turbulence driven by stochastic forcing2 a. pierens et al.: on the dynamics of resonant...

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Astronomy & Astrophysicsmanuscript no. astroph c© ESO 2018June 4, 2018

On the dynamics of resonant super-Earths in disks withturbulence driven by stochastic forcing

A. Pierens1,2, C. Baruteau3,4, and F. Hersant1,2

1 Universite de Bordeaux, Observatoire Aquitain des Sciences de l’Univers, BP89 33271 Floirac Cedex, France2 Laboratoire d’Astrophysique de Bordeaux, BP89 33271 Floirac Cedex, France

e-mail:[email protected] Department of Astronomy and Astrophysics, University of Santa Cruz, CA 95064, United States4 DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB30WA, United Kingdom

Preprint online version: June 4, 2018

ABSTRACT

Context. A number of sytems of multiple super-Earths have recently been discovered. Although the observed period ratios are generally farfrom strict commensurability, the radio pulsar PSRB1257+ 12 exhibits two near equal-mass planets of∼ 4 M⊕ close to being in a 3:2 meanmotion resonance (MMR).Aims. We investigate the evolution of a system of two super-Earthswith masses≤ 4 M⊕ embedded in a turbulent protoplanetary disk. The aimis to examine whether or not resonant trapping can occur and be maintained in presence of turbulence and how this depends on the amplitudeof the stochastic density fluctuations in the disk.Methods. We have performed 2D numerical simulations using a grid-based hydrodynamical code in which turbulence is modelled as stochasticforcing. We assume that the outermost planet is initially located just outside the position of the 3:2 mean motion resonance with the inner oneand we study the dependance of the resonance stability with the amplitude of the stochastic forcing.Results. For systems of two equal-mass planets we find that in disk models with an effective viscous stress parameterα ∼ 10−3, dampingeffects due to type I migration can counteract the effects of diffusion of the resonant angles, in such a way that the 3:2 resonance can possiblyremain stable over the disk lifetime. For systems of super-Earths with mass ratioq = mi/mo ≤ 1/2, wheremi(mo) is the mass of the innermost(outermost) planet, the 3:2 resonance is broken in turbulent disks with effective viscous stresses 2×10−4

. α . 1×10−3 but the planets becomelocked in strongerp + 1:p resonances, withp increasing as the value forα increases. Forα & 2 × 10−3, the evolution can eventually involvetemporary capture in a 8:7 commensurability but no stable MMR is formed.Conclusions. Our results suggest that for values of the viscous stress parameter typical to those generated by MHD turbulence, MMRs betweentwo super-Earths are likely to be disrupted by stochastic density fluctuations. For lower levels of turbulence however,as is the case in presenceof a dead-zone, resonant trapping can be maintained in systems with moderate values of the planet mass ratio.

Key words. accretion, accretion disks – planets and satellites: formation – hydrodynamics – methods: numerical

1. Introduction

To date, about 25 extrasolar planets with masses less than10 M⊕ and commonly referred to as super-Earths have beendiscovered (e.g.http://exoplanet.eu). Although two ofthem, Corot-7b (Leger et al. 2009; Queloz et al. 2009) and GJ1214b (Charbonneau et al. 2009) were detected via the transitmethod, most of them were found by high-precision radialvelocity surveys. It is expected that the number of observedsuper-Earths will considerably increase in the near futurewiththe advent of space observatories Corot and Kepler.Interestingly, Kepler team has recently announced the dis-covery of∼ 170 multi-planet systems candidates (Lissauer etal. 2011), although these need to be confirmed by follow-upprograms. Previous to Kepler results, four multi-planet systemscontaining at least two super-Earths had been detected around

PSR B1257+12, HD 69830, GJ 581 and HD 40307. Forthe sytems around main-sequence stars (HD 69830, GJ 581,HD 40307), the observed period ratios between two adjacentlow-mass planets are quite far from strict commensurability.However, the planetary system that is orbiting the radio pulsarPSR B1257+12 exhibits two planets with masses 3.9 M⊕ and4.3 M⊕ in a 3:2 mean motion resonance (Konacki & Wolszczan2003). Papaloizou & Szuszkiewicz (2005) showed that, for thissystem, the existence of such a resonance can be understood bya model in which two low-mass planets with mass ratio closeto unity undergo convergent type I migration (e.g. Ward 1997;Tanaka et al. 2002) while still embedded in a gaseous laminardisk until capture in that resonance occurs. More generally,these authors found that, for more disparate mass ratios andprovided that convergent migration occurs, the evolution of a

http://arxiv.org/abs/1103.4923v1

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2 A. Pierens et al.: On the dynamics of resonant super-Earthsin disks with turbulence driven by stochastic forcing

system of two planets in the 1− 4 M⊕ mass range is likely toresult in the formation of high first-order commensurabilitiesp+ 1:p with p ≥ 3. Studies aimed at examining the interactionof many embryos within protoplanetary disks also suggestthat capture in resonance between adjacent cores throughtype I migration appears as a natural outcome of such asystem (McNeil et al. 2005; Cresswell & Nelson 2006). This,combined with the fact that the majority of super-Earths arefound in multiplanetary systems (Mayor et al. 2009), wouldsuggest that systems of resonant super-Earths are common.The fact that most of the multiple systems of super-Earthsobserved so far do not exhibit mean motion resonances maybe explained by a scenario in which strict commensurabilityis lost due to circularization through tidal interaction with thecentral star as the planets migrate inward and pass through thedisk inner edge (Terquem & Papaloizou 2007).

Moreover, it is expected that in presence of strong disk tur-bulence, effects arising from stochastic density flucuations willprevent super-Earths from staying in a resonant configuration.It is indeed now widely accepted that a source of anomalousviscosity due to turbulence is required to account for theestimated accretion rates for Class II T Tauri stars, which aretypically ∼ 10−8 M⊙yr−1 (Sicilia-Aguilar et al. 2004). Theorigin of turbulence is believed to be related to the magneto-rotational instability (MRI, Balbus & Hawley 1991) for whicha number of studies (Hawley et al. 1996; Brandenburg etal. 1996) have shown that the non-linear outcome of thisinstability is MHD turbulence with an effective viscous stressparameterα ranging between∼ 5× 10−3 and∼ 0.1, dependingon the magnetic field amplitude and topology.So far, the effects of stochastic density fluctuations in thedisk on the evolution of two-planet systems has received littleattention. Rein & Papaloizou (2009) developed an analyticalmodel and performed N-body simulations of two-planetsystems subject to external stochastic forcing and showed thatturbulence can produce systems in mean motion resonancewith broken apsidal corotation, explaining thereby the resonantconfiguration of the HD 128311 system. Adams et al. (2008)examined the effets of turbulent torques on the survival of res-onances using a pendulum model with an additional stochasticforcing term. They found that mean motion resonances aregenerally disrupted by turbulence within disk lifetimes.Lecoanet et al. (2009) extended this latter work by consideringdisk-induced damping effets and planet-planet interactions.They found that systems with sufficiently large damping canmaintain resonances and suggested that two-planet systemscomposed of a Jovian outer planet plus a smaller inner planetare likely to remain bound in resonance.

