Chaotic diffusion of small bodies in the Solar System

Chaotic diffusion of small bodies in the Solar System Kleomenis Tsiganis (tsiganis@obs-nicefr) and Alessandro Morbidelli Observatoire de la Côte d Azur - Nice, CNRS, BP 4229, 06304 Nice Cedex 4, France Abstract Near Earth Asteroids (NEAs) are small bodies that orbit around the Sun, in the vicinity of the Earth Since the orbits of NEAs are chaotic, catastrophic collisions with our planet can occur Most NEAs originate from the main asteroid belt Resonances, between the orbital frequencies of an asteroid and those of the major planets, are responsible for generating chaotic motion and transporting asteroids towards the near-earth space Assessing the long-term effects of different types of resonance on the replenishment of the NEAs population is the main goal of our research project In this paper we focus on one class of resonances: mediumhigh-order mean motion resonances with Jupiter 1 INTRODUCTION During the last 20 years, our perception of the dynamical evolution of our Solar System changed drastically; from a well-tuned stable system, envisioned by Newton and Laplace, to a dynamically vivid chaotic world sculpted by collisions, migration, and ejection of bodies It is now widely accepted that chaotic, rather than regular, motion prevails in the Solar System The planets themselves follow weakly chaotic orbits, as was recently shown [1-2] This complicated dynamical behavior sets important constraints on our theories of formation and long-term evolution of planetary systems, as well as on the habitability of terrestrial-like planets Our planetary system consists of much more bodies than just the major planets and their satellites A huge number of small bodies,asteroids or comets, exhibit highly irregular motion, owing to the gravitational perturbations exerted on their orbits by the major planets Collisions between the proto-earth and water-rich small bodies may be responsible for the delivery of water to our planet [3] The Moon itself is thought to be the by-product of an early collision between the Earth and a Martian-sized body; a much felicitous incident, if we take into account that the Moon prevents the Earth s spin axis from suffering chaotic episodes that would not have allowed our planet to develop stable climate (or life) [4] Such violent events, with catastrophic consequences for life, can still occur The biggest threat comes from Near Earth Asteroids (NEAs) These objects follow chaotic orbits, whose dynamical lifetime is short ( years) [5] compared to the age of the Solar System ( years), but long enough to perform millions of revolutions in the vicinity of the Earth Given these short dynamical lifetimes, it is evident that NEAs must originate from another region and their population is somehow sustained in a sort of steady-state [6] Figure 1: Distribution of main-belt asteroids in the plane Several gaps are visible, at the location of the main mean motion resonances Mars-crossing objects are located above the solid line Most asteroids in the Solar System are concentrated in a big reservoir, known as the main asteroid belt Their orbits are (to a first approximation) elliptic and lie between those of Mars and Jupiter, with semi-major axes in the range (Fig 1) The eccentricities, and inclinations,!, of their orbital planes (with respect to the ecliptic) are generally small Their orbits are perturbed by the planets, mostly Jupiter which has by far the largest %$& mass (" Solar masses) Chaotic motion is born in the vicinity of resonances, between the orbital frequencies of an asteroid and the planets Different types of resonance occur, the main ones being (i) mean motion resonances with Jupiter, (ii) secular resonances between the frequencies of precession of the asteroid orbit and those of the planetary orbits, and (iii) three-body mean motion resonance between the orbital frequencies of an asteroid and two planets (see [7] for a detailed description) We differentiate between strong (low-order mean motion or secular) resonances and weak (high-order mean motion or threebody) resonances Low-order resonances are responsible for shaping the main belt [7-8] (see also Fig 1) They can increase the eccentricity of an asteroid s orbit up to Marscrossing or even Earth-crossing values on a short time scale ( ( years) Mars-crossers can in turn evolve to NEAs, due to a complicated interplay between resonances and close encounters with Mars, which drives them slowly towards the Earth [9] Weak resonances on the other hand generate a slow chaotic diffusion [10], whose long-term effects on the production of NEAs, as well as on the spreading of

