Newton’s three-body problem explained - Fabio Pacucci
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The experiment ran by Laskar and Gastineau in 2009 about the stability of the Solar System is very surprising. Since ancient astronomers started to look at the sky, thousands of years ago, the motion of the planets seems very stable and, in a way, reassuring. Despite this, extreme events happen all the time in the Universe: stars explode, planets get bombarded by asteroids, entire planetary systems get disrupted by black holes. Why then, from our own limited perspective on the Earth, the heavens seem so peaceful?
The search for a solution to the three-body system is one of the most famous mathematical problems applied to astronomy and it is known since centuries. Although generally not known in its mathematical details, it is certainly part of common knowledge and recently was even chosen as a title for a world-famous science fiction novel by Liu Cixin: “The Three-Body Problem”.
It is very interesting to focus on the history of this problem, as well as its early applications. What seems a mathematical oddity is, in fact, a real-world trouble. Just think about the simplest and probably most useful example of a three-body system: the Sun-Earth-Moon system. Newton, in his monumental 1687 opera “Philosophiae Naturalis Principia Mathematica”, introduced the theory of gravitation and applied it to the Sun-Earth-Moon system. Of course, he couldn’t find a full solution, but he developed a method named “lunar theory” to calculate the approximate position of the Moon. This task was crucially important for navigation: in open seas, with no landmarks to track one’s location, a successful orientation relies on an accurate estimation of latitude and longitude. The determination of the latitude is in general very straightforward and based on the measurement of the position of the Sun or the Polar Star in the sky. On the contrary, the determination of the longitude is an extremely complex task, achievable, for example, with calculations based on the measurements of the Moon’s position. The problem was so economically important for navigation that several countries, including Britain, Spain and the Netherlands, established extremely remunerative rewards for whoever solved the problem (the wonderful book by Dava Sobel “Longitude” offers a remarkable account of this scientific enterprise). The lunar method to estimate the longitude was used, for example, by the explorer Amerigo Vespucci in 1499 to prove that Brazil was definitely not India, as conjectured by Columbus in 1492 (have a look at this very nice account of this story).
Centuries passed by and, despite much progress brought by first-class mathematicians such as Lagrange, Laplace and Euler, the solution to this problem was missing. At the end of the XIX century Poincare proved that there are no closed solutions to the general 3-body problem: it’s not possible to write down a solution with a finite number of terms. Why is it impossible? As briefly explained in the TED-Ed lesson, there are too many unknowns. For each of the three bodies, we have 6 unknowns: three numbers to specify its position and three numbers to specify its velocity. We thus have 6x3=18 unknowns for the whole system. Poincare proved that a 3-body system, in a 3D space, expresses only 12 constant quantities, or integrals of motion, useful to determine the unknowns. As the unknowns are more, the system is unsolvable.
Are we at a loss? Not really. Since 1765 solutions have been found when the three bodies occupy some special spatial configurations. For example, Lagrange found a family of exact solutions when the bodies occupy the vertices of equilateral triangles. You can see some very nice representations of these special solutions in this article. While remarkable, these solutions do not solve the general problem.
In 1912 the mathematician Sundman proved that an exact solution to the general problem exists, although not in a finite form: it is an infinite summation. This solution is not practical, as one needs to compute an unimaginable number of terms in the summation to obtain a reasonable approximation. This article describes in more details why this solution is, in practice, unusable.
A more practical way to solve the problem is with numerical methods. The infinitesimal quantities present in the original set of equations are replaced with finite, although small, quantities. The resulting, simplified, equations are then solved iteratively. These methods are fast and able to accurately approximate the real solution. For example, numerical methods on the most powerful computers of the 1960s were used to compute the orbit of the NASA’s spacecraft Voyager on its Grand Tour around the Solar System. Read this inspiring story of innovation on this article (with video!) by the BBC.
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