An analysis of the Hygiea asteroid family orbital region

Transcription

1 MNRAS 431, (2013) Advance Access publication 2013 April 3 doi: /mnras/stt437 An analysis of the Hygiea asteroid family orbital region V. Carruba UNESP, Univ. Estadual Paulista, Grupo de dinâmica Orbital e Planetologia, Guaratinguetá, SP , Brazil Accepted 2013 March 8. Received 2013 March 7; in original form 2013 January 15 ABSTRACT (10) Hygiea is the fourth largest asteroid of the main belt, by volume and mass, and it is the largest member of its family, that is made mostly by low-albedo, C-type asteroids, typical of the outer main belt. Like many other large families, it is associated with a halo of objects, that extends far beyond the boundary of the core family, as detected by traditional hierarchical clustering methods (HCM) in proper element domains. Numerical simulations of the orbital evolution of family members may help in estimating the family and halo family age, and the original ejection velocity field. But, in order to minimize the errors associated with including too many interlopers, it is important to have good estimates of family membership that include available data on local asteroid taxonomy, geometrical albedo and local dynamics. For this purpose, we obtained synthetic proper elements and frequencies of asteroids in the Hygiea orbital region, with their errors. We revised the current knowledge on asteroid taxonomy, including Sloan Digital Sky Survey-Moving Object Catalog 4th release (SDSS- MOC 4) data, and geometric albedo data from Wide-field Infrared Survey Explorer (WISE) and Near-Earth Object WISE (NEOWISE). We identified asteroid family members using HCM in the domain of proper elements (a, e, sin(i)) and in the domains of proper frequencies most appropriate to study diffusion in the local web of secular resonances, and eliminated possible interlopers based on taxonomic and geometrical albedo considerations. To identify the family halo, we devised a new hierarchical clustering method in an extended domain that includes proper elements, principal components PC 1, PC 2 obtained based on SDSS photometric data and, for the first time, WISE and NEOWISE geometric albedo. Data on asteroid size distribution, light curves and rotations were also revised for the Hygiea family. The Hygiea family is the largest group in its region, with two smaller families in proper element domain and 18 families in various frequencies domains identified in this work for the first time. Frequency groups tend to extend vertically in the (a,sin(i)) plane and cross not only the Hygiea family but also the near C-type families of Themis and Veritas, causing a mixture of objects all of relatively low albedo in the Hygiea family area. A few high-albedo asteroids, most likely associated with the Eos family, are also present in the region. Finally, the new multidomains hierarchical clustering method allowed us to obtain a good and robust estimate of the membership of the Hygiea family halo, quite separated from other asteroids families halo in the region, and with a very limited (about 3 per cent) presence of likely interlopers. Key words: celestial mechanics minor planets, asteroids: general minor planets, asteroids: individual: Hygiea. 1 INTRODUCTION The Hygiea family, associated with (10) Hygiea, the fourth most massive asteroid in the main belt, is made mostly by low-albedo C-type asteroids, typical of the outer main belt. Recently, it has been suggested that close encounters with (10) Hygiea may have played vcarruba@feg.unesp.br an important role in the dynamical evolution of members of this family (Carruba et al. 2013). Numerical simulations of the orbital evolution of family members under the gravitational influence of all planets, several massive asteroids and non-gravitational forces such as Yarkovsky and Yarkovsky O Keefe Radzievskii Paddack (YORP) may help in setting constraints on the family age and original ejection velocity field. But such modelling requires good estimations of family membership that includes analysis of the local asteroid taxonomy (if available), geometrical albedos and influence C 2013 The Author Published by Oxford University Press on behalf of the Royal Astronomical Society

2 3558 V. Carruba of the local dynamics, so as to minimize the errors associated with including too many interlopers. In this work we tried to obtain a good understanding of the local dynamics, by obtaining synthetic proper elements and frequencies of asteroids in the Hygiea orbital region, with their errors. We revised the current knowledge on asteroid taxonomy, including Sloan Digital Sky Survey-Moving Object Catalog 4th release (SDSS-MOC 4; Parker et al. 2008), and we take advantage of the availability of Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) and the Near-Earth Object WISE (NEOWISE; Mainzer et al. 2011) albedo data (Masiero et al. 2011) to (i) identify asteroid family members using hierarchical clustering methods (HCM; Bendjoya & Zappalá 2002) in the domain of proper elements (a, e, sin (i)) and in the domains of proper frequencies most appropriate to study diffusion in the local web of secular resonances (Carruba & Michtchenko 2007, 2009); 1 and (ii) to try to eliminate possible interlopers based on taxonomical and geometrical albedo considerations. Since the work of Parker et al. (2008) has shown that large families have an associated halo of objects with similar SDSS-MOC 4 data that extends far beyond the border of the HCM families, we also devised a new method to identify family haloes in an extended domain that includes proper elements, principal components PC 1, PC 2 obtained based on