] M M asteroid families [11], have yet to be demonstrated In this paper we concentrate on high-order mean motion resonances between an asteroid and Jupiter, which occur when the mean revolution frequency of an asteroid, (called mean motion), and that of Jupiter,, satisfy a relation of the form, where are integers and is called the order of the resonance These resonances cover the whole range of semi-major axes We limit our study to, since low-order three-body resonances are more numerous and stronger than simple resonances with [12] Mean motion resonances exist even in the simplest version of the three-body problem (Sun - Jupiter - asteroid), where Jupiter is assumed to move on a circular orbit However, the eccentricity of Jupiter s orbit has to be taken into account In this 2-D elliptic (2DE) or 3-D elliptic (3DE) problem (depending on whether we study asteroids that have orbits co-planar to Jupiter s orbit or not), each resonance splits into a multiplet of near-by resonances which overlap with each other and generate chaotic motion [7] We note that the presence of other planets in the model can have significant consequences for resonant dynamics In a first approximation one can incorporate the secular effects by assuming Jupiter s elements to be varying according to a secular planetary theory (eg [13]) These variations will force mean motion resonances to pulsate, by generating additional near-by resonant terms We use the 2-D and 3-D secularly precessing problems (2DP and 3DP) in our numerical studies described below In the following section we present some analytical and numerical results, in the framework of the 2DE model In Section 3, we present the results of numerical experiments in the framework of the 2DE, 3DE, 2DP and 3DP models A discussion on these results and our conclusions are given in Section 4 2 RESONANT MOTION We study the dynamics of a massless asteroid (test-particle) in the Newtonian gravitational field of the Sun and Jupiter We assume that Jupiter moves on a fixed elliptic orbit around the Sun and that the particle moves in the plane defined by Jupiter s orbit In a heliocentric reference frame, the Hamiltonian of the elliptic restricted three-body problem has the form where ( &! (1) ) are the position vectors of the asteroid and Jupiter, respectively, ( ) the masses of the Sun and Jupiter, respectively, the magnitude of the heliocentric velocity vector of the asteroid and the gravitational constant In celestial mechanics we use the osculating orbital elements to characterize the perturbed elliptic orbits of asteroids: the semi-major axis,, eccentricity, and longitude of pericenter and " (the orientation angle of the ellipse) The position of the body on its orbit is given by the mean longitude, 1 We use functions of the elements, eg the modified Delaunay elements 1 In a 3-D configuration space we also need the inclination of the orbital plane, $, and the longitude of the node,% e cos(w) e cos(w) 03 02 01 0-01 -02-03 83 0515 0516 0517 0518 0519 052 0521 0522 0523 0524 aa_j 03 02 01 0-01 -02-03 31 0472 0474 0476 0478 048 0482 0484 0486 0488 049 aa_j Figure 2: The representative plane of the 83 (top) and 31 (bottom) resonances The thick V -shaped( curves show the maximal width of the resonance for & ) Several level curves of are shown in each case 1 +* 2" +3 -, 0, (2) which are canonical variables and transform the Hamiltonian to * 54 6 1 7* 83:98 8* 1 83 (3) As usual, primed elements refer to Jupiter The system of units is chosen such that, ; and are all equal to 1 In these units, the revolution period of Jupiter is equal to < ) and its mean motion is The ratio of Jupiter s mass to the total mass of the system is " and A@CBED8FHG CI I J@KBLD8FHG >%? Since Jupiter is assumed to move on a fixed ellipse, ", and Using canonical perturbation theory we can construct a simpler averaged Hamiltonian, suitable for studying motion in the vicinity of a resonance [7] We start by taking the expansion of 4 (see [14]) The lowest-degree term for each combination of angles takes the form 6N >OCP ORQ @CBF ST for the 2DE problem, or @CBF[ST[ T ; 1 U 1 WV 6N O P O Q X>YZP[X Y\Q2] T U 1 U 1 ; _^ ; ^ V (4) (5)

