sign in sign up
<<
Home , Kirkwood gap

Nesvorný et al.: Dynamics in Mean Motion Resonances 379

Regular and Chaotic Dynamics in the Mean-Motion Resonances: Implications for the Structure and Evolution of the Belt

D. Nesvorný Southwest Research Institute

S. Ferraz-Mello Universidade de São Paulo

M. Holman Harvard-Smithsonian Center for Astrophysics

A. Morbidelli Observatoire de la Côte d’ Azur

This chapter summarizes the achievements over the last decade in understanding the effect of mean-motion resonances on asteroid orbits. The developments from the beginning of the 1990s are many. They range from a complete theoretical description of the secular dynamics in the mean-motion resonances associated with the Kirkwood gaps to the discovery of the three- body resonances and slow chaotic phenomena acting throughout the . Consequences arising from these results have required remodeling the process of asteroid delivery to the - crossing orbits.

1. INTRODUCTION namics in the main MMRs with , taking into account the precession of Jupiter’s orbit ( and Morbidelli, It is evident from the plethora of studies of mean-motion 1995). This supplied convincing evidence that the orbital resonances (MMRs) in the last decade that major advances evolution of simulated bodies toward the -crossing have been made in this area, and these advances have broad orbits is driven by chaotic secular resonances (Morbidelli consequences for our understanding of . An atten- and Moons, 1995). A brief account of this new development tive observer of this branch of dynamical would is given in section 4. have noted the following progress. The 2:1 MMR with Jupiter at 3.27 AU is a unique case, It was shown that bodies in the main MMRs with Jupiter because here the chaotic secular resonances are located at (3:1, 4:1, 5:2) can reach very high orbital eccentricities high eccentricities (Morbidelli and Moons, 1993). Yet in the (Ferraz-Mello and Klafke, 1991; Klafke et al., 1992; Saha, early 1990s, only a few asteroids were known to exist on 1992). This result suggested that resonant bodies can be resonant orbits, in contrast to the large nonresonant popula- transported to -, Earth-, and -crossing orbits and tions on either side of the 2:1 MMR. The studies devoted then be efficiently extracted from the resonances due to the to the question of the long-term stability of the resonant larger mass of the two latter . The cited works con- orbits were marked by the evolution of the numerical meth- firmed the pioneering findings of Wisdom (1982, 1983, 1985) ods (Wisdom and Holman, 1991; Levison and Duncan, 1994) on the Kirkwood gap at the 3:1 MMR with Jupiter and and the computer speed. Important progress was made when showed that very high eccentricities are frequent outcomes the globally chaotic nature of orbits in the 2:1 MMR was of the dynamics in MMRs on million-year timescales. demonstrated (Ferraz-Mello, 1994b). Later works (Morbi- The progress in numerical modeling revealed another delli, 1996; Nesvorný and Ferraz-Mello, 1997; Ferraz-Mello surprising alternative: Resonant asteroids can fall into the et al., 1998a) showed that this resonance is an intermedi- (Farinella et al., 1993, 1994), as a consequence of their ate case between the unstable MMRs and stable 3:2 MMR eccentricity approaching unity. The current state of our with Jupiter, the latter hosting the Hilda group. The 2:1 understanding of the removal of resonant bodies from the MMR possesses a core with dynamical lifetime comparable 3:1 MMR is that 65–70% are extracted by Earth and Venus, to the age of the , where currently nearly 30 and 25–30% go directly into the Sun (Gladman et al., 1997). small asteroids are known (the Zhongguo group). We de- The innovative seminumerical treatment of classical vote section 5 to this subject. perturbation methods introduced by Henrard (1990) later A major breakthrough, already signaled by the studies allowed a global and realistic description of the secular dy- of the slow chaos in the 2:1 MMR, was made concerning

379 380 Asteroids III

ϖ Ω λ ϖ Ω the phenomenon referred to as “stable chaos.” It was pre- , , j, j, j are orbital angles in the usual notation (index viously noted that a large number of asteroids have strongly j goes over N planets). σ σ chaotic orbits yet are stable on long intervals of time (Milani The MMR occurs when k,k,l,l,m,m = 0, where k,k,l,l,m,m σ and Nobili, 1992; Milani et al., 1997). The main reason for is the time derivative of k,k,l,l,m,m given by equation (1). In ϖ ϖ such behavior was revealed by the discovery of the so-called the case of asteroid motion, the secular frequencies , j, Ω Ω λ λ three-body resonances, i.e., MMRs, where the commensu- , j are small compared to the orbital frequencies , j. σ Σ λ λ rability of orbital periods occurs between an asteroid and For this reason, k,k,l,l,m,m = 0 is approximately kj j + k = two planets [mainly Jupiter and ; (Murray et al., 1998, 0, which can be solved for the resonant semimajor axis (ares) Nesvorný and Morbidelli, 1998)]. The subsequent modeling in the Keplerian approximation. This means that resonant σ of these resonances showed that they represent narrow but conditions k,k,l,l,m,m = 0 with unique k,k but different strongly chaotic layers densely intersecting the asteroid belt l,l,m,m hold at about the same semimajor axis. Thus, each (section 6.1). (k,k) MMR may be thought to be composed from several The slow chaotic evolution observed in numerical simu- resonant terms with different l,l,m,m (this structure is called lations in the narrow MMRs in the outer asteroid belt was the “resonant multiplet”). explored by Murray and Holman (1997). This study cor- In practice, there are two important cases: (1) the two- rectly showed that the mechanism driving the chaotic diffu- body MMR, when index k has only one nonzero integer k j1 sion lies in the “multiplet structure” of the narrow MMRs where j1 denotes, in the asteroid belt, either Jupiter or Mars (section 6.2). (a few two-body resonances with Saturn and the Earth occur The study of the dynamics in the inner belt (2.1–2.5 AU) but they are of minor importance), and (2) the three-body revealed surprising instabilities of asteroid orbits. Numerical MMR, when index k has two nonzero integers, k and k , j1 j2 simulations showed that many asteroids currently on non- corresponding to two planets (mainly to pairs Jupiter–Saturn planet-crossing orbits with large eccentricities evolve to and Mars–Jupiter). Without loss of generality we assume Mars-crossing orbits within the next 100 m.y. (Migliorini that k > 0. j1 et al., 1998, Morbidelli and Nesvorný, 1999). The respon- The notation of MMRs that we adopt in the following sible resonances in this case are the MMRs with Mars, pre- text is that a (k,k) two-body MMR with Jupiter (j1 = 5) is viously thought unimportant because of the small mass of k5J:–k, and with Mars (j1 = 4) is k4M:–k, where k4, k5, and this planet. The MMRs with Mars were identified as a major k are integers defining the resonant angle in equation (1). source of the large near-Earth asteroids (NEAs; section 6.3). In this notation, 2J:1 is a MMR with Jupiter, where an - They were also conjectured to disperse the asteroid families oid has the of one-half that of Jupiter. More- in the inner asteroid belt (Nesvorný et al., 2002). over, if it is clear from the text which planet is considered, After introducing basic notation (section 2) and showing we drop the letter indicating the planet (2J:1 becomes 2:1 the global structure of MMRs in the asteroid belt (section 3), as frequently used in the literature). Concerning the three- we follow the above historical overview. Section 2 is di- body resonances we denote k5J:k6S:k for the MMRs with rected toward the reader who wishes to gain some basic in- Jupiter and Saturn, and k4M:k5J:k for the MMRs with Mars sight in the mathematical methods developed and utilized in and Jupiter. (This notation attempts to generalize the clas- studies of MMRs. Perspectives are given in the last section. sical notation, i.e., the minus sign in k :–k for two-body j1 MMRs, to the case of three-body MMRs and MMRs with 2. NOTATION AND BASIC TERMINOLOGY planets other than Jupiter.) The equations of motion of an asteroid in the presence A mean-motion resonance (MMR) occurs when an as- of a MMR can be conveniently written in the Hamiltonian teroid has an orbital period commensurate with the orbital formalism. In such formalism, the equations of motion de- period of one or more of the planets. To fix the notation, we rive from a function of the canonical variables (the Hamil- define tonian). This formulation is useful because it allows us to use rigorous methods for treating problems where an inte- N grable system is coupled with small perturbations (in the σλλ=++∑ kk k,k,l,l,m,m j j current context, the asteroid’s motion about the Sun is per- j = 1 (1) turbed by the planets). N N ϖϖ++ΩΩ + The classical expression for the Hamiltonian of a small ∑∑lljj mmjj j ==11j body evolving under the gravitational force of the Sun and N planets is where k, l, m and k = (k1, …, kN), l = (l1, …, lN), m = (m1, ∑ 1 N …, mN) are integers with zero sum: k + l + m + (kj + lj + =−− ∑ µ m ) = 0 (because of the invariance of interaction by rota- jj j 2a j = 1 tion, only such combinations exist; this and other condi- (2) rr. tions on integer coefficients required by symmetries of the =−1 j j ∆ 3 gravitational interaction are known as D’Alembert rules), j rj and k ≠ 0 and ||k|| ≠ 0 (i.e., MMRs are related to the fast λ ∆ orbital frequencies, unlike the secular resonances). Here , where j = |r – rj| and r.rj are the heliocentric positions of Nesvorný et al.: Dynamics in Mean Motion Resonances 381 the small body and planet j respectively, and a is the semi- tion (4)). This is an autonomous Hamiltonian of 3(N + 1) major axis of the small body. (We adopt units where the degrees of freedom with the general form product of the gravitational constant and the mass of the Sun µ =+µµ +2 + µ3 is 1.) Here, 1/2a is the heliocentric Keplerian part and j j 01 2 () (5) µ is the perturbation exerted by the planet j having mass j. Appropriate canonical variables of the Hamiltonian where equation (2) are the so-called modified Delaunay variables 1 N =− +∑ ()ngsΛΛΛ + + (6) λΛ 0 Λ2 jj jgjj js = L 2 j = 1