In this paper we present the results of hydrodynamicalsimulations of systems composed of two planets in the 1− 4M⊕ mass range embedded in a protoplanetary disk in whichturbulence is driven by stochastic forcing. Planets undergoconvergent migration as a result of the underlying type Imigration and we consider a scenario in which the initialseparation between the planets is slightly larger than thatcorresponding to the 3:2 resonance. The aim of this work is to

investigate whether or not resonant trapping can occur and bemaintained in turbulent disks and how the stability of the 3:2resonance depends on the amplitude of the turbulence-induceddensity fluctuations. We find that for systems of equal-massplanets the 3:2 resonance can be maintained provided thatthe level of turbulence is relatively weak, corresponding to avalue for the effective viscous stress parameter ofα . 10−3.In models with mass ratiosq = m/mo ≤ 1/2 however,wheremi (mo) is the mass of the inner (outer) planet, the 3:2resonance is disrupted in presence of weak turbulence but theplanets can become eventually locked in higher first-ordercommensurabilities. For a level of turbulence correspondingto α ∼ 5× 10−3 however, MMRs are likely to be disrupted bystochastic density fluctuations.

This paper is organized as follows. In Sect. 2, we describethe hydrodynamical model and the numerical setup. In Sect.3,we use a simple model to estimate the critical level of turbu-lence above which the 3:2 resonance would be unstable. InSect. 4 we present the results of our simulations. We finallysummarize and draw our conclusions in Sect. 5.

2. The hydrodynamical model

2.1. Numerical method

In this paper, we adopt a 2D disk model for which all thephysical quantities are vertically averaged. We work in anon-rotating frame, and adopt cylindrical polar coordinates(R, φ) with the origin located at the position of the centralstar. Indirect terms resulting from the fact that this frameisnon-inertial are incorporated in the equations governing thedisk evolution (Nelson et al. 2000). Simulations were per-formed with the FARGO and GENESIS numerical codes. Bothcodes employ an advection scheme based on the monotonictransport algorithm (Van Leer 1977) and include the FARGOalgorithm (Masset 2000) to avoid timestep limitation due tothe Keplerian orbital velocity at the inner edge of the grid.The evolution of each planetary orbit is computed using afifth-order Runge-Kutta integrator (Press et al. 1992) and bycalculating the torques exerted by the disk on each planet.We note that a softening parameterb = 0.6H, whereH is thedisk scale height, is employed when calculating the planetpotentials.

The computational units that we adopt are such that theunit of mass is the central massM∗, the unit of distance is theinitial semi-major axisai of the innermost planet and the unitof time is (GM∗/a3

i )−1/2. In the simulations presented here, we

useNR = 256 radial grid cells uniformly distributed betweenRin = 0.4 andRout = 2.5 andNφ = 768 azimuthal grid cells.Wave-killing zones are employed forR < 0.5 andR > 2.1 inorder to avoid wave reflections at the disk edges (de Val-Borroet al. 2006).

In this work, turbulence is modelled by applying at eachtime-step a turbulent potentialΦturb to the disk (Laughlin & al.

A. Pierens et al.: On the dynamics of resonant super-Earths in disks with turbulence driven by stochastic forcing 3

2004, Baruteau & Lin 2010) and corresponding to the superpo-sition of 50 wave-like modes. This reads:

Φturb(R, φ, t) = γR2Ω

250∑

k=1

Λk(R, φ, t), (1)

with:

Λk = ξke− (R−Rk)2

σ2k cos(mkφ − φk −Ωktk) sin(πtk/∆tk). (2)

In Eq. 2,ξk is a dimensionless constant parameter randomlysorted with a Gaussian distribution of unit width.Rk andφk

are, respectively, the radial and azimuthal initial coordinatesof the mode with wavenumbermk, σk = πRk/4mk is the radialextent of that mode andΩk denotes the keplerian angularvelocity atR = Rk. Rk andφk are both randomly sorted witha uniform distribution whereasmk is randomly sorted witha logarithmic distribution betweenmk = 1 and mk = 96.Following Ogihara et al. (2007), we setΛk = 0 if mk > 6 inorder to save computing time. Each mode of wavenumbermk

starts at timet = t0,k and terminates whentk = t − t0,k > ∆tk,where∆tk = 0.2πRk/mkcs, cs being the sound speed, denotesthe lifetime of mode with wavenumbermk. Such a value for∆tkyields a turbulence with autocorrelation timescaleτc ∼ Torb,whereTorb is the orbital period atR = 1 (Baruteau & Lin2010).In Eq. 1,γ denotes the value of the turbulent forcing param-eter which controls the amplitude of the stochastic densityperturbations. In the simulations presented here, we usedfour different values forγ namely:γ = 6 × 10−5, 1.3 × 10−4,1.9 × 10−4, 3 × 10−4. Given thatγ is related to the effectiveviscous stress parameterα and the disk aspect ratioh = H/Rby the relationα = 1.4× 102(γ/h)2 (Baruteau & Lin 2010), thelatter values forγ correspond toα 2× 10−4, 10−3, 2× 10−3,5 × 10−3 respectively. Inviscid simulations withα = 0 werealso performed for comparison.

In calculations with high values ofγ, viscous stresses aris-ing from turbulence can eventually lead to a significant changein the disk surface density profile over a few thousand orbits.This is also observed in three dimensional MHD simulationsin which turbulence is generated by the MRI (Papaloizou &Nelson 2003). For lower values ofγ, such an effect also oc-curs but over a much longer timescale. In order to examinehow this affects the results of the simulations, we have per-formed additional simulations in which the initial surfaceden-sity profile is restored on a characteristic timescaleτm. We fol-low Nelson & Gressel (2010) and solve the following equationat each timestep:

∂Σ

∂t= −Σ − Σinit

τm(3)

whereΣinit is the initial disk surface density and whereτm wasset toτm = 20 orbits. Such a value is shorter than the vis-cous timescale but longer than both the dynamical timescaleatthe outer edge of the disk and the lifetime of the mode withwavenumberm = 1. The results of such simulations are dis-cussed in Sect. 4.1.4.

