Migration into a Companion's Trap: Disruption of Multiplanet Systems in Binaries
aa r X i v : . [ a s t r o - ph . E P ] S e p Migration into a Companion’s Trap:Disruption of Multiplanet Systems in Binaries
Jihad R. Touma and S. Sridhar Department of Physics, American University of Beirut, PO Box 11–0236, Riad El-Solh, Beirut11072020,Lebanon Raman Research Institute,Sadashivanagar,Bangalore560 080,India
Most exoplanetary systems in binary stars are of S–type, and consist of one or more planetsorbiting a primary star with a wide binary stellar companion. Gravitational forcing of asingle planet by a sufficiently inclined binary orbit can induce large amplitude oscillations ofthe planet’s eccentricity and inclination through the Kozai-Lidov (KL) instability [1, 2]. KLcycling was invoked to explain: the large eccentricities of planetary orbits [3]; the family ofclose–in hot Jupiters [4, 5]; and the retrograde planetary orbits in eccentric binary systems[6, 7]. However, several kinds of perturbations can quench the KL instability, by inducingfast periapse precessions which stabilize circular orbits of all inclinations [3]: these couldbe a Jupiter–mass planet, a massive remnant disc or general relativistic precession. Indeed,mutual gravitational perturbations in multiplanet S–type systems can be strong enough tolend a certain dynamical rigidity to their orbital planes [8]. Here we present a new and fasterprocess that is driven by this very agent inhibiting KL cycling. Planetary perturbationsenable secular oscillations of planetary eccentricities and inclinations, also called Laplace–Lagrange (LL) eigenmodes [9]. Interactions with a remnant disc of planetesimals can makeplanets migrate, causing a drift of LL mode periods which can bring one or more LL modesinto resonance with binary orbital motion. The results can be dramatic, ranging from excita-tion of large eccentricities and mutual inclinations to total disruption. Not requiring specialphysical or initial conditions, binary resonant driving is generic and could have profoundlyaltered the architecture of many S–type multiplanet systems. It can also weaken the multi-planet occurrence rate in wide binaries, and affect planet formation in close binaries.
The fiducial system has two planets on initially coplanar orbits around a solar mass primary star:an interior M Jup planet on a circular orbit with an initial semi-major axis of a i in = 5 AU , andan exterior M ⊕ planet with initial semi–major axis a i out between and
11 AU and eccentric-ity e i out = 0 . . The binary is also a solar mass star with semi–major axis a b >
100 AU andcorresponding period T b and angular frequency n b . Planetary migration driven by scattering ofplanetesimals has a long and productive history in relation to solar system archeology [10–12]. Itis a complex process, as discussed in the Supplementary Notes. Here we use it in a simple man-ner: the outer planet is allowed to migrate outward due to interactions with a planetesimal disc,with its semi–major axis having a prescribed form, with characteristic timescale τ . For the solarsystem, there are plausible arguments that the migration time τ > yr [11], with a lower bound τ > yr arguably needed to recover the properties of the Neptune Trojans [13]; we assume1 . × yr < τ < × yr . The physics of the problem appears clearest in a secular settingwherein the fast planetary (but not the binary) orbital motions are averaged over, turning a pointmass planet into a shape and orientation changing Gaussian wire [9]. We present a numericalsimulation with a state–of–the–art, N –wire algorithm [14], and develop a mathematical model tounderstand the results.In the N –wire experiment the binary orbit was coplanar and circular, with a b = 1000 AU,and a period T b = 22 .
36 Kyr . The initial periods of the two LL eigenmodes are .
53 Myr (aslow mode determined mainly by the massive inner planet) and .
25 Kyr (a fast mode reflectingmainly the precession of the outer planet). Outward migration of the outer planet slows down thefaster LL mode until its period approaches the binary period T b = 22 .
