Planetary Migration and Eccentricity and Inclination Resonances in Extrasolar Planetary Systems
aa r X i v : . [ a s t r o - ph . E P ] J u l Planetary Migration and Eccentricity and Inclination Resonances in ExtrasolarPlanetary Systems
Man Hoi Lee , and Edward W. Thommes ABSTRACT
The differential migration of two planets due to planet-disk interaction can result incapture into the 2:1 eccentricity-type mean-motion resonances. Both the sequence of 2:1eccentricity resonances that the system is driven through by continued migration andthe possibility of a subsequent capture into the 4:2 inclination resonances are sensitiveto the migration rate within the range expected for type II migration due to planet-diskinteraction. If the migration rate is fast, the resonant pair can evolve into a family of2:1 eccentricity resonances different from those found by Lee (2004). This new familyhas outer orbital eccentricity e & . .
5, asymmetric librations of both eccentricityresonance variables, and orbits that intersect if they are exactly coplanar. Althoughthis family exists for an inner-to-outer planet mass ratio m /m & .
2, it is possibleto evolve into this family by fast migration only for m /m &
2. Thommes & Lissauer(2003) have found that a capture into the 4:2 inclination resonances is possible onlyfor m /m .
2. We show that this capture is also possible for m /m & e maynot be able to reach the values needed to enter either the new 2:1 eccentricity resonancesor the 4:2 inclination resonances. Thus, if future observations of extrasolar planetarysystems were to reveal certain combinations of mass ratio and resonant configuration,they would place a constraint on the strength of eccentricity damping during migration,as well as on the rate of the migration itself.
1. INTRODUCTION
Extrasolar planet searches have to date yielded about 33 systems with multiple planets, and atleast 8 of these systems have a pair of planets known or suspected to be in mean-motion resonances. Department of Physics, University of California, Santa Barbara, CA 93106. Department of Earth Sciences and Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong. Department of Physics, University of Guelph, Guelph, ON N1G 2W1, Canada. θ = λ − λ + ̟ (1) θ = λ − λ + ̟ , (2)and hence the secular apsidal resonance variable θ SAR = ̟ − ̟ = θ − θ , (3)librate about 0 ◦ , which mean that the periapses are nearly aligned and that conjunctions of theplanets occur when both planets are near periapse. In the above equations, λ and λ are themean longitudes of the inner and outer planets, respectively, and ̟ j are the longitudes of peri-apse. There are three other systems with planets in 2:1 resonances: HD 82943 (Mayor et al. 2004;Ferraz-Mello et al. 2005; Lee et al. 2006; Beaug´e et al. 2008), HD 128311 (Vogt et al. 2005), and HD73526 (Tinney et al. 2006; S´andor et al. 2007), although it should be noted that a pair of planets in1:1 resonance is a plausible alternative for at least HD 82943 and HD 128311 (Go´zdziewski & Konacki2006). In addition, the HD 45364 (Correia et al. 2009), 55 Cancri (Marcy et al. 2002; McArthur et al.2004), HD 60532 (Desort et al. 2008; Laskar & Correia 2009), and HD 202206 (Correia et al. 2005)systems have planets that are in 3:2, 3:1, 3:1 and 5:1 resonances, respectively. There are un-certainties in converting this data into the fraction of multiple-planet systems with mean-motionresonances. Some of the suspected resonant pairs may not be confirmed eventually (see, e.g.,Fischer et al. 2008 for 55 Cancri). On the other hand, the number of resonant pairs that remainundetected could be quite large, because the radial velocity variation due to two planets in reso-nance (in particular, 2:1) could be indistinguishable from that due to a single planet for certainplanetary mass ratio and orbital eccentricities, given the precision levels of the existing radial ve-locity surveys (Anglada-Escud´e et al. 2008; Giuppone et al. 2009). Nevertheless, the existing dataindicate that ∼
20% of multiple-planet systems have mean-motion resonances.Mean-motion resonances can be easily established during planet formation by the convergentmigration of planets due to interactions with the circumstellar gas disk. Two giant planets thatare massive enough to open gaps in the disk individually can clear out the disk material betweenthem rather quickly, and the outer planet is forced to migrate inward by the disk material out-side its orbit (and the inner planet outward if there is any disk material left inside its orbit)(Bryden et al. 2000; Kley 2000). Both hydrodynamic and three-body simulations (with imposedmigration for the latter) have shown that the convergence of the orbits naturally leads to captureinto mean-motion resonances (Bryden et al. 2000; Kley 2000; Snellgrove et al. 2001; Lee & Peale2002; Nelson & Papaloizou 2002; Papaloizou 2003; Thommes & Lissauer 2003; Kley et al. 2004,2005; Lee 2004). 