Model mi (M⊕) mo (M⊕) Σ0 hG1 3.3 3.3 2× 10−4 0.05G2 3.3 3.3 4× 10−4 0.05G3 3.3 3.3 2× 10−4 0.04G4 1.6 1.6 2× 10−4 0.05G5 1.6 3.3 2× 10−4 0.05

Table 1. Parameters used in the simulations

2.2. Initial conditions

In this paper, we adopt a locally isothermal equation ofstate with a fixed temperature profile given byT = T0R−β

whereβ = 1 and whereT0 is the temperature atR = 1. Thiscorresponds to a disk with constant aspect ratioh and for mostof the simulations, we chooseT0 so thath = 0.05. The initialsurface density profile is chosen to beΣinit (R) = Σ0R−σ withσ = 0.5 and we have performed simulations withΣ0 = 2×10−4

andΣ0 = 4 × 10−4. Assuming that the radiusR = 1 in thecomputational domain correponds to 5.2 AU, such valuesfor Σ0 correspond to disks containing 0.02 M⋆ and 0.04 M⋆respectively of gas material interior to 40 AU. No kinematicviscosity is employed in all the runs presented here.

The inner and outer planets initially evolve on circular or-bits atai = 1 andao = 1.33 respectively, which correspondsto a configuration for which the outermost planet is initially lo-cated just outside the 3:2 MMR with the inner one. For mostmodels, we focus on equal-mass planets withmi = mo ≤ 3.3M⊕, wheremi (mo) is the mass of innermost (outermost) planet.However, we have also considered one case in which the planetmass ratioq = mi/mo is q = 1/2. The parameters for all modelswe conducted are summarized in Table 1. Given that the type Imigration timescaleτmig,p of a planet with massmp, semimajoraxisap and on a circular orbit with angular frequencyΩp canbe estimated in the locally isothermal limit by (Paardekooperet al. 2010):

τmig,p = (1.6+ β + 0.7σ)−1 M⋆mp

M⋆Σ(ap)r2

ph2Ω−1p , (4)

we expect that equal low-mass planets embedded in our diskmodel will undergo convergent migration and become eventu-ally trapped in the 3:2 resonance. For a larger initial separationbetween the two planets, capture in 2:1 resonance may also oc-cur. However, test simulations have shown that unless the diskmass is very low, differential migration is not slow enough forthe planets to become trapped in that resonance. This justifiesour assumption that the planets are initially located just outsidethe 3:2 resonance. We also comment that equal-mass planetsmigrating in the type I regime will undergo convergent migra-tion provided thatσ < 3/2 whereasσ > 3/2 will lead to diver-gent migration.

3. Theoretical expectations

In this section, we consider two low-mass planets embedded ina turbulent disk and locked in ap+1:p mean motion resonance,and we derive the critical amplitude of the turbulent forcing


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Fig. 1. Time evolution of the period ratio resulting from N-bodyruns for model G1 and for six different realizations withγ =1.3× 10−4, γ = 1.9× 10−4, andγ = 3× 10−4.

γc below which the resonance would remain stable. FollowingAdams et al. (2008) and Rein & Papaloizou (2009), we assumethat only the outermost planet experiences the torques arisingfrom the disk. We also assume that the planets have near equalmass, in order to avoid the chaotic regime which comes intoplay for disparate masses (Papaloizou & Szuszkiewicz 2005).In the limit of large damping rate for the resonance and neglect-ing effects from planet-planet interaction, the asymptotic valueP of the probability for the resonance to be maintained is givenby (Lecoanet et al. 2009):

P = 4

(

1πτdDφ

)1/2

, (5)

whereDφ is the diffusion coefficient associated with the res-onant angle diffusion andτd is the damping timescale for theresonance angle. This equation is valid at late timest ≫ ω−1

0whereω0 is the libration frequency of the resonant angles, aslong ast ≫ D−1

φ andD−1φ ≫ τd. From the previous equation, we

can estimate the maximum value of the diffusion coefficient forthe sytem to remain bound in resonance with probabilityP = 1.This reads:

Dφ =16πτ−1

d . (6)

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, φ2

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Fig. 2. Upper panel:time evolution of the resonant anglesφ1 =

3λo−2λi−ωi (black) andφ2 = 3λo−2λi−ωo (red) resulting fromN-body runs for model G1 and for two different realizationswith γ = 1.3×10−4. Middle panel:same but forγ = 1.9×10−4.Lower panel:same but forγ = 3× 10−4.

As shown in Adams et al. (2008),Dφ can be related to the diffu-sion coefficientDH,o associated with the diffusion of the outerplanet’s angular momentum as:

Dφ =DH,o

9m2oω

20a4

o

. (7)

For moderate eccentricities,ω0 is given by:

ω20 = −3 j22CΩie

| j4|i with C = q0Ωiα fd(α), (8)

whereei is the eccentricity of the inner planet,q0 = m0/M⋆andΩi is the angular frequency of the innermost planet. Inthe previous equation,α = ai/ao, ( j2, j4) are integers whichdepend on the resonance being considered andfd(α) resultsfrom the expansion of the disturbing function. In the case ofthe3:2 resonance, we havej2 = −2, j4 = −1 andα fd(α) ∼ −1.54(Murray & Dermott 1999).

In Eq. 7,DH,o can be expressed in terms of both the correla-tion timescaleτc associated with the stochastic torques exertedon the outer planet and the standard deviation of the turbulenttorque distributionσt as:

DH,o = σ2t τc. (9)

As discussed in Baruteau & Lin (2010),σt takes the fol-lowing form when applied to the outermost planet:


σt = CΣoqoγa4oΩ

2o, (10)

whereqo = mo/M⋆, Σo is the value of the surface densityat the position of the outer planet,Ωo is the angular frequencyof this planet andC is a constant. For a simulation using thesame disk parameters as for model G1 and withγ = 6× 10−5,we findC ∼ 1.6 × 102, which is close to the value found byBaruteau & Lin (2010). Combining Eqs. 7, 9 and 10 gives anexpression for the diffusion coefficient associated with the dif-fusion of the resonant angleDΦ, in terms of the value for theturbulent forcingγ. This reads:

Dφ =2C2q2

dγ2Ω

3o

9πω20

, (11)

with qd = πΣoa2o/M⋆. Settingδω = ω0/Ωo, we can rewrite the

previous expression as:

Dφ =2C2q2

dγ2Ωo

9πδω2. (12)

We notice that in the case wherep = 1, δω is comparable tothe dimensionless width of the libration zone. Using the previ-ous equation together with Eq. 6, we find that the critical valuefor the turbulent forcing above which the 3:2 resonance is dis-rupted is given by:

γc ∼ 5.3× 10−2δω

qd(τdΩo)−1/2. (13)

