Secular Chaos and Its Application to Mercury, Hot Jupiters, and The

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 resonances 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 amplitude 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 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. As shown in ref. 21 and reviewed below, such precession rate, and em is proportional to the mode’s amplitude a migration mechanism can reproduce a range of observed fea- (i.e., to the eccentricities of the massive planets that participate tures of hot Jupiters, giant planets that orbit their host stars at 3 in the mode). To order of magnitude, γ ∼ nμpαp and em ∼ αpep, periods of a few days. It differs from other mechanisms that have where starred quantities correspond to the planets that dominate been proposed for migrating hot Jupiters, including disk migra- the forcing (and assuming αp 1). The solution to Eq. 3 is a sum tion (22, 23), planet scattering (24), and Kozai migration by of free and forced eccentricities: a stellar or planetary companion (20, 25, 26). γ These studies prompt us to suggest that secular chaos may play iγt igmt z = const × e + em e : [4] an important role in reshaping planetary systems after they γ − gm emerge from their nascent disks. Secular chaos causes planets’ eccentricities to randomly wander. When one of the planets The free eccentricity exists even in the absence of the mode, and attains high enough e that it suffers collision, ejection, or tidal it precesses at frequency γ. The forced eccentricity precesses at capture, the removal of that planet can then lead to a more the frequency of the mode that drives it, and its amplitude is stable system, with a longer chaotic diffusion time. Such a sce- proportional to that mode’s amplitude. Formally, it diverges at nario (e.g., ref. 1) can explain why the solar system, as well as resonance, γ = gm. However, nonlinearities alter that conclusion. many observed exoplanetary systems, are perched on the thresh- The leading nonlinear correction to Hamiltonian (1) is fourth old of instability. order in eccentricity, which changes Eq. 3 to the following: ! ! Theory of Secular Chaos dz jzj2 Linear Secular Theory. We review first linear secular theory before = iγ z 1 − − e eigmt : [5] dt 2 m 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 [This equation assumes α 1, for algebraic simplicity. If α ∼ 1, elements with canonical variables. Focusing first on two coplanar as in many extrasolar systems, the order unity numerical con- planets, their secular interaction energy is, to leading order in stants would be altered.] Hence nonlinearity reduces the fre- 2 eccentricities and dropping constant terms, quency of free precession from γ to γð1 − e =2Þ. There are a number of interesting consequences. First, even if the particle mm′ is at exact linear resonance ðγ = gmÞ, then as its eccentricity H = − G f e2 + e′2 + f ee′ cosðϖ − ϖ′Þ ; [1] a′ 2 10 changes its frequency shifts away from resonance, protecting it against the divergence that appears in linear theory (Eq. 4). following the notation of ref. 11 and dropping higher-order terms Second, if the particle is not at linear resonance it can still ap- in e (which lead to nonlinear equations). Here, unprimed and proach resonance when its eccentricity changes. With nonlinear- “ ’ primed quantities denote the inner and outer planets and fj are ity included, a resonance takes on the familiar shape of a cat s- Laplace coefficients that are functions of α ≡ a=a′ (appendix B in eye” in phase space, and a particle can librate stably in resonance ref. 11). In secular theory, the semimajor axes are constant (even (figure 1 of ref. 17). to nonlinear order), and hence may be considered as parameters. Although the nonlinear cat’s-eye protects against divergences, To derive the equations of motion for the inner planet, one danger lurks at the corner of a cat’s-eye: an unstable fixed point. replaces e and ϖ in the interaction energy with a canonically Motion due to a single resonance (e.g., Eq. 5) is perfectly regular