3 –The ubiquity of mean-motion resonances in extrasolar planetary systems and the ease of cap-ture into such resonances by convergent migration have prompted investigations into the varietyof stable mean-motion resonance configurations (Lee & Peale 2002, 2003; Beaug´e et al. 2003, 2006;Ferraz-Mello et al. 2003; Hadjidemetriou & Psychoyos 2003; Ji et al. 2003; Thommes & Lissauer2003; Lee 2004; Voyatzis & Hadjidemetriou 2005, 2006; Marzari et al. 2006; Michtchenko et al.2006). The 2:1 mean-motion commensurability has received the most attention, because it isthe most common one observed and includes the best case, GJ 876. With the exception ofThommes & Lissauer (2003), all of the works just cited have focused on systems with two planetson coplanar orbits. For small orbital eccentricities, antisymmetric configurations with θ libratingabout 0 ◦ and θ about 180 ◦ (as in the case of the Jovian satellites Io and Europa) are the only sta-ble 2:1 resonance configuration with both θ and θ librating. For moderate to large eccentricities,the Io-Europa configuration is not stable, but there is a wide variety of other stable 2:1 resonanceconfigurations, including symmetric configurations with both θ and θ librating about 0 ◦ (as in theGJ 876 system), asymmetric configurations with θ and θ librating about angles other than 0 ◦ and180 ◦ (some with intersecting orbits), and antisymmetric configurations with θ ≈ ◦ and θ ≈ ◦ (and intersecting orbits). Lee (2004) has shown that the sequence of 2:1 resonance configurationsthat a system with initially coplanar and nearly circular orbits is driven through by continued mi-gration depends mainly on the planetary mass ratio m /m , if the migration rate is sufficiently slow.However, there are stable 2:1 resonance configurations (e.g., those with θ ≈ ◦ and θ ≈ ◦ )that cannot be reached by the convergent migration of planets with constant masses and initiallycoplanar and nearly circular orbits. If real systems with these configurations are ever found, theirorigin would require a change in the planetary mass ratio m /m during migration, multiple-planetscattering in crowded planetary systems, or a migration scenario involving inclination resonances(Lee 2004).Thommes & Lissauer (2003) have studied the convergent migration of planets with non-coplanar orbits and found that, subsequent to the capture into the 2:1 eccentricity resonances, a capture intothe 4:2 inclination resonances (which are the lowest order inclination resonances at the 2:1 com-mensurability) is possible if m /m .
2. The 4:2 inclination-type mean-motion resonance variablesare φ = 2 λ − λ + 2Ω (4) φ = 2 λ − λ + 2Ω , (5)where Ω j are the longitudes of the ascending node. The simultaneous librations of φ and φ mean that the mixed resonance variable φ = 2 λ − λ + Ω + Ω = ( φ + φ ) / Throughout this paper, we often use “ θ ≈ x ” as an abbreviation for “the libration of θ about an angle x ” (andsimilarly for the other resonance variables) when we describe a resonance configuration. θ ≈ ◦ and θ ≈ ◦ configurationsmentioned above. Thommes & Lissauer (2003) showed an example with m /m = 3, which doesnot have capture into the inclination resonances and remains nearly coplanar throughout its evo-lution, but we notice that the evolution of the eccentricities and θ j is different from that found byLee (2004) when the outer orbital eccentricity e & .
45. These configurations with e & .
45 alsodo not correspond to any of the other eccentricity resonance configurations found by Lee (2004).As we shall see in §
3, they belong to a new family of 2:1 eccentricity resonances that can be reachedby migration if the migration rate is faster than that adopted by Lee (2004) and m /m & (cid:12)(cid:12)(cid:12)(cid:12) ˙ aa (cid:12)(cid:12)(cid:12)(cid:12) ≈ ν a = 9 . × − (cid:18) α × − (cid:19) (cid:18) H/a . (cid:19) P − (7)(Ward 1997), where a is the semimajor axis of the planet’s orbit about a star of mass m , ˙ a ≡ da/dt , ν = αH Ω is the kinematic viscosity, α is the Shakura-Sunyaev viscosity parameter, H is the scaleheight of the disk, and P = 2 π/ Ω ≈ πa / / ( Gm ) / is the orbital period. The uncertainties andradial variations in α and H/a mean that the migration rate can be at least a factor of a few faster orslower than 10 − /P . Although both Thommes & Lissauer (2003) and Lee (2004) performed three-body simulations with imposed inward migration on the outer planet only, it is difficult to determinefrom the calculations in these papers that the different results at e & .