SDSS photometric data (Nesvorný et al. 2005) and, for the first time, WISE and NEOWISE geometric albedo. Data on asteroid size distribution, light curves and rotations will also be revised for the Hygiea family. This paper is so divided: in the second section we compute synthetic proper elements for asteroids in the orbital region of (10) Hygiea, and analyse mean-motion dynamics in the area. In the third section we study secular dynamics with the approach of Machuca & Carruba (2011). In the fourth section we revised the current knowledge of asteroid taxonomy and SDSS-MOC 4 data for the region. Section 5 is dedicated to an analysis of the WISE and NEOWISE albedos and diameters data. Section 6 deals with asteroid family identification in proper element and frequency domains. In Section 7, we discuss methods for identifying members of the Hygiea family halo, and in Section 8 we discuss the cumulative size distributions of asteroids groups in the area. Section 9 deals with information on asteroids rotation rates for the Hygiea region. Finally, in Section 10 we present our conclusions. frequency modified Fourier transform method of Šidlichovský & Nesvorný (1997) for numbered and 8453 multi-opposition asteroids, whose initial conditions were obtained by the AstDyS ( Knežević and Milani 2003) on 2012 October 4. These asteroids where those whose osculating orbital elements a, e,sini had a distance of less than 0.15 from those of (10) Hygiea. Proper elements and frequencies were obtained with a 10 Myr numerical integration with a Burlisch Stoer integrator. The elements and frequencies were then computed over a 2 Myr period, and a running box method was used to produce 10 data points. The values of proper elements and frequencies were the mean of the data points, while the standard deviation was used as an approximation of the error. For asteroid families identification purposes, we eliminated from our data set of proper elements and frequencies all asteroids that were lost from the simulation either because of collision with the Sun or because they left the Solar system. We also eliminated all objects for which one of the proper elements a, e,sini or proper frequencies g and s had errors larger than those classified as pathological by Knežević & Milani (2003), i.e. a = 0.01 au, e = 0.1, sin i = 0.03 and g = s = 10 arcsec yr 1.Thisleftuswitha data set of numbered and 8163 multi-opposition asteroids with proper elements in the orbital region of (10) Hygiea, for a total of asteroids with reliable proper elements. Fig. 1 shows an (a,sini) projection of proper elements for numbered (panel A) and multi-opposition (panel B) asteroids in this region, including objects with pathological errors in proper a (shown as red full dots in the figure). Objects classified as having unstable proper a ( < a < 0.01 au) are shown as yellow full dots. Two- and three-body mean-motion resonances are shown as vertical red lines. (10) Hygiea itself is shown as a large green dot. The approximate orbital location of the other families in the region (Eos, Koronis, Veritas and Themis) is shown by their name in green characters. Among the numerous four- and five-bodies resonances that may be causing the appearance of chaos in the region, we identified the 5J:-2S:- 2U:1A four-body resonance of order 1 at a au and the -5J:-6S:6U:-1M:6A resonance of order zero at a au. But many other combinations are possible in the region, driving a non-destabilizing chaotic dynamics (see also Machuca & Carruba 2011). 2 ASTEROID PROPER ELEMENTS A first step in order to obtain dynamical asteroid families, either in the space of proper elements (Bendjoya & Zappalá 2002) or proper frequencies (Carruba & Michtchenko 2007, 2009), is to obtain a reliable set of proper elements. In this work we computed synthetic proper elements a, e, i (semimajor axis, eccentricity and inclination, respectively) and frequencies n, g, s (mean motion, precession frequencies of the argument of pericentre ω and longitude of the node, respectively) according to the procedure outlined in Knežević & Milani (2003) and Carruba (2010), based also on the 1 Frequency groups are not the same as families found in proper element domain, for which it is possible, with some approximations, to give estimates of the velocity at infinity of each members. The purpose of frequency groups or families is not to try to reconstruct the original ejection velocity field, but to look for objects that drifted from the family into a near secular resonance, for groups of asteroids that are currently inside this resonance. For more detail on this method, please see Section SECULAR DYNAMICS IN THE HYGIEA REGION The Hygiea orbital region is crossed by a rich web of secular resonances. Linear secular resonances, such as the ν 6 = g g 6 or the ν 16 = s s 6, occurs when there is a commensurability between the precession frequencies of the asteroids and of one planet (in the case above reported, Saturn). Non-linear secular resonances, such as the z 1 = ν 6 + ν 16 = g g 6 + s s 6, are usually combinations of linear secular resonance arguments, satisfying the two D Alembert laws. Following the approach of Machuca & Carruba (2011), we tried in this section to identify all the secular resonances (up to order six) with the largest population of likely resonators, i.e. asteroids to within ±0.3 arcsec yr 1 from the resonance centre (e.g. in the case of the z 1 secular resonance, a g + s kind of resonance in the notation of Machuca & Carruba 2011, likely resonators would be objects whose combination of g + s would fall to within gs = 0.3 arcsec yr 1 from the resonance centre, i.e. g + s = g 6 + s 6 ± gs ).

