Analytical model of multi-planetary resonant chains and constraints on migration scenarios
AAstronomy & Astrophysics manuscript no. laplace c (cid:13)
ESO 2018November 16, 2018
Analytical model of multi-planetary resonant chainsand constraints on migration scenarios
J.-B. Delisle , Observatoire de l’Université de Genève, 51 chemin des Maillettes, 1290, Sauverny, Switzerlande-mail: [email protected] ASD, IMCCE, Observatoire de Paris - PSL Research University, UPMC Univ. Paris 6, CNRS,77 Avenue Denfert-Rochereau, 75014 Paris, FranceNovember 16, 2018
ABSTRACT
Resonant chains are groups of planets for which each pair is in resonance, with an orbital period ratio locked at a rational value (2 / /
2, etc.). Such chains naturally form as a result of convergent migration of the planets in the proto-planetary disk. In this article,I present an analytical model of resonant chains of any number of planets. Using this model, I show that a system captured in aresonant chain can librate around several possible equilibrium configurations. The probability of capture around each equilibriumdepends on how the chain formed, and especially on the order in which the planets have been captured in the chain. Therefore, for anobserved resonant chain, knowing around which equilibrium the chain is librating allows for constraints to be put on the formationand migration scenario of the system. I apply this reasoning to the four planets orbiting Kepler-223 in a 3:4:6:8 resonant chain. I showthat the system is observed around one of the six equilibria predicted by the analytical model. Using N-body integrations, I show thatthe most favorable scenario to reproduce the observed configuration is to first capture the two intermediate planets, then the outermost,and finally the innermost.
Key words. celestial mechanics – planets and satellites: general
1. Introduction
Mean motion resonances (MMR) between two planets are a nat-ural outcome of the convergent migration of planets in a gas-disk(e.g., Weidenschilling & Davis 1985). The planets initially formfarther away from each other, and planet-disk interactions inducea migration of the planets. The period ratio between the planetsdecreases until they get captured in a MMR. The planets thencontinue to migrate whilst maintaining their period ratio at a ra-tional value (2 /
1, 3 /
2, etc.). The eccentricities increase due to theresonant interactions, until they reach an equilibrium betweenthe migration torque and the eccentricity damping exerted bythe disk. The argument of the resonance, which is a combinationof the mean longitudes of the two planets, enters into libration(oscillations around an equilibrium value).For systems of three and more planets, once a pair of planetshas been captured in a MMR, the other planets might also jointhis couple to form a chain of resonances. Each time a planetgets captured in the chain, it enters into a MMR (and thus main-tains a constant and rational period ratio) with each of the otherplanets of the chain. The eccentricities of the planets and theresonant arguments of each pair find a new equilibrium. Suchmulti-planetary resonant chains are expected from simulationsof planet migration (e.g., Cresswell & Nelson 2006). Recently,Mills et al. (2016) showed that the four planets in the Kepler-223 system are in a 3:4:6:8 resonant chain (period ratios of 4 / /
2, and 4 / ff en et al. 2012; Go´zdziewski et al.2016). This model is very similar to the studies of the Laplaceresonance between the Galilean moons, but is not well suited inthe general case. For instance, four-planet (or more) resonancesare not considered. Moreover, for some three-planet resonances,the interactions between non-consecutive planets cannot be ne-glected. For instance, in a 3:4:6 resonant chain, each planet islocked in a first-order resonance with each of the other planets.In particular, the innermost and outermost planets are involvedin a 2 / Article number, page 1 of 10 a r X i v : . [ a s t r o - ph . E P ] J un & A proofs: manuscript no. laplace nificant amplitude of libration around the equilibrium, or couldeven have some angles circulating, the position of the equilibriastill provides useful insights into the dynamics of the system. InSect. 2, I describe this analytical model, and the method I use tofind the equilibrium configurations. In Sect. 3, I apply the modelto Kepler-223. I show that six equilibrium configurations existfor this resonant chain, and that the system is observed to be li-brating around one of them. I also show that knowing the currentconfiguration of the system allows for interesting constraints tobe put on its migration scenario, and in particular on the order inwhich the planets have been captured in the chain.
