Dynamical behaviour of the primitive asteroid belt

Mon. Not. R. Astron. Soc. 293, 405 410 (1998) Dynamical behaviour of the primitive asteroid belt Adrián Brunini Observatorio Astronómico de La Plata, Profoeg, Paseo del Bosque, (1900) La Plata, Argentina Accepted 1997 September 1. Received 1997 August 5; in original form 1996 November 13 ABSTRACT In this paper we consider the dynamical evolution and orbital stability of objects in the asteroid belt. A simple physical model, including full gravitational perturbations from both giant planets, is used to compute the dynamical evolution of 1000 test particles simulating the primitive asteroids. The criterion of planet crossing (or close approach in the case of resonant objects) is used to reject particles from the simulation. 44 per cent of the particles survived for the whole time-span covered by the numerical integration ( 10 9 yr). The 4:1, 3:1 and to a lesser extent the 2:1 Kirkwood gaps are formed in 10 7 yr of evolution, representing direct numerical evidence about their gravitational origin. We found that the rms eccentricity and inclination of the sample experience a fast increase during the first 10 6 yr. The final rms eccentricity is 0.11, 60 per cent smaller than the present rms eccentricity (0.17). Nevertheless, the gravitational action of the giant planets suffices to prevent the formation of large objects, allowing catastrophic collisions and the subsequent depletion of material from this zone of the Solar system. The excited eccentricity by Jupiter and Saturn may favour mutual encounters and the further increase of the relative velocities up to their present values. Key words: celestial mechanics, stellar dynamics minor planets, asteroids. 1 INTRODUCTION The present surface density of the solar system exhibits a 10 2 10 4 -fold depletion in the region of the asteroid belt (Safronov 1969; Wetherill 1989; Hughes 1991). Several theories have been proposed to explain the absence of a planet in this zone of the solar system. Theoretical evidence suggests that the amount of mass in this region of the solar nebula was 4 M, being removed subsequently to the formation of the largest asteroids, by some mechanism not yet clearly stated (Wetherill 1989). Among the proposed theories, one of the most studied is the one involving the so-called Jupiter-scattered planetoids, residuals from the accretion of Jupiter and Saturn, that may have migrated from the region of these two massive planets to the asteroid belt zone (Safronov 1969; Weidenschilling 1977; Ip 1987). Many asteroids may have acquired large Member of the Carrera del Investigador Científico, CONICET, Profoeg. relative velocities through close encounters with these planetoids, thus stopping the runaway grow of the largest asteroids and favouring fragmentation rather than accretion (Safronov 1979; Wetherill 1989). On another hand, the Jupiter-scattered planetesimals hypothesis furnishes a way in which to explain how such a large asteroid as 2 Pallas can be found in a rather eccentric and inclined orbit. After fragmentation, the residual material in the form of dust was lost from the asteroidal zone by means of the action of non-dissipative mechanisms, effective when the protoplanetary nebula was dense in gas and dust. As an alternative to Jupiter-scattered planetoids, direct gravitational perturbations from the growing Jupiter and Saturn might, in principle, excite large relative velocities in the main belt zone (Heppenheimer 1979; Ward 1980). Early numerical simulations have shown that the perturbations of Jupiter and Saturn can effectively remove matter from the outer part of the belt (Lecar & Franklin 1973) and also from narrow bands placed at heliocentric distances, where the orbital period of the asteroids is in resonance with that of 1998 RAS

406 A. Brunini Jupiter (Binzel 1989). These bands are found in the actual distribution of asteroids and are known as Kirkwood gaps. The dynamics of resonance has been widely investigated during the last decade, and our understanding about its effects on asteroidal orbits has been largely improved, mainly as a result of the works of Wisdom (1983, 1987), who has shown that the gap at the 3:1 resonance with Jupiter may be explained by chaotic diffusion of resonant orbits through a boundary of two zones in phase-space with different modes of motion. This diffusion may pump the orbital eccentricities up to values allowing asteroid orbits to cross the orbit of Mars in short time-scales: once, an asteroid thus becomes Mars-crossing, a close encounter with Mars (usually the norm in this case) can expel the asteroid from its primitive orbit. At present, there is a generalized consensus about the chaotic origin of all Kirkwood gaps (Ferraz-Mello 1994). Nevertheless, Kirkwood gaps can only account for 15 per cent of the absence of mass in