36 Kyr . Fig.1a shows thatthe eccentricity of the outer planet e out ≃ . until a out ≃ .
89 AU . Then it is captured intoa resonance and e out begins increasing, rising to . when a out ≃
15 AU at time t ≃ T b .Capture is also apparent in the behavior of the resonant argument, φ res ( t ) = ̟ out ( t ) − n b t , where ̟ out is the apsidal longitude. From Fig.1b we see that, after a period of circulation, φ res enters intolibration at resonance passage, with libration maintained for the full duration of the simulation.More details of the capture process are given in Extended Data Figs.1(a,b).This capture phenomenon is closely related to the lunar evection resonance, that may haveplayed a significant role in shaping the early history of the lunar orbit [15, 16]. However, the LL–mode evection resonance (LLER) is a new process, so we also present an analytical model, validfor arbitrary binary eccentricity, in the Supplementary Notes. For a circular binary orbit, LLERdynamics is governed by the normal form Hamiltonian of eqn(25): H nf = δ (cid:18) ξ + η (cid:19) − α ′ (cid:18) ξ + η (cid:19) − β ′ (cid:18) ξ − η (cid:19) , where η and ξ are a canonically conjugate pair of LL mode variables, and the parameters δ , α ′ and β ′ are functions of the slowly migrating a out . The theoretical prediction for the location ofthe exact resonance is shown as the dashed red curve in Fig.1a: exact resonance is first met by azero eccentricity planet around a out ≃ .
875 AU ; the planet circulating at e out ≃ . encountersresonance a bit later (when a out ≃ .
89 AU ), then gets engulfed by a growing and migratingnonlinearly bounded resonance region. Prediction follows simulation in the mean until e out ≃ . in our 4th order model; a higher–order expansion will improve the fit between model andsimulation. The evolving topology of flows in the ( η, ξ ) phase space, along with key structuralfeatures in and around resonance are discussed in the Supplementary Notes.Whereas encounter with the LLER is certain with migration, capture in it is probabilistic, anddepends on the strength of the resonance, the migration rate and the initial planetary eccentricityat which LLER is encountered. For τ ∼ T b and initial e out = 0 . , the probability of capturein LLER exceeds one–half. At the assumed binary separation, capture becomes certain as themigration rate is slowed down by two orders of magnitude; more details are discussed in theSupplementary Notes. Capture is likely to improve in tighter S–type systems: for a b = 200 AU a in = 5 AU , evection is crossed at a out ≃ .
43 AU , with faster apse–precession and a strongerresonance, in the course of migration. The capture probability remains high for initial e out = 0 . ,even at faster migration rates. Having described the broad secular skeleton of LLER, we note thatthe full problem is richer due to the interplay of planetary mean–motion resonances (PMMR). Tostudy this, it is necessary to perform simulations that do not average over planetary mean–motions.Below we present two such N –body simulations with the open–source package MERCURY [12,17]. The first MERCURY experiment is the unaveraged version of the N –wire simulation ofFig.1, and its results are displayed in Fig.2. Signs of PMMR are apparent in Fig.2a, in the jumpsexperienced by semi–major axis of the outer planet as it migrates. The system is captured inLLER with consequent growth of the eccentricity (Fig.2b), and libration of the resonant argument(Extended Data Fig.2a). What is remarkable though, and distinct from the N –wire experiment,is the non–monotonic behavior of the mean eccentricity of the LLER–locked planet, leading toescape from LLER altogether, eventually settling at e out ≃ . . Escape from capture is due toplanetary mean motion resonances (PMMR), which enhance exchange of angular momentum. Ofthe four PMMR located near a out ≃ .
56 AU the strongest is the 4:1, with argument φ =4 λ out − λ in − ̟ out ; a short time segment of φ is shown in Extended Data Fig.2b.In the second MERCURY experiment, the binary orbit had eccentricity . and inclination ◦ , which are modest values for wide–binaries. Fig.3a shows a 3:1 PMMR exciting e out to . at t ≃
21 Myr , with capture in LLER at t ≃
72 Myr when a out ≃ .
92 AU . As earlier, passagethrough the 4:1 resonance forces the outer planet out of LLER. This is followed by another excita-tion around
120 Myr , associated with passage through a 9:2 resonance; and then through a clusterof resonances around .
63 AU . Meanwhile a out grows with jumps at the PMMR (Extended DataFig.3a), and φ res transits in and out of libration during LLER (Extended Data Fig.3b). Both a out and e out diffuse until the planet is ejected from the system altogether . Ejection is not a necessaryoutcome, but is often associated with PMMR when both a out and e out are large. In Fig.3b we followthe excitation of the mutual inclination to ◦ , due to coupling within LLER–lock, of eccentricityand inclination by a vertical resonance, which is followed by another excitation to . ◦ .LLER is a powerful and generic mechanism that can profoundly affect the architecture ofmultiplanet S–type binary systems. It can also come in different flavors. Inward migrationof the inner planet can occur through a runaway process [18], whose slower migration rate hashigher capture probability, particularly in tighter S–type systems with shorter LL periods. Inwardmigration may explain the largish eccentricities and inclinations in systems with super–Jupitersized planets on sub–AU orbits. LLER–induced disruption in moderately wide binaries ( a b < ) may be responsible for the recently reported dearth of multiplanet systems in binariesat such separations [19]. The extent to which LLER disrupts/suppresses planet formation when a b <
20 AU [20] needs to assessed within planet formation studies [21]. A multi–mass planetarysystem will have a broader spectrum of LL frequencies than a two–planet system. The richer LLERand stronger PMMR open more pathways for disruption, and could relieve an initial multiplanet3ystem of all but one of its planets.
