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# Secular chaos and its application to Mercury, hot Jupiters, and the organization of planetary systems

Edited by Adam S. Burrows, Princeton University, Princeton, NJ, and accepted by the Editorial Board October 31, 2013 (received for review May 2, 2013)

## Significance

Planets perturb one another as they orbit their star. These perturbations can build up over a long time, leading to instability and chaos, and, ultimately, to dramatic events such as interplanetary collisions. We focus here on “secular chaos,” which is the chaos that arises in the orbit-averaged equations. We explain how secular chaos works and show that it explains the chaos of Mercury's orbit. We also show that secular chaos could be responsible for the formation of “hot Jupiters,” which are Jupiter-mass planets that orbit very close to their star. Finally, we suggest that secular chaos might be important not only for Mercury and hot Jupiters, but also more generally for organizing a wide variety of extrasolar planetary systems.

## Abstract

In the inner solar system, the planets’ orbits evolve chaotically, driven primarily by secular chaos. Mercury has a particularly chaotic orbit and is in danger of being lost within a few billion years. Just as secular chaos is reorganizing the solar system today, so it has likely helped organize it in the past. We suggest that extrasolar planetary systems are also organized to a large extent by secular chaos. A hot Jupiter could be the end state of a secularly chaotic planetary system reminiscent of the solar system. However, in the case of the hot Jupiter, the innermost planet was Jupiter (rather than Mercury) sized, and its chaotic evolution was terminated when it was tidally captured by its star. In this contribution, we review our recent work elucidating the physics of secular chaos and applying it to Mercury and to hot Jupiters. We also present results comparing the inclinations of hot Jupiters thus produced with observations.

The question of the stability of the solar system has a long and illustrious history (e.g., ref. 1). It was finally answered with the aid of computer simulations (2⇓–4), which have shown that the solar system is “marginally stable”: it is chaotically unstable, but on a timescale comparable to its age. In the inner solar system, the planets’ eccentricities (*e*) and inclinations (*i*) diffuse in billions of years, with the two lightest planets, Mercury (Fig. 1) and Mars, experiencing particularly large variations. Mercury may even be lost from the solar system on a billion-year timescale (5⇓–7). [Chaos is much weaker in the outer solar system than in the inner (8⇓–10).] However, despite the spectacular success in solving solar system stability, fundamental questions remain: What is the theoretical explanation for orbital chaos of the solar system? What does the chaotic nature of the solar system teach us about its history and organization? Also, how does this relate to extrasolar systems?

For well-spaced planets that are not close to strong mean-motion resonances (MMRs), the orbits evolve on timescales much longer than orbital periods. Hence one may often simplify the problem by orbit-averaging the interplanetary interactions. The averaged equations are known as the “secular” equations (e.g., ref. 11). To linear order in masses, secular evolution consists of interactions between rings, which represent the planets after their masses have been smeared out over an orbit. Secular evolution dominates the evolution of the terrestrial planets in the solar system (5), and it is natural to suppose that it dominates in many extrasolar systems as well. This is the type of planetary interaction we focus on in this contribution.

One might be tempted by the small eccentricities and inclinations in the solar system to simplify further and linearize the secular equations, i.e., consider only terms to leading order in eccentricity and inclination. Linear secular theory reduces to a simple eigenvalue problem. For *N* secularly interacting planets, the solution consists of 2*N* eigenmodes: *N* for the eccentricity degrees of freedom and another *N* for the inclination. Each of the eigenmodes evolves independently of the others (11). Linear secular theory provides a satisfactory description of the planets’ orbits on million-year timescales. [Higher accuracy can be achieved by including the most important MMRs (11).] It is the cause, for example, of Earth’s eccentricity-driven Milankovitch cycle. However, on timescales ≳10^{7} years, the evolution is chaotic (e.g., Fig. 1), in sharp contrast to the prediction of linear secular theory. That appears to be puzzling, given the small eccentricities and inclinations in the solar system.

However, despite its importance, there has been little theoretical understanding of how secular chaos works. Conversely, chaos driven by MMRs is much better understood. For example, chaos due to MMR overlap explains the Kirkwood gaps in the asteroid belt (12), and chaos due to the overlap of three-body MMRs accounts for the very weak chaos in the outer solar system (8). Because chaos in the solar system is typically driven by overlapping resonances [e.g., see review (13)], one might reason that the secular chaos of the inner solar system is driven by overlapping “secular resonances.” Laskar (14, 15) and Sussman and Wisdom (16) identified a number of candidate secular resonances that might drive chaos in the inner solar system by examining angle combinations that alternately librated and circulated in their simulations. However, there are an infinite number of such angle combinations, and it is not clear which are dynamically important—or why (13, 16).

