Resonances of Multiple Exoplanets and Implications for Their Formation
AAccepted for ApJ Letters 2014 May 29
Preprint typeset using L A TEX style emulateapj v. 5/2/11
RESONANCES OF MULTIPLE EXOPLANETS AND IMPLICATIONS FOR THEIR FORMATION
Xiaojia Zhang ∗ , Hui Li , Shengtai Li , Douglas N. C. Lin Accepted for ApJ Letters 2014 May 29
ABSTRACTAmong ∼
160 of the multiple exoplanetary systems confirmed, about 30% of them have neighboringpairs with a period ratio ≤
2. A significant fraction of these pairs are around mean motion resonance(MMR), more interestingly, peak around 2:1 and 3:2, with a clear absence of more closely packedMMRs with period ratios less than 4:3, regardless of planet masses. Here we report numerical sim-ulations demonstrating that such MMR behavior places important constraints on the disk evolutionstage out of which the observed planets formed. Multiple massive planets (with mass ≥ . M Jup )tend to end up with a 2:1 MMR mostly independent of the disk masses but low-mass planets (withmass ≤ M ⊕ ) can have MMRs larger than 4:3 only when the disk mass is quite small, suggestingthat the observed dynamical architecture of most low-mass-planet pairs was established late in thedisk evolution stage, just before it was dispersed completely. Subject headings: planet-disk interactions — planetary systems - protoplanetary disks INTRODUCTION
Many multiple planetary candidates were discoveredby Kepler’s transit search. It has been pervasively sug-gested that the majority of them are indeed genuinemultiple-planet systems (Lissauer et al. 2012; Batalhaet al. 2013). A significant fraction of adjacent pairs ofplanets are in or near mean motion resonances (MMRs)with period ratios around 2:1, 3:2, 5:3, etc. (Lissaueret al. 2011b), although members of most multiple sys-tems do not have nearly commensurable orbits (Mayoret al. 2009).The MMR of exoplanet systems has been well stud-ied under various perspectives (Goldreich 1965; Wisdom1980; Henrard 1982; Weidenschilling & Davis 1985; Ogi-hara & Ida 2009; Ogihara et al. 2010; Lee et al. 2013). Awidely adopted scenario is that resonant pairs capturedeach other on their mutual MMRs through convergentmigration (Bryden et al. 2000; Kley 2000; Lee & Peale2002b). Migration mechanisms include tidal interactionbetween very short period planets and their host stars(Schlaufman et al. 2010) as well as protoplanet interac-tion with their natal disks (Lin et al. 1996). For planetswith periods longer than a few days, MMRs provide sup-porting evidence of the disk migration scenario.Gas-giant planets with sufficient mass to open gapsin their natal disks undergo Type II migration (Lin &Papaloizou 1986). Most
Kepler candidates have muchlower masses and undergo Type I migration (Goldreich& Tremaine 1980; Ward 1997; Paardekooper et al. 2011;Kretke & Lin 2012). In either case, orbital convergencewould occur if the inward migration of a planet catchesup with that of its siblings closer to the host star or ifa planet is trapped at some special disk region and itssiblings migrate toward it. Convergent migration leads * E-mail: [email protected] Department of Astronomy and Astrophysics, University ofCalifornia, Santa Cruz, CA 95064, USA Los Alamos National Laboratory, Los Alamos, NM 87545,USA Institute for Advanced Studies, Tsinghua University, Beijing,China to the possibility of resonant capture (Lee & Peale 2002a;Kley et al. 2004).Many multiple planet systems are indeed locked inor are close to MMRs. The current database on
The Extrasolar Planets Encyclopaedia
Website(http://exoplanet.eu) contains 157 confirmed multipleplanet systems with ∼
225 neighboring pairs. This sam-ple includes 118 systems of 2 planets, 22 systems of 3planets, 9 systems of 4 planets, 4 systems of 5 planets and4 systems of 6 planets. Figure 1 shows the distributionof these adjacent pairs as a function of their period ra-tios, focusing on systems with a period ratio around andless than 4. Overall, ∼
