Is the Solar System Stable?

It’s Not That Simple

Newton was the first to recognize the physical nature of planetary orbits. His law of universal gravitation—describing the attraction between objects and its relationship to their mass and distance—beautifully explained the order of planetary motion in the solar system. Using Newton’s laws, one can predict a planet’s future trajectory based on its current position, velocity, and all the gravitational forces acting upon it. The Sun accounts for 99.8% of the mass in the solar system, so as a very good approximation, each planet’s orbit can be described as an independent ellipse with the Sun at one focus.

If everything could be controlled this way, every planet’s orbit would remain unchanged forever. However, there are also tiny mutual attractions between the planets. As it turns out, these small effects do not always remain “low-key.” Given enough time, they can accumulate in complex and unexpected ways to produce overwhelming effects.

Even in Newton’s time, observations of planetary motion had reached a very high level. Kepler proved that the peculiar trajectories of planets in the sky could be explained by a simple ellipse in three-dimensional space. But it wasn’t until three generations later, in Newton’s era, that astronomers determined planetary trajectories with enough precision to reveal deviations from perfect ellipses.

Newton knew that the attraction between planets would change their orbits, and he was particularly eager to explain a notable characteristic of the orbits of Jupiter and Saturn. Throughout the 16th and 17th centuries, astronomers found that Jupiter was slowly spiraling inward, while Saturn was gradually moving outward. If this trend continued for tens of thousands of years, the entire solar system would fall into crisis. Despite great effort, Newton could not explain this phenomenon with his theory. The mathematics involved was simply too daunting. In a letter, he implicitly admitted failure: “...to define these motions by exact laws admitted of easy calculation and considering all the influences on (planetary) motion at once exceeds, if I am not mistaken, the force of any human mind.”

Newton’s failure to explain the “stray” trajectories of Jupiter and Saturn provided great motivation for the most outstanding mathematicians of the 18th century. In 1776, Laplace solved this problem. He proved that the orbits of Jupiter and Saturn oscillate around an average value with a period of thousands of years. This is caused by the fact that five orbits of Jupiter around the Sun take almost exactly the same time as two orbits of Saturn. This “resonance” allows the perturbations they exert on each other to accumulate continuously for hundreds of years. A series of seemingly negligible influences, if timed correctly, can produce very substantial long-term effects.

After establishing his theory, Laplace could effectively reverse the solar system to predict the positions of ancient planets in the sky. His results matched Babylonian observations from 2,000 years ago with startling accuracy. According to his own account, “On March 1, 228 BC, at 4:23 AM (Paris time), Saturn was located two fingers below the star \gamma Virginis.” This success in determining Saturn’s position across thousands of years gave him extreme confidence in the correctness of his theory and undoubtedly led to Laplacian Determinism—the idea that if you know the current position and velocity of every particle in the universe, you can know their entire future exactly.

The Dream of Predictability

Under Laplace’s formulas, the solar system was completely stable. His mathematical framework did not allow for any long-term evolution of planetary periods, and Laplace believed he had proven that planetary orbits were “unbreakable.”

However, by the mid-19th century, cracks began to appear in this precision machine of the solar system. Astronomers gradually began to understand that the basic approximation methods in Laplace’s theory made it inapplicable over spans longer than a few thousand years. Le Verrier (who accurately predicted the existence of Neptune based on its perturbations of Uranus) warned that the solar system might not be as precise as Laplace claimed. Within a few decades, the problem became increasingly urgent. A prestigious international competition was established, offering a gold medal and a prize of 2,500 Kronor from King Oscar II of Sweden to the first person to prove the stability of planetary orbits.

Frustratingly, the proof submitted by Poincaré, which ultimately won, showed that the problem was unsolvable. Poincaré proved that even a simple system containing only one star and two planets is “non-integrable,” meaning no formula exists that can always tell you the positions of these two planets—and by extension, no “yes” or “no” answer can be given regarding the long-term stability of their orbits.

Poincaré’s work was nearly 100 years ahead of its time. He was the first to hint at chaos theory and what is now widely known as the “butterfly effect”—where a tiny perturbation in a system can produce overwhelming effects. (The term comes from findings in atmospheric modeling, where even a butterfly in Brazil flapping its wings can change global weather months later.) Poincaré’s work proved that the deterministic orbits of the solar system cannot be extrapolated to infinity, and any conclusions about the fate of the planets must be expressed in terms of probability.

Brute Force Solutions

In the modern era, three things have revitalized this once-obscure field of celestial mechanics. Space flight requires extremely high precision for planetary positions and spacecraft trajectories. Supercomputers allow for high-precision brute-force simulations of future planetary motion by repeatedly calculating positions and velocities over many small time steps. Third, the discovery of extrasolar planets (including multi-planet systems with signs of resonance and chaos) has rekindled interest in the fate of the solar system.