3 An analysis of the Hygiea family region 3559 Figure 1. Panel A: an (a,sini) projection of proper elements for numbered asteroids in the region of the Hygiea family. Yellow circles display asteroids with standard deviation on a σ a between and 0.01, while red circle shows asteroids with σ a larger than Panel B: an (a, sin(i)) projection of multi-opposition asteroids in the same region. Table 1 displays a list of resonances for the region of (10) Hygiea, ordered as a function of the resonance type as defined in Machuca & Carruba (2011). The first column shows the resonance argument, the second column the resonance centre (or the value of the combination of planetary frequencies that is present in the resonant argument) and the third column displays the number of likely resonators. Unless a large population of asteroids is observed, for the sake of brevity we do not list resonances involving Uranus frequencies g 7 and s 7. 2 One may notice the very large (884 bodies) and distinct population of asteroids in the 2ν 6 2ν 5 + ν 7 g-type of resonance. Large numbers of likely resonators are observed in the 2ν 5 2ν 6 + ν 16 s-type resonance, and in three g + s-type resonances (the ν 6 + ν 16,the2ν 6 ν 5 + ν 16 and the 2ν 6 ν 5 + ν 17 resonances). With the exceptions of the strong 3ν 6 2ν 5 g-type resonance and of the ν 5 + 2ν 16 g + 2s-type of resonances, all other secular resonances tend to have relatively small (less than 40 objects) populations of likely resonators. Metrics in the domains of (n, g, s), (n, g, g + s) and(n, g, g + 2s) could therefore be the most appropriate ones to study diffusion in secular resonances for the region of (10) Hygiea. Fig. 2, panel A, shows a (g, g + s) projection of the asteroids with known proper elements in our sample. Blue lines identify the location of the secular resonances listed in Table 1 with a population of likely resonators larger than 15. Fig. 2, panel B, displays an (a, sin (i)) projection of the likely resonators population listed in Table 1. Since past experience showed us that clumps and families in frequency domain tend to be found at the crossing of one or more secular resonances (Carruba 2009, 2010), we looked for a population of likely resonators belonging to more than one secular resonance, but we could not find any in the Hygiea area. In the next sections we will try to understand in more detail how dynamical evolution in secular resonances shaped the form of the asteroid families observed in the region. 2 Because of the proximity of the values of g 5 and g 7, resonances involving g 5 values are close in the orbital domain to resonances involving g 7 values. Characteristics multiplet structures involving combinations of g 5 and g 7 values are usually associated with many of the main resonances involving the g 5 frequency. See Carruba & Michtchenko (2007) and Machuca & Carruba (2011) for a more in depth discussion of this phenomenon. 4 COMPOSITIONAL ANALYSIS: SPECTRAL TAXONOMY AND SDSS-MOC 4 DATA As a preliminary step in the analysis of asteroids in the Hygiea orbital region as defined in Section 2 we reviewed the current knowledge about taxonomic classification. Using the data present in the three major photometric/spectroscopic surveys (ECAS, SMASS and S3OS2), 3 we identified objects also in our set of asteroids with obtained proper elements. We also consulted the JPL HORIZONS ephemeris system for data on asteroids taxonomy, available at horizons-system/ and accessed on 2012 November 19. We found 112 C-type complex asteroids (10 of these objects are known B-type asteroids), 35 DT bodies, 30 L-types, 15 objects in the S-complex and 56 in the X complex, for a total of 243 objects out of asteroids with reliable proper elements, which gives a fraction of 0.7 per cent with known taxonomic properties. Fig. 3 shows an (a, sin (i)) projection of asteroids with known taxonomy in the Hygiea region. We can see a predominance of C-type objects in the regions of the Themis, Veritas, and Hygiea family (with a few X- and S-type interlopers). A clear D-, T- and L-type agglomeration is observable in the region of the Eos family core,asalsodiscussedinmothé-diniz, Roig & Carvano (2005). To obtain further information on the taxonomy of asteroids in this region, we then turned our attention to the SDSS-MOC 4 (Ivezić et al. 2001). For the purpose of deriving very reliable inferences about asteroid surface compositions, multiband photometry is not as precise as spectroscopy. However, SDSS data are very important, since this survey includes about two orders of magnitude more objects than available spectroscopic catalogues. Nesvorný etal. (2005) showed that the SDSS-MOC is an useful data set to study general statistical variations of colours of main belt asteroids, but that caution is required to interpret colours in individual cases. These authors used an automatic algorithm of principal component analysis (PCA) to analyse SDSS photometric data and to sort the 3 ECAS: Eight-Color Asteroid Survey (Zellner, Tholen & Tedesco 1985; Tholen 1989); SMASS: Small Main-Belt Asteroid Spectroscopic Survey (Xu et al. 1995; Bus & Binzel 2002a,b); S3OS2: Small Solar System Objects Spectroscopic Survey (Lazzaro et al. 2004).