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Acknowledgments
We are grateful to Scott Tremaine and the Institute for Advanced Study for hosting usin the early stages of our collaboration.
Author Contributions
J.R.T. and S.S. identified the process, developed and analyzed mathematical mod-els, and wrote paper and supplements. J.R.T. performed and analyzed numerical experiments, producingfigures in article and supplements.
Author Information
The authors declare no competing financial interests. Correspondence and requestsfor materials should be addressed to J.R.T. ([email protected]).
Capture into the Laplace–Lagrange Evection Resonance.
The fiducial N –wire ex-periment was performed with exponential migration, a out ( t ) = a i out exp[ t/τ ] , with a i out = 10 AU and τ = 10 T b . (a) Growth of e out when it is captured in the migrating LLER. The dashed red lineis the prediction from the analytical 4th–order theory presented in Supplementary Notes. (b) φ res transitions from circulation to libration around ◦ when captured in LLER.igure 2Figure 2: LLER with PMMR for coplanar circular binary orbit.
The first MERCURY experi-ment was conducted with damped migration, a out ( t ) = a fin − ∆ a exp[ − t/τ ] , with a fin = 15 AU , ∆ a = 3 AU and τ = 10 yr = 4500 T b . (a) Migration of a out with signs of PMMR. (b) LLER isencountered around a out ≃ .
88 AU at t ≃
15 Myr , with initial growth of e out during capture,then decay because of interruption by PMMR, and ultimately escape from LLER.7igure 3Figure 3: LLER with PMMR for inclined and eccentric binary orbit.
The second MERCURYexperiment was conducted with exponential migration a out ( t ) = a i out exp[ t/τ ] , with a i out = 10 AU and τ = 4 . × yr = 2 × T b . (a) The 3:1 PMMR around
21 Myr excites e out to about . .LLER is encountered around a out ≃ .
92 AU , at t ≃
72 Myr . (b) Mutual inclination first excitedto ◦ , then to . ◦ . 8 upplementary Notes A Simple Analytical Model of the Laplace–Lagrange Evection Resonance
We present an analytical model of the Laplace–Lagrange Evection Resonance (LLER) in a two–planet system orbiting the primary star, with the companion star orbiting in the same plane asthe planets. The main general result is eqn.(21) for the secular Hamiltonian for the two Laplace–Lagrange (LL) modes (given in the mode action–angle variables to 4th order in the planetary ec-centricities) when they are forced by a binary orbit of arbitrary eccentricity. In order to understandthe fiducial N –wire simulation reported in the main text we specialize to a circular binary orbit.Then the secular Hamiltonian reduces to the normal form , H nf of eqn.(25). Simple computationswith H nf provide (a) a graphic narrative of the unfolding of LLER in phase space (Extended DataFig.4); (b) characteristics of the LLER islands including measures of the adiabaticity (ExtendedData Fig.5); (c) the dependence of capture probability on the initial planetary eccentricity andnon–adiabaticity.The Hamiltonian governing the secular dynamics of planets of mass m and m can bewritten as: H sec = − Gm m (cid:28)(cid:28) | r − r | (cid:29)(cid:29) − GM B m (cid:28) | r b ( t ) − r | − r b ( t ) · r r b ( t ) (cid:29) − GM B m (cid:28) | r b ( t ) − r | − r b ( t ) · r r b ( t ) (cid:29) . (1)Here “ << >> ” means that the expression inside is to be averaged over the Kepler orbits of bothplanets, and “ < > ” means that the