Origin of the Basaltic Asteroid 1459 Magnya: A Dynamical and Mineralogical Study of the Outer Main Belt 1

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1 Icarus 158, (2002) doi: /icar Origin of the Basaltic Asteroid 1459 Magnya: A Dynamical and Mineralogical Study of the Outer Main Belt 1 T. A. Michtchenko IAG, Universidade de São Paulo, São Paulo, SP , Brazil tatiana@astro.iag.usp.usp.br D. Lazzaro Observatório Nacional, Rio de Janeiro, RJ , Brazil S. Ferraz-Mello IAG, Universidade de São Paulo, São Paulo, SP , Brazil and F. Roig Observatório Nacional, Rio de Janeiro, RJ , Brazil Received December 20, 2001; revised March 12, 2002 The recent discovery of a relatively small basaltic asteroid in the outer main belt with no apparent link to (4) Vesta raised several hypotheses on its origin. We present the results of a dynamical and mineralogical study of the region near (1459) Magnya intended to establish its origin. The dynamical analysis shows that the region is filled with high-order two-body and three-body mean motion resonances and nonlinear secular resonances, which can lead to slow chaotic diffusion. The mineralogical analysis has not identified any other asteroid with a composition similar to Magnya, nor the presence of fragments that could be securely related to the catastrophic disruption of a differentiated parent body. The various scenarios for the origin of Magnya are also discussed in the face of both the results presented here and recently published results. c 2002 Elsevier Science (USA) Key Words: asteroids; dynamics; spectroscopy; resonances. 1. INTRODUCTION Asteroid (4) Vesta has been known as the unique large object of the main belt showing a basaltic crust. This basaltic crust was inferred by McCord et al. (1970) and confirmed in all subsequent work (McFadden et al. 1977, Binzel et al. 1997). The presence of this crust demonstrates that this object has undergone extensive 1 Based on observations made with the 1.52 m telescope at the European Southern Observatory (La Silla, Chile) under the agreement with the CNPq/ Observatório Nacional. geochemical differentiation and resurfacing. Its composition is also similar to that of basaltic achondrite meteorites, specifically the eucrites, diogenites, and howardites (HED). Due to this similarity, Vesta has been the center of an intense debate over the past years, trying to establish whether it is the parent body of these meteorites or not (Drake 2001). While the global surface composition of Vesta is very similar to that of eucrites, Vesta itself was not believed to be the immediate source of these meteorites (Wasson and Wetherill 1979). There seemed to be no mechanism sufficiently efficient to transport fragments to the Earth in the time scale of the inferred cosmic ray exposure age of these meteorites. However, since the discovery of near-earth asteroids (NEA) with similar composition (Cruikshank et al. 1991, Wisniewski 1991, Hicks and Grundy 1995, Hicks et al. 1996), the problem of the delivery of HED meteorites to Earth was partially solved: they would be fragments of these near-earth asteroids. Subsequent work has been devoted to establishing if these near-earth asteroids could themselves be fragments of Vesta. This hypothesis was confirmed by the identification of a Vesta dynamical family (Williams 1989, 1992, Zappalà et al. 1990, 1995) and the discovery that these objects have a surface composition similar to Vesta (Binzel and Xu 1993, Binzel et al. 1999, Burbine et al. 2001). Dynamical works have also shown (Migliorini et al. 1997, Marzari et al. 1996) that the 3/1 mean resonance and the ν 6 secular resonance can indeed transport fragments to near-earth orbits. The traditional scenario is the following: Great impacts excavated the surface of Vesta and produced a swarm of small fragments. Part of them /02 $35.00 c 2002 Elsevier Science (USA) All rights reserved.

2 344 MICHTCHENKO ET AL. were injected into the 3/1 mean-motion resonance and the ν 6 secular resonance. Both resonances pumped up the eccentricities of these fragments, which were thus ejected due to close encounters with terrestrial planets. Most of these fragments fell directly into the Sun or escaped from the Solar System, but part of them remained in near-earth orbits. Further collisions ejected fragments into Earth-colliding orbits, which became the basaltic achondritic meteorites recovered on Earth. However, the above scenario still has some problems that need to be addressed and solved. The first of them is that the transport time of objects from the 3/1 and ν 6 resonances to NEA orbits is too rapid to be compatible with the cosmic ray exposure time observed in basaltic achondrite meteorites (Migliorini et al. 1997, Gladman et al. 1997). This problem could be partially solved considering a slow dissipation of the fragments after a collisional breakup, such as the Yarkowsky effect (Vokrouhlicky and Farinella 1998, 2000), which would increase the time elapsed between the ejection from Vesta and the entrance into the resonances. Another intriguing problem is that the spectra of the members of Vesta dynamical family, and of asteroids in the vicinity of Vesta having a V-type surface composition (hereafter we will arbitrarily use the denomination Vestoids for all these asteroids), do not match exactly the spectrum of Vesta itself. Burbine and Binzel (1997) showed that the spectra of Vestoids have a redder slope, which has been interpreted as a space weathering effect (Hiroi et al. 1995, Hiroi and Pieters 1998, Pieters et al. 2000) probably associated to the size of the particles in the regolith layer (Burbine et al. 2001). Another problem in the HED and Vesta s spectra is the presence of a small absorption feature at nm, not detected on all of the Vestoids (Vilas et al. 2000, Hiroi et al. 2001). The recent identification of a small basaltic asteroid in the outer belt, (1459) Magnya (Lazzaro et al. 2000), raised a new possibility for the source of V-type NEAs and HED, as well as new problems. First, the presence of a basaltic surface implies in an extensive geochemical differentiation and resurfacing but, according to our current understanding (Ruzicka et al. 1997), such processes should not occur on small-size objects (recall that Magnya has a diameter of 30 km according to IRAS survey). This suggests a catastrophic disruption of a basaltic object with a diameter similar to that of Vesta (around 500 km), with Magnya a remnant of that collisional event. However, Magnya is not related to any of the nearby dynamical families (Williams 1992, Zappalà et al. 1995). The nearest family identified by Zappalà et