4 3560 V. Carruba Table 1. Main secular resonances in the Hygiea region, frequency value and number of resonant asteroids. Resonance argument Frequency value Number of resonant asteroids (arcsec yr 1 ) g resonances 3ν 6 ν ν 6 2ν 5 + ν ν 6 ν ν 6 2ν s resonances 2ν 5 2ν 6 + ν g + s resonances 2ν 5 ν 6 + ν ν 5 + ν ν 6 + ν ν 6 ν 5 + ν ν 6 ν 5 + ν g + 2s resonances ν 5 + 2ν ν 6 + 2ν ν 7 + 2ν ν 6 + 2ν g s resonances 2ν 6 ν 5 ν g 2s resonances ν 5 2ν ν 6 2ν g + s resonances 2ν 5 + ν ν 5 + ν 6 + ν ν 6 + ν g s resonances 2ν 6 ν objects into different taxonomic classes. In particular, PCA can be used to derive linear combinations of the five SDSS colours (u, g, r, i, z), in order to maximize the separation between a number of different taxonomic classes in SDSS data. Two large separated complexes were found in the PCA first two components: the C/X complex and the S complex, with various subgroups identified inside the complexes (Nesvorný et al. 2005). Here we followed Nesvorný et al. (2005) approach (see also Novaković, Cellino & Knežević 2011) to obtain the first two principal components for the 4436 asteroids that have both SDSS-MOC 4 data and were listed in our set of objects with synthetic proper elements. 4 Fig. 4 shows the location of 4436 asteroids with SDSS- MOC 4 data in the principal components plane. The 19 asteroids included in the SDSS-MOC 4 having known spectral types and proper elements are shown with the same symbols used in Fig. 3. The inclined dashed line separates the C/X complexes from the S one, whose distribution is compatible with what observed in pre- 4 We must remark that, because u-band measurements are generally afflicted by large errors some authors prefer not to use this data for determining principal components PC 1 and PC 2. Since our goal here is to start investigating the Hygiea family taxonomy and halo properties we limited our analysis to the Novaković et al. (2011) approach. But other methods for finding PC 1 and PC 2 are possible and will be tested in future works. vious works. In particular, we found that all asteroids with known spectral types in the SDSS-MOC 4 data that belong to the C/X complex (C-, B- and X-types, in our case) are found on the left of the dashed line in Fig. 4, and all asteroids belonging to the S complex (S-, D-, T- and L-types, in our case) to the right, as it should be expected. What other information can be achieved by SDSS-MOC 4 data? We selected asteroids that belonged to either the CX-complex or the S-complex in Fig. 4 and we plotted them in the (a, sin (i)) plane (see Fig. 5). We found that, as discussed in Mothé-Diniz et al. (2005), the Eos and Koronis families have a majority of members in the S-complex, while the Hygiea, Veritas and Themis families have most of their members in the CX-complex. An extended halo of CX-complex asteroids is observed around the core of the Hygiea family. Several interlopers from the S-complex are also observed in the Themis and Hygiea region, which confirms our taxonomic analysis. Eliminating these objects from the list of family members will be the subject of next sections. 5 WISE/NEOWISE ALBEDOS AND DIAMETERS Diameters of asteroids are obtained using the relationship (Harris & Lagerros 2002) D = 1329 km pv 10 H/5, (1) where H is the absolute magnitude, defined as the apparent magnitude the body would have 1 au from the Sun and the observer, at 0 phase angle, and p V is the visible geometric albedo. While information on asteroid absolute magnitude is generally available for hundreds of thousands of asteroids, until recently only about two thousands asteroids had reliable values of geometric albedos (see Tedesco et al for a discussion on albedo values of asteroids). Initial results from the WISE (Wright et al. 2010), and the NEOWISE (Mainzer et al. 2011) enhancement to the WISE mission recently allowed to obtain diameters and geometric albedo values for more than Main Belt asteroids (Masiero et al. 2011), increasing the sample of objects for which albedos values were known by a factor of 50. Masiero et al. (2011) showed that, with some exceptions such as the Nysa Polana group, asteroid families typically show a characteristic albedo for all members, and that a strongly bimodal albedo distribution was observed in the inner, middle and outer portions of the Main Belt. To investigate the physical properties of asteroids in the region of the Hygiea family, we obtained the full table of best fits for main-belt asteroids from the Pass 1 processed cryogenic survey data, available at accessed on 2012 November 10, and selected the asteroids present in our sample of proper elements, obtained in Section 2. We found 8073 numbered objects in the Hygiea orbital region with both proper elements and WISE albedo data. Fig. 6 shows an (a, sini) projection of proper elements for numbered asteroids in the region of the Hygiea family, with different colour codes for the asteroids diameters (panel A) and geometric albedo p V (panel B). The Hygiea and Themis families are dominated by a population of small bodies, with the presence of only a handful of asteroids with D > 20 km [(10) Hygiea itself being one of the exceptions]. More interesting are the data on the asteroid albedo: one can clearly notice the difference between high albedos objects, associated with the Eos family in the left-hand top corner of Fig. 6, panel B, and darker objects, in the regions of the Hygiea, Themis and Veritas