45 for m /m = 3 are due todifferent migration rates, because they imposed migration in different ways. Thommes & Lissauer(2003) adopted H/a ∝ a / so that ˙ a is independent of a (with ˙ a = − − AU yr − for mostcalculations), and they imposed the migration in such a way that the migration does not slowdown after the capture of an inner planet into resonance. Lee (2004) adopted constant H/a andperformed calculations with ˙ a /a = − − /P and − − /P , imposed in such a way that themigration slows down by a factor β/ ( β + m /m ), where β = a /a ≈ − / , after the capture ofan inner planet into 2:1 resonance. In this paper we examine systematically the effects of differentmigration rates (within the range expected for type II migration) using three-body integrationswith migration imposed in the same way.We also examine systematically the effects of different eccentricity damping rates during mi-gration. Significant eccentricity damping can prevent the eccentricities from reaching high enoughvalues for capture into the new 2:1 eccentricity resonances or the 4:2 inclination resonances. Thereis significant uncertainty in both the sign and the magnitude of the net effect of planet-disk in-teraction on the orbital eccentricity of the planet because of sensitivity to the distribution of diskmaterial near the locations of the Lindblad and corotation resonances (Goldreich & Sari 2003;Ogilvie & Lubow 2003). However, hydrodynamic simulations of two planets orbiting inside anouter disk have shown eccentricity damping of the outer planet, with K = | ˙ e /e | / | ˙ a /a | ∼ K for capture intothe 4:2 inclination resonances is between 2 and 5 for m /m = 1.In § § m /m &
2, although thenew family exists for m /m & .
2. In §§ m /m & m /m . K = | ˙ e /e | / | ˙ a /a | for capture into the 4:2 inclinationresonances is of the order of unity. Our conclusions are summarized and discussed in §
2. NUMERICAL METHODS AND INITIAL CONDITIONS
We consider systems consisting of a central star of mass m , an inner planet of mass m , and anouter planet of mass m , with m /m between 0 . m + m ) /m =10 − . For the migration calculations starting with non-resonant orbits, the planets are initially oncircular orbits, with the ratio of the orbital semimajor axes β = a /a = 1 / β ≈ − / ), and the outer planet is forced to migrate inward.The calculations presented in § §§ i mu = 0 . ◦ ,where the initial invariable plane is used as the z = 0 reference plane. Thommes & Lissauer (2003)have found that the entry into the inclination resonances is not strongly influenced by the initialvalue of i mu , as long as it is . ◦ .The three-body integrations with imposed migration are performed using the code described inLee (2004), which is a modified version of the symplectic integrator SyMBA (Duncan et al. 1998).The outer planet is forced to migrate inward with a migration rate of the form ˙ a /a ∝ P − .The migration slows down by a factor β/ ( β + m /m ), where β ≈ − / , after the capture of aninner planet into 2:1 resonance (see paragraph with eq. [7]). The input and output are in Jacobiorbital elements, and we apply the forced migration to the Jacobi a (and eccentricity dampingto the Jacobi e for the calculations with eccentricity damping). To characterize the new familyof 2:1 eccentricity resonances, there are also calculations in § m /m (and nomigration). The modified SyMBA code used for these calculations is also described in Lee (2004).
3. A NEW FAMILY OF 2:1 ECCENTRICITY RESONANCES
We begin with migration calculations of coplanar orbits without eccentricity damping. Weconsider m /m = 0 .
1, 0 .
3, 0 .
9, 1 .
0, 1 .
5, 2 .
3, 2 .
65, 3, 5, and 10 (same as in Lee 2004) and ˙ a /a = − . − − −
4, and − × − /P . Figures 1 and 2 show the evolution of the semimajor 6 –axes a j , eccentricities e j , and eccentricity-type resonance variables θ j for the calculations with( m + m ) /m = 10 − , m /m = 3 and ˙ a /a = − . × − /P and − × − /P , respectively.For ˙ a /a = − . × − /P (Fig. 1), the sequence of 2:1 resonance configurations after resonancecapture — from ( θ , θ ) ≈ (0 ◦ , ◦ ) at small eccentricities to asymmetric librations of both θ and θ at moderate to large eccentricities — is identical to that found by Lee (2004) for | ˙ a /a | ≤ − /P , but with larger libration amplitudes for faster migration rate. The system eventuallybecomes unstable at t/P , = 3 . × , where P , is the initial outer orbital period. For thefaster migration rate of ˙ a /a = − × − /P (Fig. 2), the sequence of resonance configurationsis identical to that shown in Figure 1 (but with larger libration amplitudes) when e . .
45 (and t/P , . e & .
45. The differences between the configurations in Figures 1 and 2 at e & .
45 are mostobvious in the plots of e and θ . We confirm that the configurations in Figure 2 with e & .
45 arestable resonance configurations by integrating the configurations at t/P , = 7000 and 10 forwardwith migration turned off and finding stable libration of θ j , with the centers and amplitudes oflibration nearly identical to those just before the migration is turned off. Migration calculations with different m /m and ˙ a /a show that a system can enter the newfamily of 2:1 eccentricity resonances by fast migration if m /m &
2. For ( m + m ) /m = 10 − ,the transition occurs between ˙ a /a = − × − /P and − × − /P for m /m = 2 .
65 and 3,and between ˙ a /a = − × − /P and − × − /P for m /m = 2 .