averaging is to be performed over the Kepler orbit of eitherplanet 1 or 2, as the case may be. Let ( a , a , a b ) and ( e , e , e b ) be the semi–major axes and theeccentricities of planet 1, planet 2 and the binary orbit, respectively. Let g and g be the periapseangles of the orbits of planets 1 and 2, and θ b ( t ) be the polar angle to the location of the binarystar. In the absence of planetary migration, secular dynamics conserves all the semi–major axes. e b is constant, because the binary is assumed to be in a given Kepler orbit. The secular Hamiltoniangoverns the dynamics of the quantities ( g , g ; e , e ) . We assume that a ≤ a ≪ a b , and expandthe binary potential to quadrupolar order: for i = 1 , , (cid:28) | r b ( t ) − r i | − r b ( t ) · r i r b ( t ) (cid:29) = a i r b (cid:20) e i + 152 cos (2 g i − θ b ) (cid:21) + . . . . (2)However, a multipolar expansion of the interaction between planets 1 and 2 may not be appropriate,because their semi–major axes may be of comparable magnitudes. Therefore we assume that both9 ≪ and e ≪ and expand to fourth order in the eccentricities: (cid:28)(cid:28) | r − r | (cid:29)(cid:29) = a a n c e + c e e cos ( g − g ) + c e + c e + c e e cos ( g − g ) + c e e + c e e cos (2 g − g )+ c e e cos ( g − g ) + c e + . . . o , (3)where the c mnl are functions of ( a /a ) , and can be written in terms of Laplace coefficients [9].When eqns. (2) and (3) are substituted in eqn. (1), we obtain the secular Hamiltonian as a functionof the four dynamical quantities ( g , g ; e , e ) . However, these quantities are not canonicallyconjugate variables. Therefore we define new canonical coordinates ( q , q ) , and their canonicallyconjugate momenta ( p , p ) by, q i = − q m i ( GM A a i ) / [1 − (1 − e i ) / ] sin g i ,p i = + q m i ( GM A a i ) / [1 − (1 − e i ) / ] cos g i . (4)Expressing (2) and (3) in terms of the variables ( q i , p i ) , the secular Hamiltonian of eqn. (1) can bewritten as the sum of three terms: H sec = H LL + H bin + H non . (5)Here H LL is the Laplace–Lagrange Hamiltonian that consists of all the time–independent quadraticterms, H bin is the time–dependent driving due to the binary motion, and H non is the nonlinear partthat has all the time–independent fourth order terms. H LL = α (cid:0) q + p (cid:1) + β ( q q + p p ) + α (cid:0) q + p (cid:1) , (6)where the coefficients α , β and α are defined by, α = − r GM A " m a / c a + 38 M B a / a b (1 − e b ) / ,β = − r Gm m M A a / c a / ,α = − r GM A " m a c a / + 38 M B a / a b (1 − e b ) / . (7) H LL determines the two LL modes of oscillations of the eccentricities and periapses of the twoplanets. It has contributions from planetary interactions and the orbit–averaged binary quadrupole.To quadratic order the purely time–dependent binary forcing is represented by: H bin = α b ( t ) (cid:2) q + p (cid:3) + γ b ( t ) (cid:8) q p sin [2 θ b ( t )] + (cid:0) q − p (cid:1) cos [2 θ b ( t )] (cid:9) + α b ( t ) (cid:2) q + p (cid:3) + γ b ( t ) (cid:8) q p sin [2 θ b ( t )] + (cid:0) q − p (cid:1) cos [2 θ b ( t )] (cid:9) , (8)10here the constants α b ( t ) , γ b ( t ) , α b ( t ) and γ b ( t ) are defined by, α b ( t ) = − r GM A M B a / " r b ( t ) − a b (1 − e b ) / , γ b ( t ) = 158 r GM A M B a / r b ( t ) ,α b ( t ) = − r GM A M B a / " r b ( t ) − a b (1 − e b ) / , γ b ( t ) = 158 r GM A M B a / r b ( t ) . (9)When the binary orbit is circular, r