Math proves that the orbits of planets have always been unstable. Blame inter-planetary gravity.

If you are comfortably used to think that the planets in our Solar System will revolve around the home star forever as they do today – or at least until the Sun changes its status – please stop thinking this way.

Mathematicians from three countries have proven what some researchers have been suspicious for a long time: star systems with planets circling around have a small but real chance to go unstable. And our Solar System is not an exception.

The idea was first formulated in 2009, by Paris Observatory astronomers. They built a detailed computational model of our Solar System and ran thousands of numerical simulations in order to project the planets’ motions billions of years into the future, as described in the science journal Nature. 

A Padua University mathematical physicist, Gabriella Pinzari, reached the same conclusion independently, the same year.

In most of those simulations — which varied Mercury’s starting point over a range of just under 1 meter — everything proceeded as expected. In 99% of the cases, the planets continued to revolve around the sun, tracing out ellipse-shaped orbits that looked more or less the way they have been throughout our star observation history, describes this process the science review outlet QuantaMagazine. 

In around 1% of the time, things went literally rogue.

The shape of Mercury’s orbit changed significantly. Its elliptical trajectory gradually flattened, until the planet either plummeted into the sun or collided with Venus. Sometimes, its behavior destabilized other planets as well: Mars, for instance, might be ejected from the solar system, or it might crash into Earth. Venus and Earth could, in a slow, cosmic dance, exchange orbits several times before eventually colliding.

The simulations published by French astronomers Jacques Laskar and Mickaël Gastineau in 2009 suggest that there’s a small but real chance of things going sideways. They stressed that those simulations, though important, aren’t the same as a mathematical proof and can’t be completely precise. Even a small imprecision might lead to very different outcomes in the course of billions of simulated years, they said.

And now, for the first time, the mathematical support of this theory exists.

Marcel Guàrdia, Jacques Fejoz, and Andrew Clarke, three mathematicians from the Barcelona University, the Paris Dauphine University, and the Leeds University, respectively, published a series of studies on this topic in the Cornell University’s science paper platform ArXiv to demonstrate that instability inevitably arises in a model of planets orbiting a star. 

Marcel Guàrdia and Jacques Fejoz. Credit: QuantaMagazine

Since Isaac Newton formulated his laws of motion and gravity in the 17th century, mathematicians and astronomers only considered the gravitational forces of the Sun, but neglected the inter-planetary gravity. But once you take into account the gravitational attraction between the planets themselves, everything gets more complicated, because no one would be able to provide explicit calculations regarding the planets’ positions and velocities over long periods of time.

The mathematicians realized this by adding an extra planet to the model of a star with two planets, which they initially took as a basis of research. In the new 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 the conditions that result in instability.

Over time, it was possible for the first planet’s 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.

The system could go wildly unpredictable in the long term, the three authors concluded. Read more on this topic here (PDF, 1 MB).