V ] 3 T " for the 3DE problem, where M F 6N is D a function of the semimajor axis), N, and X! The permissible combinations of angles are those which obey the d Alembert rules, ie T[ T (for the 3-D case, T[ T A and even) When " the asteroid is in the resonance with Jupiter Kepler s 3rd law gives S the nominal location of the resonance, ie U & & [ In the 2DE problem, there appear combinations for every order resonance, which are all nearly-resonant when " and vary much slower than all other combinations of, Thus, we can average over the fast angle 4 and keep only the resonant terms in the expansion also contains secular terms, which do not depend on the mean longitudes We keep only the leading secular terms, of degree and respectively Thus, the averaged Hamiltonian takes the form 3 * M $ Y 3 ] Y @CBFKS 8, 1 @CBF 1 1 1 1 V (6) For resonant or- where, and M are functions of bits, the semi-major axis has small-amplitude oscillations about Thus, apart from the Keplerian part, the semimajor axis is assumed to be constant, when changing from the orbital elements to the canonical variables, while, and M s are substituted by their numerical values The averaged Hamiltonian may be further simplified, by applying a series of canonical transformations (see also [15]), leading to the final form the new variables being 6*, Y! * " M @KBEF T (7) ; where is the opposite of the free longitude of perihelion, " 2 is the free eccentricity (see [14]), $, and the M s are functions of and The free elements are a set of action-angle variables for the secular problem, ie a model where 4 only the 2nd-degree secular terms of the expansion of are retained Then, is the first approximation of the frequency of the perihelion, Note that the Hamiltonian is similar to that of a pendulum ) harmonic oscillator ( ) coupled to a slow (ie ( ), in a non-linear fashion The unperturbed location,, the width%, and the frequency,& T, of small-amplitude oscilla- T tions of each resonance of the resonant multiplet, are given by (8) Figure 3: Surface of section for the 83 resonance The borders of the narrow chaotic zone are well reproduced by the trace of the separatrix (thick curve) of the modulated pendulum T (% *) +++ -, 0 P +++ (& 21 43 (9) 3 where ( M and is the value of at the nominal resonance location Thus, the separation between the resonances is 5 6 The amount of chaos generated in the vicinity of the resonant multiplet is controlled by the mutual overlap ratios, ie% 5 We calculated the values of the M s and (at ), using O7 Y a code written by M Sidlichovsky and kindly provided to us by D Nesvorný Our results show that, for all resonant multiplets studied ( ), the widths of the resonances are much larger than their mutual separations and 98:8 Therefore, the resonances are lying almost on top of each other, the dynamics of the mean motion resonant multiplet becoming similar to those of a slowly modulated pendulum [7] We may treat this problem in the adiabatic approximation, where the slow degree of freedom ( ) is frozen and the characteristics of the fast pendulum ( ) depend parametrically on the values of ( ) In this approximation the separatrix of the pendulum (ie the curve that separates librations from circulations of the resonant angle, ) expands and contracts at a rate approximately equal to, causing orbits in the separatrix-swept zone to change their oscillation mode erratically (from libration to circulation), thus becoming chaotic We calculated the maximum width of the separatrix (ie at the stable equilibrium of the resonance) and constructed the portrait of each resonance in the so-called representative plane [16], ie the plane to which almost all orbits of @CBF come arbitrarily close This is the ( ) plane for being and ), and the resonant angle ; equal to or ), depending on whether is odd or even, respectively Figure (2) shows the results for the 83 and 31 resonances On the same plane we have superimposed several level curves of ; Note that, depending on, each level curve may have one or more branches It is par-