p = –ϖ P = L – G (3) Here, µ denotes the largest of the masses of the involved Λ Λ Λ planets, and j, g , s are the canonical momenta conju- Ω j j λ ϖ Ω q = – Q = G – H gate to the proper angles j, j, j. To first order in µ, the elimination of the fast orbital where L = a , G = Le1 − 2 and H = G cos i are the usual angles by the Lie-series canonical transformation (see, e.g., λ Delaunay variables (e and i are the eccentricity and inclina- Hori, 1966) is equivalent to averaging equation (5) over j, λ λ λ tion of the small body respectively). The perturbation j can j = 1, …, N, maintaining only the terms in which , j , j σ 1 2 be written as a function of the orbital elements of the small appear in resonant combinations k,k,l,l,m,m [corresponding body and planet j [see Moons (1993) for such an expression to the (k,k) MMR in question]. This procedure is straight- with general validity]. The Hamiltonian equation (2) is then forward if one utilizes the resonant variables in equation (5) expressed in terms of variables defined in equation (3). To realistically account for the motion of the planets, the λλλ+++++ kkkkkkpjj jj() j j time evolution of the planetary orbital elements, provided by σ = 11 22 1 2 S = P kkk++ the planetary theory (Laskar, 1988; Bretagnon and Simon, jj12 1990), must be substituted in j. If we denote the planetary orbital elements by index o (as osculating) to distinguish λλλ++ ++ kkjj jj k kkkjj them from the (the latter being de- ν = − 11 22 N = 12 Λ ++PQ kkk++ k λ ϖ Ω jj12 noted by aj, ej, ij, j, j, j), the general form of the quasi- periodic evolution of the planetary elements is kkkkkkqλλλ+++++() o =+ Ψ jj11 jj 22 j 1 j 2 aaj jvv∑ Acos σ = z Sz = Q v kkk++ jj12 λλo =+∑ B cosΨ j jvv k v λ j1 (4) j ΛΛ=− Λ o ιϖo =+ ιϖ Ψ 1 jj11k eej expj jj exp∑ C vv cos v o o k iiexpιιΩΩ=+ exp∑ D cos Ψ λ ΛΛ=−j2 Λ j j jj vv j2 jj v 22k Ψ Σ λ ϖ Ω where v = j(rj j + sj j + tj j), the multiindex v denoting λ , j ≠ j , j Λ , j ≠ j , j (7) ι − j 1 2 j 1 2 different values of integers rj, sj, tj, and = 1. By definition, the proper angles evolve linearly with time, where, in the case of a two-body MMR with the j1-th planet, with fixed frequencies provided by the planetary theory. We k = 0. The Hamiltonian equation (2) written in canonical j2 will denote by nj, gj, sj the orbital, perihelion, and node variables defined by equation (7) is then simply averaged λ frequencies respectively. In fact, because Av, Bv, Cv, Dv in over j, j = 1, …, N. The resulting Hamiltonian of the two- equation (4) are small, the new j — functions of the plan- body MMR is etary proper elements — are at the first approximation iden- N 1 kj tical to the original functions. There will, however, appear 2BR =− +1 ngsΛΛΛ +∑ () + + 2 jjgjs1 jj important terms at second and higher orders in the plan- 2Λ k = j 1 (8) etary masses generated by the substitution of equation (4) µσνσϖ(,SNS , , , , ,ϖ ,,)Ω + (µ2 ) in equation (2). A number of approximations of equation (4) 1 zz are used in the literature, varying from the planar model where o λo λ with one planet on the circular orbit [aj = aj, j = j = njt, N e = i = 0, where n = ()/1 +µ a3 and t is time] to more 1 2π j j j jj µ =− ∑ µλ∫ d (9) 1 π jjj0 realistic ones. 2 j = 1 To summarize, it is understood at this point that although ϖ ϖ ϖ Ω Ω Ω we do not explicitly show such an expression, the Hamil- and = ( 1, …, N), = ( 1, …, N). tonian equation (2) is a function of equation (3) and the In the case of a three-body MMR, 1 in equation (9) con- ϖ σ ν Ω σ ν planets’ proper elements (through the substitution in equa- tains only terms dependent on = – – and/or = – z – , 382 Asteroids III because depends solely on the variables of the small body jor axis interval, because three integers (k , k , k) allow for j j1 j2 and j-th planet. The resonant terms of three-body MMRs a larger number of combinations than two integers (k , k). j1 appear at second- and higher-order terms in µ. These terms In some sense, the larger density of three-body MMRs in are generated by the substitution of the planetary orbital the orbital space compensates for their smaller widths, so elements (equation (4)) (the so-called “indirect” contribu- that both two- and three-body MMRs are important for as- tion) and by iterating the Lie-series transformation elimina- teroid dynamics. tion of fast orbital angles to higher orders in µ (the so-called “direct” contribution). This procedure was described in Nes- 3. OVERLAPPING MEAN-MOTION vorný and Morbidelli (1999). The resonant Hamiltonian of RESONANCES AND THE GLOBAL the three-body MMR is STRUCTURE OF THE ASTEROID BELT + 1 knjj kn jj Let us consider the two-body resonances only. Figure 1 3BR =− + 11 22Λ + 2Λ2 k shows the structure of resonant trajectories for the 3J:2 N (Figs. 1a,b), 3J:1 (Fig. 1c), 5J:2 (Fig. 1d) and 5M:9 ∑ ()gsΛΛ++ (Figs. 1e,f). The model used for the Jupiter’s MMRs jgjj js = (Figs. 1a–d) is an approximation of equation (8) assuming j 1 (10) coplanar orbits and a circular orbit for Jupiter. In such case µσ+++νσ νϖϖ Ω + 1(,SNS ,z , ,,,)z the first-order resonant Hamiltonian is

µσνσϖ2 (,SNS , , , , ,ϖ ,Ω )+ (µ3 ) k2 2 zz PC =−5 ()NS −−2 + 2k2 (12) −+µσ Note that equations (8) and (10) have 2N + 3 degrees of nN551() S (,,)SN freedom (i.e., N less than equation (5)). σ Thus, we learn an important difference between the two- It does not depend on Sz, z because of the coplanarity, and ν and three-body MMRs, which is that the magnitudes of the does not depend on because 5 in equation (2) is invariant resonant terms are proportional to the planetary mass and by rotation around an axis perpendicular to the orbital plane. to the planetary mass squared respectively. As the planetary Consequently, masses in our solar system are <10–3 (in solar units), this  kk+  shows that a typical two-body MMR is expected to have a =−−+2 5 = Na11 e const larger effect on an asteroid’s orbit than a typical three-body k MMR would have. To document this fact, let us consider and trajectories can be obtained as level curves of the Ham- σ an Hamiltonian with only one k,k,l,l,m,m (i.e., fixing k,k,l,l, iltonian in equation (12). m,m, and ignoring terms in equations (8) and (10) with In Fig. 1, two N = const manifolds are shown for the 3J:2 other than this multiindex). Such a Hamiltonian accounts (first-order MMR, ares = 3.97 AU) corresponding to eccen- for a single isolated multiplet term of the (k,k) MMR. The tricities under (Fig. 1a) and above (Fig. 1b) the Jupiter- phase portrait of trajectories of a single multiplet term is crossing limit. The Jupiter-crossing limit is e = 0.31. Reso- basically equivalent to the phase portrait of trajectories of nant orbits with e > 0.31 may intersect the orbit of Jupiter. a pendulum. The width in semimajor axis, represented by The bold line in Fig. 1b is where collisions take place. There the maximal extent of the pendulum separatrixes, is are two equilibria in Fig. 1a: the stable one at σ = 0 (libra- tion center) and the unstable one at σ = π. The trajectories β connected to the unstable equilibrium are called separa- ∆aa= 8 3/2 (11) res 3 trixes. The trajectories enclosing the stable equilibrium are the resonant ones. They are characterized by oscillations of where β is the coefficient in front of the cosine of the spe- σ about 0 (so-called “libration”). Above the Jupiter-crossing cific multiplet term in the Fourier expansion of the resonant limit (Fig. 1b), the trajectories near the libration center are parts in equations (8) and (10) (Murray et al., 1998; Nes- protected from collisions. This happens due to the so-called vorný and Morbidelli, 1999). In the case of the two-body “resonant phase-protection mechanism,” which guarantees MMR, this coefficient is ∝ µ P(e, e , i, i ), where P is a that conjunctions with Jupiter occur when the resonant as- j1 j1 j1 polynomial in e, ej , i, ij with the total power of each term teroid is near the perihelion of its orbit, i.e., far from Jupiter. ρ 1 1 being at least = |kj + k| (called the resonant order). From The 3J:1 MMR (second order) has two libration centers 1 ∝ π π equation (11), the width of the two-body resonance is then at /2 and 3 /2 (Fig. 1c) and occurs closer to the Sun (ares = µ aP3/2 1/2. For the three-body MMR, ρ = |k + k + k| 2.5 AU), where collisions with Jupiter cannot happen on j1 res j1 j2 ∝ µ 3/2 1/2 and its width is aPres . Consequently, the two-body elliptic heliocentric orbits. The 5J:2 MMR (third order) has, MMR is generally larger than the three-body MMR due to in addition to the libration center at 0, two other centers at the mass factor. On the other hand, there are typically more 2π/3 and 4π/3 (Fig. 1d). In general, the two-body MMRs three-body MMRs than two-body MMRs within a semima- interior of the planet’s orbit (|k | > |k|) have libration cen- j1 Nesvorný et al.: Dynamics in Mean Motion Resonances 383