b = a b and e b = 0 ; then the coefficients α b and α b both vanish.The nonlinear, time–independent, fourth–order nonlinear terms are gathered together in: H non = η (cid:0) q + p (cid:1) + κ ( q q + p p ) (cid:0) q + p (cid:1) + ρ (cid:0) q + p (cid:1) (cid:0) q + p (cid:1) + κ ( q q + p p ) (cid:0) q + p (cid:1) + λ ( q q + p p ) + η (cid:0) q + p (cid:1) , (10)where the constants η , κ , ρ , κ , λ and η are defined by, η = − m M A m a (cid:18) c − c (cid:19) + 332 M B a M A m a b (1 − e b ) / ,κ = − m / a / M A m / a / (cid:18) c − c (cid:19) , ρ = − a / M A a / (cid:0) c − c (cid:1) ,κ = − m / a / M A m / a / (cid:18) c − c (cid:19) , λ = − a / M A a / c ,η = − m a M A m a (cid:18) c − c (cid:19) + 332 M B a M A m a b (1 − e b ) / . (11) H non determines the response of the LL modes to the resonant forcing by the binary. The sec-ular Hamiltonian, defined by eqns. (5)—(11), governs the dynamics of the LL modes, in the 4–dimensional phase space, ( q , q ; p , p ) . Below we write each of its three terms, H LL , H bin and H non , in terms of the action–angle variables of the LL modes. Define new canonical variables, ( Q , Q ; P , P ) by: Q = q cos χ + q sin χ , Q = − q sin χ + q cos χ ; P = p cos χ + P sin χ , P = − p sin χ + p cos χ , (12)where χ is such that tan (2 χ ) = β/ ( α − α ) . In the new variables, H LL of eqn (6) is: H LL = (cid:2) α cos χ + β sin χ cos χ + α sin χ (cid:3) (cid:0) Q + P (cid:1) + (cid:2) α sin χ − β sin χ cos χ + α cos χ (cid:3) (cid:0) Q + P (cid:1) . (13)11e now define action–angle variables for the LL modes, ( J , J ; ψ , ψ ) , by: Q = √ J sin ψ , P = √ J cos ψ , Q = √ J sin ψ , P = √ J cos ψ . (14)Then H LL = ω J + ω J , (15)is in canonical form where ω = ( α + α ) + ( α − α ) cos (2 χ ) + β sin (2 χ ) ,ω = ( α + α ) − ( α − α ) cos (2 χ ) − β sin (2 χ ) , (16)are the mode frequencies. We use eqns. (12) and (14) to work out H bin in terms of action–anglevariables for the LL modes. Dropping the fourth order terms in eqn. (8), we have H bin = µ ( t ) J + µ ( t ) J + µ ( t ) p J J cos ( ψ − ψ ) + ν ( t ) J cos [2 ψ + 2 θ b ( t )]+ ν ( t ) J cos [2 ψ + 2 θ b ( t ))] + ν ( t ) p J J cos [ ψ + ψ + 2 θ b ( t )] , (17)where the new coefficients, µ ( t ) , µ ( t ) , µ ( t ) , ν ( t ) , ν ( t ) and ν ( t ) , are defined by µ ( t ) = + 2 α b ( t ) cos χ + 2 α b ( t ) sin χ ,µ ( t ) = + 2 α b ( t ) sin χ + 2 α b ( t ) cos χ ,µ ( t ) = 2 [ α b ( t ) − α b ( t )] sin (2 χ ) ,ν ( t ) = − γ b ( t ) cos χ − γ b ( t ) sin χ ,ν ( t ) = − γ b ( t ) sin χ − γ b ( t ) cos χ ,ν ( t ) = − γ b ( t ) − γ b ( t )] sin (2 χ ) . (18) Lastly, we write H non in terms of LL–modal variables by usingeqns. (12) and (14) to susbtitute for ( q , q ; p , p ) in terms of ( ψ , ψ ; J , J ) in eqn. (10). Ofthe many terms, those proportional to cos ( ψ − ψ ) and cos (2 ψ − ψ ) can be dropped when ω and ω are well–separated (i.e. non–degenerate, as in the example explored in the body of thearticle), because the angle–dependent terms are oscillatory and do not contribute significantly tothe dynamics. Therefore, nonlinear part of the Hamiltonian for nondegenerate modes can be takenas, H n . d . non = ξ J + ξ J + ξ J J , (19)12here the coefficients, ξ , ξ and ξ are given by ξ = 4 η cos χ + 2 κ cos χ sin(2 χ ) + ρ sin (2 χ )+ 2 κ sin χ sin(2 χ ) + λ sin (2 χ ) + 4 η sin χ ,ξ = 4 η sin χ − κ sin χ sin(2 χ ) + ρ sin (2 χ ) − κ cos χ sin(2 χ ) + λ sin (2 χ ) + 4 η cos χ ,ξ = 4 η sin (2 χ ) − κ sin(4 χ ) + 4 ρ cos (2 χ )+ 2 κ sin(4 χ ) + 2 λ cos(4 χ ) + 4 η sin (2 χ ) . (20)