al. (1995) is that of (137) Meliboea. This family is clustered at a proper semimajor axis of AU, with no apparent extension that could include (1459) Magnya, at 3.15 AU. Moreover, the C-type taxonomic classification of Meliboea (Zellner et al. 1985), interpreted to be analogous to relatively unheated carbonaceous chondrite material, is incompatible with the required thermal history to form Magnya s basaltic material. A similar problem concerning the missing Psyche family was addressed recently by Davis et al. (1999). (16) Psyche is an M-type asteroid traditionally considered to be the exposed core of a collisionally disrupted differentiated body. However, no observed dynamical family is associated with it. According to these authors, the missing family cannot be explained on the basis of observational bias generated by collisional evolution. Therefore, (16) Psyche is unlikely to be related to the breakup of a differentiated body. On the other hand, the nonexistence of the Magnya family does not exclude the hypothesis of the catastrophic fragmentation of a large basaltic body in the region near Magnya. In fact, the preliminary dynamical analysis of this region (Lazzaro et al. 2000) revealed the presence of several mean-motion resonances, which give rise to chaotic behavior and could help to disperse such hypothetical family. However, it is not yet clear if the chaotic diffusion rates are sufficiently large to drive objects out of this region over time intervals smaller than the age of the Solar System. Therefore, we decided to perform a more complete study of the hypothesis of the catastrophic disruption of a large body through the dynamical and mineralogical analyses of the phase space region around Magnya. This paper is structured as follows. In Section 2, we present the spectral analysis method, which describes the procedures involved, the input data, and other important details. The dynamical analysis of the neighboring region of Magnya, together with the estimate of the diffusion time scale, is presented in Section 3. Section 4 presents the results of the simulation of a hypothetical Magnya family and its dynamical evolution over 500 Myrs. The mineralogical analysis of the Magnya region is given in Section 5, which tries to identify other possible remnants of a large basaltic asteroid. Some hypotheses on the origin of Magnya, and the evolution of fragments to near-earth orbits, are discussed in the last section. 2. SPECTRAL ANALYSIS METHOD The spectral analysis method (SAM) has already been used to study the asteroidal and planetary motion, showing efficiency, sensitivity, and simplicity in implementation (Lazzaro et al. 2000, Michtchenko et al. 2001, Michtchenko and Ferraz-Mello 2001). The method is based on the well-known properties of power spectra (Powell and Percival 1979) and involves two main steps. The first one is the numerical integration of the adequately chosen model, with online filtering of the short-periodic terms. The second step consists of the spectral analysis of the output of the numerical integrations and the construction of dynamical maps. In this section we describe the features of the method and apply it to the study of dynamics in the outer asteroid belt. In order to not introduce unnecessary complexity, and also to minimize the computational cost, the choice of the model requires a careful previous analysis of the dynamical problem. This is necessary because the computing times needed for the construction of dynamical maps are significant. To study the dynamics in the outer asteroid belt, a reliable model is to consider

3 ORIGIN OF THE BASALTIC ASTEROID 1459 MAGNYA 345 the asteroids as massless particles and to take into account the gravitational perturbations of the four major planets only. We recall that the perturbations due to terrestrial planets and Pluto have a negligible effect on the asteroidal motion in the outer asteroid belt. Moreover, the indirect effects of the terrestrial planets (e.g., when their masses are added to the mass of the Sun) cause only small shifts (of the second order in the mass ratio, roughly ) in the location of the asteroidal mean-motion resonances (Murray and Holman 1999). The exact equations of asteroidal motion, accounting for the perturbations of the major planets, were numerically integrated using the accurate RA15 integrator (Everhart 1985). The initial conditions of the simulation were uniformly distributed in the space of osculating orbital elements. The resolution of the dynamical map provided by SAM increases with increasing number of conditions on the chosen grid. However, a large number of initial conditions also increases the computational cost, and a good compromise is necessary between the available computational resources and the grid resolution. In this work, a grid of initial conditions was defined in the (a,e)-plane of semi-major axis and eccentricity, within the limits 3.05 AU a 3.25 AU ( a = AU) and 0.0 e 0.4 ( e = 0.005), respectively. The initial inclination and angular orbital elements of the test particles were fixed at the present values of Magnya at JD (I 15 ). The initial positions of the planets were chosen at the same epoch. In this way, our grid of initial conditions covered almost all the outer asteroid belt between the 9/4 and 2/1 mean-motion resonances with Jupiter. The total time span of the simulations has been carefully chosen to allow the detection of the main features of the asteroidal dynamics. The numerical integrations were performed over a time interval of 1 Myr, which roughly corresponds to 200,000 orbital revolutions of the asteroids. This time interval also corresponds to about 50 whole periods of circulation of the asteroidal perihelia and nodes. Thus, the chosen time interval was large enough to allow an accurate and efficient averaging of the long-period effects, and also to detect the occurrence of both mean-motion and secular resonances in the outer asteroid belt. Further details will be presented in Section 2.1. The application of the digital filtering procedure is an essential step in the construction of dynamical maps. The typical output of a long numerical integration consists of the time series of osculating orbital elements, which include both short- and longperiodic terms. Since we are interested only in the long-term features of the asteroidal behavior, the information about the short-term oscillations is unnecessary. Moreover, these shortperiod terms generate data output that is too large and make the identification of the long-term oscillations inefficient. For this reason, the time series of the asteroids osculating