3J:2, e = 0.2 (a) 3J:2, e = 0.4 (b) nances with kj = 1, the separatrix distance is measured at σ c, which must be determined beforehand as an extreme of equation (12). Iterating the algorithm over levels of N = const, the resonant width ∆a can be determined as a func- 4 4 tion of e. Figure 2 shows the MMRs in the asteroid belt. It is an ex- tension of a similar result obtained by Dermott and Murray Semimajor Axis (AU) Semimajor Axis (AU) (1983), but accounting for all MMRs with ∆a > 10–4 AU. 3.5 3.5 0 2 σ 46 02σ 46We also plot in this figure the asteroids with magnitudes up 3J:1, e = 0.2 (c) 5J:2, e = 0.2 (d) to the current level of completeness (Jedicke and Metcalfe,

2.54 2.86 1998). The orbital distribution of these bodies is thus un-

2.52 affected by observational biases. In the region of the 2J:1, 2.84 the resonant orbital elements of all known resonant objects, 2.5 2.82 regardless of size, are shown.

2.48 2.8 To correctly interpret this figure, one should be aware of the now-classical result of Chirikov (1979) known as the Semimajor Axis (AU) Semimajor Axis (AU) 2.46 2.78 “resonance overlap criterion.” This criterion affirms that 02σ 46 02σ 46whenever two resonances with similar sizes overlap, the 5M:9, e = 0.15, dvarpi = π (e) 5M:9, e = 0.15, dvarpi = 0 (f) corresponding resonant orbits become chaotic. The same criterion was used by Wisdom (1980) to show the onset of chaos in the vicinity of a planet because of the overlap of

2.255 2.255 the first-order MMRs with Jupiter. This criterion can be used in the current context to explain why there are so few asteroids located above the threshold in e for the resonance

Semimajor Axis (AU) Semimajor Axis (AU) overlap. Moreover, we know from the experience obtained 2.25 2.25 in the last few years with the secular dynamics inside the 0246 0246 σ σ MMRs that MMRs themselves are usually characterized by chaotic dynamics (see next section). In resonances like the Fig. 1. Trajectories at MMRs. (a,b) 3J:2 below (a) and above 3J:1 and 5J:2 MMRs (at 2.5 and 2.82 AU), and many oth- (b) the Jupiter-crossing limit; (c) 3J:1 MMR; (d) 5J:2 MMR; ers, the chaos causes large-scale instabilities: The resonant ϖ ϖ π ϖ ϖ (e,f) 5M:9 MMR with – 4 = (e) and – 4 = 0 (f). On x- bodies evolve to high-e orbits and are removed from reso- axis, σ is defined by equation (7). nant orbits by encounters with Mars or Jupiter (Gladman et al., 1997). Such resonances are associated with the Kirk- wood gaps in the asteroid distribution. Smaller MMRs, albeit ters at σ = 2j π/ρ if ρ is odd and at σ = (2j + 1)π/ρ if ρ strongly chaotic, do not generate strong instabilities when- c 1 c 1 is even. The exterior resonances have the libration centers ever eccentricities are not near a planet-crossing limit. Large at σ = (2j + 1)π/ρ if k ≠ 1. Conversely, if k = 1, the struc- first-order MMRs are characterized by either marginal in- c 1 j1 j1 ture of the exterior resonance is characterized by so-called stabilities (like 2J:1 at 3.27 AU) or possess cores in which asymmetric librations where none of the 2ρ libration centers orbital lifetimes exceed the solar system age (such as the π ρ π ρ is at 2j1 / or (2j1 + 1) / for most values of N (Message, 3J:2 MMR at 3.97 AU and the low-e region in the 4J:3 1958; Beaugé, 1994). MMR at 4.29 AU, where is located). An interesting case is that of the two-body MMRs with The basic impression one has from Fig. 2 is that the Mars, due to the large modulation of their widths with the MMRs delimit the orbital space inhabited by the observed evolution of perihelion longitudes and eccentricities. Fig- asteroids in a and e. Due to chaos and instabilities generated ures 1e and 1f show the trajectories near the 5M:9 MMR by MMRs, this is quite intuitive. Most of the job in remov- (ares = 2.253 AU) in the coplanar model with Mars on a ing the material from the belt was apparently done by sev- fixed elliptic orbit (e4 = 0.065), assuming two values of peri- eral large MMRs with Jupiter (as 3J:1, 5J:2, 7J:3, 9J:4, 7J:4, ϖ ϖ π ϖ ϖ helion longitudes: – 4 = in Fig. 1e and – 4 = 0 in 5J:3, etc.) and many narrow MMRs with Mars. It is not Fig. 1f. In general, when ϖ – ϖ = π, the MMRs have their more than a curiosity that a few MMRs with Saturn and j1 maximum widths. For other phases of the secular angles, Earth also exist in the inner and central asteroid belts (6S:1 the sizes of resonant islands are smaller. at 2.89 AU, 7S:1 at 2.61 AU, 2E:7 at 2.305 AU, 2E:9 at The resonant width (i.e., the width of the libration island) 2.73 AU, etc.). can be conveniently estimated by measuring the distance We have accounted solely for two-body MMRs in Fig. 2. between separatrixes for σ = 0 (for interior resonances of This happens to be a good approximation when consider- odd order) or σ = π/ρ (for interior resonances of even order ing the gross structures of the asteroid belt in a and e, but ≠ and exterior resonances with kj 1). For the exterior reso- becomes less acceptable when one’s objective is to under- 384 Asteroids III

Inner and Central Belts 4J:17J:2 8J:3 6S:1 9J:4 5S:1

3J:1 7J:3 5J:2

0.4 2J:1

0.3 Zhongguo

0.2 Eccentricity

0.1

0 2 2.5 3 3.5

Outer Belt 4S:1 3S:1

3J:2 4J:3

0.4

0.3 Hildas

0.2 7J:4 5J:3 Eccentricity

0.1

Thule 0 3.5 4 4.5 5 Semimajor Axis (AU)

Fig. 2. Global structure of the MMRs in the asteroid belt. There are four different gray shades denoting the regions of resonant mo- tion with planets: light gray for Jupiter MMRs, intermediate for Saturn MMRs, and dark gray for Mars MMRs. Each resonance cor- responds to one V-shaped region except the large first-order MMRs with Jupiter, which have particular shapes. Some of the resonances are labeled. For some Jupiter MMRs the projection of separatrixes on the a, e plane is shown by black lines; for 2J:1, 3J:2, and 4J:3, these lines are bold. We also show the proper (dots; Milani and Knezevic, 1994) or orbital elements (crosses; Bowell et al., 1994) of asteroids with magnitudes up to the completeness level. In case of the group of small asteroids in the 2J:1 MMR (arrow indicates 3789 Zhongguo), the resonant elements are plotted (asterisks; Roig et al., 2002). Other resonant asteroids are the Hilda group in the 3J:2 MMR and 279 Thule in the 4J:3 MMR. The orbits above the dashed lines are planet-crossing.

stand the dynamics of real bodies. The three-body reso- Ferraz-Mello and Klafke, 1991; Morbidelli and Moons, nances are usually as narrow as two-body resonances with 1995; Gladman et al., 1997). We briefly review how this Mars (<10–2 AU), but generally being of a low order, they can be theoretically justified, restricting for simplicity to the have nonnegligible sizes down to small eccentricities. More- case where the asteroid and planets have coplanar orbits, over, the three-body MMRs are numerous. although elliptic and precessing [see Morbidelli and Moons (1993) for a discussion of the inclined case]. We start from 4. CHAOTIC SECULAR DYNAMICS IN THE the resonant Hamiltonian equation (8), that we rewrite as MEAN-MOTION RESONANCES WITH 2BR = PC + where PC is given in equation (12), and JUPITER: THE KIRKWOOD GAPS = 2BR – PC. The function can be considered as a perturbation of PC, because (for D’Alembert rules) the The bodies inside the wide MMRs with Jupiter undergo former is proportional to the planetary eccentricities. important secular dynamics, in particular for what concerns Because PC is integrable, we introduce the Arnold ac- the evolution of e (Wisdom, 1982; Yoshikawa, 1990, 1991; tion-angle variables in the MMR region where σ librates. Nesvorný et al.: Dynamics in Mean Motion Resonances 385