2. Secular Hamiltonian for nondegenerate LL modes with binary driving:
Gathering togetherwith the expressions in eqns. (15), (17) and (19), we have the secular Hamiltonian in the desiredmode variables: H sec = [ ω + µ ( t )] J + [ ω + µ ( t )] J + ξ J + ξ J + ξ J J + µ ( t ) p J J cos ( ψ − ψ ) + ν ( t ) J cos [2 ψ + 2 θ b ( t )] ++ ν ( t ) J cos [2 ψ + 2 θ b ( t ))] + ν ( t ) p J J cos [ ψ + ψ + 2 θ b ( t )] . (21)There are resonances between the binary and the LL modes, when n b is commensurate with any ofthe frequencies ω , ω or ( ω + ω ) / . The set of resonances is particularly rich for an eccentric binary orbit. When the binary orbit is circular , the coefficients µ i ( t ) all vanish, and the ν i becometime–independent. Then eqn.(21) simplifies to: H circ = ω J + ω J + ξ J + ξ J + ξ J J + ν J cos (2 ψ + 2 n b t )+ ν J cos (2 ψ + 2 n b t ) + ν p J J cos ( ψ + ψ + 2 n b t ) , (22)where the coefficients, ν , ν and ν , are given by ν = − r GM A M B a b (cid:16) a / cos χ + a / sin χ (cid:17) ν = − r GM A M B a b (cid:16) a / sin χ + a / cos χ (cid:17) ν = − r GM A M B a b (cid:16) a / − a / (cid:17) sin (2 χ ) . (23) It turns out that ω and ω are negative, giving rise to three types of LLER, for n b ≃ | ω | , or n b ≃ | ω | , or n b ≃ | ( ω + ω ) / | . The possibilities are extremely rich, so we focus on the caserelevant to the fiducial N –wire experiment described in the main text.13 . Normal form Hamiltonian for the N –wire experiment: In the N –wire experiment, the innerplanet has m = 10 M jup , a = 5 AU with initial e = 0 , and g = 0 ; the outer planet has m = 10 M ⊕ , a < .
89 AU (here it is initially at
11 AU ) with e = 0 . and g = 0 . The binarycompanion is also a solar mass star, on a circular orbit with semi-major axis a b = 1000 AU andperiod T b = 22 , .
69 AU . The two LL mode frequencies are ω = − . × − rad/yr (a periodof 4.73 Myrs) and ω = − . × − rad/yr (and a period of , .
29 yr ). Here we study LLERwhen n b ≃ | ω | . Since ω and ω are well–separated in magnitude, it is clear that n b cannot beclose to either | ω | or | ( ω + ω ) / | . Then the driving terms proportional to cos (2 ψ + 2 n b t ) and cos ( ψ + ψ + 2 n b t ) are oscillatory, and can be dropped. Hence H circ is effectively independentof the angle ψ , which implies that J = J ≃ constant. Therefore the resonant Hamiltonian fordescribing LLER of the second mode takes the simple form, H m2 = ( ω + n b + ξ J ) J + ξ J + ν J cos (2 ϕ ) . (24)In the absence of planetary migration, this is a time–independent 1 degree–of–freedom Hamilto-nian in the canonically conjugate variables, ϕ = ψ + n b t and J , and the dynamics is obviouslyintegrable. This Hamiltonian is typical of 2nd–order resonance models, and can be further reducedto a normal form in the new canonical variables, ξ = √ J cos( ϕ ) and η = √ J sin( ϕ ) : H nf = δ (cid:18) ξ + η (cid:19) − α ′ (cid:18) ξ + η (cid:19) − β ′ (cid:18) ξ − η (cid:19) , (25)where δ = ( ω + n b + ξ J ) , α ′ = − ξ , β ′ = − ν . (26)The normal form Hamiltonian has a long history in solar–system dynamics (see [22] and referencestherein). Of relevance to our problem is lunar evection , the resonance between the precession ofthe peripase of the Moon’s orbit around an oblate Earth, and the mean motion of a massive outerperturber, the Sun [15].