elements have been smoothed by digital filtering, to remove the shortperiod oscillations (those of the order of the asteroidal orbital period). The filtering procedure was implemented online with the numerical integration as described in detail by Michtchenko and Ferraz-Mello (1995). The second step of the method used is the spectral analysis of the output of the numerical integrations. The orbital paths of the test particles were Fourier-transformed using the standard FFT algorithm. We extracted the information contained in the power spectra of the orbital elements and stored it for further construction of dynamical maps, averaged dynamical maps, and determination of secular resonances as described in the following Spectral Number and Dynamical Map Fourier transform allows us to distinguish between regular and chaotic motion, because regular and irregular trajectories behave very differently and have different kinds of transforms. The regular trajectories are conditionally periodic, such that any orbital element ele(t) has a time dependence of the form: ele(t) = m A m exp(2πi m νt). (1) Here, ν is a frequency vector whose components are the independent frequencies of motion, and m is an arbitrary integer vector. When the independent frequencies are constant in time, the spectral composition of the regular motion may be obtained from its Fourier transform. For any smooth function ele(t), the amplitudes A m decrease rapidly with m, so that the sum in (1) is dominated by a few terms. Therefore, the spectrum of regular motion is characterized by a countable (and generally small) number of frequency components. It consists of the lines associated to the independent frequencies whose number is equal to the number of degrees of freedom of the dynamical system and also to those corresponding to higher harmonics and linear combinations of the independent frequencies. The half-width of each line is of the order of ν = 1, where T is the time-length of integration. Then, T the total integration time T defines the resolution of the Fourier transform: the longer is T, the smaller is ν, and the finer the details in the Fourier spectrum that can be distinguished. For sufficiently large T, each spectral peak may be approximated by the Dirac δ-function. In the case of chaotic motion, the independent frequencies of the dynamical system vary with time, and the irregular trajectories are not conditionally periodic. The Fourier transform of the orbital elements is not a sum over Dirac δ-functions. Consequently, the power spectrum of chaotic motion is not discrete, showing broadband components. If the variation of the independent frequencies is large and fast enough to detect it over the chosen timespan T, the power spectrum yields a large amount of spectral peaks. In Fig. 1, we illustrate the power spectra of regular and chaotic motions. The power spectrum of the semi-major axis of Magnya s orbit is shown on the top panel. The bottom panel shows the power spectrum of a neighboring asteroid, (16029) 1999 DQ6, which is located close to Magnya in the space of the proper elements. Although both spectra seem to be similar,

4 346 MICHTCHENKO ET AL. frequencies is below the Fourier transform resolution (defined by the time span), the spectral analysis method is unable to detect chaos. In this work, the total integration times were chosen to be large enough as to distinguish chaos generated by both two-body mean-motion resonances and three-body resonances up to order 10. Higher order resonances should appear in the dynamical map just by extending the integration time. We also note that the mean-motion resonances strongly affect the semi-major axis variation, which justifies our choice of the asteroidal semi-major axes as the basis for the spectral number calculation. On the other hand, to analyze the chaoticity introduced by secular resonances, it would be more appropriate to use an action-like variable dependent on the eccentricity or the inclination. Since the precession periods of the asteroidal perihelion and node are at least 1 order of magnitude larger than the oscillation periods associated with mean-motion resonances, the integration time in this case should be significantly increased. Once the spectral numbers N were determined for all the initial conditions on the grid, we plot them on the plane of initial conditions using a shading scale. The calculated values of N, in the range from 1 to 80, were coded by a gray level scale that varied logarithmically from white (log N = 0) to black (log N = 1.9). Figure 2 shows the dynamical map of the outer main belt, FIG. 1. Power spectra of the semi-major axis. Top: Magnya s orbit obtained over 1 Myr (spectral number N = 8). Bottom: Orbit of DQ6 (spectral number N = 80). the enlargement of the region around one spectral peak (zoom boxes) shows the qualitative difference between them. Indeed, the power spectrum of Magnya contains a small number of welldefined spectral lines, while the power spectrum of 1999 DQ6 contains a broadband of unresolved spectral lines. In each power spectrum, we can determine the number of peaks that are above the arbitrarily defined noise, and the value so obtained will be called the spectral number N. In other words, the spectral number is the number of substantial peaks in the power spectrum of the asteroidal semi-major axis. In this work, we consider substantial those peaks with an amplitude larger than 5% of that of the largest peak. According to this definition, the spectral number N corresponding to Magnya s orbit is equal to 8. For 1999 DQ6 s orbit, this number is much larger, and, in this case, we assign to the spectral number the value 80, arbitrarily defined as an upper limit of N. The spectral number N can be used to qualify the chaoticity of asteroidal motion in the following way: small values of N correspond to regular motion; large values indicate the onset of chaos. It should be noted, however, that an orbit classified as regular can appear as chaotic if a larger time span is used in the integrations. Indeed, if the diffusion rate of independent FIG. 2. Dynamical map of the outer asteroid belt. The values of the spectral number N, in the range from 1 to 80, are coded by gray levels that vary logarithmically from white (log N = 0) to black (log N = 1.9), and plotted on the (a,e)-plane of initial osculating orbital elements. We used a grid of initial conditions, with a = AU and e = The lighter regions indicate regular motion, whereas the darker regions indicate chaotic motion. The actual position of Magnya on the (a,e)-plane of initial conditions is indicated by a plus sign.