5 An analysis of the Hygiea family region 3561 Figure 2. Panel A: a (g, g + s) projection of the asteroids with known proper elements in our sample. Blue lines identify the location of the secular resonances listed in Table 1 with a population of likely resonators larger than 15. Panel B: an (a, sin (I)) projection of the likely resonators population listed in Table 1. Figure 3. An (a, sin (i)) projection of asteroids with known taxonomy in the Hygiea region. L-type asteroids are shown as cyan diamonds, S-type as magenta asterisks, X-type as red crosses, C-type as green circles and Dand T-type asteroids as blue stars. Red lines display the location of the main mean-motion resonances, and the green full dot shows the orbital location of (10) Hygiea itself. families. The few high albedos objects found at inclinations lower than the typical values of Eos family members seems to be associated with migration in mean-motion resonances such as the 9J:-4A, the 1J:1S:3A or the 2J:5S:-2A or to migration in non-linear secular resonances such as the z1, the 3ν 6 2ν 5 or the 2ν 5 2ν 6 + ν 16. How reliable are the WISE data on asteroid albedos and diameters? To answer this question we computed percentile errors, defined as the fraction between the value of the error and the central value, for diameters and geometric albedos. We found that per cent of the objects had errors in diameters less than the mean value of the perceptual errors (equal to 25 per cent), had errors less than two mean perceptual errors (but larger than one mean value), 0.11 per cent had errors less than three mean perceptual errors (and larger than two mean values) and 0 per cent had errors larger than three mean perceptual errors. Concerning albedos, per cent had errors less than the mean value of 27 per cent, per cent had errors less than two mean values, 8.74 per cent had errors less Figure 4. Location of 4436 asteroids with SDSS-MOC 4 data in the principal components plane. The 19 asteroids included in the SDSS-MOC 4 having known spectral types and proper elements are shown with the same symbols used in Fig. 3. than three times the mean value and 3.54 per cent had errors higher than three times. Overall, WISE diameter values have smaller perceptual errors than geometric albedos ones that are known with lesser precision (perceptual errors on pv may reach 200 per cent). The relative large errors on pv will need to be accounted for when analysing these properties, especially in the context of other physical properties of asteroid family members. We will further discuss this problem in other sections of this paper. 6 A S T E RO I D FA M I LY I D E N T I F I C AT I O N Having obtained proper elements, taxonomic properties and albedos of asteroids in the Hygiea orbital region, we are now ready to start determining dynamical groups in the area. For this purpose we first look in the domain of proper elements (a, e, sin (i)) using the hierarchical clustering method HCM. An analysis in the frequency

6 3562 V. Carruba Figure 5. An (a, sin (i)) projection of asteroids belonging to the CXcomplex (green circles) and S-complex (magenta asterisks) according to SDSS-MOC 4 data. domains (Carruba & Michtchenko 2007, 2009) most appropriate to study diffusion in the main secular resonances in the region will be the subject of the next subsection. 6.1 Asteroid groups in proper element space In identifying asteroid families in the space of proper elements two parameters are fundamental: the cut-off distance at which the family members are defined, d0, and the minimum number of objects Nmin for a cluster to be considered significant. Beauge & Roig (2001) define a nominal distance cut-off as the average minimum distance between all the neighbouring asteroids in the same region of the asteroid belt. The value of Nmin is defined by Zappala et al. (1995) as (2) Nmin = N0 + 2 N0, Figure 7. The average number N0, Nmin and the maximum number max (Ni ) of asteroids as a function of the velocity cut-off in the region of the Hygiea family. where N0 is the average number of orbits within a sphere of radius d0 at every point of the proper element space. A cluster with a number of objects larger than this critical value is called a clump, while a family is a cluster with a number of members larger than 2.5Nmin. The nominal distance velocity cut-off as defined in