70 pairs (i.e., ∼ / both planets’ masses > . M Jupiter and below ∼ M ⊕ (designated as gas-giant and low-mass-planetsystems respectively) are also shown in Figure 1. Mostgas giants are discovered by the radial velocity method.Although their mass determinations are lower limits, thisobservational bias does not affect their statistics. Mostmultiple low-mass-planet systems are Kepler candidates.Their mass has been estimated by using an empirical for-mula to convert a planet’s radius to mass (Lissauer et al.2011b,a).Figure 1 confirms the previously known results thatthere are statistically clusters of systems in or near the2:1 and 3:2 MMRs (Lissauer et al. 2011b; Fabrycky &Kepler Science Team 2012). Since the confirmed Ke-pler multi-exoplanetary systems measured with TTVsare biasing the samples toward pairs near resonances,the peaks may be overrepresented, but one can still findthis statistical clustering feature by including all pairs ofKepler candidates (Goldreich & Schlichting 2014). Theexcess near the 3:2 MMR is mostly associated with low-mass-planet systems, whereas that near the 2:1 MMRcontains both low-mass-planet and gas-giant systems.If these systems formed through resonant capture dur-ing planetary migration, it appears that their asymp-totic MMR is a sensitive function of planet mass, orbitalcharacteristics, and relative migration rates (Ogihara &Kobayashi 2013). Gas-giant pairs such as GJ876 (Laugh- a r X i v : . [ a s t r o - ph . E P ] J un Zhang et al. 2014
Fig. 1.—
Top-bottom : distribution of period ratios of adja-cent pairs from all the confirmed multiple-planet systems (top) ,just the low-mass-planet systems with both planet masses ≤ M ⊕ (middle) , and just the gas-giant systems with both planet masses ≥ . M Jup (bottom) . The vertical dashed lines indicate the first-order MMR at 2:1, 3:2, 4:3, 5:4, 6:5 and 7:6, from right to left re-spectively. Note that only pairs with a period ratio ≤ lin et al. 2001; Rivera & Lissauer 2001; Laughlin et al.2005) and HD82943 (Lee et al. 2006; Tan et al. 2013)cluster mostly around the 2:1 MMR because the TypeII migration rate is relatively slow compared with thelibration timescale for the 2:1 MMR.However, Type I migration is considerably faster thanType II migration (Lin & Papaloizou 1986). This allowsmigrating pairs to bypass wider resonances, with slowerlibration timescales (Murray & Dermott 1999), and endup in more tightly packed configurations. It is notewor-thy that there is a deficit of both low-mass-planet andgas-giant pairs with period ratios smaller than 4:3. Most Kepler planetary candidates have orbital periods withina few months, and the possibility of not detecting a tran-siting companion with similar periods is small. Thus, theobserved deficit of closely packed resonant pairs and thepreferential concentration of lower-mass pairs near the3:2 MMR relative to the 2:1 MMR are statistically sig-nificant. METHOD
In order to reproduce these observations in terms ofthe resonant-capture scenario, we use a two-dimensional(2D) module of the LA-COMASS package developedat Los Alamos to investigate the orbital evolution ofmultiple planets in a protostellar disk. This 2D hy-drodynamical polar grid code solves the continuity andisothermal Navier-Stokes equations for gas inside a quasi-Keplerian disk subject to the gravity of a one-solar-masscentral star. The motion of multiple planets is calculatedwith a fourth-order Runge-Kutta solver. The evanescentboundary condition has been implemented in the code toprovide wave killing zones at each edge of the disk (deVal-Borro et al. 2006). This code has been used exten- sively for planet-disk interaction studies (e.g., Li et al.(2009)).In our simulations, the 2D disk is modeled within theradial range of [0 . R , . R ] where R is the distanceunit in the code. We investigate two main configura-tions: a disk with multiple gas giants and a disk withmultiple 10 M ⊕ planets. For the migration of gas giants,since their Type II migration does not sensitively dependon the disk structure (Lin & Papaloizou 1986) , we adoptmodels with smooth density, temperature and viscosityprofiles. However, the pace and direction of the Type Imigration does depend sensitively on the disk propertiesand they may be trapped in a disk near the magneto-spheric truncation radius, the inner edge of ”dead zone”or other locations such as the boundary between the vis-cous heating and irradiation heating region (Kretke &Lin 2012).For the migration of low-mass planets, we adopt a two-zone disk model based on the α prescription for viscosity.In order to approximate a disk structure in which themagnetohydrodynamic turbulence is prevalent through-out the inner region and the mostly neutral outer regionswith a “dead” midplane zone, we adopt a high valueof α vis = 0 .
