Death by Black Hole: And Other Cosmic Quandaries - Neil deGrasse Tyson (2014)

SECTION 5. WHEN THE UNIVERSE TURNS BAD

ALL THE WAYS THE COSMOS WANTS TO KILL US

Chapter 28. CHAOS IN THE SOLAR SYSTEM

Science distinguishes itself from almost all other human endeavors by its capacity to predict future events with precision. Daily newspapers often give you the dates for upcoming phases of the moon or the time of tomorrow’s sunrise. But they do not tend to report “news items of the future” such as next Monday’s closing prices on the New York Stock Exchange or next Tuesday’s plane crash. The general public knows intuitively, if not explicitly, that science makes predictions, but it may surprise people to learn that science can also predict that something is unpredictable. Such is the basis of chaos. And such is the future evolution of the solar system.

A chaotic solar system would, no doubt, have upset the German astronomer Johannes Kepler, who is generally credited with the first predictive laws of physics, published in 1609 and 1619. Using a formula that he derived empirically from planetary positions on the sky, he could predict the average distance between any planet and the Sun by simply knowing the duration of the planet’s year. In Isaac Newton’s 1687 Principia, his universal law of gravity allows you to mathematically derive all of Kepler’s laws from scratch.

In spite of the immediate success of his new laws of gravity, Isaac Newton remained concerned that the solar system might one day fall into disarray. With characteristic prescience, Newton noted in Book III of his 1730 edition of Optiks:

The Planets move one and the same way in Orbs concentric, some inconsiderable Irregularities excepted, which may have arisen from the mutual actions of…Planets upon one another, and which will be apt to increase, till the system wants a Reformation. (p. 402)

As we will detail in Section 7, Newton implied that God might occasionally be needed to step in and fix things. The celebrated French mathematician and dynamicist Pierre-Simon Laplace had an opposite view of the world. In his 1799-1825 five-volume treatise Traité de mécanique céleste, he was convinced that the universe was stable and fully predictable. Laplace later wrote in Philosophical Essays on Probability (1814):

[With] all the forces by which nature is animated…nothing [is] uncertain, and the future as the past would be present to [one’s] eyes. (1995, Chap. II, p. 3)

The solar system does, indeed, look stable if all you have at your disposal is a pencil and paper. But in the age of supercomputers, where billions of computations per second are routine, solar system models can be followed for hundreds of millions of years. What thanks do we get for our deep understanding of the universe?

Chaos.

Chaos reveals itself through the application of our well-tested physical laws in computer models of the solar system’s future evolution. But it has also reared its head in other disciplines, such as meteorology and predator-prey ecology, and almost anyplace where you find complex interacting systems.

To understand chaos as it applies to the solar system, one must first recognize that the difference in location between two objects, commonly known as their distance, is just one of many differences that can be calculated. Two objects can also differ in energy, orbit size, orbit shape, and orbit inclination. One could therefore broaden the concept of distance to include the separation of objects in these other variables as well. For example, two objects that are (at the moment) near each other in space may have very different orbit shapes. Our modified measure of “distance” would tell us that the two objects are widely separated.

A common test for chaos is to begin with two computer models that are identical in every way except for a small change somewhere. In one of two solar system models you might allow Earth to recoil slightly in its orbit from being hit by a small meteor. We are now armed to ask a simple question: Over time, what happens to the “distance” between these two nearly identical models? The distance may remain stable, fluctuate, or even diverge. When two models diverge exponentially, they do so because the small differences between them magnify over time, badly confounding your ability to predict the future. In some cases, an object can be ejected from the solar system completely.

This is the hallmark of chaos.

For all practical purposes, in the presence of chaos, it is impossible to reliably predict the distant future of the system’s evolution. We owe much of our early understanding of the onset of chaos to Alexander Mikhailovich Lyapunov (1857-1918), who was a Russian mathematician and mechanical engineer. His 1892 PhD thesis “The General Problem of the Stability of Motion” remains a classic to this day. (By the way, Lyapunov died a violent death in the chaos of political unrest that immediately followed the Russian Revolution.)