In the 1980s, many astronomers used supercomputers to study the subtleties of solar system dynamics. One of the results of these calculations was the direct demonstration that the solar system is indeed chaotic. More importantly, the time span for which effective orbital prediction is possible is only a few million years—far less than the age of the solar system. For example, even if the current positions of the planets could be determined with an error smaller than the diameter of an atom, you still could not accurately predict their positions 100 million years from now. We have no way of knowing for sure whether New Year’s Day in the year 100,000,000 AD will occur in winter or summer, or if Earth will still be orbiting the Sun.

In 1995, Jacques Laskar, a “successor” to Le Verrier at the Paris Observatory, published the results of an interesting numerical experiment. With the help of computer algebra, he developed an approximation method that could push planetary motion forward in time. The computer program implementing this method contained over 150,000 terms describing the subtle gravitational interactions between planets, allowing him to advance the evolution of the solar system in steps of 200 years. He thus became the first person to study the various possible evolutionary processes of the solar system over a span of more than a billion years.

So, are there results indicating that planets in the solar system might break their long-standing orbits?

After evolving forward for 500 million years, Laskar’s program showed no obvious trend toward disaster. The giant planets—Jupiter, Saturn, Uranus, and Neptune—showed no signs of instability. (We now believe they are stable over a span of a quadrillion years.) However, the terrestrial planets exhibited subtle chaos and orbital drift, with Mercury in particular having the potential to go rogue.

Laskar therefore conducted a specific experiment to study Mercury’s motion. To do this, he examined his 500-million-year calculations and selected the moment when the eccentricity of Mercury’s elliptical orbit reached its maximum. Then, he used the orbital configuration at that moment as the initial condition for four other nearly identical solar system models, in which only the Earth’s orbit had microscopic differences that were completely unobservable. When he evolved these four models forward in time, they initially moved in lockstep, but after a few million years, the butterfly effect began to manifest, and inevitable differences appeared between the models.

After all four models completed 500 million years of evolution, Laskar would again check the planetary orbits and select the moment when Mercury’s eccentricity was at its maximum. He would then use this as the initial condition for four new numerical simulations, repeating the process.

After repeating this a dozen times, a large number of possible outcomes were generated. Laskar found that in some of these outcomes, Mercury’s orbit would stretch and reach a dangerous point where it intersected the orbit of Venus. Intersecting orbits usually lead to disaster: collisions, close encounters where tidal forces shred one or both planets, or the complete ejection of a planet from the solar system.

For Earth, the outcomes ranged from experiencing more frequent asteroid collisions to direct destruction.

The Picture Today

Laskar’s 1994 results provided the first strong evidence that orbital chaos could occur even without external influence during the 6 billion years remaining for the Sun as a hydrogen-burning star. However, some important questions remained. How much did the repeated selection of seemingly most unstable orbits “help”? If a simulation were run from start to finish, how long could the solar system last? What mechanism causes Mercury’s orbit to become unstable? Finally, if the subtle effects of General Relativity were considered in the calculations, would the results be different?

These questions have recently been answered. Several independent groups conducted numerical simulations and discovered the solar system’s only fatal weakness—the mechanism that makes Mercury go rogue. It is precisely due to Jupiter’s gravitational influence that Mercury is constantly in danger.

Gravitational forces between planets cause a series of deviations from perfect ellipses. The most significant of these perturbations is orbital precession. When an orbit precesses, the orientation of its elliptical major axis changes, so the planet’s perihelion moves slowly but steadily in a clockwise or counter-clockwise direction. Mercury’s current precession rate is 0.16^\circ per year, while Jupiter’s is 0.23^\circ per year. But numerical simulations show that over very long periods, gravitational interactions can significantly increase Mercury’s precession rate. Most dramatically, if Mercury’s precession rate approaches Jupiter’s, it triggers a secular resonance, forcing Mercury’s orbital precession to synchronize with Jupiter’s.

The formation of secular resonance brings other troubles. Over millions of years, Jupiter gradually extracts angular momentum from Mercury’s orbit. This effect is almost unnoticeable on the massive Jupiter, but for the much smaller Mercury, its orbital eccentricity can increase to a degree that triggers disaster.

Today’s computers have developed to the point where thousands of solar system simulations can be performed, each without using approximation methods and including General Relativity as well as the influence of bodies like the asteroid Ceres and the Moon. In the summer of 2009, Laskar and his collaborator Mickael Gastineau published the results of the most extensive study to date, involving a total of 2,501 sets of solar system simulations.