Following Henrard (1990), the transformation has the form pendent of ψσ, the action Jσ is now a constant of motion. Thus, equation (14) describes the secular dynamics inside 2π 1 the MMR, namely the evolution of the eccentricity (through ψσ = t JSdσ = ∫ σ Tσ 2π Jν) due to the motion of the perihelia of the asteroid and ν ϖ (13) the planets (– and j respectively). ψν = ν – f(ψσ, Jσ, Jν)Jν = N The mean frequency of the longitude of perihelion can be computed as where the integral is done over a trajectory of S, σ for N = ∂ SEC const (Fig. 1), Tσ is a period of the trajectory, and f is a −=ψν (15) periodic function of ψσ, with null average. ∂Jν We then write PC and as functions of these variables. PC ψ By construction, now depends only on the new action A first-order perihelion occurs when ν + ψ ϖ variables Jσ, Jν. By averaging over σ, we obtain an gj = 0, where gj is the frequency of j [for numeric values Hamiltonian of the form of gj see, e.g., Laskar (1988)]. It often occurs in MMRs with Jupiter that the secular resonances associated with the g5 and g6 frequencies are SEC =+∑g Λ PC(,)JJσν + (,,JJ σνψϖ ν ,)ϖ (14) jgj located close to each other. To study the effects of the inter- j action between these two resonances, we construct a two- ϖ ϖ ϖ where = ( 1, ..., N). Because this Hamiltonian is inde- resonance model by retaining from equation (14) the har-

1.9 1.9

1.8 1.8

1.7 1.7 N N

1.6 1.6

1.5 1.5

1.4 1.4 0123456 0123456 q q

1.9 1.9

1.8 1.8

1.7

N N 1.7

1.6 1.6

1.5 1.5

0123456 0123456 q q

σ Fig. 3. Sections of the dynamics of equation (16) computed at 6 = 0 for the 3J:1 MMR. The four panels correspond to increasing ϖ ϖ ( −−2 ) values of Jσ, the latter being related to the amplitude of oscillation of a. The label q stands for – 5, while N = Jν = ae31 . At the center of the 3J:1, the values N = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 correspond to e = 0.2, 0.55, 0.72, 0.84, 0.92, 0.97 respectively. In each panel, the lower limit on the N axis is the value that identifies the separatrix of the MMR. Consequently, all the curves that seem to exit from the bottom border of the panels correspond to trajectories that hit the separatrix of the MMR during their secular evolu- tion, and are therefore expected to be chaotic. From Moons and Morbidelli (1995). 386 Asteroids III

σ ψ ϖ σ ψ monic of with arguments 5 = ν + 5 and/or 6 = ν + 5. TRANSIENT AND STABLE POPULATIONS ϖ 6. Thus, we consider the Hamiltonian OF THE 2J:1 AND 3J:2 MEAN-MOTION RESONANCES 5,6 =++ggΛΛ 56gg56 (16) In the 2J:1 and 3J:2 MMRs, simple models with Jupiter’s PC + 5,6 σσ (,)JJσν (,,JJ σν 56 , ) orbit fixed also show a high-e regime of motion associated ν with the 5 resonance. In this case, however, the high-e re- This is a nonintegrable Hamiltonian that must be studied gime is separated from low e by regular orbits, which are ro- σ numerically. The dynamics can be represented on the 5, bust and persist in the models that account for g6 (Henrard σ Jν plane, for instance through a section at 6 = 0. The sec- and Lemaître, 1987; Morbidelli and Moons, 1993; Ferraz- tions depend parametrically on Jσ, roughly corresponding Mello, 1994b; Michtchenko and Ferraz-Mello, 1995; Moons to the libration amplitude of a (Aa). et al., 1998). Consequently, unlike in the 3J:1, 4J:1, 5J:2, In the case of the 3:1 MMR with Jupiter (Fig. 3), most and 7J:3 MMRs, the secular dynamics do not explain why of the phase space is covered by a chaotic region, which the observed orbital distribution of asteroids should display extends up to e = 1 (top borders of the panels). Only the a gap at the place of the 2J:1 MMR (the Hecuba gap, see orbits with small Aa and e (the smooth curves at the bottom Fig. 2), where only a few tens of small resonant asteroids of the two top panels of Fig. 3) still have regular dynam- reside. Conversely, the 3J:2 MMR hosts some 260 resonant ics. These orbits, however, periodically reach Mars-crossing asteroids known at the time of writing this text (the Hilda eccentricities (e > 0.3); the encounters with Mars give im- group), with about 30 bodies exceeding 50 km in diameter. pulsive changes to a and e, which, although generally very This puzzling difference, known as the “2:1 vs. 3:2 para- small, effectively modify Aa. At large Aa, the chaotic region dox,” led some authors to investigate the possibility of ν ν generated by the overlap of the 5 and 6 resonances ex- opening the Hecuba gap during the primordial stages of the tends to all eccentricities (bottom panels of Fig. 3), and the solar system formation by invoking effects that can mutu- asteroid can therefore rapidly and chaotically evolve to very ally displace the resonances and amplitudes of asteroids large e. This combination of large-scale chaos of the secu- (Henrard and Lemaître, 1983). It became clear later that lar dynamics and weak martian encounters explains the be- although plausible, such effects are not strictly required to havior of 3:1 resonant asteroids observed in numerical inte- explain the lack of asteroids in the 2J:1 MMR. grations of the full equations of motion (see Morbidelli et An important difference between the 2J:1 and 3J:2 MMRs al., 2002), and explains the formation of a gap in the aster- was noted by Ferraz-Mello (1994a,b). He computed the oid distribution. maximum Lyapounov characteristic exponents (LCE) [the The same happens in many other major MMRs with maximum LCE measures the rate of divergence of nearby Jupiter that are associated with a Kirkwood gap. Moons and orbits and is an indicator of chaos (Benettin et al., 1976)] Morbidelli (1995) have shown that also the 4J:1, 5J:2, and of a number of resonant orbits in both MMRs. It turned out 7J:3 MMRs are dominated by the chaotic region generated that in the 2J:1 MMR, LCE ~10–5–10–3.5 yr–1, while in the ν ν –5.5 –1 by the overlap of the 5 and 6 resonances, so that the aster- 3J:2 MMR, LCE <10 yr . This indicated that orbits in oids in these MMRs can also reach very large e on a time- the 2J:1 MMR are chaotic on short timescales (Morbidelli, scale of a few million years. 1996), at variance with the 3J:2 MMR where most trajec- Large evolutions of e in many MMRs are typically tories are only weakly chaotic. ν caused by the secular resonance 5 itself. The structure of The comparative study (Nesvorný and Ferraz-Mello, 1997) ν the 5 resonance can be computed in models neglecting g6 of the MMRs employing frequency map analysis (Laskar, (Ferraz-Mello and Klafke, 1991; Klafke et al., 1992; Moons 1988, 1999) provided additional clues. Plate 1 shows how and Morbidelli, 1995). These models show that by the effect the magnitude of the chaotic evolution varies in the orbital ν ϖ ϖ of 5, the eccentricity suffers large variations, while – 5 space of the 2J:1 and 3J:2 MMRs. The color coding repre- π δϖ δϖ oscillates around 0 or . In such models, however, only sents the value of log10| |, where is the relative change bodies with certain initial orbits can evolve from low to very of the perihelion frequency per 0.2 m.y. This quantity is a high eccentricities (Klafke et al., 1992). Other parts of the powerful indicator of the rate of chaotic evolution (chaotic orbital space are characterized by regular motion bounded diffusion) suffered by orbits in the integration timespan. The at moderate e. This regular motion almost completely van- results were extrapolated to larger time intervals assuming ishes when g6 is accounted for (as shown above; Fig. 3). a chaotic random walk of orbits (see Ferraz-Mello et al., Consequently, in a globally chaotic environment, orbits 1998a). ν evolve according to the underlying structure of 5, and In regions corresponding to the smallest magnitudes of despite their initial location in the orbital space, reach very the chaotic diffusion (deep blue), the orbits evolve relatively large e. This is usually accomplished by a series of small by less than 10% on 1 G.y. Such orbits should be stable over ν ν transitions because of the interaction of 5 and 6 and a few the age of the solar system. Conversely, in regions where ν δϖ large events, when e increases due to the effect of 5 (Mor- log10| | > –2.5 (red, yellow), the perihelion frequency of a bidelli and Moons, 1995). resonant body should typically change by more than 100% Nesvorný et al.: Dynamics in Mean Motion Resonances 387