2. Model
I consider a planetary system with n planets (which I denotewith indices 1 , ..., n from the innermost to the outermost) orbit-ing around a star (index 0). I assume that the system is coplanarand is locked in a chain of resonances. In such a resonant chain,each pair of planets is locked in a MMR. For two planets i < j , Idenote by k j , i / k i , j the resonant ratio, such that k j , i n j − k i , j n i ≈ , (1)where n i ( n j ) is the mean motion of planet i ( j ). I also introducethe degree of the resonance between planet i and planet jq i , j = k j , i − k i , j . (2)At low eccentricities, resonances of a lower degree have astronger influence on the dynamics of the system.In order to study the dynamics of these resonant chains, Igeneralize to n planets the method developped in the case of two-planet resonances (Delisle et al. 2012, 2014). The Hamiltonianof the system takes the form (Laskar 1991) H = − n (cid:88) i = G m m i a i + (cid:88) ≤ i < j ≤ n (cid:32) −G m i m j || r i − r j || + ˜ r i . ˜ r j m (cid:33) , (3)where G is the gravitational constant, m i is the mass of body i , a i is the semi-major axis, r i the position vector, and ˜ r i thecanonically conjugated momentum of planet i (in astrocentriccoordinates, see Laskar 1991). The first sum on the right-handside of Eq. (3) is the Keplerian part of the Hamiltonian (planet-star interactions), while the second sum is the perturbative part(planet-planet interactions).In the coplanar case (which I assume here) the system has 2 n degrees of freedom (DOF), with 2 DOF (4 coordinates) associ-ated to each planet. As for two-planet resonances (e.g., Delisleet al. 2012), the number of DOF can be reduced by using theconservation of the total angular momentum (1 DOF), and byaveraging over the fast angles (1 DOF). Therefore, the problemcan be reduced to 2( n −
1) DOF. Even with these reductions,the phase space is still very complex, especially for systems ofmany planets such as Kepler-223 (chain of 4 planets, 6 DOF),and the problem is, in most cases, non-integrable. In this study, Ifocus on finding the fixed point of the averaged problem, whichprovides useful insight into the dynamics of the system, and es-pecially into the values around which the angles of a resonantsystem should librate. The method described in the following isa generalization of the method presented in Delisle et al. (2012)which focuses on finding the fixed points for two-planet MMR.I denote by λ i and (cid:36) i the mean longitude and longitude ofperiastron of planet i (in astrocentric coordinates), respectively.The actions canonically conjugated to the angles λ i and − (cid:36) i are the circular angular momentum Λ i and the angular momentumdeficit (AMD, see Laskar 2000) D i , respectively. These actionsare defined as follows Λ i = β i √ µ i a i , (4) D i = Λ i − G i = Λ i (cid:18) − (cid:113) − e i (cid:19) , (5)where G i = Λ i (cid:113) − e i is the angular momentum of planet i , β i = m i m / ( m + m i ), µ i = G ( m + m i ). At low eccentricitiesthe deficit of angular momentum D i is proportional to e i . TheHamiltonian (Eq. (3)) can be expressed using these action-anglecoordinates H = − n (cid:88) i = µ i β i Λ i + (cid:88) ≤ i < j ≤ n H i , j ( Λ i , Λ j , D i , D j , λ i , λ j , (cid:36) i , (cid:36) j ) , (6)where the first sum is the Keplerian part, which depends onlyon Λ i (or equivalently a i ), and H i , j is the perturbation betweenthe planets i and j , which depends on the eight coordinates as-sociated to i and j . I follow the method described in Laskar &Robutel (1995) to compute H i , j as a power series of the eccen-tricities (or equivalently of √ D i and (cid:112) D j ), and a Fourrier seriesof the angles, where the coe ffi cients are functions of Λ i and Λ j (i.e., of the semi-major axes). For a system that is close to theresonance or resonant, the semi-major axes remain close to thenominal resonant values (Kepler’s third law) a i a j ≈ a i , a j , = (cid:32) k i , j k j , i (cid:33) / (cid:32) µ i µ j (cid:33) / . (7)I introduce ∆Λ i = Λ i − Λ i , , (8)where Λ i , = β i √ µ i a i , , (9)and expand the Keplerian part at degree 2, and the perturbativepart at degree 0 in ∆Λ i H = n (cid:88) i = n i , ∆Λ i − n i , Λ i , ∆Λ i + (cid:88) ≤ i < j ≤ n H i , j ( D i , D j , λ i , λ j , (cid:36) i , (cid:36) j ) , (10)where n i , is the nominal mean motion of planet i , such that n i , n j , = k j , i k i , j . (11)The perturbative part does not depend on Λ i anymore, but is sim-ply evaluated at Λ i , .In order to perform the reductions associated to the conser-vation of angular momentum and to the averaging, I first changethe system of coordinates. For the sake of readability, I presentthe general case (with any number of planets, in any resonanceof any degree) in Appendix A, and take here the example of asystem of four planets in a 3:4:6:8 resonant chain (as is the case Article number, page 2 of 10.