5 ORIGIN OF THE BASALTIC ASTEROID 1459 MAGNYA 347 plotted on the plane of initial osculating (a,e). Since large values of N indicate the onset of chaos, the shading scale is related to the degree of stochasticity of the initial conditions: lighter regions on the dynamical maps correspond to regular motion; darker tones indicate increasingly chaotic motion. The domains of chaotic motion appear in Fig. 2 as black inclined stripes of variable width, and they are associated with mean-motion resonances. There are both two-body meanmotion resonances with Jupiter and three-body mean-motion resonances with Jupiter and Saturn. The former correspond to critical combinations of the form m λ + m J λ J 0, and the latter ones to m λ + m J λ J + m S λ S 0. Here, λ, λ J, and λ S denote the mean motions of the asteroid, Jupiter and Saturn, respectively, and m, m J, and m S are integer. The order of the resonance is given by m + m J + m S and define the width of the corresponding stripe in Fig. 2. The mean-motion resonances were identified using Kepler s third law. The main ones are labeled on the top of the graph by the symbol m J /m, in the case of two-body resonances with Jupiter, and the symbol m J :m S :m, in the case of threebody resonances. The present position of Magnya on the (a,e)- plane of osculating initial conditions is marked by a plus sign. The large-scale chaos visible at the right-hand side of the graph is associated with the 2/1 mean-motion resonance with Jupiter Averaged Dynamical Map Although the dynamical map obtained in the previous section reveals the main features of the dynamics in the region under study, it suffers from limitations coming from the use of a grid in the space of instantaneous osculating orbital elements. This fact does not allow us, for example, to superimpose over this grid the position of the real asteroids, because asteroidal motions in this space present large variations. Then, in order to analyze the distribution of the real objects in the region under study, we need to determine some kind of proper elements. We have chosen the averaged orbital elements, defined as the mean values of the smoothed time series resulting from the numerical integrations, over the entire time span. By formal definition, the mean value of an oscillatory signal over a given time span is provided by the amplitude of the spectral peak of frequency equal to 0, i.e., the term A 0 in Eq. (1). Therefore, the FFT algorithm applied to the time series of the smoothed orbital elements automatically yields the averaged values of the semi-major axis: eccentricity and inclination. The averaged orbital elements corresponding to each initial condition on the grid of osculating elements can be plotted on the (ā,ē)- and (ā,ī )-planes, where ā, ē, and Ī are the averaged semi-major axis, eccentricity, and inclination, respectively. In this way, the grid of uniformly distributed osculating initial conditions is transformed by SAM into the maps of averaged elements, which we refer to as averaged dynamical maps. In Fig. 3, we present the averaged dynamical maps of the outer asteroid belt, corresponding to the initial grid of Fig. 2. Figure 3 also shows the averaged elements of Magnya s orbit, calculated in the same way. These are marked on the (ā,ē)-plane (left) and (ā,ī )-plane (right) by a plus symbol. FIG. 3. Averaged dynamical maps of the outer asteroid belt, corresponding to the grid of initial osculating elements in Fig. 2, on the (ā,ē)-plane (left) and on the (ā,ī )-plane (right). The actual position of Magnya is indicated by a plus sign, and the location of 35 members of Meliboea family is shown by star symbols. There are three groups of the secular resonances labeled by 1, 2, and 3. The critical combinations of the frequencies of the asteroidal perihelion and node, g and s, and the planetary fundamental frequencies, g 5, g 6, g 7, s 6, and s 7, are: for group 1, g + s g 7 s 6 and g + s g 5 s 6 ; for group 2, g + s g 7 s 7, g + s g 5 s 7 and g + s g 6 s 6 ; and for group 3, g + 2g 5 3s 6, g + 2g 7 3s 6 and g + g 5 3s 6 + g 7.