the asteroid belt (Wetherill 1989). Mean motion resonance is not the only phenomenon related to orbital instability in the Solar system. It has been recently shown by Laskar (1994) that the inner planets in the Solar system exhibit large-scale chaotic behaviour. Furthermore, several numerical simulations have shown that essentially the same occurs in the outer Solar system (Holman & Wisdom 1993; Levison 1991; Levison & Duncan 1993). Numerical studies of the long-term dynamical evolution of objects in the outer Solar system have shown that most orbits in the region between the four giant planets become planet-crossing after 10 5 to 10 7 yr of evolution from an initially near-circular orbit (Levison & Duncan 1993). In particular cases, some objects are able to survive for longer times, but only in regions associated with some mean motion or secular resonance with the major planets (Holman & Wisdom 1993). It is therefore expected that a substantial fraction of the planetesimals from the original population in the asteroid belt region were not able to survive for the entire age of the Solar system, as this dynamical zone of the Solar system presents a rich and complex chaotic structure (Ferraz-Mello 1994; Froeschlé & Morbidelli 1993). In the present paper, we are mainly concerned with the ability of bodies in the main asteroid belt to survive for a long time after the accretion of Jupiter and Saturn, and with the dynamical structure of the main belt after a certain dynamical equilibrium is reached. To study this problem, we have made a numerical simulation, the details of which are explained in the next section. In Section 3 we present the main results of the simulation, and the last section is devoted to our conclusions. 2 NUMERICAL SIMULATION A long-term numerical simulation of the dynamical evolution of a number of bodies simulating the original asteroids may furnish important clues about the number of asteroids present at the end of the accretion epoch, as well as the dynamical structure of the primitive asteroid belt. This kind of investigation may provide support for the theories of the origin of asteroids, contributing to the establishment of appropriate initial conditions for these theories. Recently, Ries (1996) has investigated the depletion of the outer asteroid belt (semimajor axis 3.4 au), by means of the numerical integration of the orbits of 3000 test particles in the framework of the planar, elliptic, restricted three-body problem, and spanning 38 000 yr. The work of Ries (1996) is thus directly related to our investigation, encouraging much longer integrations in the more realistic three-dimensional problem. Direct numerical integration of the equations of motion of a large number of particles for the age of the Solar system is, however, a very difficult task, mainly because of the short orbital periods in the inner Solar system (typically one to two orders of magnitude shorter than in the outer Solar system). The most complete numerical simulations of the evolution of swarms of particles in the main belt of asteroids have been by Wetherill (1992), but they are based on the method of Arnold (1965) (essentially they are Monte Carlo schemes), including the effect of external perturbations only through approximations to the real dynamical problem. In these simulations, the main role of the major planets was to eject those bodies with aphelia near the orbit of Jupiter, and the effect of resonances was included ad hoc in the form of a random walk diffusion in eccentricity and inclination. Our purpose is, however, to investigate the dynamical behaviour of particles in the asteroid belt region in a more self-consistent way, from the dynamical point of view. We have thus simulated the dynamical evolution of a swarm of 1000 massless bodies, originally distributed with semimajor axes between 2 and 4 au from the Sun. The mean longitude, argument of the perihelion and longitude of the ascending node of the particle were chosen at random in the interval (0, 2π). The semimajor axes a were generated according to a distribution of probabilities of the form a 1 s. We adopted s 2 for the exponent value, in agreement with the current theories of the radial mass distribution in the protoplanetary disc (Weidenschilling 1977; Tremaine 1990). Jupiter and Saturn were assumed to be already formed, therefore they were included as solid bodies, with their present mass and orbital parameters (extracted from the American Ephemeris and Nautical Almanac 1996). Full gravitational interactions between both planets was considered, as were the perturbations by both planets on the massless particles. Although it is believed that the original mass in the asteroid belt was of order about 4 M (Hughes 1991), the particles in our simulation do not perturb the planets, nor were mutual perturbations among them considered. These are points to be more carefully studied in future investigations. Saturn was included in the model because it is recognized that its perturbations on the orbit of Jupiter have important consequences on the dynamical evolution of asteroids, mainly at some mean motion resonances (Ferraz-Mello 1994) as the 2:1 resonance. The orbital eccentricities were generated at random in the interval (0, 0.005) and the inclinations were also generated at random in the interval 0 sin (i) 0.005. These values are representatives of a very cold initial swarm. As