In absence of turbulent forcing, we expect the amplitude ofthe resonant angles to scale asΩ−1/2

o (Peale 1976). Accordingto Rein & Papaloizou (2009), this implies that the dampingtimescale of the libration amplitudeτd is twice the migrationtimescaleτmig of the whole system, namely that composed ofthe two planets locked in resonance and migrating inward to-gether. In that case, the previous equation becomes:

γc ∼ 3.7× 10−2δω

qd(τmigΩo)−1/2. (14)

In the limit wheremi ∼ 0, we would haveτmig = τmig,o

whereτmig,o is the migration timescale of the outer planet whichis given by Eq. 4 withh = 0.05,σ = 0.5 andβ = 1. In that casethe expression forγc becomes for our disk model:

γc ∼ 3.5× 10−2δωq1/2o q−1/2

d h−1. (15)

In order to check the validity of the previous expression forγc, we have performed a few N-body runs using a fifth-orderRunge Kutta method. In these calculations, the forces arisingfrom type I migration are not determined self-consistentlybutmodelled using prescriptions for both the migration rateτap andeccentricity damping rateτep of the planets. Forτap, we used:

τap =

(

1+ (ep/h)5

1− (ep/h)4

)

τmig,p, (16)

whereep is the planet eccentricity and whereτmig,p is given byEq. 4 and where the numerical factor accounts for the modifi-cation of the migration rate at large eccentricities (Papaloizou& Larwood 2000). Forτep we used (Tanaka & Ward 2004):

τep =Kh2

0.78

(

1+ 0.25(ep/h)3)

τmig,p. (17)

In the last equation,K ∼ 1.7 is a constant which was cho-sen in such a way that the eccentricity damping rate obtainedin N-boby runs gives good agreement with that resulting fromhydrodynamical simulations. Following Rein et al. (2010),wemodel effects of turbulence as an uncorrelated noise by per-turbing at each time step∆t the velocity componentsvi,p ofeach planet by∆vi,p =

√2D∆tξ whereξ is a random variable

with gaussian distribution of unit width.D is the diffusion coef-ficient which should vary as the planets migrate but which wasfixed here to a constant value ofD = σ2

t τc/a2o.

In Fig. 1 we show the time evolution of the orbital period ra-tio for N-body simulations with parameters corresponding tomodel G1 and for six different realizations withγ = 1.3 ×10−4, 1.9 × 10−4, 3 × 10−4. For this model, we estimateδω ∼1.9 × 10−3 (see Sect. 4.1.4) which leads toγc ∼ 2.5 × 10−4

using Eq. 15. From Fig. 1, it appears that capture in the 3:2MMR occurs for most of the realizations withγ ≤ 1.9× 10−4.For two specific realizations of each value forγ we considered,the time evolution of the resonant anglesφ1 = 3λo − 2λi − ωi

andφ2 = 3λo − 2λi − ωo associated with the 3:2 resonance,whereλi (λo) andωi (ωo) are respectively the mean longitudeand longitude of pericentre of the innermost (outermost) planet,is displayed in Fig. 2. Although the angles can eventually cir-culate for short periods of time, it is clear that the 3:2 MMRremains stable on average forγ ≤ 1.9×10−4. Forγ = 3×10−4,we find that the planets pass through the 3:2 resonance in twoof the six realizations performed while the four other can even-tually involve temporarily capture in the 3:2 MMR. In thesecases however, the lifetime of the resonance does not exceedafew hundred orbits, as can be seen in the lower panel of Fig. 2which displays forγ = 3 × 10−4 the time evolution ofφ1 andφ2 for two realizations in which the period ratio remains closeto that corresponding to the 3:2 MMR. Therefore, the resultsof these N-body calculations suggest that the 3:2 MMR is onlymarginally stable for such a value ofγ, which is consistent withthe aforementioned analytical estimation ofγc ∼ 2.5× 10−4.

4. Results of hydrodynamical simulations

For equal mass planets (q = 1), results of hydrodynamical sim-ulations suggest that capture in 3:2 resonance can occur in tur-bulent disks for which the level of turbulence is relativelyweak.For systems withq ≤ 1/2 however, it appears that trapping inthe 3:2 resonance is maintained only provided that the disk isclose to being inviscid.

4.1. Models with q = 1

For inviscid simulations with equal low-mass planets, the abil-ity for the two planets to become trapped in the 3:2 resonancedepends mainly on the planets’ relative migration rate whichscales ash−2. For model G3 (h = 0.04), we find that capturein 3:2 resonance does not occur in that case due to the relativemigration timescale being shorter than the libration period cor-responding to that resonance. For other models withh = 0.05however, it appears that the system can enter in a 3:2 commen-surability which remains stable for the duration of the simula-tion, which generally covers∼ 104 orbits atR= 1. This occurs


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Fig. 3. Upper left (first) panel: time evolution of planet semi-major axes for model G1 and for the different values ofγ weconsidered namely forγ = 0 (black line),γ = 6× 10−5 (red line),γ = 1.3× 10−4 (blue),γ = 1.9× 10−4 (green) andγ = 3× 10−4

(orange).Upper right (second) panel: time evolution of the inner planet eccentricity.Third panel:time evolution of the outerplanet eccentricity.Fourth panel:time evolution of the period ratio. Simulations were performed with GENESIS.

not only for laminar disks, but also for turbulent disks providedthat the value for the turbulent forcing is not too large. Forex-ample, we find that the 3:2 commensurability is maintained inmost of the turbulent runs withγ ≤ 1.3×10−4 in models G1 andG2 whereas in model G4 this occurs provided thatγ ≤ 6×10−5.Below we describe in more details the results of the simulationswith q = 1 and we use model G1 to illustrate how the evolutiondepends on the value for the forcing parameterγ.

4.1.1. Orbital evolution

The time evolution of the planets’ semi-major axes, eccentrici-ties and period ratio corresponding to model G1 and for one re-alization of the different values ofγ we considered are depictedin Fig. 3. In each case, the period ratio is observed to initiallydecrease, suggesting that early evolution involves convergentmigration of the two planets. Not surprisingly, a tendancy forthe planets to undergo a monotonic inward migration is ob-served at the beginning of the simulations with the lowest val-ues ofγ whereas these are more influenced by stochastic forc-ing for γ ≥ 1.9×10−4. This is due to the fact that the amplitudeof the turbulent density fluctuations is typically strongerthanthat of the planet’s wake for simulations withγ ≥ 1.9× 10−4,as shown in Fig. 4 which displays snapshots of the perturbedsurface density of the disk for different values ofγ.The semimajor axis evolution also reveals a clear tendancyfor lower migration rates with increasingγ, which is an effect