6 348 MICHTCHENKO ET AL. The interpretation of the averaged dynamical maps becomes easier when preceded by some considerations about the main features of the asteroidal motion. The conditionally periodic regular motion is characterized by a set of independent frequencies that are constant in time. The amplitudes of these frequencies are also constant in time. In the domains of regular motion, the slight changes in the initial conditions lead to the small variation of the independent frequencies and of their amplitudes. Consequently, the averaged elements suffer continuous variation when the initial conditions are gradually changed. In this way, the regions of regular motion are dominated by smoothed curves of points in the averaged map. On the other hand, in the case of chaotic motion, the diffusion processes cause the variation of proper asteroidal frequencies in time, together with their amplitudes. Therefore, the mean values of the orbital elements obtained over different time intervals will be different. Depending on the magnitude of the chaotic process, small changes in the initial parameters can produce a large variation of the proper frequencies and averaged orbital elements. If the chosen time span and diffusion magnitude are large enough, the method is able to detect the dispersion of the averaged orbital elements, and the domains of irregular motion are dominated by the erratic scatter of points. Finally, in the case of resonant motion, the averaging makes all the resonant particles to appear at the libration centers, whereas the rest of the resonant space appears to be empty. All these dynamical features can be found in the averaged dynamical maps shown in Fig. 3. The regions of regular motion appear as domains of smoothed curves of points. The resonant particles are distributed along the resonant libration centers, indicating the exact location of these resonances. They appear as vertical bands of different widths depending on the order of the resonance. They are labeled at the top of the graph. The meanmotion resonances overlap at eccentricities above 0.3 and give rise to chaotic motion. This region of the outer asteroid belt appears to be a continuous sea of strongly chaotic motion. Another dynamical property reflected in the averaged dynamical maps is the occurrence of secular resonances in the outer asteroid belt. The features observed in Fig. 3 (left) as narrow stripes cutting the (ā,ē)-plane, and, in Fig. 3 (right), as large empty bands on the (ā,ī )-plane, are groups of overlapping secular resonances. The main ones are labeled by the corresponding critical combination of the frequencies of the asteroidal perihelion and node, g and s, with the fundamental frequencies of planetary theories, g 5, g 6, g 7, s 6, and s 7 (see Section 2.3). It should be noted that the secular resonances do not appear in the dynamical maps of Fig. 2, because the total integration time was not large enough to detect the diffusive effects of the long-term acting secular resonances. On the other hand, these resonances are visible on the averaged maps due to the effect of excitation of the asteroidal eccentricities and inclinations that reflects in the averaged orbital elements. Thus, the averaged dynamical maps indicate the exact location of the secular resonances on the space of the proper elements and allow the estimation of their widths. Finally, the advantage of the averaged dynamical maps is that the proper elements of real asteroids can be easily plotted over them. This is especially useful if we want to compare the distribution of asteroidal families with respect to the web of resonances in the region under study. As an example, the distribution of 35 members of Meliboea family are shown in Fig. 3 (left) by star symbols. The proper elements of these objects were taken from the database of Milani and Knežević ( and the family was identified by the hierarchical clustering method (HCM) at a cutoff level of 110 m s 1. This family has been early identified by Zappalà et al. (1990), and it is the closest one to the present position of Magnya. We will come back to this point in Section Proper Frequencies and Secular Resonances The spectral analysis allows the efficient identification of the main long-term oscillations contained in the variation of the asteroidal orbit. In Fig. 4, we present the change of the main oscillation modes in the eccentricity (top) and inclination (bottom), using the initial osculating semi-major axis as a parameter. The initial values of the other orbital elements were fixed at those of Magnya. The proper frequencies of the perihelion longitude (top) and node (bottom) are shown by large points, while their linear combinations with the fundamental frequencies of the planetary motion are shown by smaller points. Over FIG. 4. The variation of the main oscillation modes in eccentricity (top) and inclination (bottom), using the initial osculating semi-major axis as a parameter. The proper frequencies, g and s, are shown by large points; their linear combinations with the planetary fundamental frequencies are shown by small points. Discontinuities in the frequency evolution are associated with the passages through the mean-motion resonances. The large-scale chaos on the righthand side is associated with the 2/1 mean motion resonance with Jupiter.

7 ORIGIN OF THE BASALTIC ASTEROID 1459 MAGNYA 349 the domains of regular motion, the value of the proper frequencies varies continuously when the semi-major axis is gradually varied. When the mean-motion resonances are approached, the frequency evolution shows a discontinuity, characterized by the erratic scatter of values. The large-scale chaos at the right-hand sides of both graphs is associated with the 2/1 mean-motion resonance with Jupiter. It is known that the mean-motion resonances are the main source of chaotic behavior of the asteroids. However, any attempt to discuss the dynamical structure of the asteroid belt must take into account also the secular resonances. The main linear secular resonances (of order 2), such as g g 5, g g 6, and s s 6, have been recognized in the asteroid belt for a long time (Williams and Faulkner 1981). But much less attention has been paid to the study of secular resonances of higher order. To our knowledge, these resonances have only been discussed in the work by Milani and Knežević (1992, 1996). In the computation of the asteroidal proper elements, these authors affronted comparatively large errors whenever a secular resonance was encountered. This occurred not only for the asteroids actually in deep resonance (with a critical argument in a libration), but also for those in shallow resonance (critical argument circulating, but with a very long period). Using their analytical algorithm, they obtained the location in the asteroid belt of the nonlinear secular resonances up to order 4. The nonlinear secular resonances are those associated with a critical combination of the form: m ν 0, where m is a vector of integer coefficients and ν = (g, s, g 5, g 6, g 7, s 6, s 7 )isa vector of the fundamental frequencies. Here, g and s are the asteroidal proper modes of the perihelion and node, respectively, and g 5, g 6, g 7, s 6, and s 7 are the fundamental frequencies of the planetary theories. For the resonances of order 6, the critical combinations contain six frequencies, that is, i m i =6; moreover, i m i = 0 by the D Alembert s rule, and the number of frequencies