The famous three-body problem is a physics challenge that stretches back to Newton. It's never been truly solved because, as was proven in 1899, it can't be. However, just because there is no complete solution to a problem doesn't mean we can't produce better and better approximations. The latest improvement on three-body solving involves a favorite mathematicians' tool: the drunken walk.
Once Newton showed how gravity worked he set about working out how the gravitational fields of two massive objects would affect each other, allowing the prediction of orbits infinitely far into the future. The world is not so simple, however. Every planet and even asteroid in the Solar System tugs on each other. Some do it so gently their influence can be ignored, but for many purposes, others are big enough or close enough that their influence matters, creating three, four, or five-body problems. Newton and many of the greatest minds that followed him attempted to find solutions to how such interactions progress.
The discovery that orbits where the gravity of three or more objects affect each other are inherently unpredictable in the long term led to endless taunts about physicists having encountered the limits of their powers. However, it also inspired the field of chaos theory, which has been crucial in many scientific endeavors, particularly meteorology. Finding ways to improve solutions, while knowing they can never be perfect, has proven a major boon for science, and a new paper in Physical Review X claims to take this further.
Technion-Israel PhD student Yonadav Ginat realized that where complete certainty is impossible, statistical approaches can be useful. Even if you don't know where competing gravitational forces will send objects caught in a three-body tug of war, you may be able to predict outcomes probabilistically, and then decide which paths are so unlikely you can ignore them.
To do this, Ginat and his supervisor Professor Hagai Perets applied the mathematical modeling known as the “drunkard's walk”. Originally based around a theoretical individual whose steps are individually unpredictable, but biased in one direction, the drunkard's walk has become widely used in statistics and economics. A simple example involves an individual standing near a cliff edge who for each step has a two in three chance of moving towards safety, and a one-third risk of stepping towards disaster.
To demonstrate the power of the process, Ginat modeled systems of three stars to see how likely it was that one would be ejected, the stellar equivalent of stepping into the void.
Most of the galaxy's stars are in binary pairs. Single stars like the Sun are rare, but triple systems are rarer still. Where they exist they're usually temporary arrangements, unless one star is so much more massive the others behave like planets. It's long been understood interactions between the stars usually cause one's ejection, but Ginat and Perets present the probabilities under different distances between the two closer stars. Their model reveals a series of close encounters before one star (not always the initial outsider) is yeeted into the great beyond and can even incorporate factors such as tides that have previously usually been discarded as too complex.
"We came up with the random walk model in 2017, when I was an undergraduate student," Ginat said in a statement. "I took a course that Prof. Perets taught, and there I had to write an essay on the three-body problem. We didn't publish it at the time, but when I started a Ph.D, we decided to expand the essay and publish it."
The work will improve our understanding of dense star clusters, where black holes and neutron stars may jostle up against giant stars, sometimes creating detectable gravitational waves. It could also be useful to spacecraft seeking stability amid gravitational jostling.