in less than 1 G.y. Such an evolution should be enough to the instabilities in the 2J:1 MMR were significantly accel- δϖ destabilize the orbit. Thus, the orbits with log10| | > –2.5 erated (inset in Plate 1). The effect of the Great Inequality are expected to be unstable. Despite the large uncertainties may be analytically modeled as overlap between the 2J:1 involved in the extrapolation from million-year to billion- and 5S:1 MMRs (Morbidelli, 2002). year timescales, Plate 1 provides a convincing argument suggesting at least 1–2 orders in magnitude shorter lifetimes 6. NUMEROUS NARROW MEAN-MOTION in the 2J:1 than in the 3J:2 MMR. The lifetimes in the most RESONANCES DRIVING THE stable regions of the 2J:1 MMR were estimated to be on CHAOTIC DIFFUSION the order of 109 yr (Morbidelli, 1996; Nesvorný and Ferraz- Mello, 1997). The narrow MMRs (three-body, high-order two-body, Recently, these results has been put on firmer ground by and resonances with Mars) in the asteroid belt are usually direct simulation of 50 test bodies in the 2J:1 MMR over not powerful enough to destabilize bodies continuously 4 G.y. (Roig et al., 2002). This simulation showed that the resupplied into them by the Yarkovsky force (Bottke et al., most stable region of the 2J:1 MMR is characterized by 2002b) and collisions (Morbidelli et al., 1995; Zappalà et marginal instabilities: There is about a 50% probability that al., 2000). Consequently, the narrow MMRs do not open a body started at 3.2 < a < 3.3 AU, 0.2 < e < 0.4, small i, gaps in the asteroid distribution and are populated by many ϖ ϖ σ and – J = = 0 escapes from the resonance in 4 G.y. asteroids at present. In the following three sections, we dis- because of the diffusive chaos. Thus, the lack of a larger cuss the three-body MMRs (section 6.1), narrow two-body asteroid population in the 2J:1 MMR is at least partially due MMRs (section 6.2), and MMRs with Mars (section 6.3). to the slow removal of primordial population in the last 4 G.y. The group of resonant asteroids, now accounting for 6.1. Three-Body Mean-Motion Resonances some 30 small bodies known as the Zhongguo group, is localized in a very small region in the resonance where life- The growing evidence that many asteroids move on cha- times generally exceed 4 G.y. otic orbits (Milani and Nobili, 1992; Milani and Knezevic, The origin of the Zhongguo group is unknown. This 1994; Holman and Murray, 1996, Šidlichovský and Nes- group has a steep size distribution, which would be expected vorný, 1997; Knezevic and Milani, 2000) has challenged the from the collision injection at the breakup of the parent view of the asteroid belt as an unchanging, fossil remnant body of the (Morbidelli, 1996). The large of its primordial state. For example, the asteroid ejection velocities needed for such injection, an offset in e at a = 3.174 AU has a Lyapunov time (defined as an inverse between the Themis family and the Zhongguo group, and of the LCE) ≈10,000 yr, showing unpredictability of the possibly incompatible spectral type of (3789) Zhongguo, orbit over >105-yr intervals. It was moreover noted that e however, suggest this to be an unlikely origin (Roig et al., and i of this object chaotically evolve outside the borders of 2002). The Zhongguo group seems to be an order of mag- the Veritas family (of which 490 Veritas is the largest frag- nitude more depleted than what would be expected from a ment) in 50 m.y. Based on this fact, the age of the Veritas dynamically eroded population similar to the Hilda group. family was hypothesized to be of this order (Milani and Assuming Zhongguos to be dynamically primordial bodies Nobili, 1992). in the 2J:1 MMR, their additional depletion could have been In our current understanding, 490 Veritas is one among achieved by collision processes or during the primordial many objects in the asteroid belt residing in the three-body stages of the solar system formation. MMRs. Convincing demonstration of this fact was provided The slow diffusive chaos in the 2J:1 MMR is where the by estimates of the LCE for large samples of real and ficti- concept of a resonance between an asteroid and two per- tious asteroids (Nesvorný and Morbidelli, 1998; Murray et turbing planets first appeared. As originally pointed out al., 1998; Morbidelli and Nesvorný, 1999). Figure 4 shows (Ferraz-Mello, 1997; Michtchenko and Ferraz-Mello, 1997; the profile of the LCE computed for test bodies initially at Ferraz-Mello et al., 1998b) the slow chaos in the 2J:1 MMR e = 0.1 and i = 0. The peaks correspond to chaotic regions. is probably generated by commensurabilities between the Some of the main peaks may be identified with two-body λ periods of the “Great Inequality,” i.e., the period of 2 5 – MMRs (3J:1 at 2.5 AU, 5J:2 at 2.82 AU, 7J:3 at 2.96 AU, λ ≈ σ λ λ ϖ 5 6 ( 880 yr), and of = 2 5 – – (300–500 yr). Sym- etc.), while most of the narrow peaks are related to three- plectic maps of the 2J:1 MMR found the central region less body MMRs. Moreover, many two-body MMRs with Mars chaotic when the effect of the Great Inequality was switched occur in the inner asteroid belt (2.1–2.5 AU). It has been off. The effect of this resonance was also put into evidence shown that a large fraction of real asteroids reside in narrow in direct numerical simulations by artificially changing the resonances (Nesvorný and Morbidelli, 1998). 490 Veritas σ Great Inequality period to 440 yr. In such situation, which evolves in the 5J:–2S:–2, the angle 5,–2,–2,0,0,–1,0,0,0 having could have occurred when Jupiter and Saturn were slightly irregular oscillations about 0 correlated with oscillations of closer to each other during the primordial migration phase semimajor axis about ares = 3.174 AU. (Fernandez and Ip, 1984), the “beat” between the periods This result raised questions concerning the implications of the Great Inequality and of σ was approximately 1:1, and of such a complex chaotic structure of the asteroid belt. 388 Asteroids III

(a) –2 ) –1

–3 8M:13 11M:18 14M:23 5J:–3S:–1 16M:27 10M:17 11J:–9S:–2 7M:12 11M:19 4M:7 4J:–1S:–1 9M:16 7J:2 9J:–5S:–2 6M:11 7M:13 5J:–4S:–1 8M:15 10J:3 9J:–6S:–2 2J:3S:–1 4J:–2S:–1 6J:–7S:–1 1M:2

ν 1M:–1J:–7 13M:22 4J:–2U:–1 3M:5 5M:9 –4 6 1M:1J:–2

–5 log(LCE) (LCE in yr

–6 2.052.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 Semimajor Axis (AU) (b) –2 ) 5J:2 –1 3J:1 –3 7J:–10S:–1 7J:–3S:–2 11J:–6S:–3 6J:–1S:–2 11J:4 9J:–2S:–3 7J:–4S:–2 8J:3 3J:–1S:–1 11J:–2S:–4 10J:–6S:–3 9J:–10S:–2

–4

–5 log(LCE) (LCE in yr

–6 2.5 2.55 2.6 2.65 2.7 2.75 2.8 Semimajor Axis (AU) (c) –2 ) –1 6S:1 4J:–4S:–1 7J:3 5J:–1S:–2 7J:–6S:–2 9J:–11S:–2 8J:–3S:–3 9J:4 12J:–2S:–5 11J:5 3J:–2S:–1 5J:–7S:–1 13J:6 12J:–3S:–5 9J:–1S:–4 15J:7 8J–4S:–3 17J:8 3J:3S:–2 5J:–2S:–2 7J:–7S:–2 7J:–2S:–3

–3 11J:–3S:–4

2J:1

–4 2J:1S:–1 12J:5

–5 log(LCE) (LCE in yr

–6 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 Semimajor Axis (AU)

Fig. 4. The estimate of the LCE computed for test bodies initially placed on a regular grid in semimajor axis, with e = 0.1. (a) The gravitational perturbations provided by the terrestrial planets were taken into account in addition to the perturbations of four giant planets. (b,c) Accounted only for four giant planets. The figure shows many peaks that rise from a background level. The background value of about 10–5.3 yr–1 is dictated by the limited integration timespan (2.2 m.y.); increasing the latter, the background level gener- ally decreases. The peaks reveal the existence of narrow chaotic regions associated with MMRs. The main resonances are indicated above the corresponding peaks. From Morbidelli and Nesvorný (1999).

Asteroids were shown to escape from the main belt to Mars- jectured, many asteroid families could have been created and Jupiter-crossing orbits due to a slow chaotic evolution as small groupings and later were dynamically dispersed of eccentricities in narrow resonances. Realistic modeling by chaotic diffusion in narrow MMRs and by the Yarkovsky including the Yarkovsky effect demonstrated that the narrow effect. This mechanism may eventually solve the paradox resonances are an important source of the near-Earth-object between hydrocode simulations of breakup events suggest- population (Migliorini et al., 1998; Morbidelli and Nes- ing creation of tightly grouped collision swarms, and the vorný, 1999; Bottke et al., 2002a). Recently, narrow MMRs observed, largely dispersed asteroid families. were also hypothesized to disperse the asteroid families in Theoretical modeling of the three-body MMRs revealed e and i, possibly creating their often asymmetric shapes several clues to understanding the behavior of asteroids in (Nesvorný et al., 2002). If this process is as general as con- numerical simulations. The resonant Hamiltonian of sev- Nesvorný et al.: Dynamics in Mean Motion Resonances 389

eral three-body MMRs with Jupiter and Saturn in the pla- amplitude librations (in years) is nar model has been computed by Murray et al. (1998) and by Nesvorný and Morbidelli (1999). Following the notation a 1 of the latter work, the Hamiltonian of the k J:k S:k MMRs T = res (19) 5 6 β (equation (10)) can be written as a Fourier expansion k 3