3, 5, and 10. However, if˙ a /a is as fast as − × − /P , the libration amplitudes are sufficiently large that the systembecomes unstable soon after entering the new family. Calculations with twice the total planetarymass [( m + m ) /m = 2 × − ] show that the critical migration rate for entry into the new familyis roughly proportional to ( m + m ) /m .To find the small-libration-amplitude (or near exact resonance) counterpart for this new familyand to determine the range of m /m for which this family exists, we take the large-libration-amplitude configuration at t/P , = 7000 in Figure 2 and adjust the orbital parameters to obtain asmall-libration-amplitude configuration with m /m = 3, e = 0 . e = 0 . θ = 1 ◦ , and θ =98 ◦ . This small-libration-amplitude configuration is used as the starting point for two calculationsin which m /m is increased or decreased slowly to find a sequence of configurations with different m /m . The results are shown in Figure 3, with the calculations with d ln( m /m ) /dt = − − /P , and 10 − /P , along the positive and negative time axis, respectively. The inner eccentricity e increases (and outer eccentricity e decreases) with decreasing m /m , and the system becomesunstable when m /m is decreased to about 0 .
2. The resonance configuration for a given m /m from Figure 3 is then used as the starting point for slow inward ( ˙ a /a = − − /P ) and outward Forced migration causes offsets in the libration centers of the resonance variables (Lee 2004;Murray-Clay & Chiang 2005). Both the offsets and the libration amplitudes increase with the migration rate. Theoffsets are typically much smaller than the libration amplitudes for the asymmetric configurations but could be no-ticeable for, e.g., the ( θ , θ ) ≈ (0 ◦ , ◦ ) configuration that the system is first captured into (compare Figs. 1 and2). a /a = 10 − /P ) migration calculations to search for other resonance configurations with thesame m /m . (The slow rate of change in m /m or a in these calculations ensures that thelibration amplitudes and offsets remain small.) Figure 4 a shows the loci in the e - e plane of thestable resonance configurations from the migration calculations with m /m = 0 .
3, 1, 3, and 10.The initial conditions (dashed lines in Fig. 3) are indicated by the triangles in Figure 4 a , and theresults from inward (outward) migration extend above (below) the triangles. In addition to thecalculations shown in Figure 3, we perform several calculations in which different configurationsalong the locus shown in Figure 4 a for m /m = 0 . m /m isdecreased. In all cases, the system becomes unstable when m /m is decreased to about 0 .
2. Thusthe new family of 2:1 eccentricity resonances does not appear to exist for m /m . . e & . .
5, asymmetric librations, andintersecting orbits, and it is distinct from any of the families found by Lee (2004). In Figures 4 b – d ,we compare the new family (labeled IV) with the families (labeled I–III) found by Lee (2004) for m /m = 3, 1, and 0 .
3, respectively. Sequence I is the sequence reached by slow migration of planetswith constant masses and initially nearly circular orbits; sequence II was found by a combination ofcalculations in which m /m is changed and slow migration calculations; and sequence III consistsof configurations with ( θ , θ ) ≈ (180 ◦ , ◦ ). For m /m = 3 (Fig. 4 b ), sequences I and IV comeclose to each other. In addition, like sequence IV, the configurations in sequence I with e & . m /m = 1 (Fig. 4 c ), sequences I and IV do not come close to each other.For m /m = 0 . d ), although sequences I and IV come relatively close to each other atlarge e , the configurations in sequence I at large e have ( θ , θ ) ≈ (0 ◦ , ◦ ) and non-intersectingorbits, and it is not possible to jump to sequence IV with asymmetric librations and intersectingorbits. Because the combination that sequences I and IV come close to each other and that someconfigurations along sequence I have asymmetric librations and intersecting orbits occurs only for m /m &
2, we can understand why it is possible to enter the new family by fast migration onlyfor m /m &
2, even though the new family exists for m /m & . m /m = 0 .
54, 1, and 1 .
86. The stable periodic orbits are resonance configurations with zerolibration amplitudes. In the range of m /m studied by Voyatzis & Hadjidemetriou (2005) (i.e., m /m . .
75, Lee 2004), sequence II has loci similar to those shown in Figures 4 c and 4 d , andVoyatzis & Hadjidemetriou (2005) found that sequences II and IV (the new family) are connectedto each other by a sequence of unstable periodic orbits. 8 –
4. THE 4:2 INCLINATION RESONANCES
We consider next migration calculations of non-coplanar orbits without eccentricity damping.We perform calculations with initial mutual orbital inclination i mu = 0 . ◦ and migration rate˙ a /a = − . − . . . . , − × − /P . Figures 5, 6, and 7 show the results for m /m = 3and ˙ a /a = − − . − . × − /P , respectively. In each figure, we plot the evolution ofthe inclinations i j and the inclination-type resonance variables φ jj , as well as a j , e j , and θ j . Theevolution of φ is nearly identical to that of φ , which means that Ω − Ω = ( φ − φ ) / ◦ initially due to our choice of the initial invariable plane as the z = 0 reference plane) is closeto 180 ◦ throughout and the ascending nodes are nearly antialigned.The fast migration calculation shown in Figure 5 is similar to that in Figure 2, but with non-coplanar orbits. The e j and θ j evolve as in the planar case shown in Figure 2 and enter the newfamily of eccentricity resonances when e & .