2of6 | www.pnas.org/cgi/doi/10.1073/pnas.1308261110 Lithwick and Wu Downloaded by guest on September 29, 2021 (nonchaotic). However, if there is a second resonance nearby in phase space—in particular, if the separatrices enclosing two SPECIAL FEATURE 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 con- tinue to treat Mercury as massless, which is an adequate ap- proximation (17).] We denote these g and s, respectively. In linear theory, g = − s = γ. However, to leading nonlinear order, these frequencies are modified in the manner described above to the following: e2 g = γ 1 − − 2i2 [6] 2 i2 s = γ −1 − 2e2 + [7] 2 Fig. 2. Secular resonances that lead to Mercury’s orbital chaos, showing that Mercury’s current orbit is close to the overlap of two secular resonances (17). Each of these frequencies can be in resonance with one of (i.e., g = g5 with s = s2). To plot the lines at the center of these resonances, we — ASTRONOMY the other 13 planetary modes two for each planet, excluding assume the current values of g5 and s2, and that Mercury remains with e = i. the zero frequency inclination mode that defines the invariable The widths of the resonances, as calculated in ref. 17, are shaded. The widths plane. (Another resonance—the Kozai resonance—occurs at depend not only on Mercury’s orbital parameters, but on the other planets’ high enough i so that g = s, although at such high i’s, our expan- as well. To plot these widths, we scale all of the planets’ e’s and i’sbythe sion to leading nonlinear order is suspect.) Two solar system same factor (κ; see right axis) relative to their current values. We do this to modes have frequencies close to Mercury’s linear eccentricity illustrate how the solar system is perched on the threshold of chaos. For this ( precession rate (i.e., to γ): the Venus-dominated e-mode (fre- plot, the widths are only correct within a factor of 2 because the exact width (see ref. 17) depends on the trajectory of an orbit in e-i space. quency g2) and the Jupiter-dominated e-mode (g5). In addition, one mode’s frequency lies close to Mercury’s linear inclination −γ precession rate (i.e., to ): the Venus-dominated i-mode (s2). how Mercury is perched on the threshold of chaos. We speculate One can visualize this by imagining moving Mercury’s a, holding = = γ below as to how Mercury might have ended up in such a seem- its e i 0. Because is a function of a, the linear secular ingly unlikely state. resonances occur at particular values of a. The three aforementioned resonances lie ∼20% away from Mercury’s actual a at Hot Jupiters = = e i 0 (Fig. 2). Because Mercury is at some distance from those The first batch of extrasolar planets that were discovered were resonances to linear order, it is at first sight surprising that they “hot Jupiters” (28, 29). It is now clear that ∼1% of solar-type can play an important role in Mercury’s evolution. However, stars are orbited by Jovian giant planets with periods of ∼3d.In Eqs. 6 and 7 show that the locations of these resonances move comparison with this pileup of hot Jupiters at small separation as Mercury’s e and i are increased. In fact, two of them (g5 and s2) ’ (30–33), there is a deficit of gas giants with periods between 10 overlap very close to Mercury s current orbital parameters. “ ” The overlapping of those two resonances is the underlying cause and 100 d [the period valley (34, 35)] before the number picks of Mercury’s chaos. Even though Mercury has relatively small e up and rises outward again [see reviews (36, 37)]. and i, its proximity to two secular resonances drives its orbit to According to conventional theories of planet formation, hot be chaotic. Jupiters could not have formed in situ, given the large stellar To make the above theory more precise, one must calculate tidal field, high gas temperature, and low disk mass to be found the widths of the resonances, which are sketched in Fig. 2. If the so close to the star. It is therefore commonly thought that these resonant widths are negligibly small, Mercury would have to lie planets are formed beyond a few astronomical units and then