The Three-body Problem from Pythagoras to Hawking
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This book, written for a general readership, reviews and explains the three-body problem in historical context reaching to latest developments in computational physics and gravitation theory. The three-body problem is one of the oldest problems in science and it is most relevant even in today’s physics and astronomy.
The long history of the problem from Pythagoras to Hawking parallels the evolution of ideas about our physical universe, with a particular emphasis on understanding gravity and how it operates between astronomical bodies. The oldest astronomical three-body problem is the question how and when the moon and the sun line up with the earth to produce eclipses. Once the universal gravitation was discovered by Newton, it became immediately a problem to understand why these three-bodies form a stable system, in spite of the pull exerted from one to the other. In fact, it was a big question whether this system is stable at all in the long run.
Leading mathematicians attacked this problem over more than two centuries without arriving at a definite answer. The introduction of computers in the last half-a-century has revolutionized the study; now many answers have been found while new questions about the three-body problem have sprung up. One of the most recent developments has been in the treatment of the problem in Einstein’s General Relativity, the new theory of gravitation which is an improvement on Newton’s theory. Now it is possible to solve the problem for three black holes and to test one of the most fundamental theorems of black hole physics, the no-hair theorem, due to Hawking and his co-workers.
three-body problem. Turku, Finland Austin, TX St. Petersburg, Russia Grenada, West Indies St. Petersburg, Russia Tokyo, Japan September 2015 Mauri Valtonen Joanna Anosova Konstantin Kholshevnikov Aleksandr Mylla¨ri Victor Orlov Kiyotaka Tanikawa ThiS is a FM Blank Page Contents 1 Classical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impossible Problems to Solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to Solve the Three-Body
act as small rockets which tend to take the comet off its course in somewhat unpredictable ways. Lexell and the Discovery Uranus Interesting changes in orbits of comets may be caused by the perturbation of Jupiter. In 1770 Charles Messier discovered a new comet which came almost straight at the Earth and passed by us within just over two million kilometers, six times the distance to the Moon. In the scale of the Solar System, this was very close indeed! At best the comet moved the diameter of
concluded that Mayer’s star was not a star at all but an Lexell and the Discovery Uranus 63 Fig. 3.7 William Herschel (left, Engraving by James Godby, published by Frederick Rehberg. Shows Herschel against background of stars in Gemini where Uranus was discovered in 1781. Credit: Institute of Astronomy, University of Cambridge) and Anders Lexell (right, Granite relief by Sofia Saari, Credit: University of Turku) unknown planet. It had not vanished but had moved in the sky, as planets do with
down, and therefore the time unit used in astronomical measurements is getting longer. If measured using uniform time, the speed of the Moon in the sky is actually slowing down. The effect was first explained by Laplace, but he found a different reason. He noticed that the eccentricity of the Earth’s orbit around the Sun is getting less, moving toward a circle, due to the effect of other planets on the orbit of our planet. And as we noted above, the orbit of the Moon is affected by the
then, it is still clear that the detailed evolution of the n-body problem is not calculable even though statistically the system is well in-hand. Hierarchical Triples Most of the triple stars in the universe are hierarchical simply because otherwise the system is unstable and breaks up into a binary and an escaping third star. The criterion for the stability of the three-body system has therefore been of interest. 140 7 Three Body Problem in Perspective The instability can lead to the escape