45. There is no capture into the inclination resonancesor excitation of the inclinations. Note, however, that the circulation of the inclination resonancevariables φ jj changes from prograde to retrograde at about the same time as the entry into the newfamily of eccentricity resonances. The evolution in this figure is that found by Thommes & Lissauer(2003) in their simulation with m /m = 3.As in the planar case, the non-coplanar calculations with m /m = 3 and ˙ a /a slower than − × − /P do not enter the new family of eccentricity resonances. However, unlike the planarcase, the evolution for ˙ a /a slower than − . × − /P is qualitatively different from that for˙ a /a = − . − × − /P . Figure 6 shows a calculation with a slow migration rate ( ˙ a /a = − . × − /P ). The system is initially captured into the 2:1 eccentricity resonances only, andthe initial evolution after capture is similar to that in the planar calculation with slow migrationshown in Figure 1. But starting at t/P , ≈ . × (when e ≈ . φ jj change very slowly for about 6000 P , , and the inclinations increase rapidly. It is likelythat this slow change of φ jj is associated with the proximity to the separatrix of the inclinationresonances, since the circulation/libration period is infinite on the separatrix. We can understandqualitatively the almost exponential growth in the inclinations by noting that the lowest orderinclination resonance terms at the 2:1 commensurability in the disturbing potential Φ are secondorder and proportional to i cos φ , i i cos φ , and i cos φ . Thus the lowest order terms for di j /dt ∝ i − j ∂ Φ /∂ Ω j are proportional to i j sin φ jj and i k sin φ (where k = 2 for j = 1 and viceversa), which can result in exponential growth if φ jj and φ are not equal to 0 ◦ or 180 ◦ andchange very slowly. At t/P , ≈ . × , both φ and φ are captured into resonance andlibrate about 110 ◦ , and the inclinations increase slowly due to the continued migration forcing thesystem deeper into inclination resonances. We note that the simultaneous librations of θ j and φ jj affect the values of θ j and e j during this phase (compare Figs. 1 and 6). As in the case of theeccentricity resonances, asymmetric libration of φ jj about an angle other than 0 ◦ or 180 ◦ is possiblewhen the inclinations are not small and di j /dt is not dominated by the lowest order terms in the 9 –disturbing potential. We confirm that the configuration at, e.g., t/P , = 8 . × in Figure 6is indeed in stable inclination resonances by taking that configuration as the starting point for athree-body integration without forced migration and finding stable libration of φ jj about 110 ◦ andno secular change in i j throughout that integration. In Figure 6 the system eventually evolves outof the inclination resonances at t/P , ≈ . × , and the eccentricity resonances switch to the θ ≈ ◦ and θ ≈ ◦ configuration. As mentioned in §
1, Thommes & Lissauer (2003) have seensimilar switching to the θ ≈ ◦ and θ ≈ ◦ configuration in their simulations with m /m = 1.In contrast to the overall evolution on the migration timescale, the time spent in the phasewith φ jj changing slowly and i j increasing rapidly is nearly independent of the migration rate forslow migration. Thus this phase takes up a larger and larger fraction of the total evolution timewith increasing migration rate, and there is no longer a phase with φ jj clearly in resonance if themigration rate ˙ a /a is as fast as − . − × − /P . (Even faster migration rate would resultin entry into the new family of eccentricity resonances and no inclination excitation, as discussedabove.) For ˙ a /a = − . × − /P (Fig. 7), the rapid inclination excitation phase occurs from t/P , ≈ . × to 2 . × . Then φ jj , as well as θ , alternate between libration and circulationfor about 6000 P , , before φ jj change to circulation only and the eccentricity resonances to the θ ≈ ◦ and θ ≈ ◦ configuration, with θ nearly circulating but spending most of its timearound 180 ◦ . The oscillations of the inclination resonance variables φ jj between t/P , ≈ . × to 2 . × in Figure 7 might lead one to think that φ jj are in resonance and librating aboutequilibrium values and that the rapid increase in the inclinations is due to continued migrationforcing the system deeper into inclination resonances. However, this would be inconsistent withour earlier observation that the duration of this phase is nearly independent of the migration ratefor slow migration. To show that this rapid inclination excitation is in fact not due to migrationforcing, we take the configuration at t/P , = 2 . × in Figure 7 as the starting point for athree-body integration without forced migration. The results are shown in Figure 8. As we cansee, the inclinations continue to increase rapidly for about 4000 P , even without forced migration.Furthermore, the evolution of all the plotted variables for the first 10 P , in Figure 8 withoutmigration is similar to that between t/P , = 2 . × and 3 . × in Figure 7 with migration.Figure 8 also shows us what would happen if the migration stops due to, e.g., disk dispersal whenthe system is in the phase with rapid inclination excitation. The inclinations would continue toincrease for a while, and the inclination resonance variables would eventually end up in large-amplitude libration (alternating with circulation to varying degree).Figure 9 summarizes the results for ( m + m ) /m = 10 − and different m /m and ˙ a /a . For In the limit of the circular, planar, restricted, three-body problem, one can identify the terms that give riseto asymmetric libration for the n :1 (not just 2:1) exterior resonance as coming from the indirect part of the dis-turbing potential, and there is a qualitative physical explanation based on the indirect acceleration imparted on thetest particle over a synodic period (Pan & Sari 2004; Murray-Clay & Chiang 2005). This type of analysis has notbeen generalized to either the planar two-planet problem with two resonance variables θ and θ or the inclinationresonances.