4. Planetary migration:
Numerical simulations of planetary migration with gaseous discs havereported a wide variety of behaviour — planets opening gaps, clearing out inner discs, stalling intheir migration, reversing migration; multiple planets undergoing divergent migration, undergoingconvergent migration, or getting captured into mean motion resonances [23–25] — but we donot explore this here. We consider planetary migration driven by scattering of planetesimals .This process is believed to have taken place in the solar system, and to have left its signaturein the dynamical properties of minor bodies, as well as spin and orbital features of the planetsthemselves [10, 11, 26–28]. The planetary system is considered fresh out of the evaporation ofthe gaseous disc, with a remnant disc of surviving planetesimals which, in the course of theirdynamical stirring, then scattering, by the planets is expected to drive migration. The timescaleof migration is set by the inner boundary, mass and size distribution in the remnant disc [11, 27].For the solar system, there are plausible arguments for it being on the order of a few times years [11], with a lower bound of a few yr set by exercises which seek to recover propertiesof Neptune Trojans with planetary migration [13]. We cannot commit to any particular timescale14ithout careful (and largely numerical) treatment of mean motion resonances and their stirring ofa preexisting disc into planet crossing orbits. We have thus assumed a range of timescales yr to × yr , which is about to , binary orbital periods. The direction of migration onthe other hand is largely determined by the mass and location of the perturbing planets.
5. Dynamics of LLER: As a increases from an initial value of
11 AU due to planetary migration,the control parameters ( δ, α ′ , β ′ ) acquire slow time dependence, making H nf of eqn.(25) a 1.5degree–of–freedom system. In the N –wire fiducial system, δ is an increasing function of time,starting with a negative value − . × − and then transitioning to positive values around a =11 .
884 AU ; β ′ ∼ − is always positive; α ′ = 2 . initially, and decreases while remainingpositive over the relevant range of a . The variation of ( δ, α ′ , β ′ ) results in non trivial changesin the topology of the instantaneous global phase portraits of H nf in the ( η, ξ ) plane. As can beseen in the four panels of Extended Data Fig.4, the origin (0 , — corresponding to a circularorbit — is always an equilibrium point. Since α ′ > , the origin is initially stable because δ < − β ′ . As δ increases in the course of migration, it goes unstable for δ ≥ − β ′ , which happensat a = 11 .
875 AU . This first bifurcation gives rise to two stable equilibria at ( ± η c , , where η c = [( δ + β ′ ) /α ′ ] / . These are the centres of LLER with librating orbits around them; seeFigs.S1(a, b). Post–encounter, as δ continues to increase, the centres drift apart and the islandsgrow, capturing into LLER any trajectory that comes their way. Stability is restored to the originfor δ ≥ + β ′ (at a = 11 .
89 AU ). This second bifurcation gives rise to two unstable equilibriaat (0 , ± ξ un ) where ξ un = [( δ − β ′ ) /α ′ ] / ; see Extended Data Figs.4(c, d). As δ continues togrow, the basin of circulating orbits around the origin also grows, squeezing the LLER islands andcapturing some of their librating orbits. Extended Data Fig.5a shows the evolution of η c , and theextrema of the separatrix. In Extended Data Figs.5(b,c) we map the evolution of η c to that of theplanetary eccentricities. The dashed red curve in Fig. 1a of the main text is obtained from e ( t ) ofExtended Data Fig.5b .When ( δ, α ′ , β ′ ) vary slowly compared to the libration period around LLER, we are in the adiabatic regime . At any time, there is a maximum eccentricity, e max ( t ) , that is reached by theseparatrix; let e c be the maximum value of all the e max ( t ) . Capture is certain if LLER is encoun-tered when e < e c . In our problem, e c ≃ . , corresponding to the the onset of the secondbifurcation shown in Extended Data Fig.4c. If e > e c at resonance encounter, capture is notcertain. The probability of capture can be computed analytically [22], and is given by the ratioof (a) the rate of increase of the area of the libration zone, to (b) the rate of increase of the sumof the areas of the libration and circulation zones. Note that the circulation zone has zero areafor − β ′ /α ′ ≤ δ/α ′ ≤ β ′ /α ′ , hence capture is certain with δ/α ′ increasing past − β ′ /α ′ . Whenthe variation is non–adiabatic , outcomes are not easily predictable from the instantaneous phaseportraits. Then capture and escape must be quantified through computations with H nf for differentinitial conditions of the planet.
