Secular chaos and its application to Mercury, hot SPECIAL FEATURE Jupiters, and the organization of planetary systems Yoram Lithwicka,b,1 and Yanqin Wuc aDepartment of Physics and Astronomy and bCenter for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, Evanston, IL 60208; and cDepartment of Astronomy and Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H4 Edited by Adam S. Burrows, Princeton University, Princeton, NJ, and accepted by the Editorial Board October 31, 2013 (received for review May 2, 2013) In the inner solar system, the planets’ orbits evolve chaotically, by including the most important MMRs (11).] It is the cause, for driven primarily by secular chaos. Mercury has a particularly cha- example, of Earth’s eccentricity-driven Milankovitch cycle. How- otic orbit and is in danger of being lost within a few billion years. ever, on timescales ≳107 years, the evolution is chaotic (e.g., Fig. Just as secular chaos is reorganizing the solar system today, so it 1), in sharp contrast to the prediction of linear secular theory. has likely helped organize it in the past. We suggest that extra- That appears to be puzzling, given the small eccentricities and solar planetary systems are also organized to a large extent by inclinations in the solar system. secular chaos. A hot Jupiter could be the end state of a secularly However, despite its importance, there has been little theo- chaotic planetary system reminiscent of the solar system. How- retical understanding of how secular chaos works. Conversely, ever, in the case of the hot Jupiter, the innermost planet was chaos driven by MMRs is much better understood. For example, Jupiter (rather than Mercury) sized, and its chaotic evolution was chaos due to MMR overlap explains the Kirkwood gaps in the terminated when it was tidally captured by its star. In this contri- asteroid belt (12), and chaos due to the overlap of three-body bution, we review our recent work elucidating the physics of sec- MMRs accounts for the very weak chaos in the outer solar sys- ular chaos and applying it to Mercury and to hot Jupiters. We also tem (8). Because chaos in the solar system is typically driven by present results comparing the inclinations of hot Jupiters thus pro- overlapping resonances [e.g., see review (13)], one might reason duced with observations. that the secular chaos of the inner solar system is driven by overlapping “secular resonances.” Laskar (14, 15) and Sussman ASTRONOMY planetary dynamics | extrasolar planets and Wisdom (16) identified a number of candidate secular res- onances that might drive chaos in the inner solar system by ex- he question of the stability of the solar system has a long and amining angle combinations that alternately librated and circulated Tillustrious history (e.g., ref. 1). It was finally answered with in their simulations. However, there are an infinite number of such the aid of computer simulations (2–4), which have shown that the angle combinations, and it is not clear which are dynamically im- solar system is “marginally stable”: it is chaotically unstable, but portant—or why (13, 16). on a timescale comparable to its age. In the inner solar system, In ref. 17, we developed the theory for secular chaos, and the planets’ eccentricities (e) and inclinations (i) diffuse in bil- applied it to Mercury, the solar system’s most unstable planet. lions of years, with the two lightest planets, Mercury (Fig. 1) and We demonstrated how the locations and widths of general sec- Mars, experiencing particularly large variations. Mercury may ular resonances can be calculated, and how the overlap of the even be lost from the solar system on a billion-year timescale (5– relevant resonances quantitatively explains Mercury’s chaotic 7). [Chaos is much weaker in the outer solar system than in the orbit. This theory, which we review below, shows why Mercury’s inner (8–10).] However, despite the spectacular success in solv- motion is nonlinear—and chaotic—even though e’s and i’s re- ing solar system stability, fundamental questions remain: What main modest. It also shows that Mercury lies just above the is the theoretical explanation for orbital chaos of the solar sys- threshold for chaotic diffusion. tem? What does the chaotic nature of the solar system teach us A system of just two secularly interacting planets can be cha- about its history and organization? Also, how does this relate to otic if their eccentricities and inclinations are both of order unity extrasolar systems? (18–20). In systems with three or more planets, however, there is For well-spaced planets that are not close to strong mean- a less stringent criterion on the minimum eccentricity and inclination motion resonances (MMRs), the orbits evolve on timescales much longer than orbital periods. Hence one may often simplify Significance