In ref. 17, we developed the theory for secular chaos, and applied it to Mercury, the solar system’s most unstable planet. We demonstrated how the locations and widths of general secular resonances can be calculated, and how the overlap of the relevant resonances quantitatively explains Mercury’s chaotic orbit. This theory, which we review below, shows why Mercury’s motion is nonlinear—and chaotic—even though *e*’s and *i*’s remain modest. It also shows that Mercury lies just above the threshold for chaotic diffusion.

A system of just two secularly interacting planets can be chaotic if their eccentricities and inclinations are both of order unity (18⇓–20). In systems with three or more planets, however, there is a less stringent criterion on the minimum eccentricity and inclination required for chaos, and the character of secular chaos is more diffusive. This diffusive type of secular chaos promotes equipartition between different secular degrees of freedom. During secular chaos, the angular momentum of each planet varies chaotically, with the innermost planet being slightly more susceptible to large variations (21). If enough angular momentum is removed from that planet, its pericenter will approach the star. In addition, if that planet resembles Jupiter, tidal interaction with its host star may then remove its orbital energy, turning it into a hot Jupiter. Hot Saturns or hot Neptunes may also be produced similarly. As shown in ref. 21 and reviewed below, such a migration mechanism can reproduce a range of observed features of hot Jupiters, giant planets that orbit their host stars at periods of a few days. It differs from other mechanisms that have been proposed for migrating hot Jupiters, including disk migration (22, 23), planet scattering (24), and Kozai migration by a stellar or planetary companion (20, 25, 26).

These studies prompt us to suggest that secular chaos may play an important role in reshaping planetary systems after they emerge from their nascent disks. Secular chaos causes planets’ eccentricities to randomly wander. When one of the planets attains high enough *e* that it suffers collision, ejection, or tidal capture, the removal of that planet can then lead to a more stable system, with a longer chaotic diffusion time. Such a scenario (e.g., ref. 1) can explain why the solar system, as well as many observed exoplanetary systems, are perched on the threshold of instability.

## Theory of Secular Chaos

### Linear Secular Theory.

We review first linear secular theory before introducing nonlinear effects. The equations of motion may be derived from the expression for the energy, which we shall label *H* because it turns into the Hamiltonian after replacing orbital elements with canonical variables. Focusing first on two coplanar planets, their secular interaction energy is, to leading order in eccentricities and dropping constant terms,*e* (which lead to nonlinear equations). Here, unprimed and primed quantities denote the inner and outer planets and *f*_{j} are Laplace coefficients that are functions of *e* and ϖ in the interaction energy with a canonically conjugate pair [e.g., the Poincaré variables *α*). The solution to this linear set of equations is a sum of two eigenmodes, each of which has a constant amplitude and a longitude that precesses uniformly in time. The theory may be trivially extended to *N* planets, leading to *N* eigenmodes. It may also be extended to linear order in inclinations, which leads to a second set of *N* eigenmodes. The equations for *z*, but with

### Overlap of Secular Resonances Drives Secular Chaos.

The linear equations admit the possibility for a secular resonance, which occurs when two eigenfrequencies match. Consider a massless particle perturbed by a precessing mode. In anticipation of application to the solar system, one may think of the test particle as Mercury, and the mode as the one dominated by Jupiter. Eq. **2** implies that the particle’s *z* is governed by the following:*γ* is the particle’s free precession rate, *g*_{m} is the mode’s precession rate, and *e*_{m} is proportional to the mode’s amplitude (i.e., to the eccentricities of the massive planets that participate in the mode). To order of magnitude, **3** is a sum of free and forced eccentricities:*γ*. The forced eccentricity precesses at the frequency of the mode that drives it, and its amplitude is proportional to that mode’s amplitude. Formally, it diverges at resonance,

The leading nonlinear correction to Hamiltonian (1) is fourth order in eccentricity, which changes Eq. **3** to the following:*γ* to **4**). Second, if the particle is not at linear resonance it can still approach resonance when its eccentricity changes. With nonlinearity included, a resonance takes on the familiar shape of a “cat’s-eye” in phase space, and a particle can librate stably in resonance (figure 1 of ref. 17).