-B. Delisle: Analytical model of resonant chains for Kepler-223). I introduce new canonically conjugated anglesand actions as follows (see Eqs. (A.1) and (A.2)) φ = λ + λ − λ , L = Λ ,φ = λ + λ − λ , L = Λ + Λ ,φ = λ − λ , Γ = Λ + Λ + Λ + Λ ,φ = λ − λ , G = G + G + G + G ,σ = λ − λ − (cid:36) , D ,σ = λ − λ − (cid:36) , D ,σ = λ − λ − (cid:36) , D ,σ = λ − λ − (cid:36) , D . (12) G is the total angular momentum, and is a conserved quantity; Itscanonically conjugated angle ( φ ) does not appear in the Hamil-tonian. The angle φ is the only fast angle, and the averagingof the Hamiltonian is done over this angle. Therefore, its conju-gated action ( Γ ) is constant in the average problem. The averag-ing is simply done by discarding all terms that depends on φ inthe Fourrier expansion of the Hamiltonian. The angle φ is theargument of the Laplace resonance between the three innermostplanets. The angle φ is the argument of the Laplace resonancebetween the three outermost planets.The two outer planets (3 and 4) may seem to play an impor-tant role in Eq. (12), but this is only due to the arbitrary choiceof canonical coordinates (many other choices are possible). Anytwo-planet resonant angle can be expressed as a combination ofthe angles of Eq. (12). The arguments of the 4 / σ and σ . The argumentsof the 3 / σ − φ = λ − λ − (cid:36) and σ − φ = λ − λ − (cid:36) . The arguments ofthe 4 / σ − φ − φ = λ − λ − (cid:36) and σ − φ − φ = λ − λ − (cid:36) . Arguments ofresonances between non-consecutive pairs can also be expressedin the same way. For instance, for the 2 / σ − φ − φ = λ − λ − (cid:36) and σ − φ − φ = λ − λ − (cid:36) . For a system of n planetscaptured in a resonant chain, φ i ( i ≤ n −
2) and σ i ( i ≤ n ) (and alltheir linear combinations) librate around equilibrium values. Allthe actions also oscillate around equilibria. These equilibriumvalues correspond to stable fixed points of the average problem.For this example, I expand the perturbative part at first orderin eccentricities ( √ D i ), and obtain an expression of the form H = − (cid:88) i = n i , Λ i , ∆Λ i + C , (cid:112) D cos( σ − φ − φ ) + C , (cid:112) D cos( σ − φ − φ ) + C , (cid:112) D cos( σ − φ − φ ) + C , (cid:112) D cos( σ − φ − φ ) + C , (cid:112) D cos( σ − φ ) + C , (cid:112) D cos( σ − φ ) + C , (cid:112) D cos( σ − φ ) + C , (cid:112) D cos( σ − φ ) + C , (cid:112) D cos( σ ) + C , (cid:112) D cos( σ ) , (13)where the first term of Eq. (10) vanishes (because Γ is constant),and C i , j are constant coe ffi cients that depend on the masses andnominal semi-major axes ( a i , ). I provide explicit formulas for I restrict this study to first order in the planet-star mass ratio, whichmeans that three-planet terms (of order two in the mass) are neglected. the case of the 3:4:6:8 resonant chain in Appendix B. Since theHamiltonian (Eq. (13)) is developed at first order in eccentric-ities, only first-order resonances appear. In particular, the 3 / ∆Λ i should be replaced inEq. (13) by ∆Λ = ∆ L , ∆Λ = ∆ L − ∆ L , ∆Λ = ∆ L − ∆ L + (cid:15) ) , ∆Λ = ∆ L + (cid:15), (14)where (cid:15) = D − δ = n (cid:88) i = ∆Λ i , (15)which measures the distance of the system to the exact reso-nance, where D = n (cid:88) i = D i , (16)is the total deficit of angular momentum, and δ = n (cid:88) i = Λ i , − G (17)is the nominal total deficit of angular momentum (at exact res-onance). Since Λ i , and G are constants, δ is also a conservedquantity and can be used as a parameter instead of G . Theother parameter Γ does not appear explicitly in the Hamilto-nian, but is hidden in the values of Λ i , , a i , , and n i , . Indeed,the nominal semi-major axis ratios are fixed at the resonant val-ues (Eq. (7)), but Γ sets the global scale of the system ( Γ = Λ , + Λ , + Λ , + Λ , , for a 3:4:6:8 chain). The valueof Γ does not influence the dynamics of the system appart fromchanging the scales of distance, time, and energy (Delisle et al.2012). Therefore, one only need to vary the value of δ/ Γ to studythe evolution of the phase space (in particular the positions offixed points).For a given value of δ/ Γ , the fixed points are found by solv-ing the following system of equations˙ σ i = ∂ H ∂ D i = i ≤ n ) , ˙ D i = − ∂ H ∂σ i = i ≤ n ) , ˙ φ i = ∂ H ∂ L i = i ≤ n − , ˙ L i = − ∂ H ∂φ i = i ≤ n − . (18)This is a system of 4( n −
1) equations, with 4( n −
1) unknowns(2( n −
1) DOF), which in general possesses a finite number ofsolutions. These solutions can correspond to elliptical (stable)fixed points or hyperbolic (unstable) ones. To assess the stabilityof fixed points, I compute the eigenvalues of the linearized equa-tions of motions around the fixed point. Stable fixed points havepurely imaginary eigenvalues, while the eigenvalues around un-stable fixed points have a non-zero real part.