related to the nodes (s, s 6, s 7 ) must be even. Therefore, having the values of the asteroidal proper frequencies over the grid of initial conditions, and knowing the precise values of the fundamental planetary frequencies (Nobili and Milani 1989), we can calculate the location of the nonlinear secular resonances which occur in the region under study. Using SAM, we have identified the location of the secular resonances up to order 6 in the region of the phase space around Magnya. Further details are given in Section 3. To close this section, it should be noted that the spectral analysis method, although powerful, suffers from the same limitations that characterize all methods used for detecting the chaoticity of motion. Their common shortcoming is the nonexistence of an exact correlation between the indicator of chaos provided by the methods and the macroscopic instability of motion. The large values of the spectral number N, as well as the large frequency variations or the short Lyapunov times, can be taken as estimates of robust chaos. However, once an indication of chaos is obtained, very long-term precise integrations of the dynamical model are necessary in order to assess the significance of this chaos. In this work, we have performed very long-term integrations over a few sets of initial conditions, and the relevant results are detailed in Section DYNAMICAL MAPS OF THE MAGNYA REGION A strong observational evidence for the relatively common occurrence of collisional disruption events in the asteroid belt is given by the existence of several dynamical families located at different heliocentric distances (Zappalà et al. 1995). The breakup hypothesis for the origin of a V-type Magnya (crustal material) assumes the collisional disruption of a differentiated parent body. This implies that a dynamical family composed by metallic, mantle, and crustal material should be formed. However, there is no observed dynamical family associated with Magnya. We can think in two different explanations for this intriguing missing-family problem. On one hand, our current understanding of the formation of dynamical families is quite incomplete, and the present models of the internal structure of differentiated asteroids could even be wrong. On the other hand, there are some dynamical mechanisms acting in the region of the hypothetical Magnya s family that could disperse the members of a big family over time intervals comparable to the age of the Solar System. In this way, we would not be able to identify the family from the presently observed orbits. Assuming the second hypothesis, we study the dynamics in the region around Magnya, in order to detect the possible mechanisms responsible for the eventual erosion of Magnya family. We arbitrarily defined a Magnya region as the region of the space of asteroidal osculating orbital elements in the range 3.12 AU <a <3.16 AU, 0.15<e <0.35, and 0 < I <25. Using SAM, we constructed the dynamical maps of the Magnya region, which are shown in Fig. 5. Figure 5 (top) shows the map on the (a,e)-plane of osculating initial conditions. This map was obtained using a points grid with a = AU and e = The initial inclinations of the test particles were set to the actual value of the Magnya s inclination (I 15 ). Figure 5 (bottom) shows the (e,i )-plane obtained with a points grid with e = and I = 0.5. The initial semi-major axes of the test particles are equal to that of Magnya. In both figures, the initial values of the other angular elements of the test particles were also set to those of Magnya. The osculating position of Magnya is marked by a plus symbol. Figure 6 shows the averaged dynamical map of the Magnya region. The positions of real objects in the Magnya region are plotted on the (ā,ē)-plane using symbols. The numbered asteroids in that region were extracted from the April 2001 version of the asteroid database of Lowell Observatory (ftp://ftp.lowell.edu/ pub/elgb). Averaged values of the semi-major axis and eccentricity of these real objects were calculated over 600,000 years using the same procedure described in Section 2.2. The position of Magnya is marked by a plus symbol. The high-order two-body and three-body mean-motion resonances occurring in the Magnya region are labeled on the top of

8 350 MICHTCHENKO ET AL. while the secular frequencies suffer a relative change of 10 2 to 10 3 per million years. This indicates the possibility of an important chaotic diffusion both in eccentricity and inclination. However, very long-term integrations need to be done in order to estimate the actual diffusion rates. Applying SAM to the Magnya region, we have also detected the occurrence of numerous nonlinear secular resonances. In some cases, these resonances lie very close one to another and form groups with more or less parallel overlapping components. Four of such groups, composed by secular resonances up to order 6, have been detected in the neighborhood of Magnya. They are plotted by continuous curves on the (a,e)- and (e,i )-planes in Fig. 5 and labeled from B I to B IV. The components of each group are listed in Table I. The long-term stability of asteroidal orbits evolving inside nonlinear secular resonances is still an open problem, but, from previous studies (Milani and Knežević 1992), it is clear that these resonances would induce long-term instabilities in the asteroidal motion. To assess the actual effects of the instabilities induced by both mean-motion and secular resonances, we have performed very long-term simulations of Magnya together with seven other test particles close to it. The initial osculating orbital elements of the test particles were set to those of Magnya, except the eccentricities which varied from 0.2 to 0.25 (the initial eccentricity of Magnya at epoch was 0.237). We must note that the initial eccentricities were chosen in such a way that none of the test particles started inside any mean-motion resonance. The orbital paths were followed over 1 Gyr. The time series of the osculating elements were smoothed by digital filtering to remove the FIG. 5. Dynamical maps of the Magnya region defined in Section 3. Top: (a,e)-plane of the osculating elements (the initial inclination was fixed at the current value of the Magnya s inclination); Bottom: (e,i )-plane (the initial semi-major axis was fixed at the current value of the Magnya s semi-major axis). The domains of chaotic motion (dark regions) are associated with the mean motion resonances and are labeled on the top panel by corresponding symbol. The bands of the secular resonances are indicated by the symbols from BI to BIV (see Table I). The actual position of Magnya is shown by a plus sign. the graph. We use the notation m J /m for two-body resonances, or m J :m S : m for three-body resonances, with m, m J and m S integers. There are several of these resonances in the Magnya region, all of them characterized by chaotic motions. In fact, the real objects, which are presently involved in any of these mean-motion resonances (see Fig. 6), exhibit chaotic behavior. From previous studies (Nesvorný and Morbidelli 1998), it is known that typical Lyapunov times associated with these mean-motion resonances are of the order of to 10 5 years, FIG. 6. Averaged dynamical maps of the Magnya region. The distribution of the real objects are shown by a star symbol. Two bands, BI and BII, of the secular resonances are observed. The position of Magnya is indicated by a plus symbol.