Although the nonlinear cat’s-eye protects against divergences, danger lurks at the corner of a cat’s-eye: an unstable fixed point. Motion due to a single resonance (e.g., Eq. **5**) is perfectly regular (nonchaotic). However, if there is a second resonance nearby in phase space—in particular, if the separatrices enclosing two different cats’-eyes overlap—chaos ensues (figure 2 of ref. 17). Chaos due to the overlapping of secular resonances drives Mercury’s long-term evolution and may well be one of the key drivers of the long-term evolution of planetary systems.

## Mercury

The theory described above for coplanar secular chaos was first developed in ref. 27. However, to explain secular chaos in the solar system, one must extend it to include nonzero inclinations, which we did in ref. 17. Mercury has two free frequencies, one for its eccentricity (*z*) and one for its inclination (ζ). [We continue to treat Mercury as massless, which is an adequate approximation (17).] We denote these *g* and *s*, respectively. In linear theory, *i* so that *i*’s, our expansion to leading nonlinear order is suspect.) Two solar system modes have frequencies close to Mercury’s linear eccentricity precession rate (i.e., to *γ*): the Venus-dominated *e*-mode (frequency *g*_{2}) and the Jupiter-dominated *e*-mode (*g*_{5}). In addition, one mode’s frequency lies close to Mercury’s linear inclination precession rate (i.e., to −*γ*): the Venus-dominated *i*-mode (*s*_{2}). One can visualize this by imagining moving Mercury’s *a*, holding its *e* = *i* = 0. Because *γ* is a function of *a*, the linear secular resonances occur at particular values of *a*. The three aforementioned resonances lie ∼20% away from Mercury’s actual *a* at **6** and **7** show that the locations of these resonances move as Mercury’s *e* and *i* are increased. In fact, two of them (*g*_{5} and *s*_{2}) overlap very close to Mercury’s current orbital parameters. The overlapping of those two resonances is the underlying cause of Mercury’s chaos. Even though Mercury has relatively small *e* and *i*, its proximity to two secular resonances drives its orbit to be chaotic.

To make the above theory more precise, one must calculate the widths of the resonances, which are sketched in Fig. 2. If the resonant widths are negligibly small, Mercury would have to lie precisely at the region of overlap, which would be unlikely. One also has to account for higher-order (combinatorial) resonances, the most important one of which is

Fig. 3 compares Mercury’s orbital evolution in an *N*-body simulation of the solar system (blue points) with the prediction based on a simplified model that includes only the *g*_{5} and *s*_{2} forcing terms. Mercury’s true orbit traces the boundary of chaos as predicted by the model, illustrating that the model suffices to explain Mercury’s chaos. More dramatically, it also illustrates how Mercury is perched on the threshold of chaos. We speculate below as to how Mercury might have ended up in such a seemingly unlikely state.

## Hot Jupiters

The first batch of extrasolar planets that were discovered were “hot Jupiters” (28, 29). It is now clear that ∼1% of solar-type stars are orbited by Jovian giant planets with periods of ∼3 d. In comparison with this pileup of hot Jupiters at small separation (30⇓⇓–33), there is a deficit of gas giants with periods between 10 and 100 d [the “period valley” (34, 35)] before the number picks up and rises outward again [see reviews (36, 37)].

According to conventional theories of planet formation, hot Jupiters could not have formed in situ, given the large stellar tidal field, high gas temperature, and low disk mass to be found so close to the star. It is therefore commonly thought that these planets are formed beyond a few astronomical units and then are migrated inward. Candidate migration scenarios that have been proposed include protoplanetary disks (e.g., refs. 22, 23), Kozai migration by binary or planetary companions (e.g., refs. 20, 25, 26), scattering with other planets in the system (e.g., ref. 24), and secular chaos (21, 38). A critical review of these mechanisms is given in ref. 21.

Here, we present a short description of the secular chaos scenario. Moreover, now that the orbital axis (relative to the stellar rotation axis) of some 60 hot Jupiters has been measured, we compare the observed distribution against that produced in a unique suite of secular chaos simulations.

### Simulations and Results.

Our fiducial planetary system is composed of three giant planets *e*’s and *i*’s might arise from, e.g., planet scattering or collisions.

