Is the Solar System Stable?

The stability of the solar system is one of the oldest problems in theoretical physics, dating back to Isaac Newton. After Newton discovered his famous laws of motion and gravity, he used these to determine the motion of a single planet around the Sun and showed that the planet followed an ellipse with the Sun at one focus. However, the actual solar system contains eight planets, six of which were known to Newton, and each planet exerts small, periodically varying, gravitational forces on all the others.

The puzzle posed by Newton is whether the net effect of these periodic forces on the planetary orbits averages to zero over long times, so that the planets continue to follow orbits similar to the ones they have today, or whether these small mutual interactions gradually degrade the regular arrangement of the orbits in the solar system, leading eventually to a collision between two planets, the ejection of a planet to interstellar space, or perhaps the incineration of a planet by the Sun. The interplanetary gravitational interactions are very small—the force on Earth from Jupiter, the largest planet, is only about ten parts per million of the force from the Sun—but the time available for their effects to accumulate is even longer: over four billion years since the solar system was formed, and almost eight billion years until the death of the Sun.

Newton’s comment on this problem is worth quoting: “the Planets move one and the same way in Orbs concentrick, some inconsiderable Irregularities excepted, which may have arisen from the mutual Actions of Comets and Planets upon one another, and which will be apt to increase, till this System wants a Reformation.” Evidently Newton believed that the solar system was unstable, and that occasional divine intervention was required to restore the well-spaced, nearly circular planetary orbits that we observe today. According to the historian Michael Hoskin, in Newton’s world view “God demonstrated his continuing concern for his clockwork universe by entering into what we might describe as a permanent servicing contract” for the solar system.

Other mathematicians have also been seduced into philosophical speculation by the problem of the stability of the solar system. Quoting Hoskin again, Newton’s contemporary and rival Gottfried Leibniz “sneer[ed] at Newton’s conception, as being that of a God so incompetent as to be reduced to miracles in order to rescue his machinery from collapse.” A century later, the mathematician Pierre Simon Laplace was inspired by the success of celestial mechanics to make the famous comment that now encapsulates the concept of causal or Laplacian determinism: “An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such an intelligence.”

Many illustrious mathematicians and physicists have worked on this problem in the three centuries since Newton, including Vladimir Arnold, Boris Delaunay, Carl Friedrich Gauss, Andrei Kolmogorov, Joseph Lagrange, Laplace, Jürgen Moser, Henri Poincaré, Siméon Poisson, and others. Several “proofs” of stability have been announced in the course of these labors; these have all been based on approximations that are not completely accurate for our own solar system and thus do not prove its stability. Nevertheless, research on this problem has led to many new mathematical tools and insights (perturbation theory, the KAM theorem,etc.) and inspired the modern disciplines of nonlinear dynamics and chaos theory.

The long-term behavior of the solar system is also relevant to a variety of other issues. Particle accelerators such as the Large Hadron Collider must guide protons for over a hundred million orbits, a problem similar in several respects to maintaining the planets on stable orbits for the lifetime of the solar system. The delivery of meteorites to Earth from their birthplace in the asteroid belt is driven by the long-term evolution of asteroid orbits due to forces from Jupiter and other planets. The primary mechanism that drives climate change and ice ages on timescales of tens of thousands of years is the periodic variation in the Earth’s orbit due to forces from the other planets. The discovery of hundreds of extrasolar planetary systems in the last two decades raises the tantalizing possibility that some or all of their properties are determined by the requirement that these systems have been stable for billions of years. In a different arena, some astronomers argue that Robert Frost’s famous poem “Fire and Ice” was inspired by the possible fates of the Earth at the demise of the solar system.

The most straightforward way to solve the problem of the stability of the solar system is to follow the planetary orbits for a few billion years on a computer. All of the planetary masses and their present orbits are known very accurately and the forces from other bodies—passing stars, the Galactic tidal field, comets, asteroids, planetary satellites, etc.—are either easy to incorporate or extremely small. There are two main challenges. The first is to devise numerical methods that can follow the motions of the planets with sufficient accuracy over a few billion orbits; this was solved by the development in the 1990s of symplectic integration algorithms, which preserve the geometrical structure of dynamical flows in multidimensional phase space and thereby provide much better long-term performance than general-purpose integrators.The second challenge was the overall processing time needed to follow planetary orbits for billions of years; this was solved by the exponential growth in speed of computing hardware that has persisted for the last five decades. At the present time, following planetary systems over billion-year intervals is difficult mostly because it is a serial problem—you have to follow the orbits from 2011 to 2020 before you can follow them from 2021 to 2030—whereas most of the computational speed gains of the last few years have been achieved by parallelization, the distributing of a computing problem among hundreds or thousands of processors that work simultaneously.

So what are the results? Most of the calculations agree that eight billion years from now, just before the Sun swallows the inner planets and incinerates the outer ones, all of the planets will still be in orbits very similar to their present ones. In this limited sense, the solar system is stable. However, a closer look at the orbit histories reveals that the story is more nuanced. After a few tens of millions of years, calculations using slightly different parameters (e.g., different planetary masses or initial positions within the small ranges allowed by current observations) or different numerical algorithms begin to diverge at an alarming rate. More precisely, the growth of small differences changes from linear to exponential: at early times, the differences in position at successive time intervals grow as 1 mm, 2 mm, 3 mm, etc., while at later times they grow as 1 mm, 2 mm, 4 mm, 8mm, 16 mm, etc. This behavior is the signature of mathematical chaos, and implies that for practical purposes the positions of the planets are unpredictable further than about a hundred million years in the future because of their extreme sensitivity to initial conditions. As an example, shifting your pencil from one side of your desk to the other today could change the gravitational forces on Jupiter enough to shift its position from one side of the Sun to the other a billion years from now. The unpredictability of the solar system over very long times is of course ironic since this was the prototypical system that inspired Laplacian determinism.

Fortunately, most of this unpredictability is in the orbital phases of the planets, not the shapes and sizes of their orbits, so the chaotic nature of the solar system does not normally lead to collisions between planets. However, the presence of chaos implies that we can only study the long-term fate of the solar system in a statistical sense, by launching in our computers an armada of solar systems with slightly different parameters at the present time—typically, each planet is shifted by a random amount of about a millimeter—and following their evolution. When this is done, it turns out that in about 1 percent of these systems, Mercury’s orbit becomes sufficiently eccentric so that it collides with Venus before the death of the Sun. Thus, the answer to the question of the stability of the solar system—more precisely, will all the planets survive until the death of the Sun—is neither “yes” nor “no” but “yes, with 99 percent probability.”

There remain two intriguing facts that lead to a plausible speculation. First, the future time required for the loss of Mercury is rather similar, within a factor of five or so, to the past time at which the solar system was born. Second, the solar system is nearly “full,” in the sense that there are few places where we could insert an additional planet without causing immediate instability. Both of these facts are explained naturally if the solar system began with more planets, in a configuration that was unstable on timescales much smaller than its current age. As time passed, the system lost more and more planets and thereby gradually self-organized into a more and more stable state. In this process, the time required to lose the next planet would quite naturally be a few times the current age. There would be few fossil traces of these lost siblings of the Earth.