Now, in three papers that together exceed 150 pages, Guàrdia and two collaborators have proved for the first time that instability inevitably arises in a model of planets orbiting a sun.

“The result is really very spectacular,” said Gabriella Pinzari, a mathematical physicist at the University of Padua in Italy. “The authors proved a theorem that is one of the most beautiful theorems that one could prove.” It could also help explain why our solar system looks the way it does.

Four Pages and a New Story

Centuries ago, it was already clear that interactions among the planets could have long-term effects. Consider Mercury. It takes approximately three months to travel around the sun on an elliptical path. But that path also slowly rotates — one degree every 600 years, a full rotation every 200,000. This kind of rotation, known as precession, is largely a result of Venus, Earth and Jupiter pulling on Mercury.

But research in the 18th century by mathematical giants like Pierre-Simon Laplace and Joseph-Louis Lagrange indicated that, precession aside, the size and shape of the ellipse are stable. It wasn’t until the late 19th century that this intuition started to shift, when Henri Poincaré found that even in a model with just three bodies (say, a star orbited by two planets), it’s impossible to compute exact solutions to Newton’s equations. “Celestial mechanics is a delicate thing,” said Rafael de la Llave, a mathematician at the Georgia Institute of Technology. Alter the initial conditions by a hair — for example, by shifting the assumed position of one planet by a mere meter, as Laskar and Gastineau did in their simulations — and over long timescales the system can look very different.

In the three-body problem, Poincaré found a tangle of possible behaviors so complicated that at first he thought he’d made a mistake. Once he accepted the truth of his results, it was no longer possible to take the solar system’s stability for granted. But because working with Newton’s equations is so difficult, it wasn’t clear if the behavior of the solar system might be complicated and chaotic only on a small scale — planets might end up in different positions within a predictable band, for instance — or if, as Guàrdia and his collaborators would eventually prove in their own model, the size and shape of orbits might change so much that planets could conceivably crash into each other or travel off to infinity.

Then, in 1964, the mathematician Vladimir Arnold wrote a four-page paper that established the right language for framing the problem. He found a specific reason why key variables in a dynamical system might change in a big way. First, he cooked up an artificial example, a strange blend of a pendulum and a rotor that didn’t remotely resemble anything you’d encounter in nature. In this toy model, he proved that, given enough time, certain quantities that usually stay constant can change by large amounts.

Arnold then conjectured that most dynamical systems should exhibit this kind of instability. In the case of the solar system, this might mean that the orbital shapes, or eccentricities, of certain planets could potentially shift over billions of years.

But while mathematicians and physicists eventually made a lot of progress on proving that instability arises in general, they struggled to show it for celestial models. That’s because the gravitational effect of the sun is so overwhelmingly strong that many features of the clockwork planetary model persist even when you consider the additional forces exerted by the planets. (In this context, Newtonian mechanics gives such a good approximation of reality that these models don’t need to consider the effects of general relativity.) Such inherent stability makes instability difficult to detect.

Could parameters that stayed so stable in computations done by Laplace, Lagrange and others really change significantly? “You have to handle an instability which is extremely weak,” said Laurent Niederman of Paris-Saclay University. The usual methods won’t catch it.

Numerical simulations offered hope that the hunt for such a proof was not in vain. And there were preliminary proofs. In 2016, for instance, de la Llave and two colleagues proved instability in a simplified celestial mechanics model consisting of a sun, a planet and a comet, where the comet was assumed to have no mass and therefore no gravitational effect on the planet. This setup is known as a “restricted” n-body problem.

The new papers tackle a true n-body problem — showing that instability arises in a planetary system where three small bodies revolve around a much larger sun. Even though the size and shape of the orbits might spend a long time oscillating around fixed values, they will eventually change dramatically.

This had been expected — it was widely believed that stability and instability coexist in this kind of model — but the mathematicians were the first to prove it.

The Ultimate Instability

Together with Jacques Fejoz of the University of Paris Dauphine, Guàrdia first attempted to prove instability in the three-body problem (one sun, two planets) in 2016. Though they were able to show that chaotic dynamics arose in the flavor of Poincaré, they couldn’t prove that this chaotic behavior corresponded to large and long-term changes.

Andrew Clarke, a postdoc studying under Guàrdia, joined them in September 2020, and they decided to give the problem another go, this time adding an extra planet to the mix. In their model, three planets revolve around a sun at increasingly large distances from each other. Crucially, the innermost planet starts out orbiting at a significant tilt relative to the second and third planets, so that its path practically forms a right angle to theirs.

This inclination allowed the mathematicians to find initial conditions that result in instability.

They showed the existence of trajectories that led to pretty much any possible eccentricity for the second planet: Over time, it was possible for its ellipse to flatten until it almost looked like a straight line. Meanwhile, the orbits of the second and third planets, which had started out in the same plane, could also end up perpendicular to each other. The second planet could even flip a full 180 degrees, so that while all the planets might at first have moved clockwise around the sun, the second one ended up moving counterclockwise. “Imagine that you look forward a million years, and Mars is going the opposite way,” said Richard Montgomery of the University of California, Santa Cruz. “That would be weird.”

“You cannot avoid very wild orbits, even in this simple setting,” Niederman said.

Even so, the sizes of the orbits stayed stable. That’s because in this model, the planets move around the sun very quickly compared to how long it takes for their orbits to precess — allowing the mathematicians to gloss over the “fast” variables related to the planets’ motions. “It’s tedious to think about what’s happening every year if what you’re really interested in is what’s happening over a thousand years,” Moeckel said. Oscillations in the size of each ellipse (measured in terms of its long radius, or semimajor axis) average out.

This wasn’t surprising. “Common knowledge says that the inclination and the eccentricity should be more unstable than the semimajor axis,” Guàrdia said. But then he and his colleagues realized that if they placed the third planet even farther away from the sun, they might be able to add more instability into their model.

This new system and the equations that governed it were more complicated, and the mathematicians weren’t certain they’d be able to get any results. But “it was too much to ignore,” Clarke said. “If there was a chance of showing semimajor axes could drift, then I mean, you have to pursue that.”

Laskar, who has led much of the numerical work on instability in the solar system, said that if you superimposed this kind of solar system on our own, you might see the first planet nestled right up against the sun, the second planet where Earth would be, and the third planet all the way out at the Oort Cloud, at our solar system’s outer limits. (As a result, he added, this represents a “very extreme situation” — one he doesn’t necessarily expect to find in our own galaxy.)

The greater a planet’s distance from the sun, the longer it takes to complete an orbit. In this case, the third planet is so far away that the precession of the two inner planets occurs at a faster rate. It is no longer possible to average out the motion of the last planet — a scenario Lagrange and Laplace didn’t consider in their accounts of the solar system’s stability. “This will change completely the structure of the equation,” said Alain Chenciner, a mathematician also at the Paris Observatory. There were now more variables to worry about.

Clarke, Fejoz and Guàrdia proved that the orbits can grow arbitrarily large. “They finally get the size of the orbit to increase, as opposed to just the shape or something like that,” Moeckel said. “That’s the ultimate instability.”

Even though these changes accumulated very slowly, they still occurred more quickly than one might have expected — suggesting that in a realistic planetary system, changes might accumulate over hundreds of millions of years, rather than billions.