Article number, page 3 of 10 & A proofs: manuscript no. laplace
3. Application to Kepler-223
In this section, I apply my model to the four planets orbiting ina 3:4:6:8 resonant chain around Kepler-223. In Sect. 3.1, I fo-cus on the comparison between the positions of the stable fixedpoints and the observed configuration of the system (values ofresonant angles, eccentricities). In Sect. 3.2, I derive constraintson the order in which the planets have been captured in the reso-nant chain, from the observation of the equilibrium around whichthe system is currently librating.
I follow the method described in Sect. 2, and expand the Hamil-tonian at first order in eccentricities (Eq. (13)), to solve for thepositions of the fixed points (Eq. (18)) as a function of the pa-rameter δ/ Γ (see Appendix C for more details). I only considerhere the stable (elliptical) fixed points that correspond to the li-bration in resonance. In the case of Kepler-223, I find six familiesof stable fixed points (parameterized by δ/ Γ ), corresponding tosix possible areas of libration for the system. I show in Fig. 1 thepositions of these fixed points, and compare them with the ob-served values (taken from Mills et al. 2016). As shown by Millset al. (2016), the TTVs constrain the values of φ and φ verywell, while the eccentricities are only roughly determined. Thetwo-planet resonant angles σ i (which depend on the longitudesof periastron) are not well constrained (Mills et al. 2016), so thetheoretical values cannot be compared to the observations. As δ/ Γ increases, the eccentricities increase, but the angles ( φ i , σ i )remain constant. Since the best observational constraints are onthe values of φ and φ , I compare in Fig. 2 the observed valuesof these angles, with the six possible equilibrium values (whichare independent of δ/ Γ ). The observations correspond very wellto a libration of the system around one of the six configurations(the red one). This result confirms that the system is capturedin the resonant chain, and that my model is correct in first ap-proximation. It is interesting to wonder why the system was cap-tured around this particular configuration and not one of the fiveothers. This might simply be by chance (probability of 1 / As explained in Sect. 1, when two planets are captured in aMMR, their eccentricities increase until they reach an equilib-rium (e.g., Delisle et al. 2012), their period ratio remains lockedat the resonant value, and the arguments of the resonance librate.For a resonance between two planets, and at low eccentricities,the equilibrium configuration is unique. When additional planetsjoin the chain, the system evolves toward a new equilibrium (ec-centricities, two-planet resonant angles, Laplace angles). How-ever, when more than two planets are involved in the chain, therecan be more than one equilibrium (see Sect. 3.1), and the prob-ability of capture around each of the new equilibria is not nec-essarily equal. Moreover, this probability might depend on theorder in which the planets are captured in the chain. Indeed, theplanets that are already captured in resonance have their eccen-tricities excited, and the angles associated to the already formedresonances are librating around an equilibrium, while planetsthat are not yet captured should have lower eccentricities, andtheir mean longitudes should be randomly distributed. There-
Table 1.
Statistics of capture of Kepler-223 around the six possible equi-librium configurations (see Figs. 1 and 2), as a function of the order inwhich the planets have been captured in the resonant chain. order ABC ACB BAC BCA CAB CBA mean . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . Notes.
A stands for the capture of planets 1,2 in the 4 / / / σ confidence interval) around each equilibrium. The mean values( right column) are computed assuming an equal probability for eachcapture order. The system is currently observed around the red equilib-rium (the corresponding row is highlighted in red). fore, the initial conditions (angles, eccentricities) at the momentof the capture of a planet in the chain greatly depend on whichplanets are already in the chain.In this section, I investigate how the order in which the plan-ets are captured in the resonant chain influences the probabilityof capture around each of the six equilibria found in the caseof Kepler-223 (see Sec. 3.1). This problem is very complex (thephase space has 6 DOF). In particular, each time an additionalplanet joins the chain, the system might cross one or several sep-aratrices before being captured around one of the new equilibria.I do not attempt to treat this problem analytically, but rather nu-merically estimate the probabilities of capture by running 6000N-body simulations including prescriptions for the migration,with varying initial conditions. I use the same integrator as inDelisle et al. (2015), and set constant timescales for the migra-tion torque and the eccentricity damping of each planet. Themigration timescales ( T mig. , i ) are set to 10 , 3 × , 2 × ,and 1 . × yr (from the innermost to the outermost planet).The eccentricity damping timescales ( T ecc. , i ) are set such that foreach planet T mig. , i / T ecc. , i =
50. The planets start with circular andcoplanar orbits. The innermost planet is at 1 AU, and λ =
0, andthe other planets’ semi-major axes and mean longitudes are ran-domly drawn. The mean longitudes are drawn from a uniformdistribution between 0 and 2 π . The semi-major axis ratio a / a is uniformly drawn in the range [1 . , . a / a in the range[1 . , . a / a in the range [1 . , . × yr.Among the 6000 simulations, 5758 are captured in the3:4:6:8 resonant chain, while 242 are captured in other chains(or not captured). All of the 5758 captured simulations are ob-served to librate around one of the six equilibria predicted by myanalytical model. This confirms that this first-order model is cor-rectly describing the dynamics of the resonant chain. For eachsimulation, I check in which order the planets were captured,and around which equilibrium the system ended. Article number, page 4 of 10.-B. Delisle: Analytical model of resonant chains . . . e $ − $ ( d e g ) . . . . e $ − $ ( d e g ) . . . . e $ − $ ( d e g ) φ ( d e g ) − − δ/ Γ × − . . . . e − − δ/ Γ × − σ ( d e g ) − − δ/ Γ × − φ ( d e g ) Fig. 1.