10 – m /m & φ jj are captured into libration after a phase with φ jj changing slowly and i j increasing rapidly.The inclination resonance configuration is symmetric with φ jj librating about 180 ◦ in the regionlabeled S with m /m . . m /m = 1),and it is asymmetric with φ jj librating about an angle other than 0 ◦ or 180 ◦ in the region labeledA with m /m & . m /m . φ jj clearly in resonance (e.g., Fig. 7). The inclination excitation can be partial, with the mutualinclination reaching a maximum of ∼ ◦ or less, if m /m is large (in particular m /m = 10).In the phase with simultaneous librations of the eccentricity and inclination resonance variables,we can see from the definitions of θ j (eq. [1]–[2]) and φ jj (eq. [4]–[5]) that the arguments of periapse ω j = ̟ j − Ω j = θ j − φ jj / m /m . . φ jj , ω j librateabout ± ◦ (i.e., the periapse is on average 90 ◦ ahead of or behind the ascending node), while for m /m & . φ jj , the libration of ω j is also asymmetric.
5. EFFECTS OF ECCENTRICITY DAMPING
As we mentioned in §
1, sufficient eccentricity damping can prevent the eccentricities from reach-ing high enough values for inclination excitation and/or capture into the inclination resonances.In order to study the effects of eccentricity damping, we repeat the non-coplanar calculations in § K = | ˙ e /e | / | ˙ a /a | ,ranging from 0 .
25 to 8.We consider first the calculations with slow migration ( ˙ a /a below the solid line in Fig.9). Figure 10 shows the evolution of the mutual inclination i mu for m /m = 0 .
3, 1 .
5, and 5 . a /a = − . × − /P , and different K . As K increases from zero, the system enters the rapidinclination excitation phase and the subsequent capture into the 4:2 inclination resonances laterand later, because the eccentricities grow slower and slower. However, when K exceeds a criticalvalue, the eccentricities never reach high enough values for inclination excitation and capture intoinclination resonances. The critical value of K is ≈ . m /m . . ≈ . m /m ≈ . .
5, and ≈ . m /m & .
65 (Fig. 11).For faster migration rate, the effects of eccentricity damping on the evolution of the systemcan be more complicated. For example, the calculation shown in Figure 12 is similar to thatin Figure 7 ( m /m = 3 and ˙ a /a = − . × − /P ) but with K = 0 .
25. In this case, theeccentricity damping results in clear libration of the inclination resonance variables φ jj after therapid inclination excitation phase. Nevertheless, the critical value of K as a function of m /m shown in Figure 11 also summarizes the results for migration rate up to ˙ a /a = − × − /P , if 11 –it is interpreted as the critical value for inclination excitation, which may or may not be followedby a phase with φ jj clearly in resonance. For ˙ a /a = − × − /P (the maximum migration ratestudied), the critical value of K is modified at large m /m , with none of the calculations with m /m ≥
6. CONCLUSIONS
We have investigated the effects of different migration rates on the capture into and evolution ineccentricity and inclination resonances at the 2:1 mean-motion commensurability by the convergentmigration of two planets. We focused on systems with orbits that are initially slightly inclined withrespect to each other. The system is first captured into the sequence I of 2:1 eccentricity resonancesfound by Lee (2004), the same as in the case of exactly coplanar orbits. If the migration rate is fastand m /m &
2, the subsequent evolution is also identical to the coplanar case, with the eccentricityresonances entering a new family (sequence IV), and there is no inclination excitation or captureinto inclination resonances. The new family of 2:1 eccentricity resonances (with e & . . m /m & . m /m &
2. If the migrationrate is slow, the system subsequently enters a phase with the 4:2 inclination resonance variables φ jj changing slowly and the inclinations increasing rapidly, before it is captured into 4:2 inclinationresonances. The inclination resonance configuration is symmetric, with φ ≈ φ ≈ ◦ , if m /m . . m /m & .
5. For intermediate migration rate (and fast migrationrate if m /m . φ jj clearly in resonance. We have also studied the effects of different eccentricity damping rates duringmigration and found that the maximum value of K = | ˙ e /e | / | ˙ a /a | for inclination excitation(which may or may not be followed by a phase with φ jj clearly in resonance if the migration rateis not slow) ranges from ≈ . m /m & .