" Figure 4: Surface of section for the 31 resonance The homoclinic orbit crosses the separatrix of the pendulum and the chaotic zone extends to Mars-crossing values of ticularly interesting that, for resonances with, there exist level curves which are closed (see Fig 2) Even if such a level curve intersects the separatrix, chaotic motion will be restricted to small eccentricities, due to energy conservation This means that low-eccentricity chaotic orbits cannot access the high-eccentricity regions and are therefore confined at low values of In the non-averaged problem, chaotic orbits cannot be eternally confined However, the jumps between different energy levels are controlled by the magnitude of the remainder that is disregarded when constructing the Hamiltonian (6) Apart from resonances which are close to other, lower-order, resonances, the effects of the remainder should be miniscule for times comparable to the age of the solar system The equations of motion resulting from Eq (6) (for different resonances) were numerically integrated, using a 2nd-order symplectic implicit scheme [17], Surfaces of @KBEF section for the 83 and 31 resonances (,, F D2, at H) ) are shown in Figs (3)- (4) Regular orbits appear as smooth curves on a surface of section, while chaotic orbits appear as a cloud of points We have also superimposed the trace of the separatrix of the pendulum, as calculated in the adiabatic approximation, for different values of ( ) The 83 and 31 resonances are representative of the two types of mean motion resonance that exist in the main belt In the 83 case, a narrow chaotic zone at moderate values of is found Chaotic orbits are confined in this narrow zone and low-eccentricity orbits cannot reach the high-eccentricity region The borders of the chaotic zone are almost tangent to the analytically calculated separatrix In the 31 case, however, the topology of the surface of section is very different Low-order resonances can force the frequency of perihelion to become zero This leads to a corotation resonance, when, which appears as a set of fixed points on the surface of section In Fig (4) the fixed points are located at "?, (, stable) and ", ( ), unstable) In the integrable approximation there exists a homoclinic curve, composed of two knots which join smoothly at the unstable point Each knot encircles one of the two stable islands (like the ones shown in Fig 4) Chaotic motion is generated in the vicinity of the homoclinic orbit If the homoclinic orbit intersects the separatrix of the pendulum, the two chaotic regions merge and large-scale chaos sets in We note though that the adiabatic approximation cannot well reproduce the inner borders of the chaotic zone Note that now almost circular orbits can random-walk to the higheccentricity region In [16] it was shown that asteroids in the low-order mean motion resonances, associated with the Kirkwood gaps, can escape from the main belt because of this mechanism It is of course necessary to check now (i) the behavior of orbits in the non-averaged 2-D elliptic problem and (ii) how the above results may change by including additional perturbations in our model, such as the third spatial dimension and the secular perturbations of Jupiter s orbit due to its interaction with Saturn 3 ADDITIONAL PERTURBATIONS We numerically integrated the orbits of a carefully selected sample of fictitious asteroids, within all four models described above All integrations were performed with a mixed variable symplectic integrator [18], as it is implemented in the SWIFT package [19] 31 Numerical experiments The initial conditions were selected, by considering particles initially placed in the vicinity of several mean motion resonances of the 2DE model We studied resonances up to order in the central belt ( ) and for the outer belt ( ); 22 resonances are studied in total For each resonance we integrated the orbits of 60 particles, 30 with free eccentricity " and 30 with " We set " " ), so that the resonance s width is at maximum The mean longitude was selected so that the critical angle " corresponded to the stable equilibrium Finally, the initial value of was varied, with respect to the nominal resonance location, within a range that agrees with our calculations for the width of the resonance A short term ( yrs) integration was performed, in order to select 10 chaotic orbits in each resonance for a 1 Gyrs integration The selection was based on the behavior of the critical angle For highorder inner-belt resonances it was not always easy to find chaotic orbits; the width of the chaotic domain can be very small In such cases, slowly circulating or librating (but with large libration amplitude) orbits were selected After selecting our sample of 220 fictitious asteroids we verified, by means of Lyapunov exponent 2 estimates, that more than of our selected particles indeed follow chaotic orbits The orbits of these 220 fictitious asteroids were first integrated, for a time corresponding to years, in the framework of the 2DE model The orbital elements of Jupiter were set to % AU, E? (present epoch), and I (at ) The same particles were 2 The maximal Lyapunov exponent is the mean rate of exponential divergence of near-by orbits, in a chaotic domain