Location of the (six) fixed points for the Kepler-223 resonant chain (4 planets), as a function of the parameter δ/ Γ (see Sect. 2). I plot theeccentricities of the planet ( left ), the di ff erence of longitudes of periastron between two successive planets, as well as the two-planet resonant angle σ ( center ), and the Laplace angles ( right ). Each fixed point is represented by the same color in all the plots. The gray bands (on the left and right columns) show the observed parameters of the planets (taken from Mills et al. 2016). For the eccentricities ( left ), the gray bands correspond to the1 − σ uncertainties. For the Laplace angles, they correspond to the observed libration amplitudes (which are well constrained by TTVs, see Millset al. 2016). The observed positions of the planets (and especially the Laplace angles), are consistent with a libration around the red fixed point(see also Fig. 2). The results are shown in Table 1. I use ‘A’ to denote the cap-ture of the two innermost planets (1,2) in the 4 / / / . / ffi cult to reach (low probability) ex-cept in the cases ACB and CAB (see Table 1). This means thatthe two innermost planets and the two outermost ones must first be captured in two independent two-planet resonances, and thenthe two pairs join to form the four-planet resonant chain.Since the system is currently observed around the red equi-librium (see Figs. 1 and 2), it is interesting to look at the corre-sponding capture probabilities. The most favorable case is BCA,with a probability of 52% to capture the system around the ob-served equilibrium (Table 1). This case corresponds to a captureof planets 2 and 3 in the 3 / Article number, page 5 of 10 & A proofs: manuscript no. laplace φ (deg)090180270360 φ ( d e g ) Fig. 2.
Comparison of the observed libration amplitudes of the Laplaceangles in the Kepler-223 system (gray rectangle, values taken fromMills et al. 2016), with the positions of the (six) fixed points as deter-mined from the analytical model. I use the same colors as in Fig. 1. Theobservations are consistent with a libration around the red fixed point. observed configuration is only 4 . .
4. Discussion
In this article I describe an analytical model of resonant chains.The model is valid for any number of planets involved in thechain, and for any resonances (of any order). In particular, I use itto determine the equilibrium configurations around which a res-onant chain librates. I show that contrarily to two-planet MMR,multiple equilibria may exist, even at low eccentricities, whenthree or more planets are involved.I specifically study the case of the four planets aroundKepler-223 which have been confirmed to be captured in a3:4:6:8 resonant chain (using TTVs, see Mills et al. 2016). Us-ing the analytical model expanded at first order in eccentricities,I show that six equilibrium configurations exist for this system,and the planets might have been captured around any of these sixequilibria. However, the capture probabilities are not the samefor each equilibrium, and depend on the order in which the plan-ets have been captured in the chain. Using N-body integrationsincluding migration prescriptions, I show that the scenario themost capable of reproducing the observed configuration of thesystem is to first capture the intermediate planets (2 and 3) inthe 3 / .
1% of the sim-ulations. It should be noted that several hypotheses are made tocompute these statistics. The planets are initially outside the res-onances, with period ratios slightly higher than the resonant val-ues. The migration and eccentricity damping timescales are fixedfor each planet, such that the period ratio between each pair de-creases (convergent migration). The planets are initially on cir-cular and coplanar orbits. I only vary the initial semi-major axes . . . . P i + / P i − k i + , i / k i , i + . . . . . . e i t (yr) × φ i ( d e g ) Fig. 3.
Example of a simulation that successfully reproduces the ob-served configuration of the Kepler-223 system. The scenario of captureis of type BCA (see Sect. 3.2). I plot the period ratio between each con-secutive pair of planets ( top ), the planets’ eccentricities ( middle ), andthe two Laplace angles ( bottom ). The two horizontal black lines in the bottom plot represent the equilibrium values of the Laplace angles ex-pected from the analytical model (see Sect. 3.1). and initial mean longitudes of the planets. The statistics wouldprobably slightly change with a di ff erent set-up for the simu-lations, but this would not change the two main results of thisstudy:1. Resonant chains can be captured around several equilibriumconfigurations (six for Kepler-223),2. observing a system around one of the possible equilibria pro-vides useful constraints on the scenario of formation and mi-gration of the planets. Article number, page 6 of 10.-B. Delisle: Analytical model of resonant chains
Other properties of the resonant chain might provide useful ad-ditional constraints on such a scenario. For instance, in the caseof Kepler-223, the two inner planets, as well as the two outerones, are in a compact 4 / / / / Acknowledgements.