Since the time of Newton, people knew that you can calculate the exact paths of two isolated objects in mutual orbit, such as a binary star system, for all of time. No instabilities there. But as you add more objects to the dance card, orbits become more and more complex, and more and more sensitive to their initial conditions. In the solar system we have the Sun, its eight planets, their 70+ satellites, asteroids, and comets. This may sound complicated enough, but the story is not yet complete. Orbits in the solar system are further influenced by the Sun’s loss of 4 million tons of matter every second from the thermonuclear fusion in its core. The matter converts to energy, which subsequently escapes as light from the Sun’s surface. The Sun also loses mass from the continuously ejected stream of charged particles known as the solar wind. And the solar system is further subject to the perturbing gravity from stars that occasionally pass by in their normal orbit around the galactic center.

To appreciate the task of the solar system dynamicist, consider that the equations of motion allow you to calculate the net force of gravity on an object, at any given instant, from all other known objects in the solar system and beyond. Once you know the force on each object, you nudge them all (on the computer) in the direction they ought to go. But the force on each object in the solar system is now slightly different because everybody has moved. You must therefore recompute all forces and nudge them again. This continues for the duration of the simulation, which in some cases involves trillions of nudges. When you do these calculations, or ones similar to them, the solar system’s behavior is chaotic. Over time intervals of about 5 million years for the inner terrestrial planets (Mercury, Venus, Earth, and Mars) and about 20 million years for the outer gas giants (Jupiter, Saturn, Uranus, and Neptune), arbitrarily small “distances” between initial conditions noticeably diverge. By 100 to 200 million years into the model, we have lost all ability to predict planet trajectories.

Yes, this is bad. Consider the following example: The recoil of Earth from the launch of a single space probe can influence our future in such a way that in about 200 million years, the position of Earth in its orbit around the Sun will be shifted by nearly 60 degrees. For the distant future, surely it’s just benign ignorance if we do not know where Earth will be in its orbit. But tension rises when we realize that asteroids in one family of orbits can chaotically migrate to another family of orbits. If asteroids can migrate, and if Earth can be somewhere in its orbit that we cannot predict, then there is a limit to how far in the future we can reliably calculate the risk of a major asteroid impact and the global extinction that might ensue.

Should the probes we launch be made of lighter materials? Should we abandon the space program? Should we worry about solar mass loss? Should we be concerned about the thousand tons of meteor dust per day that Earth accumulates as it plows through the debris of interplanetary space? Should we all gather on one side of Earth and leap into space together? None of the above. The long-term effects of these small variations are lost in the chaos that unfolds. In a few cases, ignorance in the face of chaos can work to our advantage.

A skeptic might worry that the unpredictability of a complex, dynamic system over long time intervals is due to a computational round-off error, or some peculiar feature of the computer chip or computer program. But if this suspicion were true, then two-object systems might eventually show chaos in the computer models. But they don’t. And if you pluck Uranus from the solar system model and repeat the orbit calculations for the gas giant planets, then there is no chaos. Another test comes from computer simulations of Pluto, which has a high eccentricity and an embarrassing tilt to its orbit. Pluto actually exhibits well-behaved chaos, where small “distances” between initial conditions lead to an unpredictable yet limited set of trajectories. Most importantly, however, different investigators using different computers and different computational methods have derived similar time intervals for the onset of chaos in the long-term evolution of the solar system.

Apart from our selfish desire to avoid extinction, broader reasons exist for studying the long-term behavior of the solar system. With a full evolutionary model, dynamicists can go backward in time to probe the history of the solar system, when the planetary roll call may have been very different from today. For example, some planets that existed at the birth of the solar system (5 billion years ago) could have since been forcibly ejected. Indeed we may have begun with several dozen planets, instead of eight, having lost most of them jack-in-the-box style to interplanetary space.

In the past four centuries, we have gone from not knowing the motions of the planets to knowing that we cannot know the evolution of the solar system into the unlimited future—a bittersweet victory in our unending quest to understand the universe.