler, its center of mass would be exactly in the center of the ruler because a ruler is uniform, with both ends being equally heavy. Placing one’s finger right in the middle of the ruler would balance the ruler. On the other hand, comparing a celestial body to a nonuniform object such as a hammer would place its center of mass somewhere other than its center. Trying to balance a hammer by placing a finger at its center would cause the hammer to tip over. This is because one end of the hammer is heavier than the other end. Therefore, the hammer’s center of mass is closer to the heavier end.
The center of mass also applies to the orbit of a celestial body. The body will orbit another body based on the common center of mass of the system, which is also called the barycenter. One celestial body does not technically orbit another celestial body; the two bodies actually orbit each other. The reason for this is gravity. Gravity causes the bodies to pull on each other, and there is a balance point. The balance point is the common center of mass. The common center of mass is closer to the body with the larger mass. The two bodies, therefore, orbit each other around this balance point.
A system containing more than two celestial bodies also has a common center of mass. Three celestial bodies will orbit one another around the common center of mass of all three bodies. A good example is the system of the earth, moon, and sun. The earth has more mass than the moon, so the common center of mass between the two bodies is inside the earth. The earth and moon orbit each other around this common center of mass. Similarly, the sun has more mass than the earth, so the two bodies’ common center of mass is inside the sun, and they orbit each other around this balance point.
Solutions to the Three-Body Problem
German mathematician Heinrich Bruns claimed that a general solution to the three-body problem was impossible. He further argued that the only possible solutions were those that could be used under certain conditions. Various solutions to the three-body problem have been discovered, however. Typically, the solutions fall into one of several families of solutions, including the Lagrange-Euler family, the Broucke-Hénon family, and the figure-eight family.
In the eighteenth century, mathematicians Joseph-Louis Lagrange (1736–1813) and Leonhard Euler (1707–83) identified the solutions that eventually fell under the Lagrange-Euler family. The solutions in this family feature three equally spaced bodies that rotate around in a circle. The solutions in the Broucke-Hénon family were discovered in the 1970s by American mathematician Roger Broucke and French astronomer Michel Hénon. These solutions have a complex configuration, as they contain two bodies that orbit on the inside and one body that orbits on the outside. The solutions in the figure-eight family were first alluded to in a paper by American mathematician Richard Moeckel in 1988, but were discovered by American physicist Cristopher Moore in 1993. The figure-eight solutions feature three bodies that orbit one another in a shape that resembles an eight. These solutions were later proven by French mathematician Alain Chenciner and American mathematician Richard Montgomery. Spanish mathematician Carles Simó further investigated the solutions.
In 2013, thirteen new families of solutions to the three-body problem were discovered. Two physicists from the Institute of Physics Belgrade, Milovan Šuvakov and Veljko Dmitrašinović, made the discoveries. They used a computer simulation to come up with the new solutions. They also created a new classification system for the solutions and also used a "shape-sphere" to illustrate the shape of the bodies’ orbits. One of the new solutions has the appearance of a ball of yarn.
A definitive solution to the three-body problem is technically impossible because of the random nature of gravitational interaction with three celestial bodies. However, in 2021, two Israeli researchers published their solution to the problem by calculating the probability of certain outcomes from the interaction between the celestial bodies. Their solution likens the interactions to the random steps of a drunk person’s walk. The researchers’ solution has been compared to a weather forecast done by meteorologists, who examine data and try to predict the most likely outcome.
PRINCIPAL TERMS
- common center of mass: balance point of a system of orbiting celestial bodies; also known as the barycenter.
Bibliography
Barrow-Green, June. "Oscar II’s Prize Competition and the Error in Poincaré’s Memoir on the Three Body Problem." Archive for History of Exact Sciences 48.2 (1994): 107–31. Print.
Cartwright, Jon. "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem." Science. Amer. Assoc. for the Advancement of Science, 8 Mar. 2013. Web. 15 May 2015.
Casselman, Bill. "A New Solution to the Three Body Problem." Feature Column. American Mathematical Society. Web. 15 May 2015.
"Drunken Solution to the Chaotic Three-Body Problem." Phys.org, 28 Dec. 2021, phys.org/news/2021-12-drunken-solution-chaotic-three-body-problem.html. 13 June 2022.
Gray, Jeremy John. "Poincaré, Henri." Britannica Biographies 1 Mar. 2012: 1–2. Biography Reference Center. Web. 9 June 2015.
Murzi, Mauro. "Jules Henri Poincaré (1854–1912)." Internet Encyclopedia of Philosophy. Internet Encyclopedia of Philosophy. Web. 15 May 2015.
Valtonen, M. J., and Hannu Karttunen. The Three-Body Problem. Cambridge: Cambridge UP, 2005. eBook Collection (EBSCOhost). Web. 9 June 2015.