Dynamical behaviour of the primitive asteroid belt 407 we are assuming that Jupiter and Saturn were already formed, the asteroids may have acquired some orbital excitation prior to this time. Therefore, the results of our simulation might be taken as boundaries to the real case. Some more comments about this particular point will be discussed in the next section. Following all the previous simulations (Lecar & Franklin 1973; Ries 1996), those particles that became Jupiter-crossing during the simulation were removed from the sample, except in the case of particles at mean motion resonance with Jupiter; in this case a crossing of the orbit of Jupiter may occur that still avoids close approaches. In this last case, the particles were rejected if an approach to Jupiter occurred in less than one Hill s sphere of influence. Although Mars was not included in the simulation, those particles with perihelion within the orbit of Mars (q 1.52 au) were also removed from the simulation. This criterion is based on the fact that, for a typical Mars-crossing asteroid, the mean time between encounters within one Hill s sphere of influence with Mars is 10 4 yr (Olsson-Steel 1987), i.e. much smaller than the time of orbital evolution. We are thus assuming that after a close encounter with Mars, the orbit of the asteroid would change in such a way that it would be lost from the main belt. Regarding the numerical integration scheme, Holman & Wisdom (1993) have recently developed a powerful symplectic numerical integrator, taking advantage of the particular structure of the Hamiltonian in the N-body problem. A detailed exposition of the construction and performance of symplectic algorithms may be found in Saha & Tremaine 1992. For this study, we have developed a code of the second-order symplectic integrator of Holman & Wisdom (1993). This numerical code was able to integrate the total sample of 1000 particles for a total time of 10 8 Jovian periods ( 10 9 yr), being the longer integration of main belt asteroids performed up to the present. Longer simulations are beyond the capabilities of our computational scheme. The amount of mass-loss attributable to orbital instabilities generated by mean motion resonances is about 27 per cent, in reasonably good agreement with previous estimates (Wetherill 1989). It is worth noting that the particles of the inner belt were rejected when they became Mars-crossing. However, the lifetime of these particles as members of the main belt may be longer than this time, introducing some distortion in the rate of depletion. The root mean square (rms) eccentricity of the final sample is e 2 1/2 0.11, and the rms inclination is i 2 1/2 4.5. Fig. 4 displays the behaviour of the rms eccentricity. It is worth noting that it increases suddenly, stabilizing after 10 7 yr at its final value of 0.11. Therefore it seems to be the 3 RESULTS In this section, we will present the main results of the numerical simulation described above. From the original 1000 objects, 467 were able to survive for the entire time-span covered by the numerical simualtion. Fig. 1(a) displays the distribution of the final semimajor axes a and eccentricities e of the asteroids, and Fig. 1(b) shows the distribution in the (a, i) space. Some of the main Kirkwood gaps (but not all) are visible in the sample. Figs 2(a), (b) and (c) display the distribution of asteroids in the (a, e) space after 10 6, 10 7 and 10 8 yr respectively. In Fig. 3 we show the number of survivors as a function of the time. At the end of our simulation, and according to the adopted rejection criteria, only 44 per cent of the original asteroids have survived. We have also noted that the dynamical evolution of the sample is very fast. All the main observable features in the sample at the end of the simulation were already observable at the first 10 7 yr (Fig. 2b). At this time, the 4:1, 3:1 and 2:1 Kirkwood gaps are almost formed. One asteroid in the 3:2 resonant group was visible in the sample at 10 8 yr but it was not able to survive to the end of the simulation. Figure 1. (a) Distribution of the final semimajor axes a in au, versus eccentricities e of the asteroids remaining after 10 8 Jovian periods. (b) Same as (a) but for the final semimajor axes versus inclinations in degrees.