Beauge & Roig (2001) is of 44.0 m s 1, while Fig. 7 displays the average number N0, Nmin and the maximum number max (Ni ) of asteroids as a function of the velocity cut-off, for asteroids in the region. The value of Nmin corresponding at d0 = 44.0 m s 1 is 5. As can be seen in Fig. 7, the fact that max (Ni ) is much larger than Nmin may be a hint that the local population of asteroids is dominated by the local background (in the opposite case, there would be a majority of objects with small relative distances, and max (Ni ) would not be much greater than Nmin ). Figure 6. Panel A: an (a, sin i) projection of proper elements for numbered asteroids in the region of the Hygiea family. The colour of the full dots identifies WISE diameter values, see text for a discussion of the colour code. Black dots identify asteroids with D < 5 km, blue full dots are associated with objects with 5 < D < 10 km, magenta full dots show asteroids with 10 < D < 20 km and red full dots display asteroids with D > 20 km, according to the WISE data. Panel B: an (a, sin (i)) projection of numbered asteroids in the same region. Black dots identify asteroids with pv < 0.075, blue full dots are associated with objects with < pv < 0.125, magenta full dots show asteroids with < pv < and red full dots display asteroids with pv > Green full dots in both panels display the orbital position of (10) Hygiea itself.

7 An analysis of the Hygiea family region 3563 Figure 8. Panel A: the number and differential number of members of the Hygiea family in proper element domain as a function of the velocity cut-off. Panel B: a stalactite diagram of the Hygiea family region. Using the standard distance metric d of Zappala et al. (1990) in proper element space, where the distance is given by d = na k1 a a 2 + k2 ( e)2 + k3 ( sin (i))2, (3) where n is the asteroid mean motion; x the difference in proper a, e and sin (i); and k1, k2, k3 are weighting factors, defined as k1 = 5/4, k2 = 2, k3 = 2 in Zappala et al. (1990, 1995), we computed Hygiea family members as a function of the velocity cut-off. Fig. 8, panel A, displays the number (in blue) and differential number (in green) of members of the Hygiea family in proper element domain as a function of the velocity cut-off. The numbers on the peaks of the number of family members are related to the clusters that were enclosed by the Hygiea family at higher velocity cut-offs. The vertical red line shows the value of d0 obtained with the Beauge & Roig (2001) approach. The Hygiea group collects its first members for a cut-off of 37 m s 1, it merges with another big group at 42 m s 1, collects several smaller clumps and groups at increasing cut-off (the largest group is accreted for a cut-off of 72 m s 1 ) and it merges with the local background (and with the Themis family) for a cut-off of 77 m s 1. The reader may notice that, before the Hygiea family merging with the local background, its growth of number of members as a function of increasing cut-off values is quite smooth: no significant other largest group in the region of the Hygiea family was identified in this work. To re-identify asteroid families we also constructed a stalactite diagram in the traditional way defined by Zappala et al. (1990): we start with (10) Hygiea as the first central body and identify all the bodies associated with it at dcut-off = 76 m s 1, value for which no other independent cluster of asteroids was found in the region. We then decreased the cut-off by 5 m s 1 and identified the families and clumps among the asteroids not associated with (10) Hygiea. Fig. 8, panel B, displays our results in the interval of cut-offs between 41 and 76 m s 1. Full black squares are associated with families in the region, and empty black squares are associated with clumps, according to the limits displayed in Fig. 7. Table 2 reports the families that we identified in this work at a cut-off of 66 m s 1. Table 2. Proper element based families in the region of the Hygiea group. The first column reports the lowest numbered asteroid in the group, the second the number of objects associated with the group, the third the number of object with spectral classification (see Section 4) and the fourth the number of objects with SDSS-MOC 4 data (see Section 4). Name N Nspec NSDSS-MOC 4 (10) Hygiea (5340) Burton (15755) (1992 ET5) The largest group at dcut-off = 41 m s 1 is actually not associated with (10) Hygiea but with (159) Aemilia (so that it may be more appropriate to talk of the Aemilia family rather than of the Hygiea one). The two groups merge at a cut-off of 42 m s 1, and several other small groups are enclosed by the Hygiea family at larger cutoffs. However, only two of such small groups satisfy the criteria for a reliable family outlined in Carruba (2009), i.e. that the family should at least be observable for three intervals in cut-offs: at a cutoff of 66 m s 1 we