arising from the desaturation of the horseshoe drag by turbu-lence (Baruteau & Lin 2010). Asγ increases, the disk torquesare indeed expected to increase from the differential Lindbladtorque, obtained forγ = 0, up to the fully unsaturated torque,which is maintained forα ∼ 0.16(mi/M∗)3/2h−4 (Baruteau &Lin 2010). Forh = 0.05, such a value forα corresponds toγ ∼ 1.2× 10−4. For higher values ofγ however, we expect thetorques to slightly decrease with increasingγ due to a cut-off ofthe horseshoe drag arising when the diffusion timescale acrossthe horseshoe region is smaller than the horseshoe U-turn time(Baruteau & Lin 2010).This can be confirmed by inspecting the evolution of therunning-time averaged torques exerted on both planets andwhich are presented in Fig. 5. Up to a time of approximately∼ 103 orbits and forγ ≤ 1.3×10−4, the torques are observed toincrease with increasingγ, while they decrease for higher val-ues ofγ. This is in very good agreement with the expectationthat the fully unsaturated torque is reached forγ = 1.2× 10−4.At later times however, the torques obtained in simulationswithγ ≥ 1.9× 10−4 can eventually exceed those computed in runswith γ ≤ 1.3× 10−4. We suggest that this is related to the factthat the disk surface density tends to be significantly modifiedat the planet positions at high turbulence level. This is illus-trated in Fig. 6 which shows the disk surface density profileat t = 2000 orbits for the different values ofγ we consid-ered. Here the inner and outer planets are located atai ∼ 0.98and ao ∼ 1.25 respectively. It is interesting to note that forγ = 3 × 10−4, the outer planet tends to evolve in a region of


γ=0

-2 -1 0 1 2X

-2

-1

0

1

2

Y

-0.10

-0.07

-0.03

0.00

0.03

0.07

0.10γ=6x10-5

-2 -1 0 1 2X

-2

-1

0

1

2

Y

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γ=1.3x10-4

-2 -1 0 1 2X

-2

-1

0

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2

Y

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0.00

0.07

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0.20γ=1.9x10-4

-2 -1 0 1 2X

-2

-1

0

1

2

Y

-0.20

-0.13

-0.07

0.00

0.07

0.13

0.20

Fig. 4. This figure shows, for model G1, snapshots of the perturbed surface density of the disk forγ = 0 (first panel),γ = 6×10−5

(second panel),γ = 1.3× 10−4 (third panel) andγ = 1.9× 10−4 (fourth panel).

0 2000 4000 6000 8000 10000Time (orbits)

-3.0•10-6

-2.5•10-6

-2.0•10-6

-1.5•10-6

-1.0•10-6

-5.0•10-7

Run

ning

-tim

e av

erag

ed to

rque

inner planet

0 2000 4000 6000 8000 10000Time (orbits)

-3.0•10-6

-2.5•10-6

-2.0•10-6

-1.5•10-6

-1.0•10-6

-5.0•10-7

Run

ning

-tim

e av

erag

ed to

rque

outer planet

Fig. 5. Upper panel:time evolution of the running-time av-eraged torques exerted on the inner planet for model G1 andfor the different values ofγ we considered namely forγ = 0(black line),γ = 6 × 10−5 (red line),γ = 1.3 × 10−4 (blue),γ = 1.9×10−4 (green) andγ = 3×10−4 (orange).Lower panel:same but for the outer planet.

0.5 1.0 1.5 2.0Radius

1.5•10-4

2.0•10-4

2.5•10-4

3.0•10-4

Sur

face

den

sity

Fig. 6. Disk surface density profile att = 2000 orbits for modelG1 and for the different values ofγ we considered namely forγ = 0 (black line),γ = 6 × 10−5 (red line),γ = 1.3 × 10−4

(blue),γ = 1.9× 10−4 (green) andγ = 3× 10−4 (orange).

positive surface density gradient where the corotation torque ispositive, in such a way that the torque exerted on that planetcan become higher than that exerted on the inner one. This ef-fect is responsible for the increase of period ratio observed atlate times in simulations withγ ≥ 1.9× 10−4.


100 200 300 400 500Time (orbits)

-1.5•10-4

-1.0•10-4

-5.0•10-5

0

dei /

dt

100 200 300 400 500Time (orbits)

-1.5•10-4

-1.0•10-4

-5.0•10-5

0

dei /

dt

Model G1, γ=6x10-5

Model G2, γ=6x10-5

Model G4, γ=6x10-5

Fig. 7. Upper panel:time evolution of the theoretical change ofthe inner planet eccentricitydei/dt given by Eq. 18 for modelG1 and for the different values ofγ we considered namely forγ = 0 (black line),γ = 6×10−5 (red line),γ = 1.3×10−4 (blue),γ = 1.9×10−4 (green) andγ = 3×10−4 (orange).Lower panel:same but forγ = 6× 10−5 and models G1 (red), G2 (blue) andG4 (green).

4.1.2. Eccentricity evolution

For model G1, examination of the early planets’ eccentricitiesevolution displayed in Fig. 3 shows a clear tendancy of highereccentricities with increasing the value forγ, which is in agree-ment with the expectation that turbulence is a source of ec-centricity driving (Nelson 2005). This can be confirmed by in-specting the theoretical change of the inner planet eccentricitydei/dt which can be computed using the following expression(Burns 1976):

dei

dt=

e2i − 1

2e2i

(

EE+ 2

HH

)

, (18)

whereE = −G(M⋆ + mi)/2ai is the specific energy of the

inner planet,H =√

G(M⋆ +mi)ai(1− e2i ) its specific angular

momentum,H the torque exerted by the disk andE the powerof the force exerted by the disk on the planet. The early timeevolution ofdei/dt is displayed, for this model and for the dif-ferent values ofγ we considered in the upper panel of Fig. 7. Itclearly demonstrates that, compared with the laminar run, thetheoretical rate of change ofei is higher in turbulent runs andthat it increases with increasing the value forγ.Also, inspection of the lower panel of Fig. 7 reveals that, com-pared with model G1 in whichmi = mo = 3.3 M⊕ andΣ0 = 2×10−4, the disk induced eccentricity damping is weakerin model G4 for whichmi = mo = 1.6 M⊕. This is due to thedamping of eccentricity at coorbital Lindblad resonances scal-ing linearly with planet mass (Nelson 2005). Given that thisdamping also scales with disk mass, this explains why the diskinduced eccentricity damping is stronger in model G2 in whichΣ0 = 4× 10−4.

0 2000 4000 6000 8000 10000Time (orbits)

1.495

1.500

1.505

1.510

1.515

(ao/

a i)1.

5

γ=1.3x10-4

0 2000 4000 6000 8000 10000Time (orbits)

1.495

1.500

1.505

1.510

1.515

(ao/

a i)1.