The angular momentum deficit is defined as follows (e.g., refs. 39, 40):

In Fig. 4, we observe that the three planets secularly (and diffusively) transfer angular momentum (but not energy) for almost 300 My without major mishap, until the inner planet has gradually acquired so much AMD that its eccentricity, starting at

Because the AMD is transferred to the inner planet to raise its eccentricity, the outer two planets end up with reduced eccentricities and mutual inclinations. They remain at large separations, waiting to be probed by techniques such as radial velocity, astrometry, or gravitational lensing. By getting rid of their inner companion, the remaining planets organize themselves into a more stable configuration, analogous to what would happen in the inner solar system after the loss of Mercury.

In addition to the showcase in Fig. 4, we have performed for this contribution 100 simulations with *e*’s are typical of those measured for extrasolar giant planets (not hot Jupiters). The origin of this AMD, however, is beyond the scope of this review.] The planets were initially at 3, 15, and *a*_{3} AU, with *a*_{3} uniformly distributed between 30 and 60 AU. We find that roughly 60% of these systems produce a hot Jupiter. Most of these newly formed Jupiters have orbits that are prograde relative to the original orbital plane, but some can be retrograde (Fig. 5—to be discussed in more detail below). Moreover, the time for secular chaos to excite the orbital eccentricities to tidal capture ranges from a few million years to a hundred-million years. Raising the initial AMD leads to more efficient hot Jupiter formation.

### Secular Chaos Confronting Observations.

In the following, we compare the predictions of secular chaos with observations, highlighting the distribution of spin-orbit angles.

There is a sharp inner cutoff to the 3-d pileup of hot Jupiters. They appear to avoid the region inward of twice the Roche radius (43), where the Roche radius is the distance within which a planet would be tidally shredded. New data spanning two orders of magnitude in planetary masses (and including planet radius measurements) have strengthened this claim. There are only five known exceptions lying inward of twice the Roche radius, and the rest mostly lie between twice and four times the Roche radius. Mechanisms that rely on eccentricity excitation, such as Kozai migration or planet–planet scattering, naturally produce hot Jupiters that tend to avoid the region inside of twice the Roche radius (43). However, only Kozai migration and secular chaos naturally produce a pileup just outside twice the Roche radius, as the eccentricity rise in these cases is gradual and planets are accumulated at the right location.

Hot Jupiters appear to be less massive than more distant planets (44⇓–46). Among planets discovered with the radial velocity method, close-in planets typically have projected masses (*M* sin *i*) less than twice Jupiter’s mass. However, numerous further out planets have

Secular chaos predicts that hot Jupiters may have misaligned orbits relative to the invariable plane of the system. Here, we use the stellar spin axis as the proxy for the latter, assuming that the stellar spin is aligned with the protoplanetary disk in which the planets were born. The spin-orbit angle can be probed in cases where the hot Jupiter transits its star, via the Rossiter–McLaughlin (R-M) effect (e.g., ref. 47). The sky-projected value of the stellar obliquity has been reported for some 60 hot Jupiters (Fig. 5). Although a majority of the hot Jupiters are aligned with the stellar spin, a smattering of them [especially those around hotter stars (48⇓–50)] appear to have isotropic orbits. The observed distribution can be decomposed into one that peaks at alignment and one that is isotropic (51).

Among the 60 hot Jupiters that formed in our set of 100 simulations, the vast majority have prograde orbits (with only 2 retrograde ones). This is because we initialized the simulations with 50% more AMD than the critical amount to form a hot Jupiter. In that case, when a sufficient amount of AMD has been transferred into the innermost planet to increase its eccentricity to ∼1, there is not much AMD left to excite its inclination too. In simulations with higher initial AMD, more inclined hot Jupiters tend to be produced. In our 100 simulations presented here, the spin-orbit angles are roughly Gaussian distributed with a FWHM of ∼30°. We project these angles onto the sky, assuming that the systems are randomly distributed relative to the line-of-sight (Fig. 5). The sky-projected obliquity (R-M angle) is dominated by nearly aligned planets, with R-M angle for 30% of the systems smaller than 2°, and 60% of the system within 10°. However, a significant tail, about 40% of the systems, extends to ∼50°. This may explain the observed population of prograde planets, especially considering that observational error bars tend to broaden the distribution. We note that a more coplanar mechanism like disk migration will likely produce a peak at alignment, but no tail.