are precisely at the region of overlap, which would be unlikely. One migrated inward. Candidate migration scenarios that have been also has to account for higher-order (combinatorial) resonances, proposed include protoplanetary disks (e.g., refs. 22, 23), Kozai the most important one of which is ðg − g5Þ − ðs − s2Þ. That migration by binary or planetary companions (e.g., refs. 20, 25, combinatorial resonance was identified by Laskar (14) from the 26), scattering with other planets in the system (e.g., ref. 24), and fact that it librated in his simulations for 200 My. In ref. 17, we secular chaos (21, 38). A critical review of these mechanisms is calculated the widths of the aforementioned resonances and given in ref. 21. showed that the widths match in detail what is seen in simu- Here, we present a short description of the secular chaos lations (see figures 4, 6, and 7 in that paper). scenario. Moreover, now that the orbital axis (relative to the Fig. 3 compares Mercury’s orbital evolution in an N-body stellar rotation axis) of some 60 hot Jupiters has been measured, simulation of the solar system (blue points) with the prediction we compare the observed distribution against that produced in based on a simplified model that includes only the g5 and s2 a unique suite of secular chaos simulations. forcing terms. Mercury’s true orbit traces the boundary of chaos as predicted by the model, illustrating that the model suffices to Simulations and Results. Our fiducial planetary system is composed explain Mercury’s chaos. More dramatically, it also illustrates of three giant planets ð0:5; 1:0; 1:5MJ Þ that are well spaced (1, 6,

Lithwick and Wu PNAS Early Edition | 3of6 Downloaded by guest on September 29, 2021 decreasing periapse, the Roche radius (inside of which the planet would be shredded) roughly characterizes the distance at which tides stall any further periapse decrease.] We then specify in our numerical simulation that tidal interaction with the central star removes orbital energy from the planet (42). The orbit decays inward, until the planet is tidally circularized at ∼2× Roche radius (because of angular momentum conservation). The inner planet is now captured into a “hot Jupiter.” 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 separa- tions, 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 performedpffiffiffiffiffi for this contribution 100 simulations with AMD = 1:5m1 a1, 50% over the minimum criterion. [This AMD corresponds to e; i ∼ 0:3. Such e’s are typical of those measured for extrasolar giant planets Fig. 3. The blue points show the running average of Mercury’s e2 and i2 in (not hot Jupiters). The origin of this AMD, however, is beyond an N-body simulation of the solar system that lasts 600 My. The black points the scope of this review.] The planets were initially at 3, 15, and illustrate where chaos occurs in a highly simplified model that includes only a3 AU, with a3 uniformly distributed between 30 and 60 AU. We the g5 and s2 forcing frequencies. Those points denote the time-averaged find that roughly 60% of these systems produce a hot Jupiter. results from simulations that are initialized on a regular grid; hence the regularly spaced black points denote nonchaotic orbits, and the irregularly Most of these newly formed Jupiters have orbits that are pro- spaced points denote chaotic ones. The fact that Mercury’s orbit lies within grade relative to the original orbital plane, but some can be ret- the chaotic zone shows that our highly simplified model is sufficient to ac- rograde (Fig. 5—to be discussed in more detail below). Moreover, count for Mercury’s chaos. Moreover, this model demonstrates that Mercury the time for secular chaos to excite the orbital eccentricities to is perched on the threshold of chaos. The red lines denote various secular tidal capture ranges from a few million years to a hundred-million resonances that involve the g5 and s2 frequencies. Adapted from figure 5 of years. Raising the initial AMD leads to more efficient hot ref. 17; see that paper for more detail. Jupiter formation.