5

γ=1.9x10-4

Fig. 8. Upper panel: time evolution of the period ratio formodel G1 and for four different realizations withγ = 1.3×10−4.Lower panel:same but forγ = 1.9 × 10−4. Simulations wereperformed with GENESIS.

4.1.3. Time evolution of the resonant angles

As mentioned above, we find that for model G1 a stable 3:2commensurability generally forms forγ ≤ 1.3 × 10−4, whichcan be confirmed by inspecting the upper panel of Fig. 8which displays the time evolution of the period ratio for fourdifferent realizations with this value ofγ. However, as one cansee in the lower panel of Fig. 8, such a resonance is observedto break at late times for all the realizations performed withγ ≥ 1.9 × 10−4. Comparing Fig. 8 and the two upper panelsin Fig. 1, we see that the results of these hydrodynamicalsimulations are at first sight in good agreement with those ofN-body runs. Moreover, it is interesting to note that in someruns, planets which leave the 3:2 resonance can eventuallypass through that resonance again at later times or otherscommensurabilities like the 2:1 resonance.A similar outcome is observed in model G2 which had a valueof Σ0 = 4× 10−4. This arises because both the turbulent torqueand the damping rate of the libration amplitude scale linearlywith Σ0. For model G4 however, which hadΣ0 = 2× 10−4 andmi = mo = 1.6 M⊕, the 3:2 resonance is disrupted in the runwith γ = 1.3 × 10−4 due to the damping rate being smallerfor this model. For a single realization for each value ofγ, theevolution of the period ratio for models G2 and G4 is displayedin Fig. 9.

In the two upper panels of Fig. 10 we show, for model G1and for two realizations withγ = 6× 10−5 andγ = 1.3× 10−4,the time evolution of the resonant angles associated with the 3:2


0 1000 2000 3000 4000 5000 6000 7000Time (orbits)

1.40

1.45

1.50

1.55

1.60

(ao/

a i)1.

5

model G2

0 2.0•103 4.0•103 6.0•103 8.0•103 1.0•104 1.2•104 1.4•104

Time (orbits)

1.40

1.45

1.50

1.55

1.60

(ao/

a i)1.

5

model G4

Fig. 9. Upper panel: time evolution of the period ratio formodel G2 and for the different values ofγ we considerednamely forγ = 0 (black line),γ = 6×10−5 (red),γ = 1.3×10−4

(blue),γ = 1.9×10−4 (green) andγ = 3×10−4 (orange).Lowerpanel: same but for model G4. Simulations were performedwith GENESIS.

resonance. Forγ = 6×10−5, the 3:2 resonance is established att ∼ 1800 orbits while it forms att ∼ 2000 orbits for the calcu-lation withγ = 1.3× 10−4. This is consistent with the fact that,for moderate values ofγ, migration rates tend to decrease withincreasingγ. As illustrated in the second and third panels ofFig. 3, resonant capture makes the eccentricities of both plan-ets grow rapidly before they saturate at values ofei ∼ eo ∼ 0.01in the run withγ = 6× 10−5. As discussed in Sect. 4.1.2, thesetend to be larger in the case whereγ = 1.3× 10−4, with the ec-centricities reaching peak values ofei ∼ 0.02 andeo ∼ 0.015.Not surprisingly, there is a clear trend for the amplitude ofthe resonant angles to increase with increasing the value forγ in model G1. For the run withγ = 6 × 10−5, the anglesslightly spread untilt ∼ 5 × 103 orbits before their ampli-tude continuously decrease with time. This indicates that overlong timescales damping of the resonant angles through mi-gration tends to overcome diffusion effects. In the case whereγ = 1.3 × 10−4 however, periods of cyclic variations of theresonant angles can be seen with the angles librating with highamplitude before being subsequently damped. Given that in ab-sence of turbulent forcing, the libration amplitude shouldde-crease asΩ−1/2

i (Peale 1976), we would expect the 3:2 reso-nance to be maintained, forγ ≤ 1.3 × 10−4, over timescalesmuch longer than those covered by the simulations.For two realizations withγ = 6× 10−5 andγ = 1.3× 10−4, theevolution of bothφ1 andφ2 for models G2 and G4 is diplayedin the middle and lower panels of Fig. 10 respectively. In com-parison with model G1, the resonant angles librate with slightlyhigher amplitudes in model G2 and can eventually start oscil-

0 2000 4000 6000 8000 10000Time (orbits)

1

2

3

4

5

6

φ 1, φ

2

0 2000 4000 6000 8000 10000Time (orbits)

1

2

3

4

5

6

φ 1, φ

2

0 1400 2800 4200 5600 7000Time (orbits)

1

2

3

4

5

6

φ 1, φ

2

0 1400 2800 4200 5600 7000Time (orbits)

1

2

3

4

5

6

φ 1, φ

2

5.0•103 1.0•104 1.5•1040Time (orbits)

1

2

3

4

5

6φ 1

, φ2

5.0•103 1.0•104 1.5•1040Time (orbits)

1

2

3

4

5

6

φ 1, φ

2

Fig. 10. Upper panel:time evolution of the resonant anglesφ1 = 3λo − 2λi − ωi (black) andφ2 = 3λo − 2λi − ωo (red)for model G1 withγ = 6 × 10−5 (left) andγ = 1.3 × 10−4

(right). Middle panel:same but for modelG2. Lower panel:same but for model G4.

lations between periods of circulation and libration in therunwith γ = 1.3× 10−4. Again this arises because, compared withmodel G1, the turbulent density fluctuations are stronger inthismodel. In model G4 however, the 3:2 resonance is maintainedfor only∼ 3×103 orbits in the case whereγ = 1.3×10−4, whichindicates that for this model the damping rate is too weak forthis resonance to remain stable.

4.1.4. Comparison with analytics

We now examine how the results of the simulations describedabove compare with the expectations discussed in Sect. 3. Formodel G1, we can estimate the libration frequencyω0 usingEq. 8 in conjunction with the results from this model shown inFig. 3. Adopting the inviscid simulation as a fiducial case, wehaveai = 0.9, ao = 1.17 andei = 0.01 at t ∼ 4000 orbits,which leads toδω = ω0/Ωo ∼ 1.9×10−3. Moreover, the migra-tion timescale of the system was estimated toτmig ∼ 3.3× 104

orbits from the results of this simulation. Using Eq. 14 whichgives the value forγc as a function ofτmig, we can then pro-vide an analytical estimateγana

c of the critical amplitude forthe turbulent forcing above which the 3:2 resonance should bedisrupted. For this model, this critical value is estimatedto beγana

c ∼ 1.9 × 10−4 while Eq. 15 predictsγanac ∼ 2.5 × 10−4.