Hot Jupiters also tend to be alone, at least out to a few astronomical units. From radial velocity surveys, ∼30% of planets are in multiple planet systems [including ones with radial velocity trends (31)], whereas only five hot Jupiters, i.e., fewer than 10% of hot Jupiters are known to have companions within a couple astronomical units. This relative deficit also shows up in the transit sample, where most attempts at detecting transit timing variations caused by close companions of hot Jupiters (52, 53) have been unsuccessful (e.g., refs. 54⇓⇓–57). That contrasts with the many transit timing variation detections for other kinds of planets (e.g., refs. 58⇓–60). Such an absence of close-by companions to hot Jupiters is consistent with the picture that hot Jupiters had high eccentricities in the past. Secular chaos also predicts that, in systems with hot Jupiters, there are at least two other giant planets roaming at larger distances. This is testable with ongoing long-term, high-precision radial-velocity monitoring.

Both secular chaos and Kozai migration predict that hot Jupiters are migrated after the disk dispersal. So detection of such objects around T Tauri stars can be used to falsify these theories.

## Organization of Planetary Systems by Secular Chaos

The fact that the solar system is marginally stable might be hinting at a deeper truth about how planetary systems are organized. It seems implausible that the solar system was so finely tuned at birth to yield an instability time comparable to its age today. Rather, the solar system might have maintained a state of marginal stability at all times (1). In this scenario, the stability time was shorter when the solar system was younger because there were more planets then. As the solar system aged, it lost planets to collision or ejection. Each loss lengthens the stability time because a more widely spaced system is more stable. In this way, the solar system would naturally maintain marginal stability. The precarious state of Mercury on the threshold of chaos (Fig. 3) might merely be the last manifestation of such a self-organizing process. Similarly, hot Jupiters might be the most conspicuous evidence that extrasolar systems also undergo such self-organization.

We suggest that secular chaos might be responsible to a large extent for organizing planetary systems. In secular interactions, AMD is conserved—one may think of AMD as the free energy. We conjecture that secular chaos drives systems toward equipartition of AMD, such that, on average, all secular modes have equal AMD. That would be consistent with the terrestrial planets, where the lightest planets are the most excited ones. Let us consider a possible scenario for how planetary systems evolve (see also ref. 1). Initially, planets merge or are ejected until the AMD is such that neighboring planets cannot collide in a state of equipartition. The secular evolution on long timescales is then set by fluctuations about equipartition—one planet (or more properly its mode) happens to gain a sufficiently large portion of the AMD that it merges with its neighbor, or is ejected, or approaches the star and forms a hot Jupiter. After such an event, the AMD would decrease, and the planetary system would be more stable than before. However, on a longer timescale, fluctuations can once again lead to instability. Of course, this scenario is speculative and must be tested against simulations and observations. Fortunately, the hundreds of planetary systems recently discovered provide a test bed for such explorations.

## Acknowledgments

Y.L. acknowledges support from National Science Foundation Grant AST-1109776. Y.W. acknowledges support by the Natural Sciences and Engineering Research Council and the Government of Ontario, Canada.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: y-lithwick{at}northwestern.edu.

Author contributions: Y.L. and Y.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. A.S.B. is a guest editor invited by the Editorial Board.

## References

- ↵
- ↵
- Sussman GJ,
- Wisdom J

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Murray N,
- Holman M

- ↵
- Grazier KR,
- Newman WI,
- Hyman JM,
- Sharp PW,
- Goldstein DJ

- ↵
- Hayes WB

- ↵
- Murray CD,
- Dermott SF

- ↵
- ↵
- ↵
- ↵
- Laskar J

*Chaos, Resonance, and Collective Dynamical Phenomena in the Solar System: Proceedings of the 152nd Symposium of the International Astronomical Union*, ed Ferraz-Mello S (Kluwer, Dordrecht), pp 1–16. - ↵
- Sussman GJ,
- Wisdom J

- ↵
- ↵
- ↵
- Migaszewski C,
- Goździewski K

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Wittenmyer RA,
- et al.

*ApJ*722:1854–1863. - ↵
- Marcy G,
- et al.

*Progress of Theoretical Physics Supplement*158:24–42. - ↵
- ↵
- ↵
- Laskar J

- ↵
- Ogilvie GI

- ↵
- ↵
- Hut P

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Holman MJ,
- Murray NW

- ↵
- Agol E,
- Steffen J,
- Sari R,
- Clarkson W

- ↵
- ↵
- ↵
- Hrudková M,
- et al.

- ↵
- Steffen JH,
- et al.

- ↵
- ↵
- Steffen JH,
- et al.

- ↵

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