004 for disk radius a < a crit = 0 . R and α vis = 0 .
001 for a > a crit . Typical values of R mayrange from a few stellar radii (for the inner disk bound-ary or the inner boundary of the dead zone) to a few AUs(for the interface between the viscously heated and irradi-ated disk regions). In a steady state (Kretke et al. 2009),this prescription leads to a disk surface density (Σ) andpressure enhancement across a crit . This Σ distributionaffects the direction and pace of low-mass-planets TypeI migration (Paardekooper et al. 2010, 2011). The plan-ets are placed relatively far away from R so that theirinitial migration follows the typical Type I migration.The ratio of disk scale height over disk radius is takento be a constant of h/r = 0 .
05. For all simulations, theinitial semi-axis of planet pairs are all beyond R . Theplanets are held on their initial orbits for 1000 Keplerianperiods and are released after their disk has adjustedto their perturbation. Our simulated time span is muchshorter than the Myr disk depletion timescale. The over-all change in the disk mass is modest. SIMULATION RESULTS
The top panel of Figure 2 shows the evolution of theperiod ratio of two gas giants under a combination of twodifferent mass ratios and two different sets of initial radialseparations, based on the assumption that gas giants areformed independently and separated by modest periodratios ( ∼ O r b i t a l P e r i od r a t i o mass ratio = 1:2 0 500 1000 1500 2000t (orbits)1.61.82.02.22.42.62.83.0 mass ratio = 1:1 Fig. 2.—
Top : the curves represent the period ratio evolutionof gas-giant pairs with combinations of outer-to-inner planet massratios of 0 . M J . They all settle toa period ratio of around 2:1. Bottom : the period ratio evolutionof low-mass-planet pairs in two-planet (black curve), three-planet(light gray curves) and four-planet (dark gray curves) systems. Allplanets have 10 M ⊕ . They all settle to a period ratio of around 5:4or 6:5. with extremely high accretion rate.We also simulated the evolution of initially compactgas-giant pairs (with a period ratio less than 2). Popula-tion synthesis models (Ida et al. 2013) indicate that mul-tiple gas-giant systems can form with relatively compactorbits because their progenitor embryos are separated by ∼
10 Hill’s radii before they acquire sufficient mass to ac-crete gas efficiently. Our extensive simulations show thatthe orbits of these compact gas-giant pairs are not stableagainst their intense gravitational perturbation on eachother (Zhou et al. 2007). In all cases, one of the twogas giants is scattered to large distances from its initiallocation. In the inner regions, the scattered planets mayresume their migration. If they are able to catch up withtheir companions, they would eventually settle into the2:1 resonance. The gas-giants’ resonance “barrier” at the2:1 MMR is confirmed by these simulations.Typically the Type I migration timescale of a planet ona circular orbit is τ ∼ h q − M (cid:63) / (Σ p R p )Ω − p , where h isthe aspect ratio of the disk, q is the mass ratio of planetto star, R p is the location of planets and Σ p is the localsurface density at R p . The migration rate also sensitivelydependent on the gradient of the surface density and tem-perature of the disk (Paardekooper et al. 2010, 2011). Inthis Letter, we only control the migration rate by apply-ing different surface densities. Our disk model leads tothe convergent evolution between adjacent pairs of two,three, or four low-mass-planet systems. In these simula-tions, the mass of each planet is assumed to be 10 M (cid:12) andthe gas mass within the computational domain is 2 × − that of the host star. The asymptotic period ratio of allneighboring pairs is around 5:4 or smaller. This abilityto closely pack multiple low-mass-planet systems is due to their relatively rapid converging speed.Such compact systems are rarely found among the Ke-pler planetary candidates. This apparent discrepancy be-tween observations and simulations can be reduced con-siderably with much slower converging Type I migrationrates. In order to illustrate this conjecture, we have per-formed a large set of simulations of multiple low-masssystems for a broad range of disk surface densities.In these simulations, we have chosen the central starmass M (cid:63) = 1 M (cid:12) , h/r = 0 .