Returning to Fig. 8, we see that the results of the simulations

10 A. Pierens et al.: On the dynamics of resonant super-Earths in disks with turbulence driven by stochastic forcing

0 2000 4000 6000 8000 10000Time (orbits at r=1)

1.500

1.505

1.510

1.515

1.520

1.525

(ao/a

i)1.5

γ = 6x10-5

0 2000 4000 6000 8000 10000Time (orbits at r=1)

1.500

1.505

1.510

1.515

1.520

1.525

(ao/a

i)1.5

γ = 1.3x10-4

Fig. 11. Upper panel:time evolution of the period ratio formodel G1 and for four runs withγ = 6 × 10−5 and in whichEq. 3 is solved at each timestep .Lower panel:same but forγ = 1.3× 10−4. Simulations were performed with FARGO.

performed with GENESIS suggest that 1.3 × 10−4 ≤ γc <

1.9× 10−4 for this model, which is clearly in broad agreementwith the previous analytical estimate. We note however thatboth simulations performed with FARGO and additional runs

0 2000 4000 6000 8000 10000Time (orbits)

01

2

3

4

5

6

Φ1,

Φ2

0 2000 4000 6000 8000 10000Time (orbits)

01

2

3

4

5

6

Φ1,

Φ2

0 2000 4000 6000 8000 10000Time (orbits)

01

2

3

4

5

6

Φ1,

Φ2

0 2000 4000 6000 8000 10000Time (orbits)

01

2

3

4

5

6

Φ1,

Φ2

Fig. 12. Time evolution, for model G1 and for four differentrealizations, of the resonant anglesφ1 = 3λo − 2λi −ωi (black)andφ2 = 3λo−2λi −ωo (red) forγ = 1.3×10−4 and in the casewhere Eq. 3 is solved at each timestep.

in which a roughly constant surface density profile is main-tained (see Sect. 2.1) produced slightly different results sincewe find 6× 10−5 ≤ γc < 1.3 × 10−4 in these cases. The littledifference exhibited by our two codes is apparently due to thefact that turbulence induces changes in the surface densitypro-file which are slightly different. In FARGO, the disk density atthe position of the inner planet is slightly higher comparedwithGENESIS while it is slightly lower at the position of the outerplanet.For calculations in which Eq. 3 is solved at each timestep, thetime evolution of the period ratio for four realizations withγ = 6× 10−5 andγ = 1.3× 10−4 is displayed in Fig. 11. In thatcase, all the realizations performed withγ = 6×10−5 resulted inthe formation of the 3:2 resonance whereas forγ = 1.3× 10−4,two of the four realizations resulted in capture in that resonanceby the end of the run. Forγ = 1.3 × 10−4, the time evolu-tion of the resonant angles associated with the 3:2 resonanceis displayed in Fig. 12. Compared with previous runs in whichthe surface density profile was altered by turbulence, we seethat capture in resonance tends to occur later in runs where aroughly constant surface density profile is maintained. This oc-curs because the disk density at the position of the inner planettends to be higher in runs where the initial disk surface densityprofile is restored, leading to a slower differential migration be-tween the two planets.For other models, repeating the previously decribed procedureleads to analytical estimates ofγana

c = 1.2× 10−4 for model G2andγana

c ∼ 9.3 × 10−5 for model G4. Given that the simula-tions performed with GENESIS indicate that 1.3×10−4 ≤ γc <

1.9×10−4 and 6×10−5 ≤ γc < 1.3×10−4 for models G2 and G4respectively, we see that again the previous analytical estimatescompare reasonably well with the results of our simulations.

4.2. Model with q = 1/2

For systems withq = 1/2, the results of the simulations in-dicate that the 3:2 resonance can be maintained only in caseswhere the disk is close to being inviscid. Fig. 13 shows the re-sults for model G5 in whichmi = 1.6 M⊕ andmo = 3.3 M⊕and for a single realization of the different values ofγ we con-sidered. Moving from left to right and from top to bottom thepanels display the time evolution of the planets’ semi-majoraxes, eccentricity of the inner planet, eccentricity of theouterone, and period ratio. The evolution of semi-major axes showsstrong similarities with that of model G1, with the migrationrates of the planets observed to decrease with increasing thevalue forγ, as discussed in Sect. 4.1.1.Because of stochastic density fluctuations, the planets eccen-tricities are highly variables quantities and a clear trendofhigher eccentricities for higher values ofγ is again observedat the beginning of the simulations. Forγ = 0, the time evolu-tion of the period ratio shows that trapping in the 3:2 resonanceoccurs att ∼ 103 orbits. This resonant interaction causes ec-centricity growth of both planets with the eccentricities of theinner and outer planets reaching peak values ofei ∼ 0.035 andeo ∼ 0.02 respectively, although convergence is not fully es-tablished at the end of the simulation. Comparing Figs 3 and


0 1000 2000 3000 4000 5000 6000Time (orbits)

0.8

0.9

1.0

1.1

1.2

1.3

1.4a

0 1000 2000 3000 4000 5000 6000Time (orbits)

0.00

0.01

0.02

0.03

0.04

0.05

e i

model G5

0 1000 2000 3000 4000 5000 6000Time (orbits)

0.000

0.005

0.010

0.015

0.020

0.025

e o

0 1000 2000 3000 4000 5000 6000Time (orbits)

1.401.42

1.44

1.46

1.48

1.50

1.52

1.54

(ao

/ai)1.

5

Fig. 13. Upper left (first) panel: time evolution of planet semi-major axes for model G5 and for the different values ofγ weconsidered namely forγ = 0 (black line),γ = 6× 10−5 (red line),γ = 1.3× 10−4 (blue),γ = 1.9× 10−4 (green) andγ = 3× 10−4

(orange).Upper right (second) panel: time evolution of the inner planet ecentricity.Third panel: time evolution of the outerplanet eccentricity.Fourth panel:time evolution of the period ratio (ao/ai)1.5. Simulations were performed with GENESIS.