Secular Chaos Confronting Observations. In the following, we com- 16 AU) with initially mild eccentricities and inclinations (e = 0:07 pare the predictions of secular chaos with observations, highlight- to 0.3; inclination, 4.5–20°; see table 1 of ref. 21). Such a config- ing the distribution of spin-orbit angles. uration is possible for a system that emerges out of a dissipative There is a sharp inner cutoff to the 3-d pileup of hot Jupiters. protoplanetary disk, as it avoids short-term instabilities. However, They appear to avoid the region inward of twice the Roche ra- disk–planet interactions are not yet well understood. It is also dius (43), where the Roche radius is the distance within which possible that disks always damp planets onto nearly circular co- a planet would be tidally shredded. New data spanning two e’ i’ planar orbits, in which case such sand s might arise from, e.g., orders of magnitude in planetary masses (and including planet planet scattering or collisions. The angular momentum deficit is defined as follows (e.g., refs. 39, 40): X qffiffiffiffiffiffiffiffiffiffiffiffi X pffiffiffiffiffi 1 pffiffiffiffiffi AMD ≡ m a 1 − 1 − e2cos i ≈ m a e2 + i2 ; k k k k 2 k k k k k k [8]

where the summation is over all planets. The AMD describes the deficit in angular momentum relative to that of a coplanar, circular system. When the AMD is not zero, secular interactions continuously modify the planets’ eccentricities and inclinations, preserving the total AMD (because the total angular momentum is conserved, and secular interactions do not modify the orbital energies). A system with a higher AMD will interact more strongly, Fig. 4. Formation of a hot Jupiter in our fiducial system, simulated using the SWIFT code (41) with tidal and general relativity effects added. (Left) Radial and above some critical AMD value, the evolution is chaotic (e.g., excursions of the three planets (semimajor axis, periapse, and apoapse) are Fig. 2). To produce a hot Jupiter by secular interactionsp (requiringffiffiffiffiffi shown as functions of time, with the various radii relevant for hot Jupiters → = that e1 1), the minimum AMD valuep isffiffiffiffiffi AMD m1 a1.Our marked by arrows. (Right) Planet inclinations measured relative to the sys- fiducial system has an AMD of 1:17m1 a1. This AMD value is tem’s invariable plane. All planets initially have mildly eccentric and inclined also high enough for the system to be secularly chaotic. orbits, but over a period of 300 My so much of the angular momentum in In Fig. 4, we observe that the three planets secularly (and the innermost planet can be removed that its eccentricity and inclination diffusively) transfer angular momentum (but not energy) for al- can diffusively reach order unity values. The semimajor axes remain nearly most 300 My without major mishap, until the inner planet has constant until the end, a tell-tale sign that secular interactions dominate the dynamics. At ∼300 My, the pericenter of the inner planet reaches inward of gradually acquired so much AMD that its eccentricity, starting at = : = : a few stellar radii and tidal interaction with the central star kicks in. Precessions e1 0 07, has reached e1 0 985. This corresponds to a periapse by general relativity, by tidal and rotational quadrupoles, as well as tidal dis- distance of order the Roche radius, að1 − eÞ∼ 0.015 AU. [Be- sipation, prevent the pericenter from reaching inward of the Roche radius. As cause the strength of tidal damping rises very rapidly with a result, the final hot Jupiter has a period of ∼3 d. Adapted from ref. 21.

4of6 | www.pnas.org/cgi/doi/10.1073/pnas.1308261110 Lithwick and Wu Downloaded by guest on September 29, 2021 Among the 60 hot Jupiters that formed in our set of 100 simulations, the vast majority have prograde orbits (with only 2 SPECIAL FEATURE 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 Fig. 5. The upper panel shows the sky-projected spin-orbit angles (in astronomical units. This relative deficit also shows up in the absolute value) for some 60 hot Jupiters (m sin i ≥ 0:3MJ; data taken from exoplanets.org) as a function of host star effective temperatures. The transit sample, where most attempts at detecting transit timing ASTRONOMY lower panel shows the distribution of this projected angle (green shaded variations caused by close companions of hot Jupiters (52, 53) histogram). The two red histograms show the angle distribution obtained have been unsuccessful (e.g., refs. 54–57). That contrasts with from our simulations before projection (unshaded) and after projection the many transit timing variation detections for other kinds of (shaded). Secular chaos tends to produce prograde hot Jupiters, with the planets (e.g., refs. 58–60). Such an absence of close-by com- projected obliquities peaking at alignment, and a significant tail extend- panions to hot Jupiters is consistent with the picture that hot ∼ ing to 50°. 