T L (yrs) 10 6 10 5 10 4 10 3 10 2 2-D E 2-D P 3-D E 3-D P 20 25 30 35 40 a (AU) Figure 5: The Lyapunov time of the test-particles orbits in all models For most particles, the value of < is already dictated by the 2DE perturbations As Jupiter is approached, < drops significantly, ie the orbits become more chaotic Particles above the solid line most probably follow regular orbits integrated in the framework of the 2DP model, the frequencies and amplitudes of Jupiter s precession taken from [20] Then, the particles were given an inclination of! with respect to Jupiter s orbital plane and the integration was repeated, both in the 3DE and 3DP models In all integrations the particles were stopped if they became Jupiter-crossers (JCs) or Sun-grazers (SGs) From the particles surviving the 1 Gyrs run, we recorded those that reached perihelion distances smaller than 15 AU, ie became Mars-crossers (MCs) 32 Results Our results show that the degree of stochasticity of the orbits is already determined by the least sophisticated model (2DE) The value of the Lyapunov exponent does not change significantly, as additional perturbations are taken into account (Fig 5) The global transport properties, however, are significantly modified The results of all the runs are summarized in Table 1 In the 2DE model, the inner-belt particles that reach planet-crossing eccentricities within the integration timespan are those that start from the vicinity of the lowestorder resonances, which are associated to the Kirkwood gaps Note that most particles end up encountering Jupiter, after spending a considerable amount of time as MCs The rest of inner-belt resonances do not contribute to the Marscrossing population In the outer belt, the large fraction of escapes can be attributed to the overlap between neighboring mean motion resonances, for " For the remaining particles the numerically averaged elements (as computed using a running-window averaging) change by very small amounts These numerical results confirm what was suggested by our analytical approach, concerning the existence of chaotic orbits semi-confined in high-order resonances, for times comparable to the age of the Solar System In what concerns the outer belt, the only way asteroids Model SG ( ) MC ( ) JC ( ) JC ( ) 2DE - 2 7 36 2DP 1 6 21 51 3DE 14 2 19 88 3DP 22 6 16 85 Table 1: The percentage of particles in each final state (MC, SG or JC) Columns 2-4 are for inner-belt and column 5 for outer-belt particles The numbers are given with respect to the total number of particles in the inner (140) or outer belt (80), respectively The complement (not shown) corresponds to orbits that are not strongly excited e 06 05 04 03 02 01 00 Earth-crossers Mars-crossers 41 31 52 200 225 250 275 300 325 350 375 73 a (AU) 21 initial maximum Figure 6: The initial (open circles) and maximum values (filled circles) of the time-averaged elements of the surviving particles Note the 72 particles, which become Earth-crossers The rest of the inner-belt particles are not excited enough to cross the orbit of Mars (within 1 Gyr) can escape is by encountering Jupiter The percentage of JCs ranges from to?e?, depending on the model It is easy to note that including the secular precession of Jupiter enhances chaotic transport in 2-D space, but it is the 3rd spatial degree of freedom that greatly increases the percentage of JCs Only 1-2 particles seem to follow stable orbits, so we expect the rest of the particles to escape as well, in the frame of (3DE) and (3DP) For this population, the escape time is comparable to the age of the Solar System For the inner-belt resonances, the percentage of escaping particles goes from (2DE) to (3DP) Chaotic diffusion is again much more effective in 3-D space Two snapshots of the evolution of the surviving inner-belt particles in the (3DP) model are shown in Fig (6) Each particle I is projected on the ( ) plane of averaged elements at Myrs (termed initial ) and at the moment when the eccentricity is at maximum In this model the (MC)-(SG) end state becomes the most probable (? ) High-order resonances, like the 72 at AU, can drive asteroids to the Mars-crossing region within I 8 Myrs and even lead to the Earth-crossing region ( AU) However, AU, the 1 Gyr time-scale is short to produce for Mars-crossers