9 ORIGIN OF THE BASALTIC ASTEROID 1459 MAGNYA 351 TABLE I The Groups of Nonlinear Secular Resonances in the Magnya Region Group Order 4 Order 6 BI g + s g 7 s 6 g + s + g 6 2g 7 s 7 g + s g 5 s 6 g + s g 5 + g 6 g 7 s 7 g + s 2g 5 + g 6 s 7 g + s 2g 5 + g 7 s 6 BII g + s g 7 s 7 g + s + g 5 2g 7 s 7 g + s g 5 s 7 g + s + g 5 g 6 g 7 s 6 g + s g 6 s 6 g + s 2g 5 + g 7 s 7 g + s g 5 g 6 + g 7 s 6 BIII g + s g 6 s 7 g + s + g 5 g 6 g 7 s 7 g + s + g 5 2g 6 s 6 g + s g 5 g 6 + g 7 s 7 g + s 2g 6 + g 7 s 6 BIV s + 2g 6 2g 7 s 6 s g 5 + 2g 6 g 7 s 6 s 2g 5 + 2g 6 s 6 short-term oscillations, as well as the long-term oscillations with frequencies g g 5 and g g 6. In Fig. 7, the resulting paths are superimposed over the averaged dynamical map of the Magnya region on the (ā,ē)-plane. Each path is indicated by a number from 1 to 7. The path of Magnya (labeled by M) is confined to a very small domain, which indicates a quite regular behavior. On the other hand, the remaining test particles show an appreciable diffusion, originated by both mean-motion and nonlinear secular resonances. The role of secular resonances is, in this case, to transport objects out by pushing them into the nearby FIG. 7. Evolutionary paths of seven fictitious initial conditions in the very neighborhood of Magnya. The path of Magnya is marked by M. mean-motion resonances, namely the 17/8 and 13:-6:-5 resonances. The estimated diffusion times are of the order of billion years. 4. DYNAMICAL DISPERSION OF FAMILIES The above results provide evidence to support the dispersion of an asteroidal family in the Magnya region. To thoroughly test this idea, we simulated a hypothetical Magnya family and studied its dynamical evolution over 500 Myrs. We generated initial conditions of 50 fragments resulting from the breakup of a Magnya-type parent body of 100 km in diameter. The breakup was simulated using a simple model of isotropic ejection, in which the ejection velocities were distributed following a Maxwellian and did not depend on the mass of the fragments (e.g., Farinella and Davis 1992, Petit and Farinella 1993). We chose the parameters of the model such that we ended up with the least dispersed fragments, corresponding to an average effective ejection velocity of 70 m s 1. The initial conditions were then numerically propagated considering planetary perturbations from Earth to Neptune, and proper elements ā, ē, and Ī were determined as averages of a, e, and I over 10 Myrs, and sampled every 0.1 Myr. Figure 8 shows the initial distribution of the fragments after the breakup (top) and their final distribution after 500 Myr of evolution (bottom). For the sake of comparison, the fragments are superimposed on the averaged dynamical map of Fig. 6. A remarkable dispersion of the fragments at the end of the simulation is observed. The eccentricities were significantly excited due to the chaotic diffusion along the numerous mean-motion resonances (high-order two-body resonances and low-order threebody resonances) occurring in the region. The typical evolutionary paths along these resonances are shown in Fig. 9. Magnya s path is located between the 17/8 and 13:-6:-5 resonances, at ē 0.21, and does not show any significant diffusion (compare with Fig. 7). Using the hierarchical clustering method (Zappalà et al. 1990), we found that about 20 to 30% of the original members of the family at a cutoff level of 110 m s 1 were still identified as members after 500 Myr. In other words, the family lost more than 70% of its members during the simulation due to diffusion along mean-motion resonances. Since the evolutionary paths in Fig. 9 have the properties of a quasi-random walk, the temporal dispersion (diffusion) of the family is given by the relation ( ) 5 d 2 D a = 4 a D e + 2D sin I t, (2) where d is the average distance between the members of the family (in m s 1 ); t is the time; D a, D e, D sin I are the diffusion coefficients of each proper element; and a 0 is the average semimajor axis of the family. The diffusion coefficients are determined as the variance, σ 2, of the proper elements over 500 Myrs. According to typical values of these coefficients, we have d/ t

10 352 MICHTCHENKO ET AL. Moreover, we must note that our simulation of the hypothetical Magnya family assumed that the parent body was at Magnya s present location at the time of breakup. In fact, the overall dispersion of the family could have been significantly accelerated if the parent body was initially at another, more chaotic, place. Such a situation is achieved, for example, by locating the parent body at a slightly larger eccentricity or inclination than Magnya s ones (see Fig. 5). In this case, Magnya would be a fragment of the family s periphery, while the core of the family would have been at a much more chaotic region, being rapidly dispersed. We will return to this point later in Section 6. Our present simulations of Magnya s family did not account for the possible effects of nonconservative forces, like Yarkovsky effect. Recent studies showed that Yarkovsky orbital drift (Farinella and Vokrouhlický 1999) plays an important role in spreading asteroidal families (Bottke et al. 2001, Nesvorný et al. 2002). It contributes to the mobility in semi-major axis, driving bodies into the mean-motion and secular resonances. Yarkovsky effect is actually relevant for small bodies. Simulations by Roig et al. (in preparation) showed that, for 5-km asteroids in the Magnya region, this effect causes a drift in ā of some 0.02 AU per Gyr. The interaction between Yarkovsky effect and weak resonances could have introduced a significant additional spread of Magnya s hypothetical family over the age of the Solar System. It is interesting to compare the results of our simulations of the fictitious Magnya family to the Meliboea family, which is FIG. 8. The evolution of the hypothetical Magnya family due to dynamical diffusion. Top: Initial family after the hypothetical breakup. Bottom: Final distribution of the fragments after 500 Myrs of evolution Then, we conclude that after a couple of billion years of evolution the family would be sufficiently dispersed to avoid its detection at cutoff levels under 150 m s 1. We can conclude that the numerous mean-motion and secular resonances acting in the Magnya region can produce significant diffusion and, consequently, dispersion of the hypothetical Magnya family. Indeed, the effect of these resonances can result in the loss of several family members, becoming impossible to identify the family in a reliable way. FIG. 9. Typical evolutionary paths of the fragments of our hypothetical Magnya family. Averages of a, e, I were calculated over 10 Myrs and sampled every 0.1 Myr.