408 A. Brunini Figure 3. The number of survivors as a function of the time in logarithmic scale. Figure 4. The temporal evolution of the rms eccentricity. Figure 2. The radial distribution of the asteroids after (a) 10 6, (b) 10 7 and (c) 10 8 yr of evolution. steady-state value in a collision-free belt, suggesting that this is a good initial condition for further studies. We have also computed the density distribution of relative velocities. For two planar crossing orbits (a rather good approximation in this case), averaging over all possible encounter geometries, the relative velocity at collision is expressed as V 2 1,2 V 2 1 V 2 2 O(e 1, e 2 ) 2, where V 1 and V 2 are the mean (not the circular) orbital velocities of the two particles. Fig. 5 displays the distributions of the relative velocities in km s 1. The distribution is almost Gaussian, very similar to the one observed in the main belt of asteroids (Lissauer 1993). The most probable collision velocity is 2.7 km s 1, whereas the present mean relative velocity in the asteroid belt is 5 km s 1.

Dynamical behaviour of the primitive asteroid belt 409 Figure 5. Temporal evolution of the relative velocity in km s 1. Figure 6. Distributions of the relative velocities (in km s 1 ) at the end of the simulation. As shown in Fig. 6, after only 10 7 yr of evolution, the relative velocities have already increased up to near the final values. 4 CONCLUSIONS In this paper we have performed a numerical simulation of the dynamical evolution and dynamical stability of the asteroid belt just after the accretion of Jupiter and Saturn. Our initial conditions overlap those investigated by Ries (1996), but our integration covers a much longer time-span and is three-dimensional. Although some major simplifications were included in the dynamical model, mainly because of our ignorance about many aspects of the formation process of the Solar system, we have reached some interesting conclusions. Even in the case of a very cold initial population, the asteroid belt was rapidly depleted at mean motion resonances with Jupiter (the locations of the present Kirkwood gaps). Nevertheless, the short time-scale of orbital instability for resonant objects may have some interesting consequences within the framework of our study and deserves some comment. The time-scale for the formation of Jupiter and Saturn is believed to be of the same order as the time of orbital instability of most particles at mean motion resonances, suggesting that most of these particles were removed from the belt before the complete formation of Jupiter and Saturn. Additional complications with this scenario arise from the fact that, as the growing Jupiter and Saturn acquired mass, additional perturbations must be taken into consideration, in order to account for the effect of the variation of mass. These effects at mean motion resonances are not known and should be investigated. In addition, it is possible that, as a consequence of angular momentum transfer with Jupiter-scattered planetesimals, the Sun Jupiter distance may have varied by some tenths of au (Fernández & Ip 1984), changing the position of the resonances and consequently sweeping a large number of asteroids from the belt (Ward 1980). However, a more detailed knowledge of the process of accretion of Jupiter is required in order to evaluate the importance of these effects. After only 10 7 yr of evolution the relative velocities increased up to 2.7 km s 1, preventing the further growth of large objects. This result represents an important step in our understanding of the process by which a large planet fails to accrete in the asteroid belt region: this is the first confirmation, through a direct numerical simulation, that only gravitational perturbations