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 radius measurements) have strengthened this claim. There are with ongoing long-term, high-precision radial-velocity monitoring. only five known exceptions lying inward of twice the Roche ra- Both secular chaos and Kozai migration predict that hot dius, and the rest mostly lie between twice and four times the Jupiters are migrated after the disk dispersal. So detection of such Roche radius. Mechanisms that rely on eccentricity excitation, objects around T Tauri stars can be used to falsify these theories. such as Kozai migration or planet–planet scattering, naturally produce hot Jupiters that tend to avoid the region inside of twice Organization of Planetary Systems by Secular Chaos the Roche radius (43). However, only Kozai migration and sec- The fact that the solar system is marginally stable might be ular chaos naturally produce a pileup just outside twice the hinting at a deeper truth about how planetary systems are or- Roche radius, as the eccentricity rise in these cases is gradual and ganized. It seems implausible that the solar system was so finely planets are accumulated at the right location. tuned at birth to yield an instability time comparable to its age Hot Jupiters appear to be less massive than more distant today. Rather, the solar system might have maintained a state of planets (44–46). Among planets discovered with the radial ve- marginal stability at all times (1). In this scenario, the stability locity method, close-in planets typically have projected masses time was shorter when the solar system was younger because (M sin i) less than twice Jupiter’s mass. However, numerous there were more planets then. As the solar system aged, it lost further out planets have M sin i > 2MJ (figure 5 of ref. 37). This planets to collision or ejection. Each loss lengthens the stability is expected in the context of secular chaos (but not the Kozai time because a more widely spaced system is more stable. In this mechanism). Because the minimum AMD to produce a hot Ju- way, the solar system would naturally maintain marginal stability. piter rises with the planetary mass, we expect hot Jupiters to be The precarious state of Mercury on the threshold of chaos (Fig. 3) lower mass than average. might merely be the last manifestation of such a self-organizing Secular chaos predicts that hot Jupiters may have misaligned process. Similarly, hot Jupiters might be the most conspicuous evi- orbits relative to the invariable plane of the system. Here, we use dence that extrasolar systems also undergo such self-organization. We suggest that secular chaos might be responsible to a large the stellar spin axis as the proxy for the latter, assuming that the extent for organizing planetary systems. In secular interactions, stellar spin is aligned with the protoplanetary disk in which the AMD is conserved—one may think of AMD as the free energy. planets were born. The spin-orbit angle can be probed in cases – We conjecture that secular chaos drives systems toward equi- where the hot Jupiter transits its star, via the Rossiter McLaughlin partition of AMD, such that, on average, all secular modes have (R-M) effect (e.g., ref. 47). The sky-projected value of the stellar equal AMD. That would be consistent with the terrestrial plan- obliquity has been reported for some 60 hot Jupiters (Fig. 5). ets, where the lightest planets are the most excited ones. Let us Although a majority of the hot Jupiters are aligned with the stellar consider a possible scenario for how planetary systems evolve spin, a smattering of them [especially those around hotter stars (see also ref. 1). Initially, planets merge or are ejected until the (48–50)] appear to have isotropic orbits. The observed distribution AMD is such that neighboring planets cannot collide in a state of can be decomposed into one that peaks at alignment and one that equipartition. The secular evolution on long timescales is then is isotropic (51). set by fluctuations about equipartition—one planet (or more

Lithwick and Wu PNAS Early Edition | 5of6 Downloaded by guest on September 29, 2021 properly its mode) happens to gain a sufficiently large portion scenario is speculative and must be tested against simulations of the AMD that it merges with its neighbor, or is ejected, or and observations. Fortunately, the hundreds of planetary systems approaches the star and forms a hot Jupiter. After such an event, recently discovered provide a test bed for such explorations. the AMD would decrease, and the planetary system would be ACKNOWLEDGMENTS. Y.L. acknowledges support from National Science Foun- more stable than before. However, on a longer timescale, fluc- dation Grant AST-1109776. Y.W. acknowledges support by the Natural Sciences tuations can once again lead to instability. Of course, this and Engineering Research Council and the Government of Ontario, Canada.

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