65 to ≈ . m /m ∼
1. Since the evolution issensitive to the rates of migration and eccentricity damping within the ranges expected for type IImigration due to planet-disk interaction, the discovery of extrasolar planetary systems with certaincombinations of mass ratio and 2:1 resonance geometry would place a constraint on the strengthof eccentricity damping during migration, as well as on the rate of migration itself.There are several effects of disk-planet interaction that were neglected in our analysis and mayrequire further investigations. We have focused on inward migration and eccentricity damping ofthe outer planet, because previous hydrodynamic simulations (e.g., Kley 2000; Kley et al. 2004)have shown that the disk inside the inner planet’s orbit, not just the disk material between theplanets, should be cleared rapidly. However, Crida et al. (2008) have recently shown that a betternumerical treatment of the inner disk may result in a slower depletion of the inner disk and that theeccentricity damping from the inner disk could be important in explaining the observed eccentricitiesof resonant pairs such as that in the GJ 876 system. On the other hand, when the nearby diskmass is comparable to the planet mass (i.e., in older, partially depleted disks), planets will undergo 12 –type II migration at significantly less than the disk’s viscous advection speed (Syer & Clarke 1995),so that the inner disk “outruns” the planet and eventually leaves an inner hole, no matter howsmall the inner boundary radius. The simulations of Thommes et al. (2008) suggest the majorityof planets form late enough in their parent disk’s lifetime that such holes are ubiquitous.Planet-disk interaction can also affect the orbital inclination of a planet. The net effect ofinclination damping by secular interactions and excitation by interactions at mean-motion reso-nances depends on the disk parameters, but any net damping should be on a timescale comparableto or longer than the migration timescale (e.g., Lubow & Ogilvie 2001). This is likely too slowto affect the rapid inclination excitation phase, but may result in equilibrium inclinations if thesystem is subsequently captured into inclination resonances and the inclinations are excited slowlyby continued migration.We have also neglected the secular apsidal and nodal precessions induced by the disk, whichcould change the sequence of resonance capture by changing and splitting the locations of thevarious resonances at the same mean-motion commensurability. Kley et al. (2005) have performedcoplanar three-body integrations of the GJ 876 resonant pair with additional apsidal precessionand found that the eccentricity resonances θ and θ are captured into libration in a sequencethat differs little in order or timing from the case without additional apsidal precession. Thiscan be explained by the fact that the 2:1 eccentricity resonances are first order, which meansthat the resonance-induced retrograde apsidal precession is proportional to 1 /e j and much larger inmagnitude than the disk-induced prograde precession for small e j . On the other hand, disk-inducednodal precession could have a larger effect, because the 4:2 inclination resonances are second orderand the resonance-induced nodal precession is roughly constant for small i j . Thommes & Lissauer(2003) have performed some non-coplanar calculations with additional apsidal and nodal precessionsand did not find any significant difference from the calculations without additional precessions.Although the adopted disk surface density is 5 times that of the minimum mass solar nebula, theyassumed an outer disk with an inner edge that is likely too far (20 Hill radii) from the outer planet’sorbit, and the amount of precession induced by the disk is determined primarily by the materialclosest to the planet.Adams et al. (2008) and Lecoanet et al. (2009) have recently examined the effects of turbulencein circumstellar gas disks on mean-motion resonances in extrasolar planetary systems. They havefound that stochastic perturbations due to turbulence could prevent planets from staying in resonantconfigurations and that planetary systems with mean-motion resonances should be rare. Thisappears to be inconsistent with the observational evidence discussed in §
1. One possible explanationis that these studies assumed full magnetorotational turbulence, whereas circumstellar disk modelsusually exhibit an extensive dead zone around the midplane, where the ionization fraction is lowand the disk is magnetorotationally stable due to ohmic dissipation (e.g., Gammie 1996; Sano et al.2000; Turner et al. 2007; Ilgner & Nelson 2008). For weak turbulence, the turbulence may generatelarger libration amplitudes than in smooth migration and allow, e.g., the jump from sequence I tosequence IV to occur at a slower migration rate. 13 –Finally, a better understanding of the capture into the inclination resonances is needed. Sincethe time spent in the phase with φ jj changing slowly and i j increasing rapidly is nearly independentof the migration rate for slow migration (see § j ≈ − ˙ λ + 2 ˙ λ . On theother hand, if the system is already in eccentricity resonances so that ˙ ̟ j ≈ − ˙ λ + 2 ˙ λ , then thecapture into the inclination resonances requires ˙ ω j ≈ REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
17 –Fig. 1.—
Evolution of the semimajor axes a and a , eccentricities e and e , and 2:1 eccentricity-typemean-motion resonance variables θ = λ − λ + ̟ and θ = λ − λ + ̟ for a differential migrationcalculation of coplanar orbits without eccentricity damping. The mass ratios ( m + m ) /m = 10 − and m /m = 3. The outer planet is forced to migrate inward with ˙ a /a = − . × − /P . The semimajoraxes and time are in units of the initial orbital semimajor axis, a , , and period, P , of the outer planet,respectively. The sequence of resonance configurations after resonance capture — ( θ , θ ) ≈ (0 ◦ , ◦ ) → asymmetric librations — is identical for | ˙ a /a | ≤ − /P , but with larger libration amplitudes for fastermigration rate. Fig. 2.—
Same as Fig. 1, but for the faster migration rate of ˙ a /a = − × − /P . The system enters anew family of 2:1 resonance configurations when e & .
