Dynamical Stability and Chaos

Chaos is a general term that describes a system whose motion is non-repeating over a given timescale, that is, the motion appears random. Stability describes the "boundedness" of a system; a system is stable if changes in its evolution are confined to a certain range. Therefore, one of the most fundamental features of a chaotic system is stability (for a more complete review of chaos theory, consult (Chirikov, 1979)). For example, the Solar System is a chaotic system, but it is stable in the sense that the orbits of the planets do not interchange or become unbounded, and the oscillations of orbital elements, like eccentricity, occur over a finite range. Alternatively the Solar System is unstable in the sense that the minor planets' orbits can evolve in a non-repeating manner, as was spectacularly displayed when comet Shoemaker-Levy 9 impacted Jupiter. So is the Solar System stable? It depends on the bodies in question and the timescale. The orbits of many comets are not stable on timescales comparable to the age of the Solar System, but the orbits of the planets clearly are (they're still here undergoing periodic evolution). But on longer timescales, the planets' orbits are not stable; the most unstable planet in the Solar

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System, Mercury, may be lost to the Solar System in another 1012 years (Lecar et al., 2001). The example of our Solar System elucidates an important dichotomy in chaotic systems: a system may be formally unstable, but, for all practical purposes, is stable. It is irrelevant that Mercury could collide with Venus or the Sun in 1012 years because it will be engulfed by the Sun when it enters its red giant phase in 5 x 109 years. So, practically speaking, the planets in the Solar System are on stable orbits. But from a rigorous definition from chaos theory, the planets cannot be said to be stable; the Solar System's lifetime is just less than the timescale for instabilities to arise.

For a system to be chaotic, its motion must be 1) governed by nonlinear equations, and 2) sensitive to initial conditions. These requirements are met for systems with 2 or more planets that are close enough to each other. How close is "close enough" is a subject of intense research. In linear, non-chaotic motion, two nearby trajectories diverge at a constant rate, like two balls thrown together; their random motions increase their separation at a constant rate (their relative velocity times the time). In chaotic systems two nearby trajectories diverge at an exponential rate. Take, for example, two water molecules in a stream that begin right next to each other. Although in general the water flows downhill, the paths of the molecules will eventually become divergent due to rocks, vortices, tributaries, etc. Once one molecule reaches the ocean, the other may be stuck kilometers upstream. A planetary example of chaotic motion is represented in the Kirkwood gaps in the asteroid belt (Kirkwood, 1888; Moons, 1997; Tsiganis, Varvoglis & Hadjidemetriou, 2002). These gaps result from the ejection of asteroids in resonances with Jupiter. Asteroids next to the gaps have evolved regularly (the motion is repeating) for billions of years, but those in the gap were ejected in just millions of years (Lecar et al., 2001).

Most exoplanet systems of 2 or more planets are chaotic, and we would like to know if they are dynamically stable. Several meanings of stability with regard to planetary systems have arisen that complicate discussions. A system in which no planet is ejected and the semi-major axes remain bounded for all time is known as "Lagrange stable". This definition is the preferable definition of stability, as it implies a system will behave the way it does now for all time. Unfortunately, there is no known way to prove Lagrange stability at this time (although numerical simulations may disprove it). A more subtle form of stability exists when the ordering of the planets remain constant. This type is known as "Hill stability" or "hierarchical stability" and it can be proven analytically for non-resonant, two-planet systems (Marchal & Bozis, 1982; Milani & Nobili, 1983; Gladman, 1993; Chambers et al., 1996; Barnes & Greenberg, 2006b). In this type of stability the outermost planet may escape, but not the inner; the ordering of the bodies remains constant for all time.

Unfortunately the term hierarchical has two meanings that must be explained here. In stability analyses, a system is hierarchical if it satisfies a simple equation. However the term "hierarchical" is now also employed to describe exoplanet systems for which the ratio of the semi-major axes (a/a') is low (< 0.3) (Lee & Peale, 2002; GoZdziewski & Konacki, 2004). These conflicting definitions naturally lead to the problem that a "hierarchical planetary system" may not be "hierarchically stable", if the eccentricities are large enough.

Recently it has been shown that the Hill and Lagrange boundaries may be quite close to each other (see 7.2.3 or (Barnes & Greenberg, 2006b, 2007b)). The proximity of a system to the Hill boundary may be parameterized by [3. If [3 = 1, the system is on the Hill boundary, and if [3 > 1, the system is Hill stable. (Barnes & Greenberg, 2006b) found that for two systems (47 UMa and HD 12661), the Lagrange stability boundary corresponded to [3 values of about 1.02 and 1.1, respectively. Although the expression for Hill stability is only valid for systems of 2 planets outside of resonance, many observed systems have only two known companions. Therefore [ can be calculated for the majority of observed multiple planet systems to determine their proximity to instability (Barnes & Greenberg, 2007b).

A system's sensitivity to initial conditions is often measured by the Lyapunov time. This time is a measure of the divergence between two initially nearby trajectories. The Lyapunov time does not necessarily predict the onset of irregular (non-repeating) motion. The Earth has a Lyapunov time of about 5 million years (Laskar, 1989; Sussman & Wisdom, 1988, 1992). This value does not mean that in 5 million years the Earth's orbit will begin to change wildly, it just means that 5 million years from now the Earth's position cannot be known with arbitrarily high precision. Furthermore one must not think of the Lyapunov time as a measure of the "degree" or "amount" of chaos. A system is either chaotic or it is not. For more on chaos in planetary systems see (Lecar et al., 2001).

The Lyapunov exponent has been exploited in one code in common use in dynamical analyses of exoplanets: MEGNO (Cincotta & Simo, 2000). This code determines the Lyapunov time in a grid of parameter space, and stability is inferred from this time. Although evolution from a given set of initial conditions was not proven to be stable, if the Lyapunov time is long enough, the configuration is assumed stable (again, "long enough" is not rigorously defined); the Lyapunov time is assumed to be a proxy for stability.

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