σ The position of the libration center c is determined by the 1 2 2 = α ss56s ϖϖσν+−+ signs of α and β. As α = –3λ /2a is always negative, the 1 ∑ 1,vres()eee56 cos[lll55 66 ( )] res a5 v resonant center is at 0 when β > 0, and at π when β < 0. Moreover, according to simple arguments, the Lyapunov (17) time is on the order of the libration period, so that T pro- 1 = ∑ ()α eeess56s cos[σ ] vides a rough estimate of the LCE (Benettin and Gallavotti, 2 2,vres56 k,,(,kll56 ),,, l0 0 a5 v 1986; Holman and Murray, 1996). ∆ σ Table 1 shows a, T, and c for the most important three- α where res = ares/a5, and the multiindex v denotes different body MMRs with e = 0.1. The values computed from equa- values of integers s , s , s, l , l , l. The angle σ tions (11) and (19) are in good agreement with numerical 5 6 5 6 k,k,(l5,l6),l,0,0 in equation (17) is given in terms of σ and ν (equation (7)). results (Fig. 4). In particular, 490 Veritas was numerically Denoting found to evolve in the 5J:–2S:–2 MMR with 0.004-AU 3 µ2 oscillations of a and the libration period of several 10 yr, β = 5 s both values being in agreement with the ones in Table 1. ∑ 2,v e (18) a5 v Understanding the chaos and the long-term evolution of == ll560 asteroids in the three-body MMRs requires to account for these coefficients, truncated at two lowest orders in e, are more than one multiplet term in equation (17). Murray et given in Table 1. al. (1998) and Nesvorný and Morbidelli (1999) showed that Ignoring all but one resonant multiplet term in equa- different multiplets of the three-body MMR generally over- σ tion (17) (generically denoted by *), the complete Hamil- lap, thus creating a chaotic domain at the position of the tonian equation (10) can be reduced to the Hamiltonian of MMR. Quantitative estimates of the LCE and diffusion α 2 β σ a simple pendulum = (S – N) + cos *, N represent- timescales in this domain showed that significant chaotic ing a constant parameter (Nesvorný and Morbidelli, 1999; evolutions of e and i of resonant bodies occur on long time- Murray et al., 1998). The resonant full width can then be spans (Murray and Holman, 1997; Murray et al., 1998). computed from equation (11). Similarly, the period of small Conversely, the semimajor axis of resonant bodies remains

TABLE 1. Analytic results on the three-body MMRs (from Nesvorný and Morbidelli, 1999).

Resonant Libration Period Libration Resonance Semimajor Axis Coefficient Resonance Width T (103 yr) Center β –8 σ Designation ares (AU) (× 10 ) (e = 0.1) (e = 0.1) c 4J:–1S:–1 2.2155 6.05e2 – 2.22e4 0.37 52 0 3J: 1S:–1 2.2994 3.78e3 – 1.94e5 0.1 217 0 4J:–2S:–1 2.3978 19.7e – 3.95e3 2.4 9.9 0 7J:–2S:–2 2.4479 46.0e3 – 53.0e5 0.37 33 0 7J:–3S:–2 2.5600 –4.35e2 + 5.28e4 0.39 36 π 2J: 2S:–1 2.6155 6.03e3 + 3.63e5 0.15 200 0 6J:–1S:–2 2.6192 –17.6e3 + 21.3e5 0.25 57 π 4J:–3S:–1 2.6229 –0.206 – 0.763e2 0.9 30 π 7J:–4S:–2 2.6858 0.255e – 0.356e3 0.3 49 0 3J:–1S:–1 2.7527 –7.99e + 0.360e3 1.9 17.8 π 4J:–4S:–1 2.9092 0.0203e + 0.0691e3 0.1 370 0 5J:–1S:–2 2.9864 20.9e2 – 21.3e4 1.1 19 0 8J:–3S:–3 3.0166 –10.0e2 + 24.0e4 0.8 19 π 3J:–2S:–1 3.0794 –3.06 – 15.5e2 4.5 9.9 π 6J: 1S:–3 3.1389 –22.4e4 + 143e6 0.12 132 π 8J:–4S:–3 3.1421 0.357e – 0.99e3 0.5 33 0 3J: 3S:–2 3.1705 –25.8e4 + 5.15e6 0.13 180 π 5J:–2S:–2 3.1751 45.6e – 32.3e3 5.6 4.3 0 7J:–2S:–3 3.2100 –115e2 + 273e4 2.8 5.8 π 390 Asteroids III locked in the interval spanned by the resonant multiplet and widths of the individual multiplet terms in the resonance. although behaving stochastically on short timespans, never The separation in Ψ between two such terms is given by directly evolves outside the resonant borders, unless two or βδΨ = 2εA, while the width of each multiplet is ∆Ψ = more different MMRs overlap (the case of high e in Fig. 2). 2 cIP(, )/β. The stochasticity parameter (Chirikov, 1979; sj1 Lichtenberg and Lieberman, 1983) is then 6.2. Narrow Two-Body Mean-Motion Resonances with Jupiter 22 ∆Ψ cs πβ K = π = 0 eff δΨ β εΑ (23) As discussed above, the details of the multiplet struc- ture of each two-body and three-body resonance affect the extent of the chaotic zone associated with the resonance, where c is the coefficient of a representative multiplet term s0 the typical Lyapunov time for a small body in that reso- (Murray and Holman, 1997). The orbits in the MMR will nance, and the timescale on which such a body might cha- be chaotic if Keff > Kcrit, where Kcrit ~1. The extent of the otically diffuse in the space of orbital elements. Useful chaotic zone in Ψ (and therefore in semimajor axis) is analytic models of these effects can be obtained using some roughly given by the Ψ range of those multiplet terms that simplifying assumptions (Holman and Murray, 1996; overlap. The Lyapunov time in the chaotic zone is given by Murray and Holman, 1997; Murray et al., 1998). We show an example of such modeling for a two-body resonance (see π KKK2 π also Morbidelli, 2002). This modeling can be used with T ≈+++/ln1 eff eff eff ~ (24) L εΑΑε small modifications for the three-body (Nesvorný and Mor- 424 bidelli, 1999) and Mars’ MMRs. To most simply account for the multiplet structure of a which indicates that the Lyapunov time is comparable to two-body (k,k) MMR, we restrict ourselves to the planar the precession period. Ψ elliptic three-body model. The resonant Hamiltonian of such If Keff > Kcrit, the diffusion in and I, can be described model can be written by the Fokker-Planck equation (Lichtenberg and Lieberman, 1983; Murray and Holman, 1997). The associated diffusion kk+ 1 k j1 coefficients are =− +j1 ncPΛΛ + (,)+ ∑ cP (,) Λ× 2Λ2 k js1 0 s = 0 2 2 2 2 K εΑ 1 cs πβ 1 π (20) ≈=eff 0 = 2 DΨ cs (25) cos kλλ++− k sp() k +−ksϖ πβ β εΑΑ0 ε jj11 jjj11 2 2 2 µ where c0 is a part of 1 in equation (8) independent of and angles. In equation (20), we retained only those terms of s2 π ==2 0 2 the Fourier expansion of equation (8) that are lowest order DsDIsΨ c (26) 0 0 εΑ in the eccentricities of the planet and the small body. For 2 simplicity, we arbitrarily set ϖ = 0, and drop it from equa- In the model of Murray and Holman (1997) the resonant j1 tion (20). bodies execute random walks in Ψ (or semimajor axis) and The canonical variables we use in the following slightly I (or eccentricity). These random walk are subject to bound- differ from equation (7). We define them by ary conditions. The permissible values of Ψ lie within the range of values for which the resonant multiplet terms over- 1 ψ = k λ + kλ Ψ=()ΛΛ − lap (the chaotic zone). The values of I range from I = 0 j1 j1 k 0 (circular orbit) to I = Imax, where Imax corresponds to planet φ = p I = P (21) crossing. The bounding values of Ψ and I = 0 can be mod- k eled as reflecting barriers, and the value of I = Imax as an λ ΛΛ=−j1 Λ absorbing barrier. For a resonant asteroid at I , the solution j1 jj11 0 k of the Fokker-Planck equation yields typical values of the Λ where 0 = ares is the unperturbed MMR location. Keeping removal time, TR, of the lowest-order terms in Ψ and I in the Taylor expansion of first three terms of Hamiltonian equation (20), we get IImax 0 TR ~ (27) kk+ DIII0()()max DI 1 j1 =++βΨ2 ε ψ + φ 2 AI∑ cs( I ) cos( s ) (22) 2 s = 0 Figure 5 shows removal times for outer-belt asteroids in a number of two-body MMRs. The solid symbols are the β ε ∂ ∂ Λ where and 2 A = c0/ I( 0,0) are the constants associ- results of numerical integrations; the open symbols are Λ ated with the Taylor expansion, and cs(I) = cs( 0,I). analytic predictions from equation (27). The values of I0 To apply the resonance overlap criterion to the Hamil- were determined from short numerical integrations of reso- tonian equation (22) we must compute the separation and nant asteroids, hence the scatter in the predicted removal Nesvorný et al.: Dynamics in Mean Motion Resonances 391

a (Jupiter Units) is a factor of ≈3000 less massive than Jupiter. Thus, accord- 0.660.68 0.7 0.72 0.74 ing to the discussion in section 2, the two-body MMRs with

1012 Mars should be in general a factor 3000 ~55 smaller than 13:7 the two-body MMRs with Jupiter. In the inner asteroid belt, 1011 however, this factor may be compensated for by the small distance between an asteroid and Mars. Indeed, the width 1010 11:6 of any MMRs near the planet-crossing limit becomes large. 109 According to Fig. 2 (see also Fig. 4), the MMRs with Mars 12:7 13:8 provide a dominant source of chaos in the inner asteroid belt, 108