I thank the anonymous referee for his / her useful comments.I acknowledge financial support from the Swiss National Science Foundation(SNSF). This work has, in part, been carried out within the framework of theNational Centre for Competence in Research PlanetS supported by SNSF. References
Cresswell, P. & Nelson, R. P. 2006, A&A, 450, 833Delisle, J.-B., Correia, A. C. M., & Laskar, J. 2015, A&A, 579, A128Delisle, J.-B., Laskar, J., & Correia, A. C. M. 2014, A&A, 566, A137Delisle, J.-B., Laskar, J., Correia, A. C. M., & Boué, G. 2012, A&A, 546, A71Go´zdziewski, K., Migaszewski, C., Panichi, F., & Szuszkiewicz, E. 2016, MN-RAS, 455, L104Henrard, J. 1983, Icarus, 53, 55Laskar, J. 1991, in Predictability, Stability, and Chaos in N-Body DynamicalSystems, ed. S. Roeser & U. Bastian, 93–114Laskar, J. 2000, Physical Review Letters, 84, 3240Laskar, J. & Robutel, P. 1995, Celestial Mechanics and Dynamical Astronomy,62, 193Libert, A.-S. & Tsiganis, K. 2011, Celestial Mechanics and Dynamical Astron-omy, 111, 201Mills, S. M., Fabrycky, D. C., Migaszewski, C., et al. 2016, Nature, 533, 509Papaloizou, J. C. B. 2015, International Journal of Astrobiology, 14, 291Papaloizou, J. C. B. 2016, Celestial Mechanics and Dynamical Astronomy, 126,157Papaloizou, J. C. B. & Terquem, C. 2010, Monthly Notices of the Royal Astro-nomical Society, 405, 573Ste ff en, J. H., Fabrycky, D. C., Agol, E., et al. 2012 [ arXiv:1208.3499 ]Weidenschilling, S. J. & Davis, D. R. 1985, Icarus, 62, 16 Article number, page 7 of 10 & A proofs: manuscript no. laplace
Appendix A: Change of coordinates in the generalcase
In this Appendix I describe the change of coordinates corre-sponding to Eq. (12) in the general case. I introduce the angles φ i = c i (cid:32) k i , i + q i , i + λ i + k i + , i + q i + , i + λ i + − (cid:32) k i + , i q i , i + + k i + , i + q i + , i + (cid:33) λ i + (cid:33) ( i ≤ n − ,φ n − = λ n − − λ n ,φ n = k n , n − λ n − k n − , n λ n − q n − , n ,σ i = φ n − (cid:36) i , (A.1)where the coe ffi cients c i are renormalizing factors given byEq. (A.6). The actions canonically conjugated to σ i are the AMDof the planets ( D i ), while the actions canonically conjugated to φ i are L i = i (cid:88) j = c i l i , j Λ j ( i ≤ n − , Γ = k n − , n q n − , n (cid:32) Λ n + k n , n − k n − , n (cid:32) Λ n − + k n − , n − k n − , n − (cid:32) ... + k , k , Λ (cid:33)(cid:33)(cid:33) , G = n (cid:88) i = Λ i − D i = n (cid:88) i = G i , (A.2)where l i , j = k i , i + i (cid:88) r = j q r , r + i (cid:89) s = r + k s + , s k s − , s ( j ≤ i ≤ n − . (A.3)It can be shown that the Hamiltonian does not depend on theangle φ n , and the conjugated action G (total angular momentumof the system) is thus conserved (as expected). Since φ n − is theonly fast angle in the new system of coordinates, the averagingis done over this angle, and Γ (conjugate of φ n − ) is thus constantin the averaged problem. The remaining 2( n −
1) DOF (once G and Γ are fixed), are thus defined by the angles σ i ( i ≤ n ) and φ i ( i ≤ n − D i ( i ≤ n ) and L i ( i ≤ n − φ i ( i ≤ n −
2) are the Laplace resonant an-gles between each group of three consecutive planets. The angles σ n − and σ n are the classical two-planet resonant angles betweenthe two outermost planets, and any two-planet (not necessarilyconsecutive) resonant angle can be expressed as a combinationof the angles σ i ( i ≤ n ) and φ i ( i ≤ n − Λ i , λ i , etc. as functions of L i , φ i , etc.). For the angles, the inversetransformation reads (cid:36) i = φ n − σ i ,λ i = φ n + k n − , n q n − , n n − (cid:89) j = i k j + , j k j , j + φ n − + n − (cid:88) j = i c j l j , i φ j . (A.4)Then, the Fourier expansion of the perturbative part of the aver-aged Hamiltonian exhibits angles of the form d ( k j , i λ j − k i , j λ i ) + p (cid:36) i + ( dq i , j − p ) (cid:36) j (A.5)Since all these angles should be expressed as combinations of σ i ( i ≤ n ) and φ i ( i ≤ n − ffi cients isa fraction, then the corresponding angle will not be 2 π -periodic.For instance, if one of the angles appearing in the Fourrier seriesis 3 / φ + ... , then φ = φ = π . Oneshould also make the coe ffi cients as small as possible, to avoidintroducing an unnecessary periodicity in the Hamiltonian (andthus an artificial duplication of fixed points). I thus choose therenormalizing factors c i to obtain the smallest possible integersin the Fourier expansion of the perturbative part. c i = lcm r ≤ i , r < s ( D i , r , s )gcd r ≤ i , r < s ( N i , r , s ) ( i ≤ n − , (A.6)where N i , r , s D i , r , s = k s , r l i , s − k r , s l i , r ( l i , s = s > i ) (A.7)is the fraction reduced to its simplest form. The Hamiltonian isexpanded in power series of the eccentricities, and truncated at agiven degree d max . Only the combinations for which the degree q r , s = k s , r − k r , s ≤ d max appear in the truncated Hamiltonian.Therefore, only these combinations should be considered in thecomputation of the coe ffi cient c i in Eq. (A.6).For the actions, the inverse change of coordinates reads Λ i = k i , i + c i q i , i + L i + c i − k i , i − q i − , i L i − − c i − (cid:32) k i , i − q i − , i + k i , i + q i , i + (cid:33) L i − ( i ≤ n − , Λ n − = Γ − k n − , n q n − , n ( G + D ) + c n − k n − , n − q n − , n − L n − − c n − (cid:32) k n − , n − q n − , n − + k n − , n q n − , n (cid:33) L n − , Λ n = k n , n − q n − , n ( G + D ) − Γ + c n − k n , n − q n − , n L n − , (A.8)where D = n (cid:88) i = D i = n (cid:88) i = Λ i − G (A.9)is the total deficit of angular momentum. I introduce δ = n (cid:88) i = Λ i , − G , (A.10)which is the nominal total deficit of angular momentum (at exactresonance). Since Λ i , and G are constants, δ is also a conservedquantity. I additionally define (cid:15) = D − δ = n (cid:88) i = ∆Λ i , (A.11)which measures the distance of the system to the exact reso-nance, such that ∆Λ i = k i , i + c i q i , i + ∆ L i + c i − k i , i − q i − , i ∆ L i − − c i − (cid:32) k i , i − q i − , i + k i , i + q i , i + (cid:33) ∆ L i − ( i ≤ n − , ∆Λ n − = − k n − , n q n − , n (cid:15) + c n − k n − , n − q n − , n − ∆ L n − − c n − (cid:32) k n − , n − q n − , n − + k n − , n q n − , n (cid:33) ∆ L n − , ∆Λ n = k n , n − q n − , n (cid:15) + c n − k n , n − q n − , n ∆ L n − , (A.12) Article number, page 8 of 10.-B. Delisle: Analytical model of resonant chains with ∆ L i = L i − (cid:80) j ≤ i c i l i , j Λ j , . Appendix B: Coefficients of the Hamiltonian for a3:4:6:8 resonant chain
In this appendix, I provide the expressions of the coe ffi cients C i , j that appear in the Hamiltonian of a 3:4:6:8 resonant chainat first order in eccentricities (see Eq. 13). I follow the methoddescribed in Laskar & Robutel (1995). Each coe ffi cient is a sumof two components, coming from the direct and indirect part ofthe perturbation C i , j = −G m i m j a j , (cid:115) Λ i , (cid:32) C dir i , j − G m √ µ i µ j α i , j C ind i , j (cid:33) , C j , i = −G m i m j a j , (cid:115) Λ j , (cid:32) C dir j , i − G m √ µ i µ j α i , j C ind j , i (cid:33) , (B.1)where i < j , α i , j = a i , / a j , , and C dir i , j = α − i , j b (1)3 / − b (0)3 / + α i , j b (1)3 / − α i , j b (0)3 / + α i , j b (1)3 / ≈ − . , C dir j , i = − b (1)3 / + α i , j b (0)3 / − α i , j b (1)3 / ≈ . , C ind i , j = , C ind j , i = , (B.2)for a 2 / i and j (planets pairs 1,3 and 2,4), C dir i , j = α − i , j b (1)3 / − α − i , j b (0)3 / + b (1)3 / − α i , j b (0)3 / + α i , j b (1)3 / − α i , j b (0)3 / + α i , j b (1)3 / ≈ − . , C dir j , i = − α − i , j b (1)3 / + b (0)3 / − α i , j b (1)3 / + α i , j b (0)3 / − α