can increase the random velocities of asteroids to a significant value of a few km s 1. Such a high relative velocity might prevent the formation of massive bodies in the asteroid belt (Wetherill 1989), facilitating catastrophic disruptions rather than accretion. Therefore, one of the main results of this paper is that the depletion of the primitive asteroid belt was originated by the cooperation of two mechanisms: the rapid increase of the relative velocities by the dynamical perturbations of Jupiter and Saturn, which favoured the further increase of the orbital eccentricities and inclinations by mutual encounters and collisions. A substantial number of the surviving asteroids have e 0.1 and i 5, and some of them have reached i 15 and e 0.3. Therefore, even without the presence of Jupiterscattered planetesimals, some large asteroids, such as 2 Pallas, may acquire large eccentricities and inclinations. When the formation time-scales of Jupiter and Saturn are more precisely known, a numerical simulation such as the one presented in this paper, but which includes the growing process of both giant planets, might furnish interesting details about the total mass and dynamical structure of the original asteroid belt. Important limitations of our model are as follows. 1. The model does not include the important effect of mutual perturbation among asteroids. The inclusion of this effect may increase the random component of the motion, as well as the eccentricities and inclinations favouring the diffusion of bodies to unstable resonant regions. 2. If the original asteroid belt was more massive than today [it is expected that the total amount of mass in the belt

410 A. Brunini was as large as 4 M (Hughes 1991)], the effect of the perturbations of the entire swarm on Jupiter can no longer be neglected. 3. The effect of the collision process may be very important after 10 6 yr and should be included in the model. However, a numerical integration code taking into account these effects represent a highly complex problem from the point of view of the computational time. The development of such a code is already in progress. ACKNOWLEDGMENTS I acknowledge the referee, Dr D. W. Hughes, for providing constructive critiques which greatly improved the contents of the paper. REFERENCES Arnold J. R., 1965, ApJ, 141, 1536 Binzel R. P., 1989, in Binzel P. R., Gehrels T., Matthews M. S., eds, Asteroids II. Univ. Arizona Press, Tucson, p. 3 Fernández J. A., Ip W.-H., 1984, Icarus, 58, 109 Ferraz-Mello S., 1994, AJ, 108, 2330 Froeschlé Ch., Morbidelli A., 1993, in Milani A., Di Martino M., Cellino A., IAU Symp. 160. p. 189 Heppenheimer T. A., 1979, Lunar Planetary Sci., 10, 531 Holman M., Wisdom J., 1993, AJ, 105, 1987 Hughes D. W., 1991, QJRAS, 32, 133 Ip W.-H., 1987, Gerlangs. Beitrag. Geophysik, 96, 44 Laskar J., 1994, A&A, 31, L9 Lecar M., Franklin F., 1973. Icarus, 20, 422 Levison H. F., 1991, AJ, 102, 787 Levison H., Duncan M., 1993, ApJ, 406, L35 Lissauer J. J., 1993, ARA&A, 31, 129 Olsson-Steel D. I., 1987, MNRAS, 227, 501 Ries J. G., 1996, Icarus, 121, 202 Safronov V. S., 1969, Evolution of the Protoplanetary Cloud and Formation of the Earth and Planets. Nauka, Moscow Safronov V. S., 1979, in Gehrels T., ed., Asteroids. Univ. of Arizona Press, Tucson, p. 975 Saha P., Tremaine S., 1992, AJ, 104, 1633 Tremaine S., 1990, in Lynden-Bell D., Gilmore G., eds, Baryonic Dark Matter. Kluwer, Dordrecht, p. 37 Ward W. R., 1980, Lunar Planetary Sci., 11, 1199 Weidenschilling S. J., 1977, Planet. Space Sci., 51, 153 Wetherill G. W., 1989, in Binzel P. R., Gehrels T., Matthews M. S., eds, Asteroids II. Univ. Arizona Press, Tucson, p. 661 Wetherill G. W., 1992, Icarus, 100, 307 Wisdom J., 1983, Icarus, 56, 51 Wisdom J., 1987, Icarus, 72, 241