05 and α vis = 0 . a interior/exterior to a crit . In a steady state, thedisk accretion rate is given by ˙ M = 1 . × − · f · ( AUR ) . M (cid:12) /yr , where f = Σ R M (cid:63) is the disk-to-star massratio and Σ is the disk surface density (in g/cm ) at R . The transition radius a crit = 0 . R is also depen-dent on the disk accretion rate (Kretke et al. 2009) as a crit ∝ ˙ M . . Both the disk surface density normal-ization and the trapping location (associated with R )decrease as the disk accretion rate diminishes duringdisk depletion. For example, with f = 0 . M = 3 . × − M (cid:12) /yr and a crit = 2 . AU . In our sim-ulations, there is no time evolution of the disk, exceptfor one case shown in Figure 4. In other cases, the ac-cretion rate is constant along the simulations. Varying˙ M is equivalent to varying disk mass which is equiva-lent to varying the establishing time of the planet orbitalarchitecture.From these simulations, we find that the asymptoticperiod ratios between adjacent pairs decrease towardunity as the accretion rate increases. Figure 3 showsa general trend that lower disk accretion rates lead torelatively wider spacing for low-mass-planet systems.These results suggest that the accretion rate must be < · − M (cid:12) /yr to reproduce the observed paucity. Ifthis paucity is a signature of some alternative dynam-ical process, like instability, then we can constrain theforming stage of the planetary systems from the accre-tion rate corresponding to the final period ratio largerthan the 4:3 MMR. The corresponding disk-to-star massratio in this region is < . × − . The reproduction ofthe observed enhancement of adjacent low-mass-planetpairs with 3:2 MMR requires even lower disk accretionrates ( ˙ M < − M (cid:12) /yr ) especially in systems with morethan two planets. Note that the inferred accretion ratesaround classical T Tauri stars in the Taurus and Ophi-uchus complex (Natta et al. 2006) range mostly between10 − − − M (cid:12) yr − .In order to extract constraints on the disk accretionrate from Figure 1, we need to verify that the sta-bility of compact systems may be preserved (Gladman1993). Systematic studies (Zhou et al. 2007) show thatin a gas-free environment, multiple equal-mass low-mass-planet systems with initially circular orbits separated by k = ∆ a/R H <
10 (where R H = (2 M p / M ∗ ) / a ) be-come dynamically unstable over the Gyr main sequencelifespan of the host stars. For non-resonant M p = 10 M ⊕ low-mass-planet systems, this stability criterion requiresthe period ratio to be larger than ∼ .
43, which couldaccount for the paucity of multiple systems with a pe-riod ratio between adjacent planets smaller than that ofthe 3:2 MMR. Multiple gas-giant systems with a periodratio between adjacent planets smaller than that of the Zhang et al. 2014
Fig. 3.—
Panels from top to bottom show the final period ratio ofadjacent pairs in multiple low-mass-planet systems with two, threeand four planets of 10 M ⊕ each. The horizontal lines indicate thefirst-order MMRs at 3:2, 4:3, 5:4, 6:5 and 7:6. The vertical linemarks the observed disk accretion rate for a typical young solarmass system with an age ∼ Fig. 4.—
Pairs of low-mass-planets undergo Type I migrationand capture each other within 10 orbits, as indicated by the leftpanel. Continued simulations up to 10 orbits of such systemsdemonstrate that the pair remains around the 5:4 MMR. A gradualdecrease of the disk surface density is implemented, as shown inthe insert. f = 0 . k = 6. If it is out of resonance, it would become dy-namically unstable in < orbits in the absence of gasand < orbits if it is embedded in a minimum masssolar nebula. We extend our simulation for an additional10 orbits while the disk surface density is prescribed todecrease exponentially over that timescale. The resultsin Figure 4 indicate that once the low-mass-planet pairsare captured into a tight MMR, they tend to remain inthese MMRs in the absence of major perturbations be-fore the depletion of the disk. However, the later long-term orbital evolution of planets with an absence of gasmay break the system. Although we can not exclude theinstability criterion as an contribution to the paucity ofa period ratio smaller than the 4:3 MMR, we can takethis as a clue to the formation stage. If instability isresponsible for the paucity, it might indicate that thelow-mass-planet pairs with a period ratio larger than 4:3formed in a somewhat late stage of the disk. D i sk age ( y r) Fig. 5.—
Two solid lines, from top to bottom, mark the earliestformation time for planet pairs that produce final period ratios of3:2 and 4:3, respectively. An exponentially decreasing disk accre-tion rate model is used in this calculation.