These simulations included a wealth of detail. For example, they showed that we are lucky Einstein was right. The famous effects of General Relativity add an extra 0.43 arcseconds per year to Mercury’s precession rate. This makes it much harder for Mercury’s orbit to fall into dangerous secular resonance with Jupiter. Without this General Relativity effect, the probability of Mercury’s orbit becoming unstable would be 10%. But with the influence of General Relativity, the probability of the solar system going rogue before the Sun’s death is only about 1%.

Though small, a 1% probability is still non-negligible. Laskar and Gastineau calculated several possible future orbits for the solar system, and in some, Earth’s situation is grim. In one particularly violent case, Earth suffers a devastating direct collision with Mars. In another, Mars passes within just a few hundred kilometers of Earth’s surface. Although they don’t collide, this is by no means a good thing. As Earth and Mars approach, tidal stretching and squeezing would heat Earth enough to completely melt the crust and mantle. Earth’s oceans would evaporate into a steam atmosphere surrounding a global magma ocean.

Personally, I prefer to look at the 99% rather than the remaining 1%. We now have a definitive probabilistic solution to the centuries-old problem of the solar system’s stability. In the next 50 million years, no planet will go rogue. When the Sun faces its end 6 billion years from now, the planets will indeed exhibit their unusual trajectories.

Gregory Laughlin is a professor of astronomy at the University of California, Santa Cruz. His research focuses on extrasolar planets and their discovery. Laughlin also manages Transitsearch.org, a planet-hunting program involving advanced amateur astronomers.

The Instability of Jupiter’s Moons

In 1771, Laplace realized that three of Jupiter’s four largest moons have a close relationship. In terms of time, the period it takes for Ganymede to orbit Jupiter once is exactly equal to the time it takes for Europa to orbit twice and Io to orbit four times. In honor of Laplace, astronomers call orbital resonances occurring between any three or more bodies “Laplace resonances.” Io, Europa, and Ganymede are the only known cases in the solar system.

One result of their 4:2:1 resonance is that the mutual gravitational perturbations between them keep their orbits from being perfectly circular. For Io, the closest to Jupiter, it is precisely its elliptical orbit that creates its famous tidal effects. Io’s interior is heated as a result, making it the most volcanically active place in the solar system. In 42 hours, Io completes one orbit around Jupiter, and during this process, parts of its surface rise and fall by as much as 100 meters! For comparison, the “Earth tides” caused by the Moon are less than 1 meter.

In a paper published in the June 18, 2009 issue of Nature, Valéry Lainey, Jean-Eudes Arlot, Özgür Karatekin, and Tim Van Hoolst calculated that tidal effects on Io inject energy into its interior at a rate of 90 trillion watts, a figure more than five times the capacity of all power plants on Earth. The average surface heat flow generated by Io’s unusual tidal heating is about 25 times that of Earth. These calculations are consistent with measurements of Io using infrared telescopes, which also suggest that Io’s interior is in thermal equilibrium—meaning Io’s heat comes from tides rather than radioactive decay in its core or residual heat from its formation.

Some scientists hypothesized that Io draws energy from the other Galilean moons, thereby maintaining the 4:2:1 resonance as their orbits contract. Conversely, Lainey and his colleagues used 116 years of observational data on the positions of Jupiter’s moons to conclude that the Laplace resonance between them is slowly disintegrating. They found that since 1891, Io’s orbit has moved 55 kilometers closer to Jupiter, while Europa and Ganymede have moved outward by 125 and 365 kilometers, respectively.

The figure below shows two of the several large forces acting on Io. Lainey and his colleagues proved that the inward force in the right figure is greater than the outward force in the left figure, causing Io to move toward Jupiter. Similar pairs of forces also appear on Europa and Ganymede, but the outward force dominates, so they are moving away from Jupiter.

The fact that the Galilean moons are slowly breaking out of the Laplace resonance surprised many. Lainey and his colleagues did not predict when this resonance would collapse. But when it does, Io’s orbit will become more circular, its tidal effects will greatly diminish, and volcanic activity on Io will cease.

What then? Predicting the fate of the Galilean moons requires better data and understanding. Perhaps when the Laplace resonance collapses, tidal forces from Jupiter will once again push Io outward, and Io, Europa, and Ganymede will enter a resonance state again, causing Io’s volcanoes to wake up.

Perhaps this process has already happened several times? We can neither definitively predict the future of the Galilean moons nor trace their past. For example, we do not yet know if Callisto also participates in the resonance of the other three moons. But one thing we do know is that we are lucky to be in the active period of the Galilean moons to witness this phenomenon.

Landon Curt Noll is a computer security expert at Cisco Systems and a member of the American Astronomical Society. He enjoys giving lectures at the Fremont Peak Observatory (www.fpoa.net) and adding new research results to his astronomy webpage (www.isthe.com/astro).