In the Song of Ice and Fire series (Game of Thrones for you TV viewers) there is a stark warning that “Winter is coming.” That’s because the seasons are chaotic. Winters could last months or years, and you can never be sure when it might come. The series doesn’t go into details as to why the seasons are so variable, but one proposed idea is that their planet’s orbit is chaotic.

Our own solar system is decidedly not chaotic. In fact it is so stable that even close encounters with other stars wouldn’t perturb the planets much. For this reason it’s generally thought that planetary systems around other stars would likewise be stable over billions of years. After all, if planets have a chaotic orbit, they are much more likely to collide with other worlds or escape a solar system altogether. The exoplanetary systems we’ve discovered are stable on a billion-year scale, so this seems a reasonable assumption to make.

But we also know that planets can interact with each other to create orbital resonances, and that complex gravitational interactions can cause planetary orbits to shift dramatically. We know, for example, that early in our own solar system Jupiter and Saturn were much closer to the Sun, and that orbital resonances caused them to shift to their modern distances. Still, as a solar system matures we would expect planets to become stable, and therefore not chaotic.

But new work published in the Astrophysical Journal finds that some planetary orbits might be both chaotic and long-lived. By analyzing computer simulations they found that planets can have chaotic orbits that last on the order of 10 billion years. Some of these orbits vary so widely that they couldn’t remotely be habitable, but some could have wildly varying orbits within a habitable zone. It would be as if Earth had a chaotic orbit that stayed between the distances of Venus and Mars for billions of years. In that case Earth would might have years of Winter weather when far from the Sun, or an extremely hot Summer when orbiting close to the Sun. The chaotic orbit would give a planet extreme and unpredictable seasons, but it might still be possible for life to survive and thrive.

The idea of life on such a “chaotic earth” is still quite speculative, but what’s interesting about this work is that it shows how our assumption about chaos vs. stability in planetary systems isn’t so clearly defined. If long-lived planets can have chaotic orbits, then that changes the way we look for exoplanets.

Paper: Rory Barnes et al. Long-lived Chaotic Orbital Evolution of Exoplanets in Mean Motion Resonances with Mutual Inclinations. ApJ 801 101. doi:10.1088/0004-637X/801/2/101 (2015)

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Chaos theory is often expressed by the butterfly effect.  Stated simply, the butterfly effect is the idea that the flutter of a butterfly’s wings in China can cause a chain of effects that results in a hurricane striking Florida.  It is a representation of a non-linear system, where a small event can trigger a much larger one.  Such non-linear systems are fairly common.  For example, interstellar gas clouds can be triggered by a shock wave from a supernova or collision with another gas cloud.  Thus the flicker of a dying star can trigger the formation of a stellar nursery which results in the formation of dozens or hundreds of new stars.

A common aspect of such non-linear phenomena is that the large scale results are always the same, but the small scale effects can be dramatically different. With the interstellar cloud example, the cloud has to be on a razor edge, known as the Jeans instability, where a small “push” is enough to initiate the collapse of the cloud.  If it is on that razor’s edge, then any supernova in the general area could trigger the collapse, which will result in the formation of lots of new stars. However, if supernova A triggers things, it might produce a few hundred mid-size stars.  If supernova B triggers things, it might produce a few dozen very large stars.  Any trigger can form the stellar nursery, but the specific nature of the trigger can produce a very different outcome in terms of the the size and number of stars.  It is this latter aspect that is chaotic.

You can see a similar effect in the figure below.  The black dots represent magnets, and the lines represent the path of a simple iron-bar pendulum.  When the pendulum is released, it will eventually be captured by one of the magnets. That outcome is assured.  The specific magnet that captures it, however, depends critically on the starting point of the pendulum.  As you can see, a slight change in the initial condition leads to very different outcomes.

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A recurring theme in computational astrophysics (and physics in general) is the concept of chaos.  While many aspects of the universe are ordered and predictable, other aspects are quite chaotic.  Often things lie at a fine line between the two.