Based on the results from Figures 3 and 4, we inferthat, low-mass-planet systems can capture each otherinto 4:3 and/or 3:2 MMR,s provided they undergo TypeI migration when the accretion rate ( ˙ M ) or the mass oftheir natal disks is relatively low. The results in Fig-ure 3 indicate that in order for a pair of 10 M ⊕ low-mass planets to attain an asymptotic period ratios of4:3 and 3:2, the upper limits of the disk-to-star massratios are at f = 6 . × − and f = 4 × − , respec-tively. Since the Type I migration rate is linearly propor-tional to the product of q and f , low-mass-planet pairswould approach each other’s 4:3 (or 3:2) MMRs withthe same critical relative speed if their q · f ∼ × − (or ∼ . × − ). The condition for MMR capture re-quires that the migration timescale through the charac-teristic width ( τ mig ∼ ∆ a/ ˙ a ) is longer than the librationtimescale ( τ lib ). If we take into account that the librationtime of the lowest-order MMR is ∝ q − / and ∆ a ∝ q / in the first order of the expansion (Murray & Dermott1999), the critical condition for the lowest-order of MMRcapture would be roughly independent on the planet-to-star mass ratio.In order to quantify the stage of the disk according tothe accretion rate, we use the observations of a relativelyyoung ( t a ∼ ρ Oph (Nattaet al. 2006), where the accretion rate around ∼ M (cid:12) star is about 4 × − M (cid:12) /yr . For R = 4 . AU , thecorresponding value f = 0 . M = 4 × − · e − ( t − taτD ) M (cid:12) /yr for t ≥ t a , where t indi-cates the age of the disk, t a ∼ M yr , and τ D is the disklifetime. Figure 5 shows the dependency of final periodratios on the planet masses and the MMRs formationtimes. It confirms that a relatively large period ratio(such as 3:2) requires the low-mass-planet pairs to havemigrated and captured each other late in the disk evolu-tion stage when the disk surface density was sufficientlydepleted. SUMMARY
These simulation results for multiple gas-giant and low-mass-planet systems place important constraints on theplanet formation stage with respect to the disk evolution.esonances of Multiple Exoplanets and Implications for Their Formation 5According to the core accretion scenario, the formationof gas giants must be preceded by the emergence of suffi-ciently massive ( > M ⊕ ) protostellar embryos in a gas-rich environment, presumably during the early stage ofdisk evolution. These cores, if retained in a dense disk,would either congregate and effectively merge near sometrapping radius to become cores of proto-gas-giant plan-ets or be mostly scattered into or far from their host stars.A significant “left-over” population would have producedcompact pairs with small period ratios that are not con-sistent with observations. Around stars that only bearrelatively low-mass planets, their dynamical configura-tion may be established during the advanced stages of disk evolution when the disk gas is severely depleted. Itis also possible that these low-mass planets were assem-bled over several millions of years.We thank the referee, Frederic Rasio, for helpful com-ments that improved the manuscript. We acknowledgesupport from the LDRD program and IGPP of LosAlamos National Laboratory. H.L. and D.N.C.L. alsoacknowledge support from the UC-fee program of Uni-versity of California. Simulations were carried out usingthe Institutional Computing resources at LANL.) protostellar embryos in a gas-rich environment, presumably during the early stage ofdisk evolution. These cores, if retained in a dense disk,would either congregate and effectively merge near sometrapping radius to become cores of proto-gas-giant plan-ets or be mostly scattered into or far from their host stars.A significant “left-over” population would have producedcompact pairs with small period ratios that are not con-sistent with observations. Around stars that only bearrelatively low-mass planets, their dynamical configura-tion may be established during the advanced stages of disk evolution when the disk gas is severely depleted. Itis also possible that these low-mass planets were assem-bled over several millions of years.We thank the referee, Frederic Rasio, for helpful com-ments that improved the manuscript. We acknowledgesupport from the LDRD program and IGPP of LosAlamos National Laboratory. H.L. and D.N.C.L. alsoacknowledge support from the UC-fee program of Uni-versity of California. Simulations were carried out usingthe Institutional Computing resources at LANL.