saw two other families associated with (5340) Burton and (15755) (1992 ET5) that merge with the Hygiea family at a cut-off of 77 m s 1. As in Carruba (2010) we also looked for asteroids pairs in the region. These are objects that are extremely close in proper element space and could be associated with double asteroids that recently split up. The first five pairs, in terms of distance as given by equation (3), were (51858) and (336188), (16716) and (330422), (167570) and (334640), (138651) and (157054) and (132750) and (138758), all with distances of less than 2.7 m s 1. A complete list of all the newly identified asteroid pairs found in the Hygiea region is available upon request to the first author. To understand how the Hygiea family interacts with other families and clumps in the region we also determined the local density of asteroids, following the approach of Carruba & Michtchenko (2009). Density maps will display regions characterized by strong mean-motion or secular resonances by a relatively low number of asteroids per unit bin. To quantitatively determine the local density

8 3564 V. Carruba of resonance, that are the one for which significant populations of likely resonators, also belonging to the Hygiea family in proper element domain, were found (see Section 3), the following distance metrics can be used: f = f = h 1 ( n h 0 h 1 ( n h 0 ) 2 + h 2 ( g) 2 + h 3 ( (g + s)) 2, (4) ) 2 + h 2 ( g) 2 + h 3 ( (g + 2s)) 2, (5) Figure 9. An (a sin (i)) projection of the families and clumps (see Table 2) in the region of the Hygiea family, obtained in the proper element domain. of asteroids, we computed the log 10 of the number of asteroids per unitsquareina27by27gridina (starting at a = 2.90 au, with a step of au) and sin (i) (starting at 0, with a step of 0.008). Results are shown in Fig. 9, where, superimposed to the density map, we show the orbital projection of the three families found in this work shown as plus signs. The other symbols are the same as in Fig. 1. As can be seen in Fig. 9, the two newly found groups lie in the same orbital area of the Hygiea family and may actually be considered substructures of the family rather than independent structures. Only 12 members of the Hygiea family had taxonomic information, while no such information is available for the (5430) and (15755) groups. We found a majority of members in the C/X complex, with three C-type (10, 159, 1731), three B-type (538, 5155, 5265) and four X-type (1107, 1109, 1599, 2436) asteroids. Two asteroids belonging to the T-type (1209) and L-type (100) are most likely interlopers. Concerning the SDSS-MOC 4 data, we found 668 objects in the Hygiea family, eight in the Burton group and six in the (1992 ET5) association. The great majority of the asteroids with SDSS-MOC 4 data lies in the C/X complex in the (PC 1, PC 2 )plane (see Section 4), but there is a population of 25 asteroids with values of PC 1 and PC 2 that are more compatible with an S-complex taxonomy and that are most likely interlopers (this is a percentage of 3.7 per cent, which is less than the 10 per cent of interlopers expected in a family this size according to mathematical considerations; Migliorini et al. 1995). All objects in the (5340) and (15755) belong to the C/X complex. Having found dynamical groups in the proper element domain, we then turn our attention to possible clusters in various proper frequency domains. 6.2 Asteroid groups in proper frequencies domains Asteroids close in the frequency domain may either be members of a collisional group that drifted inside a secular resonance, or objects trapped by the local dynamics in a limited region. Since in this work our goal is to investigate the effect that secular dynamics had on the local asteroid population, we will focus our efforts on studying groups in the frequency domains inside or near non-linear secular resonances, by using distance metrics appropriate for the case of each secular resonance of interest (Carruba & Michtchenko 2007, 2009). In particular, for groups in g + s, andg + 2s-type where h 0 is a normalization factor with dimensions of 1 arcsec 1, and h 1 = h 2 = h 3 = 1. We should emphasize that the objective of looking for cluster in the frequency space is not to try to reconstruct the original ejection velocity field, but to look for objects that were either members of a family that drifted into a non-linear secular resonance or for groups of asteroids that are inside such resonances. Traditional HCM in proper element domain may not recognize these bodies as members of a family. For instance, in the regions of the Eos and the Vesta families former family members of the typical K- and V-taxonomic types drifted into such resonances and are no longer recognized as part of the group by the traditional HCM. It is to address such cases that we will look for groups in the appropriate proper frequency domain. Since