i , j b (1)3 / ≈ . , C ind i , j = , C ind j , i = , (B.3)for a 3 / C dir i , j = α − i , j b (1)3 / − α − i , j b (0)3 / + α − i , j b (1)3 / − b (0)3 / + α i , j b (1)3 / − α i , j b (0)3 / + α i , j b (1)3 / − α i , j b (0)3 / + α i , j b (1)3 / ≈ − . , C dir j , i = − α − i , j b (1)3 / + α − i , j b (0)3 / − b (1)3 / + α i , j b (0)3 / − α i , j b (1)3 / + α i , j b (0)3 / − α i , j b (1)3 / ≈ . , C ind i , j = , C ind j , i = , (B.4)for a 4 / ffi cients b (0)3 / and b (1)3 / are the Laplace coe ffi cients (e.g., Laskar & Robutel1995), evaluated at α i , j . Appendix C: Fixed points at first order
In this appendix I describe in more detail how to determine theposition of the fixed points of the averaged problem, at first orderin eccentricities. In this case, the Hamiltonian takes the form H = H ( (cid:15), ∆ L i ) + (cid:88) i , j (cid:44) i C i , j (cid:112) D i cos( σ i + p i , j . φ ) , (C.1)where p j , i = p i , j are vectors of n − ffi cients(which depend on the considered resonances), and φ is the vectorof φ i ( i ≤ n − = ∂ H ∂ D i = ∂ H ∂(cid:15) + √ D i (cid:88) j (cid:44) i C i , j cos( σ i + p i , j . φ ) (C.2)0 = − ∂ H ∂σ i = (cid:112) D i (cid:88) j (cid:44) i C i , j sin( σ i + p i , j . φ ) (C.3)0 = ∂ H ∂ ∆ L i = ∂ H ∂ ∆ L i (C.4)0 = − ∂ H ∂ φ = (cid:88) i , j (cid:44) i p i , j C i , j (cid:112) D i sin( σ i + p i , j . φ ) , (C.5)from which I deduce (cid:112) D i e i σ i = − (cid:32) ∂ H ∂(cid:15) (cid:33) − (cid:88) j (cid:44) i C i , j e − i p i , j . φ , (C.6)0 = (cid:88) i , j (cid:44) i , r (cid:44) i p i , j C i , j C i , r sin (cid:16) ( p i , j − p i , r ) . φ (cid:17) . (C.7)The parameter δ/ Γ only appears in these equations through thevalue of ∂ H ∂(cid:15) . Therefore, at first order in eccentricities, all theangles, as well as the eccentricity ratios are independent of δ/ Γ .The value of this parameter only changes a factor common to alleccentricities (see Eq. (C.6)). Equation (C.7) provides a set of n − n − φ i . For instance, for a 3:4:6:8resonant chain such as Kepler-223, I obtain (see Eqs. (13) and(C.7))0 = C , C , sin(2 φ ) + C , C , sin(3 φ ) + C , C , sin(3 φ + φ ) + C , C , sin( φ ) + C , C , sin( φ + φ ) , = C , C , sin(3 φ + φ ) + C , C , sin( φ ) + C , C , sin( φ + φ ) + C , C , sin(2 φ ) + C , C , sin( φ ) . (C.8)There are trivial solutions at 0 and π , but other (asymmetric)solutions might exist. The existence of asymmetric solutionsis due to the influence of first-order resonances between non-consecutive pairs. In the case where only consecutive pairs areinvolved in resonances, Eq. (C.7) simplifies, and it can be shownthat only symmetric solutions exist. The number of solutions of ahighly non-linear set of equations such as Eq. (C.8) is not easilypredicted. Moreover, among those solutions, some correspond toelliptical (stable) fixed points, and the others to hyperbolic (un-stable) fixed points. In the case of Kepler-223, I solve for theposition of fixed points numerically, and only consider the stablefixed points. I find six possible stable solutions.Once a solution is found for φ i ( i ≤ n − σ i , andthe eccentricity ratios can easily be deduced from Eq. (C.6) e i e i σ i ≈ e (cid:88) j (cid:44) i m j m C (cid:48) i , j e − i p i , j . φ , (C.9) Article number, page 9 of 10 & A proofs: manuscript no. laplace with C (cid:48) i , j = − a / n , a max( i , j ) , √ a i , C dir i , j − C ind i , j √ α i , j , e = − n n , (cid:32) ∂ H ∂(cid:15) (cid:33) − . (C.10)For a 3:4:6:8 resonant chain such as Kepler-223, I obtain e e e i(4 λ − λ − (cid:36) ) ≈ . m m + . m m e − i2 φ , e e e i(3 λ − λ − (cid:36) ) ≈ − . m m e i3 φ + . m m + . m m e − i φ , e e e i(4 λ − λ − (cid:36) ) ≈ − . m m e i( φ + φ ) − . m m e i2 φ + . m m , e e e i(4 λ − λ − (cid:36) ) ≈ − . m m e i φ − . m m . (C.11)(C.11)