45 (and t/P , &
18 –Fig. 3.—
Evolution of the eccentricities e and e , eccentricity-type resonance variables θ and θ , andmass ratio m /m for calculations in which a configuration in the new family of 2:1 eccentricity reso-nances with m /m = 3 is used as the starting point and m /m is increased and decreased. The start-ing configuration with small libration amplitudes is obtained by adjusting the orbital parameters of thelarge-libration-amplitude configuration at t/P , = 7000 in Fig. 2. The results from the calculations with d ln( m /m ) /dt = − − /P , and 10 − /P , are plotted along the positive and negative time axis, re-spectively. The system becomes unstable when m /m is decreased to about 0 .
2. The configurations with m /m = 0 .
3, 1, 3, and 10, indicated by the dashed lines, are used as initial conditions for calculations inFig. 4 a .
19 –Fig. 4.—
Loci in the e - e plane of coplanar 2:1 resonance configurations. ( a ) Configurations in the newfamily of eccentricity resonances from inward and outward migration calculations with initial conditions( triangles , oriented to indicate the direction for inward migration) from Fig. 3 for m /m = 0 .
3, 1, 3, and10. Comparison of the new family (labeled IV) with the families (labeled I–III) found by Lee (2004) for m /m = ( b ) 3, ( c ) 1, and ( d ) 0 .
3, respectively. It is possible to enter the new family by fast migration onlyfor m /m & m /m &
20 –Fig. 5.—
Evolution of the semimajor axes a and a , eccentricities e and e , 2:1 eccentricity-type resonancevariables θ and θ , inclinations i and i , and 4:2 inclination-type resonance variables φ = 2 λ − λ + 2Ω and φ = 2 λ − λ + 2Ω for a differential migration calculation of non-coplanar orbits without eccentricitydamping. The mass ratios ( m + m ) /m = 10 − and m /m = 3, and the initial mutual orbital inclination i mu = 0 . ◦ . The outer planet is forced to migrate inward with the fast migration rate of ˙ a /a = − × − /P . The eccentricity resonances enter the new family as in the planar case shown in Fig. 2, and thereis no capture into the inclination resonances or excitation of the inclinations.
21 –Fig. 6.—
Same as Fig. 5, but for the slow migration rate of ˙ a /a = − . × − /P . The systemis initially captured into the 2:1 eccentricity resonances only. There is a phase from t/P , ≈ . × to5 . × with φ jj changing slowly and i j increasing rapidly before φ jj are captured into libration. The systemeventually evolves out of the inclination resonances at t/P , ≈ . × , and the eccentricity resonancesswitch to the θ ≈ ◦ and θ ≈ ◦ configuration.
22 –Fig. 7.—
Same as Fig. 5, but for the intermediate migration rate of ˙ a /a = − . × − /P . The rapidinclination excitation phase occurs from t/P , ≈ . × to 2 . × . Then φ jj and θ alternate betweenlibration and circulation for about 6000 P , , before φ jj change to circulation only and the eccentricityresonances to the θ ≈ ◦ and θ ≈ ◦ configuration.
23 –Fig. 8.—
Evolution for the calculation in which the configuration at t/P , = 2 . × in Fig. 7 is usedas the starting point for a three-body integration without forced migration. The inclinations continue toincrease rapidly for about 4000 P , , and the evolution of all the plotted variables for the first 10 P , issimilar to that between t/P , = 2 . × and 3 . × in Fig. 7 with migration.
24 –Fig. 9.—
Types of evolution for different m /m and ˙ a /a . The results are from migration calculationswith ( m + m ) /m = 10 − , non-coplanar orbits, and no eccentricity damping. In the region labeled E, theeccentricity resonances enter the new family, and there is no capture into inclination resonances or excitationof the inclinations. In the region below the solid line, the inclination resonance variables φ jj are capturedinto libration (symmetric in the region labeled S and asymmetric in the region labeled A) after a phase with φ jj changing slowly and i j increasing rapidly. In the unlabeled region, there is typically a rapid inclinationexcitation phase, but not a phase with φ jj clearly in resonance. Fig. 10.—
Evolution of the mutual inclination i mu for m /m = 0 .
3, 1 .
5, and 5 .
0, ˙ a /a = − . × − /P ,and different eccentricity damping ratio K = | ˙ e /e | / | ˙ a /a | . The eccentricities never reach high enoughvalues for inclination excitation and capture into inclination resonances when K exceeds a critical value.
25 –Fig. 11.—
Critical value of K as a function of m /m . The critical value is for capture into inclinationresonances for slow migration ( ˙ a /a below the solid line in Fig. 9) and for inclination excitation (whichmay or may not be followed by a phase with φ jj clearly in resonance) for migration rate up to ˙ a /a = − × − /P . Fig. 12.—
Same as Fig. 7, but with eccentricity damping ratio K = 0 .
25. In this case, the eccentricitydamping results in clear libration of φ jjjj