13, we see that the period ratio oscillates with greater ampli-tude in model G5, indicating thereby that resonant locking isweaker. This arises because, compared with model G1, the rel-ative migration rate is higher for that model. Here, it is worth-while to notice that over timescales longer that those coveredby the simulations, it is not clear whether or not the planetswillremain bound in the 3:2 resonance since for disparate planetmasses as is the case for model G5, we expect the dynamicsof the system to be close to the chaotic regime even forγ = 0(Papaloizou & Szuszkiewicz 2005).For turbulent runs, we find that the planets become temporarilytrapped in the 3:2 resonance but in each case, the final out-come appears to be disruption of that resonance. Not surpris-ingly, the survival time of the 3:2 resonance tends to increasewith decreasing the value forγ. For these realizations, the life-times of the resonance are estimated to be∼ 1500 orbits forγ = 6× 10−5, ∼ 2× 103 orbits forγ = 1.3× 10−4, ∼ 103 orbitsfor γ = 1.9×10−4. The slightly longer lifetime of the resonanceobtained in the run withγ = 1.3 × 10−4 arises because of thestochastic nature of the problem.In Fig. 14 is displayed for two runs withγ = 6 × 10−5 andγ = 1.3× 10−4 the evolution of the resonant angles associatedwith the 3:2 resonance. In comparison with models in whichq = 1, there is a clear tendancy for these to librate with higheramplitude.Therefore, for models with disparate planet masses,the origin of the disruption of the 3:2 resonance in turbulentruns is likely to be related to the combined effect of diffusionof the resonant angles plus high libration amplitudes due tothe

resonance being weaker.For γ ≥ 6 × 10−5 , we can not rule out the possibility thatthe planets would become locked in strongerp + 1:p reso-nances withp ≥ 3. To investigate this issue in more details,we have performed an additional set of simulations in which,for each value ofγ we considered, the outer planet was ini-tially located just outside the 4:3 resonance with the innerone.In cases where the 4:3 resonance was found to be unstable, weperformed an additional run but with an initial separation be-tween the two planets slightly larger than that correspondingto the 5:4 resonance. If the 5:4 resonance is not maintained,we repeat the procedure described above until a stablep+ 1:pcommensurability is eventually formed. Because performingseveral realizations for each value ofγ would require a largesuite of simulations, we have considered here only one singlerealization for each value ofγ.Fig. 15 illustrates how the established resonance depends onthe value for the turbulent forcing. Not surprisingly, a cleartrend for forming strongerp + 1:p resonances with increas-ing γ is observed. Forγ = 6 × 10−5, the system enters in astable 4:3 resonance while forγ = 1.3 × 10−4, the planetsbecome rather locked in the 5:4 resonance. For the runs withγ = 1.9× 10−4 andγ = 3× 10−4 however, the planets becometemporarily trapped in the 8:7 resonance but in each case thiscommensurability is subsequently lost with the planets under-going divergent migration. In that case, it is interesting to notethat the system is close to the stability limit since for planets inthe Earth mass range as is the case here, we expect resonance

12 A. Pierens et al.: On the dynamics of resonant super-Earths in disks with turbulence driven by stochastic forcing

overlap to occur forp ≥ 8 (Papaloizou & Szuszkiewicz 2005).Therefore, we can reasonably suggest that for such values ofγ,super-Earths with mass ratioq = mi/mo < 1/2 may not be ableto become trapped in a stable mean motion resonance and mayeventually suffer close encouters.

5. Discussion and conclusion

In this paper we have presented the results of hydrodynamicsimulations aimed at studying the evolution of a system com-posed of two planets in the Earth mass range and embeddedin a turbulent protoplanetary disk. We employed the turbulencemodel of Laughlin et al. (2004) and modified by Baruteau &Lin (2010) in which a turbulent potential corresponding to thesuperposition of multiple wave-like modes is applied to thedisk. We focused on a scenario in which the outermost planet isinitially located just outside the 3:2 resonance and investigatedhow the evolution depends on both the planet mass ratioq andthe value for the turbulent forcing parameterγ.The results of the simulations indicate that for systems withequal mass planets, a 3:2 resonance can be maintained in pres-ence of weak turbulence. For instance, in the case of two plan-ets with equal mass 3.3 M⊕, we find that the 3:2 resonance isstable in runs withγ ≤ 1.9×10−4, which corresponds to valuesfor the effective viscous stress parameter ofα . 2×10−3. Sucha value was found to compare fairly well with that resultingfrom both analytical estimations and preliminary N-body runs.For systems with planet mass ratiosq ≤ 1/2 however, it ap-pears that a 3:2 resonance can remain stable only provided thatthe disk is close to being inviscid. In turbulent disks however,the outcome depends strongly on the value forγ:i) For 6× 10−5 ≤ γ ≤ 1.3× 10−4 (equivalent to 2× 10−4

. α .

10−3) , the planets tend to become locked in strongerp + 1:presonances, withp increasing as the value forγ increases.ii) In the case whereγ ≥ 1.9×10−4 (equivalent toα ≥ 2×10−3),we find that the planets can become temporarily trapped in a 8:7commensurability, but this resonance is disrupted at latertimesand no stable resonance is formed.Given that the volume averaged stress parameter deduced fromMHD simulations is typicallyα ∼ 5 × 10−3 (Papaloizou &Nelson 2003; Nelson 2005), these results suggest that meanmotion resonances between planets in the Earth mass rangeare likely to be disrupted in the active zones of protoplanetarydisks. For relatively low levels of turbulence however, as is thecase for a dead-zone (Gammie 1996), a resonance can be main-tained for moderate values of the planet mass ratio.Such a scenario is broadly consistent with the preliminary anal-ysis of∼ 170 multi-planetary systems candidates recently de-tected by Kepler (Lissauer et al. 2011) and which suggests thatonly a few of the observed adjacent pairings are either in or neara MMR. However, examination of the slope of the cumulativedistribution of period ratios (Fig. 7 of Lissauer et al. 2011) alsoreveals an excess of planets with period ratios corresponding tothe 2:1 or 3:2 commensurabilities. In that case, it appears thatthe neighboring planet candidates have masses within 20% ofeach other. This clearly supports our findings that in disks withmoderate levels of turbulence, MMRS are stable provided themass ratio between the neighboring planets is close to unity.

0 1200 2400 3600 4800 6000Time (orbits)

1

2

3

4

5

6

φ 1, φ

2

0 1200 2400 3600 4800 6000Time (orbits)

1

2

3

4

5

6

φ 1, φ

2

Fig. 14. Time evolution of the resonant anglesφ1 = 3λo−2λi −ωi (black) andφ2 = 3λo − 2λi − ωo (red) for model G1 withγ = 6× 10−5 (left panel) andγ = 1.3× 10−4 (right panel).

0 2000 4000 6000 8000 10000Time (orbits)

1.1

1.2

1.3

1.4

1.5

(ao/

a i)1.

5

3:2

4:3

5:4

6:5

7:6

8:7

Fig. 15. Period ratio between the two planets for model G5 withγ = 0 (black line),γ = 6 × 10−5 (red),γ = 1.3× 10−4 (blue),γ = 1.9× 10−4 (green) andγ = 3× 10−4 (orange). Simulationswere performed with GENESIS.

Since turbulence has a significant impact on the capture of twoplanets in the Earth mass range, it will be of interest to exam-ine this issue using three-dimensional MHD simulations, whicheventually include the presence of a dead-zone. We will addressthis issue in a future paper.

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