(yr) overwhelming in size and number the MMRs with Jupiter. R

T 7 9:5 The dynamical interaction of asteroids with the MMRs 10 11:7 with Mars is complex. As an example, Fig. 5 shows the evo- 106 σ λ λ ϖ 13/2 8:5 lution of a, = 9 4 – 22 + 13 , and e of 41 Daphne. This 105 asteroid is temporally captured in the 3M:11 MMR when e 5:3 becomes large during the secular oscillations. The irregular 104 7:4 behavior of a and σ witnesses the chaoticity of the orbit, which has a Lyapunov time of <104 yr (Šidlichovský, 1999). 1000 3.5 3.6 3.7 3.8 3.9 Very little analytical work has been done on the subject a (AU) of the MMRs with Mars. Figures 1e–f are apparently highly idealized approximations of the real dynamics (Fig. 6). To Fig. 5. The removal times for outer-belt asteroids in a number understand the real dynamics, it is important to account for of two-body MMRs with Jupiter. The solid symbols are the re- the rich multiplet structure of the MMRs with Mars and the sults of numerical integrations; the open symbols are analytic pre- significant time-modulation of each multiplet’s width with dictions from equation (27). The arrows represent lower limits varying eccentricities (interesting MMRs with Mars are obtained from numerical integration of bodies in MMRs charac- usually of a very high order). terized by extremely slow chaotic diffusion. Here, Jupiter units The chaotic diffusion in the MMRs with Mars provide Murray and Holman are 5.2 AU. From (1997). transfer routes from the asteroid belt to Mars-crossing or- bits. This source was hypothesized to be one of the major sources of large NEAs (Migliorini et al., 1998; Morbidelli times. In most cases the results of numerical experiments and Nesvorný, 1999; Bottke et al., 2002a). The detailed agree with the analytic predictions to an order of magni- mechanism of this transfer involves the complex interac- tude. The results of these calculations indicate that for high- tion of the Yarkovsky effect and MMRs. In particular, the order MMRs in the outer belt, removal times are in some Yarkovsky effect is thought to be responsible for refilling MMRs substantially shorter than the age of the solar system MMRs by new objects; otherwise the space occupied by (the case of the 7J:4, 5J:3, and 8J:5 MMRs) and in other the active MMRs would be cleared of the asteroids, and resonances substantially longer than the age of the solar gaps in the asteroid distribution at resonant semimajor axes system (the 13J:7 and 11J:6 MMRs). would be created. Such gaps are not currently observed. Figure 5 shows that resonances like the 7J:4, 5J:3, and 8J:5 MMR are expected to correspond to gaps in the aster- 7. PERSPECTIVES oid distribution. Indeed, no large outer-belt asteroids are observed in them (Fig. 2). The outer asteroid belt, however, In the last decade, we witnessed fascinating progress seems more depleted than what would be expected from from studies of separate MMRs in simplest physical mod- the effect of chaotic resonances. This led some authors to els to the global understanding of the structure and effects suggest that depletion occurred during the primordial ra- of MMRs throughout the asteroid belt. Nevertheless, we feel dial migration of the planets (Holman and Murray, 1996; that many open problems await solution. The least devel- Liou and Malhotra, 1997). The same scenario can also oped is the modeling of narrow resonances, especially the nicely explain the presence of many asteroids in the 3J:2 three-body MMRs and the MMRs with Mars. The intricate MMR, which could have been captured by the resonance interaction of MMRs and the Yarkovsky effect is an inter- accompanying the inward migration of the Jupiter. Alter- esting subject for future studies. natively, the outer asteroid belt could have been depleted Basically, one would like to know how the probability by the effect of large Jupiter-scattered planetesimals (Petit of capture of a drifting body into narrow MMRs depends on et al., 1999). the magnitude and direction of the semimajor axis drift, and whether the phase of secular angles plays any role in the 6.3. Mean-Motion Resonances with Mars capture process. Ideally, when capture occurs, one would also like to statistically estimate the change of e and i of an It may seem surprising at a first glance that the MMRs asteroid during the time spent on a resonant orbit. Although with Mars are important in the asteroid belt because Mars some hints about these processes may be derived from re- 392 Asteroids III

(a) 2.7648

2.7646

2.7644

2.7642 Semimajor Axis (AU)

2.764 2.4 × 105 2.6 × 105 2.8 × 105 3 × 105 3.2 × 105 3.4 × 105 Time (yr) (b)

100

0 (degrees) σ –100

2.4 × 105 2.6 × 105 2.8 × 105 3 × 105 3.2 × 105 3.4 × 105 Time (yr) (c)

2 × 10–3 13/2 e 10–3

0 2.4 × 105 2.6 × 105 2.8 × 105 3 × 105 3.2 × 105 3.4 × 105 Time (yr)

σ λ λ Fig. 6. The evolution of orbital elements of the asteroid 41 Daphne. The panels show (a) the semimajor axis, (b) = 9 4 – 22 + 13ϖ, and (c) e13/2. The evolution of 41 Daphne is characterized by the interaction with the 9M:22 MMR. The width of a MMR is proportional to eρ/2 (see equation (11) and related discussion), where here ρ = 13. Consequently, the resonant width seen by the aster- oid is largely modulated with the parameter in (c). Resonant behavior occurs only when the eccentricity of 41 Daphne is large. Adapted from Šidlichovský (1999).

cent numerical simulations, the general trends in the param- We note that a large number of asteroids exist in the 3J:2 eter space are unknown. Yet the interaction of narrow MMRs MMR, while both the 2J:1 and 4J:3 MMRs are substantially with the Yarkovsky effect is important for such issues as depleted. Although some depletion of latter resonances can the origin of NEAs and the dynamical dispersion of the be explained by the chaotic processes acting in the last asteroid families. Also remaining are a few puzzles concern- 4 b.y., additional depopulation likely occurred during the ing the asteroid population of large MMRs in the asteroid formation of the asteroid belt. Is the resonant structure of belt, possibly related to the processes involved in the early the outer asteroid belt a natural outcome of the accretion stages of the solar system formation. itself (Petit et al., 1999), or possibly a sign of the primor- Nesvorný et al.: Dynamics in Mean Motion Resonances 393 dial planetary migration? We wonder if a unique answer to Ferraz-Mello S. (1997) A symplectic mapping approach to the this question can be given. Apparently, detailed modeling study of the stochasticity in asteroidal resonances. Cel. Mech. of the primordial phases is required. Dyn. Astron., 65, 421–437. Such quantitative modeling should also show whether Ferraz-Mello S. and Klafke J. C. (1991) A model for the study of the current depletion of the outer belt between the 2J:1 and very-high-eccentricity asteroidal motion. The 3:1 resonance. In Predictability, Stability and Chaos in N-Body Dynamical 3J:2 MMRs (3.4–3.9 AU) requires primordial migration Systems (A. E. Roy, ed.), pp. 177–184. NATO Adv. Stud. Inst. (Holman and Murray, 1996; Liou and Malhotra, 1997). Ser. B Phys. 272, Plenum, New York. Moreover, what effect would planetary migration have in Ferraz-Mello S., Nesvorný D., and Michtchenko T. A. (1998a) the primordial main asteroid belt? First attempts to quan- Chaos, diffusion, escape and permanence of resonant asteroids tify this effect by direct simulations (Gomes, 1997; Levison in gaps and groups. In Solar System Formation and Evolution ν et al., 2001) show that the sweeping 6 secular resonance (D. Lazzaro et al., eds.), pp. 65–82. ASP Conference Series is efficient at deleting the asteroid population. Our hope is 149. that the increasingly detailed information about the current Ferraz-Mello S., Michtchenko T. A., and Roig F. (1998b) The asteroid belt provided by observations, combined with fu- determinat role of Jupiter’s Great Inequality in the depletion of ture progress in theoretical modeling, will further reveal the Hecuba gap. Astron. J., 116, 1491–1500. signatures that primordial processes imprinted on the or- Gladman B.J., Migliorini F., Morbidelli A., Zappalà V., Michel P., Cellino A., Froeschlé C., Levison H. F., Mailey M., and bital structure of the belt, and that have been preserved until Duncan M. (1997) Dynamical lifetimes of objects injected into now. Understanding the effects of MMRs will certainly be asteroid belt resonances. Science, 277, 197–201. part of that story. Gomes R. S. (1997) Dynamical effects of planetary migration on the primordial asteroid belt. Astron.. J., 114, 396–401. REFERENCES Henrard J. (1990) A semi-numerical perturbation method for sepa- rable Hamiltonian systems. Cel. Mech. Dyn. Astron., 49, 43–67. Beaugé C. (1994) Asymmetric librations in exterior resonances. Henrard J. and Lemaître A. (1983) A mechanism of formation for Cel. Mech. Dyn. Astron., 60, 225–248. the Kirkwood gaps. Icarus, 55, 482–494. Benettin G. and Gallavotti G. (1986) Stability of motions near Henrard J. and Lemaître A. (1987) A perturbative treatment of the resonances in quasi-integrable Hamiltonian systems. J. Stat. 2/1 Jovian resonance. Icarus, 69, 266–279. Phys., 44, 293–338. Henrard J., Watanabe N., and Moons M. (1995) A bridge between Benettin G., Galgani L., and Strelcyn J. (1976) Kolmogorov en- secondary and secular resonances inside the Hecuba gap. tropy and numerical experiments. Phys. Rev. A., 14, 2338–2345. Icarus, 115, 336–346. Bretagnon P. and Simon J.-L. (1990) General theory of the Jupiter- Holman M. J. and Murray N. W. (1996) Chaos in high-order mean Saturn pair using an iterative method. Astron. Astrophys., 239, resonances in the outer asteroid belt. Astron.. J., 112, 1278. 387–398. Hori G. (1966) Theory of general perturbation with unspecified Bottke W. F., Morbidelli A., Jedicke R., Petit J.-M., Levison H., canonical variable. Publ. Astron. Soc. Japan, 18, 287. Michel P., and Metcalfe T. S. (2002a) Debiased orbital and size Jedicke R. and Metcalfe T.S. (1998) The orbital and absolute distributions of the Near-Earth Objects. Icarus, in press. magnitude distributions of main belt asteroids. Icarus, 113, Bottke W. F. Jr., Vokrouhlický D., Rubincam D. P., and Broz M. 245–260. (2002b) The effect of Yarkovsky thermal forces on the dynami- Klafke J. C., Ferraz-Mello S., and Michtchenko T. (1992) Very- cal evolution of asteroids and . In Asteroids III high-eccentricity librations at some higher-order resonances. (W. F. Bottke Jr. et al., eds.), this volume. Univ. of Arizona, IAU Symp. 152, pp. 153–158. Tucson. Knezevic Z. and Milani A. (2000) Synthetic proper elements for Bowell E. K., Muinonen K., and Wasserman L. H. (1994) A public- outer main belt asteroids. Cel. Mech. Dyn. Astron., 78, 17. domain asteroid orbit database. In Asteroids, and Meteors Laskar J. (1988) Secular evolution of the solar system over 10 (A. Milani et al., eds.), pp. 477–481. Kluwer, Dordrecht. million years. Astron. Astrophys., 198, 341–362. Chirikov B. V. (1979) A universal instability of many-dimensional Laskar J. (1999) Introduction to frequency map analysis. In Hamil- oscillator systems. Phys. Rep., 52, 263–379. tonian Systems with Three or More Degrees of Freedom (C. Dermott S. F. and Murray C. D. (1983) Nature of the Kirkwood Simó, ed.), pp. 134–150. Kluwer, Dordrecht. gaps in the asteroid belt. Nature, 301, 201–205. Lecar M., Franklin F. A., Holman M. J., and Murray N. W. (2001) Farinella P., Froeschlé C., and Gonczi R. (1993) from Chaos in the solar system. In Annu. Rev. Astron. Astrophys., the asteroid . Cel. Mech. Dyn. Astron., 56, 287–305. 39, 581–631. Farinella P., Froeschle C., Froeschle C., Gonczi R., Hahn G., Levison H. F. and Duncan M. J. (1994) The long-term dynamical Morbidelli A., and Valsecchi G. B. (1994) Asteroids falling behavior of short-period comets. Icarus, 108, 18–36. onto the Sun. Nature, 371, 315–317. Levison H. F., Dones L., Chapman C. R., Stern S. A., Duncan Fernandez J. A. and Ip W.-H. (1984) Some dynamical aspects of M. J., and Zahnle K. (2001) Could the lunar “Late Heavy the accretion of and — The exchange of or- Bombardment” have been triggered by the formation of Ura- bital angular momentum with planetesimals. Icarus, 58, 109– nus and Neptune? Icarus, 151, 286–306. 120. Lichtenberg A. J. and Lieberman M. A. (1983) Regular and Sto- Ferraz-Mello S. (1994a) Kirkwood gaps and resonant groups. IAU chastic Motion: Applied Mathematical Sciences, Vol. 38. Symp. 160, pp. 175–188. Springer, New York. Ferraz-Mello S. (1994b) The dynamics of the asteroidal 2:1 reso- Liou J. C. and Malhotra R. (1997) Depletion of the outer asteroid nance. Astron. J., 108, 2330–2337. belt. Science, 275, 375–377. 394 Asteroids III