11 ORIGIN OF THE BASALTIC ASTEROID 1459 MAGNYA 353 located in the neighborhood of Magnya (star symbols in Fig. 3). This dynamical family appears at values of a, e, and I slightly smaller than those of Magnya and shows a rather large dispersion of its members. The family accounts for 35 members, detected at a cutoff level of 110 m s 1, and it is reduced to only 10 members at a cutoff level of 100 m s 1. Meliboea family shows clear traces of the action of weak mean-motion and secular resonances in the region. Assuming that the diffusion rates around Meliboea are similar to those we estimated around Magnya, we conclude that this family is the result of a relatively recent breakup, which happened less than 1 Gyr ago. 5. MINERALOGICAL MAP OF THE MAGNYA REGION Substantial evidence for the break-up of a Magnya s parent body would be provided by finding additional V-type (basaltic) asteroids such as Magnya and olivine-rich (A-type) and metalrich (M-type) asteroids, which should be liberated from the mantle and nucleus, respectively, of a differentiated parent body. Therefore, we analyzed the compositions, as suggested by taxonomic types, in a large region Magnya: 3.05 < a < 3.25, e < 0.4 and I > 0. In this region, 604 numbered asteroid are found of which only 123 asteroids (20% of the sample) do have a taxonomic classification. Among these, almost 73% come from the S3OS2 survey (Lazzaro et al. 2001), which specifically observed asteroids in the neighborhood of Magnya trying to increase the compositional knowledge of this region. The taxonomic classification of the remaining asteroids were obtained from two other large surveys: ECAS (Zellner et al. 1985, Tholen 1989) and SMASS (Binzel et al. 2001). The taxonomic classification of these 123 asteroids is given in Table II, together with their diameter and albedo. Whenever the IRAS albedo was not available, the diameter was computed assuming the mean albedo of the taxonomic type (0.12 for S-type, 0.05 for C- and D-type). For the X-type asteroids with no albedo, it was assumed a value of 0.08, which is the mean albedo of the asteroids in this region. It is worth recalling that, in Tholen s taxonomic scheme, an X-type spectrum is degenerate and it would be identified as an E-, M-, or P-type only on the basis of the albedo: high, medium, or low, respectively. Each of these classes are interpreted in terms of very different heating histories: while the E and M classes are linked to differentiated bodies, the P class is believed to be primitive. Assuming an albedo of about 0.38 for these X-type objects (the highest albedo in the region), the diameters would decrease somehow, but the results would not change significantly. Since a high or low albedo can indicate a preferential composition, we also included in our analysis the asteroids with known IRAS albedo (Tedesco 1997). This increased the sample of asteroids with a compositional indication to 174, about 30% of the total number of asteroids in the region. The distribution of all the numbered asteroids in the (ā,ē)- and (ē,ī )-planes around Magnya is given in Fig. 10, indicating the associated mineralogical composition of the asteroids. In this figure the letter D indicates a dark composition, i.e., asteroids TABLE II Taxonomic Classification of Asteroids in the Region around Magnya Asteroid Diameter Albedo ECAS SMASS S3OS2 57 Mnemosyne S S S 95 Arethusa C Ch C 96 Aegle T T 137 Meliboea C 152 Atala 87.7 D Sl 181 Eucharis S Xk X 199 Byblis X D 250 Betina M Xk 286 Iclea CX Ch C 314 Rosalia C 328 Gudrun S 357 Ninina CX C 373 Melusina C C 375 Ursula C Xc 399 Persephone X 439 Ohio X: D 448 Natalie C 451 Patientia CU C 457 Alleghenia 30.1 X 489 Comacina C X 491 Carina C X 493 Griseldis X 501 Urhixidur X 508 Princetonia C C 509 Iolanda S S 511 Davida C C X 601 Nerthus X C C 602 Marianna C C 612 Veronika D 618 Elfriede C C 640 Brambilla G C 663 Gerlinde X X 665 Sabine X 676 Melitta XC 680 Genoveva XC X 683 Lanzia C 696 Leonora XC X 702 Alauda C B C 704 Interamnia F B C 746 Marlu P C 756 Lilliana D 760 Massinga SU S 762 Pulcova F C 768 Struveana 43.2 X X 780 Armenia X 784 Pickeringia C 786 Bredichina C 788 Hohensteina C 791 Ani C C 805 Hormuthia CX 849 Ara M 886 Washingtonia C 893 Leopoldina XF C 894 Erda X 912 Maritima C 921 Jovita C 928 Hildrum C 943 Begonia ST X 977 Philippa C X 982 Franklina 49.9 A 983 Gunila XD X