V The fact that the 3rd spatial degree of freedom seems to be more important for large-scale transport than the secular precession of Jupiter s orbit may seem surprising However, we remind the reader that the expansion of the disturbing function has resonant harmonics whose strength is F D F D proportional to Y, as well as terms whose strength is proportional to! Y S or! Y (for even values of or, respectively) For F D! (ie! ) all harmonics have comparable strength Thus, when going to a 3-D space we superimpose harmonics of equal strength that couple the! degrees of freedom This leads to an effective transport mechanism for small- and small-! orbits 4 CONCLUSIONS We presented the results of a study of chaotic diffusion of asteroids, initially placed in mean motion resonances with Jupiter Analytical results were given, in the framework of the 2-D elliptic three-body problem Numerical results on the long-term evolution of small-eccentricity chaotic orbits were also presented Based on these results we can draw some important conclusions, concerning the mechanisms that generate chaotic diffusion inside mean motion resonances and the long-term effects of this process If the secular precession of Jupiter is not taken into account, an important fraction of chaotic orbits with small eccentricities is quasi-confined in the resonances This is a direct consequence of (i) the functional form of the constant-energy surfaces and (ii) the absence of corotation resonances for high-order resonant multiplets However, the harmonics that depend on the inclination can act as a mechanism of inclination-pumping, which in turn provides a bridge towards the high-eccentricity regions and leads orbits to the Mars-crossing limit When the secular precession of Jupiter is added in the model, chaotic diffusion is enhanced However, for small eccentricities, the inclination terms still dictate the motion In our most sophisticated model (3DP) the final distribution of fictitious asteroids shows that (i) in the outer belt, only a small fraction of resonant orbits ( ) could survive for times comparable to the age of the Solar System, and (ii) of the resonant orbits reach the terrestrial planets region, the planet-crossers coming mainly from low-order resonances The high-order resonances do not seem to be so important for the production of NEAs However, a comprehensive analysis of the diffusion process and an estimate of the diffusion rate is still required, in order to explain the spreading of asteroid families and the evolution of dust particles produced in the family-forming events We expect that diffusion will be enhanced if we include in our models the short-term variations of Jupiter s orbit due to Saturn The evolution of the present work towards more sophisticated models and more extensive simulations is currently under way 6 REFERENCES [1] Sussman GJ, and Wisdom J, (1992), Science 271, 56 [2] Laskar J, (1989) Nature 361, 273 [3] Morbidelli A, Chambers J, Lunine JI, Petit JM, Robert F, Valsecchi GB, and Cyr KE, (2000), Met & Plan Sci 35, 1309 [4] Laskar J, Joutel F, and Robutel, P, (1993), Nature 361, 618 [5] Migliorini F, Michel P, Morbidelli A, Nesvorný D, and Zappala V, (1998), Science 281, 2022 [6] Morbidelli A, (1999), Cel Mech Dyn Astr 73, 39 [7] Morbidelli A, (2002), Gordon & Breach [8] Moons M, (1997), Cel Mech Dyn Astr 65, 175 [9] Michel P, Migliorini F, Morbidelli A, and Zappala V, (2000), Icarus 145, 332 [10] Ferraz-Mello S, (1999), Cel Mech Dyn Astr 73, 25 [11] Nesvroný D, Bottke W, Levison HF, and Dones L, (2003), Ap J 591, 486 [12] Nesvorný D, and Morbidelli A, (1998), Astron J 116, 3029 [13] Milani A, and Knezević Z, (1990), Cel Mech Dyn Astr 49, 247 [14] Murray CD, and Dermott SF, (2000),Cambridge University Press [15] Murray N, and Holman M, (1997), Astron J 114, 1246 [16] Wisdom J, (1983), Icarus 56, 51 [17] Yoshida H, (1993), Cel Mech Dyn Astr 56, 27 [18] Widom J, and Holman M, (1991), Astron J 102, 1528 [19] Levison H, and Duncan M, (1994), Icarus, 13 [20] Laskar J, (1990), Icarus 288, 266 5 ACKNOWLEDGEMENTS The work of K Tsiganis is supported by an EC Marie Curie Individual Fellowship (contract N HPMF-CT-2002-01972)