Message P. J. (1958) Proceedings of the Celestial Mechanics Murray N. and Holman M. (1997) Diffusive chaos in the outer Conference: The search for asymmetric periodic orbits in the belt. Astron. J., 114, 1246–1259. restricted problem of three bodies. Astron. J., 63, 443. Murray N., Holman M., and Potter M. (1998) On the origin of Michtchenko T. A. and Ferraz-Mello S. (1995) Comparative study chaos in the solar system. Astron. J., 116, 2583–2589. of the asteroidal motion in the 3:2 and 2:1 resonances with Nesvorný D. and Ferraz-Mello S. (1997) On the asteroidal popu- Jupiter — I. Astron. Astrophys., 303, 945–963. lation of the first-order resonances. Icarus, 130, 247–258. Michtchenko T. A. and Ferraz-Mello S. (1997) Escape of aster- Nesvorný D. and Morbidelli A. (1998) Three-body mean-motion oids from the Hecuba gap. Planet. Space Sci., 45, 1587–1593. resonances and the chaotic structure of the asteroid belt. Astron. Migliorini F., Michel P., Morbidelli A., Nesvorný D., and Zappalà J., 116, 3029–3037. V. (1998) Origin of multikilometer Earth and Mars-crossing Nesvorný D. and Morbidelli A. (1999) An analytic model of three- asteroids: A quantitative simulation. Science, 281, 2022–2024. body mean motion resonances. Cel. Mech. Dyn. Astron., 71, Milani A. and Knezevic Z. (1994) Asteroid proper elements and 243–263. the dynamical structure of the asteroid main belt. Icarus, 107, Nesvorný D., Morbidelli A., and Vokrouhlický D., Bottke W. F., 219–254. Broz M. (2002) The : A case of the dynamically Milani A. and Nobili A. M. (1992) An example of stable chaos dispersed collisional swarm? Icarus, 157, 155–172. in the solar system. Nature, 357, 569–571. Petit J., Morbidelli A., and Valsecchi G. B. (1999) Large scattered Milani A., Nobili A. M., and Knezevic Z. (1997) Stable chaos in planetesimals and the excitation of the small body belts. Icarus, the asteroid belt. Icarus, 125, 13–31. 141, 367–387. Moons M. (1993) On the resonant Hamiltonian in the restricted Roig F., Nesvorný D., and Ferraz-Mello S. (2002) Asteroids in three-body problem. Internal Report, 19, Facultés Universi- the 2:1 resonance with Jupiter. Dynamics and size distribution. taires de Namur, Belgique. Mon. Not. R. Astron. Soc., in press. Moons M. and Morbidelli A. (1995) Secular resonances in mean- Saha P. (1992) Simulating the 3:1 Kirkwood gap. Icarus, 100, motion commensurabilities. The 4/1, 3/1, 5/2 and 7/3 cases. 434–439. Icarus, 114, 33–50. Šidlichovský M. (1999) On stable chaos in the asteroid belt. Cel. Moons M., Morbidelli A., and Migliorini F. (1998) Dynamical Mech. Dyn. Astron., 73, 77–86. structure of the 2/1 commensurability with Jupiter and the Šidlichovský M. and Nesvorný D. (1997) Frequency modified origin of the resonant asteroids. Icarus, 135, 458–468. Fourier transform and its applications to asteroids. Cel. Mech. Morbidelli A. (1996) The Kirkwood Gap at the 2/1 commensura- Dyn. Astron., 65, 137–148. bility with Jupiter: New numerical results. Astron. J., 111, Wisdom J. (1980) The resonance overlap criterion and the onset 2453. of stochastic behavior in the restricted three-body problem. Morbidelli A. (2002) Modern Celestial Mechanics. Aspects of the Astron.. J., 85, 1122–1133. Solar System Dynamics. Gordon and Breach, London, in press. Wisdom J. (1982) The origin of Kirkwood gaps: A mapping for Morbidelli A. and Moons M. (1993) Secular resonances in mean- asteroidal motion near the 3/1 commensurability. Astron. J., motion commensurabilities. The 2/1 and 3/2 cases. Icarus, 102, 85, 1122–1133. 316–332. Wisdom J. (1983) Chaotic behaviour and the origin of the 3/1 Morbidelli A. and Moons M. (1995) Numerical evidences of the Kirkwood gap. Icarus, 56, 51–74. chaotic nature of the 3/1 mean-motion commensurability. Wisdom J. (1985) Meteorites may follow a chaotic route to earth. Icarus, 115, 60–65. Nature, 315, 731–733. Morbidelli A. and Nesvorný D. (1999) Numerous weak resonances Wisdom J. and Holman M. (1991) Symplectic maps for the N- drive asteroids toward terrestrial planets orbits. Icarus, 139, body problem. Astron. J., 102, 1528–1538. 295–308. Yoshikawa M. (1990) Motions of asteroids at the Kirkwood gaps. Morbidelli A., Zappalà V., Moons M., Cellino A., and Gonczi R. I — On the 3:1 resonance with Jupiter. Icarus, 87, 78–102. (1995) Asteroid families close to mean-motion resonances: Yoshikawa M. (1991) Motions of asteroids at the Kirkwood Gaps. Dynamical effects and physical implications. Icarus, 118, 132– II — On the 5:2, 7:3, and 2:1 resonances with Jupiter. Icarus, 154. 92, 94–117. Morbidelli A., Bottke W. F. Jr., Froeschlé Ch., and Michel P. Zappalà V., Bendjoya P., Cellino A., Di Martino M., Doressoun- (2002) Origin and evolution of near-Earth objects. In Aster- diram A., Manara A., and Migliorini F. (2000) Fugitives from oids III (W. F. Bottke Jr. et al., eds.), this volume. Univ. of Ari- the : First spectroscopic confirmation. Icarus, 145, zona, Tucson. 4–11.