Solving the three-body problem in astronomy

The ‘3-body problem’ has long been the bane of astrophysicists.

The 3-body-problem remains a sticking point for astronomers. Credit: iStock
Published: February 21, 2020 at 8:47 am
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Isaac Newton knew that if you have a system involving two objects – a single planet orbiting a star, for example – then with a little understanding of how gravity works you can calculate how both will move. Add a third object to the system, like a moon, and that predictability disappears.


Things start off alright, but even a tiny change in the starting positions of any one of the three objects soon produces wildly different predictions for what the state of things will be in the future.

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As we can never know the initial positions of the three objects to infinite precision, then this chaotic behaviour means that the state of the system in the far future (and the distant past) is hidden from us and cannot be calculated.

This ‘three body problem’, as it’s come to be known, is a big headache. In physics textbooks and university exam papers, you can have a perfectly isolated system consisting of just a star and an orbiting world, but the real Universe shuns such simplicity.

In star forming regions, in clusters of stars and galaxies, in planet formation and in the interaction of black holes, the dance objects undertake involves triple systems more often than not.

Luckily, although the three-body problem can’t be solved analytically (where a set of equations leads to a single, definitive answer), some progress can be made.

A new paper from two astronomers sheds new light on this old problem, taking a statistical approach to what might happen.

The Laser Interferometer Gravitational-Wave Observatory © LIGO
Solving the 3-body problem could be of major importance to the Laser Interferometer Gravitational-Wave Observatory © LIGO

The systems they studied are created when a nice, predictable binary star is approached by a third star, the kind of thing that must happen all the time in young clusters of stars.

For most of the time, the models show that the resulting triple system will behave as a binary with a distant, third star interacting only weakly with the two at the centre, but as that interloper swings around there come periods of time where a mad scramble ensues.

This period ends when one of the stars is thrown back out to a distance where it is just a third star, a process which repeats and repeats until a star is ejected completely.

These calculations aren’t analytical predictions of what might happen: they’re just what the computer program thinks will happen next in any given circumstance.

The clever bit is that by running many such simulations the team could get a prediction of what is likely to happen.

That will be of enormous help to astronomers working in all sorts of fields, but one group looking extremely closely will be those trying to understand the collisions of black holes that produce the gravitational waves observed by facilities such as LIGO (the Laser Interferometer Gravitational-Wave Observatory).

Trying to understand how and why such black holes might form and collide has been difficult, but if interactions with a third object can encourage black holes to eventually merge, then the solution might lie in these clever statistical solutions to one of the oldest problems in the books.

Chris was reading The NANOGrav 11-year Data Set: Constraints on Planetary Masses Around 45 Millisecond Pulsarsby EA Behrens et al. Read it online at:


This article originally appeared in the March 2020 issue of BBC Sky at Night Magazine.


Chris Lintott, University of Oxford astronomer, BBC The Sky at Night presenter.
Chris LintottAstrophysicist

Chris Lintott is a Professor of Astrophysics at the University of Oxford and a co-host of BBC's The Sky at Night.


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