A good example of this can be seen in galactic motion.  On the one hand things are quite regular.  At a broad level stars move in a generally circular path around the galactic center.  This is analogous to our solar system, where planets move in (roughly) circular orbits around the sun.  This makes it easy to make a rough model of our galaxy as a fairly uniform disk of stars.

Of course when we look more closely things are not so simple.  For one our galaxy is not a uniform disk of stars, but rather lies mainly in spiral arms.  (Why this is the case is a topic for a future post.)  Then there is the motion of individual stars and star clusters themselves.

It turns out the motion of stars can be approximately described by a simple differential equation called the Henon-Heiles equation.  Unfortunately the solution to this equation is chaotic.  In other words the solution is very dependent on a star’s initial velocity and position.  Determining precise measurements of a star’s position and velocity can be quite a challenge.  So usually we have to look at general properties of the solution rather than finding a particular solution.

The good news is that the Henon-Heiles equation as long been studied by mathematicians, so we actually know a great deal about it.  One common way to look at general solutions is to plot what is known as a Poincare map of solutions.  I’ve plotted one for the Henon-Heiles here.  A Poincare map help you determine certain aspects of a star’s motion.  For example, in the figure below you can see that the range is bounded to a particular region.  So we know the star won’t just wander off.  We can also see regions where the motion tends to cluster.  So even though we don’t know the exact motion of a star, we know its general motion.

Problems like this can’t be solved well analytically, so it is an area where computational methods shine.

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The average distance between stars in our solar neighborhood is about one parsec, or just under 4 light years.  While this is a massive distance on our scale, from a star’s perspective it isn’t overly large.  Since stars are always moving relative to each other, there are bound to have been “close encounters” with other stars over the roughly 5 billion year history of our solar system.

By close I mean distances on the order of 1/20 of a parsec or less.  The radius of our solar system is about 1/5000 of a parsec, so even such close encounters have no chance of colliding with a planet.  Still, the close approach of a star would gravitationally pull on the planets ever so slightly, which means the planets would shift in their orbits by a tiny amount.  Kind of like the flutter of a butterfly’s wings makes a tiny bit of breeze.

This raises an interesting question about the stability of our solar system.  Would it be possible for a small tug from a visiting star to throw planets out of their orbits over millions of years.  Is it possible to have a planetary “butterfly effect”?

Correlation function for Mercury

One way to test this would be to run simulations of the solar system over millions of years, but unless you want to pay for time on a supercomputer cluster that isn’t very practical.  Another way is to run a simulation over a few hundred years, and then look at how statistically similar the motion remains over time.  In mathematics, this measure of similarity is known as a correlation function.  In the figure above I’ve plotted the correlation function for Mercury over about a century.  You can see there is a periodic oscillation (which is due to the precession of Mercury’s orbit over time) but you can also see that the overall amplitude of the oscillation is decreasing over time.  This basically means that Mercury’s orbit is slightly losing its consistency over time.

If you look at the rate of that decay, you can calculate what is known as the correlation time.  This is a measure of how long (based on your model) you could expect the orbit to be stable.  In the case of Mercury this turns out to be about 30 million years.  This does not mean that Mercury will tumble out of its orbit after that time, but rather than its orbit will remain largely unchanged for at least that long.

Correlation times of the planets

If you calculate the correlation time for all the planets, you get times on the order of tens of millions to billions of years, which means our solar system seems to be pretty stable.  If you do the same calculation for Pluto, however, you get a time of only about 10,000 years.  The orbit of Pluto is not nearly as stable as those of the classical planets.

Although we only have about a dozen planets to work with (if you count Pluto, Ceres, etc.), you can look at some of the factors that make a planet more or less stable.  For example in the second figure I plotted the correlation time of the planets with the eccentricity of their orbits (how elliptical they are).  You can see that the more elliptical the orbit, the less stable the orbit appears to be.

Not unexpectedly, it turns out that the most stable planets are ones that are reasonably massive, with fairly circular orbits close to their star.  Small mass planets far from their star in elliptical orbits are much less stable.  This means that when we look for planets around other stars we would expect to find lots of large planets orbiting in circular orbits near their star, which is exactly what we are finding.

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