6. Estimates of Adiabaticity and Capture Probability:
We computed
Γ = ω lib / πr mig , where r mig = d ln η c / d t is the migration rate of the island centres, and ω lib = 2 p β ′ [ δ + β ′ ] is theibration frequency around the island centre. Γ is a measure of adiabaticity, and is plotted versus e max in Extended Data Fig.5.d. The dynamics is increasingly adiabatic for larger a , with theisland migrating to higher eccentricities. For τ ∼ T b and e max = 0 . , we have Γ ≃ . ;the migration rate is larger than the libration time, implying non–adiabatic passage (in the rangeof e max for which capture is guaranteed in the adiabatic limit). Were τ larger by a factor , wewould be in the adiabatic regime, and capture in LLER would be certain for e ≤ . (exceptingnear–zero eccentricity where adiabaticity is practically impossible). Theoretical estimates of thecapture probabilities are not well–determined in this non–adiabatic regime. Hence we integratedtrajectories with the evolving H nf for a range of initial eccentricities and uniformly distributedperiapses, and discovered that: (a) Capture is ruled out for e < . ; this outcome appearstypical of non–adiabatic passage through LLER, and was already noted in studies of the earlyhistory of the lunar orbit [15]; (b) Matters improve for larger eccentricities: more than half theplanets with e = 0 . get captured, and the capture probability rate gets closer to the adiabaticestimate with larger e at encounter. References
22. Borderies, N. & Goldreich, P. A simple derivation of capture probabilities for the J + 1 : Jand J + 2 : J orbit-orbit resonance problems.
Celestial Mechanics
Astrophys. J.
Annu. Rev. Astron. Astrophys.
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Icarus
Astron. J.
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Capture into the Laplace–Lagrange Evection Resonance.
The fidu-cial N –wire experiment was performed with exponential migration, a out ( t ) = a i out exp[ t/τ ] , with a i out = 10 AU and τ = 10 T b . (a) Phase space trajectory of the outer planet during capture. η = [1 − p − e ] sin( φ res ) and ξ = [1 − p − e ] cos( φ res ] are canonical coordinate andmomentum of the captured LL mode. Trajectory clearly reveals transition from circulation withinitial eccentricity of . (inner ring) to libration when captured in LLER (funnel from inner ringmoving toward negative η ). (b) Capture is also seen, albeit in a subdued manner, in the modestgrowth of e in during LLER.xtended Data Figure 2Extended Data Figure 2: LLER with PMMR for coplanar circular binary orbit.
The firstMERCURY experiment was conducted with damped migration, a out ( t ) = a fin − ∆ a exp[ − t/τ ] ,with a fin = 15 AU , ∆ a = 3 AU and τ = 10 yr = 4500 T b . (a) Transitions of φ res from circulationto libration during capture and back to circulation after escape. (b) A
100 Kyr time–segment of φ when still captured in evection; signature of the PMMR is evident in the repeated transitionsbetween libration and circulation.xtended Data Figure 3Extended Data Figure 3: LLER with PMMR for inclined and eccentric binary orbit.
The secondMERCURY experiment was conducted with exponential migration a out ( t ) = a i out exp[ t/τ ] , with a i out = 10 AU and τ = 4 . × yr = 2 × T b . (a) Migration of a out with signs of PMMRaround , , and
125 Myr . (b) Transitions of φ res from circulation to libration during capturein LLER, and then back to circulation due to passage through the 4:1 PMMR which forces the outerplanet out of LLER.xtended Data Figure 4Extended Data Figure 4: Phase Space with Migrating Planet.
Isocontours of H nf at differenttimes, showing bifurcations of equilibria and emergence in islands where capture is probable.Note: both ξ and η have been rescaled by a factor of p m √ GM A a to turn them into eccentricitylike variables. (a) At a = 11 AU the origin is stable with circulating orbits around it. (b) At a = 11 .
88 AU the origin has gone unstable, and two LLER islands have appeared. (c) At a = 11 .
894 AU the origin is about to go stable again. (d) At a = 13 AU we are past the secondbifurcation; there is an inner circulating zone surrounded by two libration lobes.xtended Data Figure 5Extended Data Figure 5: Characteristics of the LLER Islands: (a)
Drift of LLER centers andextremities (again η is rescaled by a factor of p m √ GM A a to turn it into an eccentricity–likevariable). (b) The eccentricity of the outer planet increases significantly during capture in LLER.This is the dashed red line in Fig.1 which is compared with the fiducial N –wire simulation. (c) Modest growth of the eccentricity of the inner planet when captured in LLER. (d)
The adiabaticityindex Γ plotted versus e max for τ = 10 T bb