the problem by orbit-averaging the interplanetary interactions. “ ” The averaged equations are known as the secular equations Planets perturb one another as they orbit their star. These (e.g., ref. 11). To linear order in masses, secular evolution con- perturbations can build up over a long time, leading to in- sists of interactions between rings, which represent the planets stability and chaos, and, ultimately, to dramatic events such as after their masses have been smeared out over an orbit. Secular interplanetary collisions. We focus here on “secular chaos,” evolution dominates the evolution of the terrestrial planets in the which is the chaos that arises in the orbit-averaged equations. solar system (5), and it is natural to suppose that it dominates in We explain how secular chaos works and show that it explains many extrasolar systems as well. This is the type of planetary the chaos of Mercury’s orbit. We also show that secular chaos interaction we focus on in this contribution. could be responsible for the formation of “hot Jupiters,” which One might be tempted by the small eccentricities and incli- are Jupiter-mass planets that orbit very close to their star. Fi- nations in the solar system to simplify further and linearize the nally, we suggest that secular chaos might be important not secular equations, i.e., consider only terms to leading order in only for Mercury and hot Jupiters, but also more generally for eccentricity and inclination. Linear secular theory reduces to organizing a wide variety of extrasolar planetary systems. a simple eigenvalue problem. For N secularly interacting planets, the solution consists of 2N eigenmodes: N for the eccentricity Author contributions: Y.L. and Y.W. wrote the paper. degrees of freedom and another N for the inclination. Each of The authors declare no conflict of interest. the eigenmodes evolves independently of the others (11). Linear This article is a PNAS Direct Submission. A.S.B. is a guest editor invited by the Ed- secular theory provides a satisfactory description of the planets’ itorial Board. orbits on million-year timescales. [Higher accuracy can be achieved 1To whom correspondence should be addressed. E-mail: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1308261110 PNAS Early Edition | 1of6 Downloaded by guest on September 29, 2021 1=2 conjugate pair [e.g., the Poincaré variables Γ ≈ mðGMpaÞ e2=2 and γ = − ϖ] and employs the usual Hamilton’s equations for Γ and γ. The same is true for the outer planet. One finds, after writing the resulting equations in terms of complex eccentricities (z ≡ eeiϖ and z′ ≡ e′eiϖ′), pffiffiffiffi pffiffiffiffi ′ d z 2f2n μ′ a′ −βμ′ a′ z 20 = i pffiffiffi pffiffiffi pffiffiffi ; [2] dt z′ a −βμ a μ a z′ 10 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 where n′ = GMp=a′ , μ = m=Mp,andβ = −f10=ð2f2Þ (for example, 0 β ≈ 5α=4 for small α). The solution to this linear set of equations is a sum of two eigenmodes, each of which has a constant am- plitude and a longitude that precesses uniformly in time. The Fig. 1. Mercury’s chaos in an N-body integration of the solar system. Black theory may be trivially extended to N planets, leading to N eigenm- shows Mercury’s total eccentricity and inclination. Green shows free e and i, odes. It may also be extended to linear order in inclinations, which iΩ which were obtained by filtering out the forcing frequencies of the other leads to a second set of N eigenmodes. The equations for ζ = ie planets in Fourier space. The wander of the green curves demonstrates that are identical to those for z,butwithβ → 1. Mercury’s orbit is chaotic. Adapted from ref. 17. Overlap of Secular Resonances Drives Secular Chaos. The linear equations admit the possibility for a secular resonance, which required for chaos, and the character of secular chaos is more occurs when two eigenfrequencies match. Consider a massless diffusive. This diffusive type of secular chaos promotes equi- particle perturbed by a precessing mode. In anticipation of ap- partition between different secular degrees of freedom. During plication to the solar system, one may think of the test particle as secular chaos, the angular momentum of each planet varies Mercury, and the mode as the one dominated by Jupiter. Eq. 2 chaotically, with the innermost planet being slightly more sus- implies that the particle’s z is governed by the following: ceptible to large variations (21). If enough angular momentum is removed from that planet, its pericenter will approach the star. dz À Á igmt In addition, if that planet resembles Jupiter, tidal interaction = iγ z − eme ; [3] dt with its host star may then remove its orbital energy, turning it into a hot Jupiter. Hot Saturns or hot Neptunes may also be where γ is the particle’s free precession rate, gm is the mode’s produced similarly.