this work is aimed at analysing the Hygiea family orbital region, we determined frequency groups that had at least one member in the Hygiea proper-element-based family, as computed in Section 6.1. We obtain maximal families in frequency domain, i.e. we computed the families for various cut-offs and kept the group at the cut-off value just before the family was merging into the local background. Results of our analysis are summarized in Table 3. We found a total of 19 frequency groups crossing the Hygiea family, 12 of which associated with (g + s) type of resonances, and seven associated with (g + 2s) resonances. Mostly, this appear as vertical diffusion strips in the (a, sin (i)) plane. A more detailed discussion for (g + s) and(g + 2s) resonance-type groups is given in the next sections (g + s) groups 47 Hygiea proper-element-based family members are 2ν 6 ν 5 + ν 16 likely resonators, and 24 are possibly inside the 2ν 6 ν 5 + ν 17 (g + s)-type resonance. We identified seven groups with members in the 2ν 6 ν 5 + ν 16 resonance and four with 2ν 6 ν 5 + ν 17 likely resonators. Fig. 10 displays an (a, sin (i)) projection of the frequency groups obtained with the distance metric defined by equation (4) (the other symbols are the same as in Fig. 1). The largest groups that we found were those associated with (104) Klymene, with 218 members, and with (538) Friederike, with 253 members. One can notice that proper frequency groups in the (n, g, g + s) domain follow vertically inclined patterns in the (a, sin (i)) space, compatible with the location of likely resonators, as found in Section 3. As often, only a very limited number of group members had taxonomic information available. We only found one C-type asteroid in the (621) group (621 Werdandi itself) and three asteroids (two C-types, 104 Klymene and 6297 (1988 VZ1) and an X-type, 517 Edith) in the (104) group, which confirms that the Hygiea region is dominated by asteroids in the CX-complex. More information was available in the SDSS-MOC 4 data, and this seems

9 An analysis of the Hygiea family region 3565 Table 3. Proper-frequency-based families and clumps in the region of the Hygiea group. We report the kind of resonance type associated with the frequency group, the lowest numbered member identification of each newly found cluster, the frequency cut-off at which the group was determined (f cut-off ), the number of members of the group (N), the number of members with known spectral types (N spec ) and the number of members with SDSS-MOC 4 data (N SDSS-MOC 4 ) that belong to either the CX- or S-complexes, as defined in Section 4. Resonance Group f cut-off N N spec N SDSS-MOC 4 CX N SDSS-MOC 4 S type and Id. (arcsec yr 1 ) name (g + s) 2ν 6 ν 5 + ν 16 (621) ν 6 ν 5 + ν 16 (2165) ν 6 ν 5 + ν 16 (3898) ν 6 ν 5 + ν 16 (5155) ν 6 ν 5 + ν 16 (12758) ν 6 ν 5 + ν 16 (28010) ν 6 ν 5 + ν 16 (35148) ν 6 ν 5 + ν 17 (104) ν 6 ν 5 + ν 17 (538) ν 6 ν 5 + ν 17 (2774) ν 6 ν 5 + ν 17 (20402) (g + 2s) ν 5 + 2ν 16 (90) ν 5 + 2ν 16 (1691) ν 5 + 2ν 16 (3132) ν 5 + 2ν 16 (4499) ν 5 + 2ν 16 (14796) ν 6 + 2ν 16 (637) Figure 10. PanelA:an(a, sini) projection of (g + s) groups in the area of the Hygiea family, associated with the 2ν 6 ν 5 + ν 16 resonance. Panel B: the same projection for (g + 2s) groups associated with the 2ν 6 ν 5 + ν 16 resonance. to confirm the results of the taxonomic analysis: with the possible exception of the (2774) and (20402) groups, all other groups are dominated by objects in the CX-complex, as defined in Section 4, with only a handful of bodies with possible S-complex classification (g + 2s) groups There are a total of 19 asteroids that are both members of the Hygiea family in proper element domain and likely resonators in the ν 5 + 2ν 16 resonance, and six family members that are possible ν 6 + 2ν 16 resonators. Fig. 11 displays the orbital location of the six groups that we found in the (a, sin (i)) plane. We found only one group associated with the ν 6 + 2ν 16 secular resonance, and, of the five clusters associated with the ν 5 + 2ν 16 resonance, two fairly large groups around (90) Antiope (with 493 members) and around (3132) Landgraf (with 109 members). As for the (g + s) groups, only a few objects had taxonomic information available, and they were all C-types: (90) Antiope in the (90) group, (1691) Oort and (3615) Safronov in the (1691) cluster

10 3566 V. Carruba Figure 11. An (a, sini) projection of (g + s) groups in the area of the Hygiea family. and (4915) Solzhenitsyn in the (3132) group. The SDSS-MOC 4 data confirmed what observed for (g + s) groups: with the possible exception of the (3132) group, again most of the asteroids with SDSS-MOC 4 data are compatible with a CX-complex taxonomy. The possible importance that this may have on the scenarios for the formation of the family halo will be discussed in the next section. 7 HYGIEA FAMILY HALO Since the work of Parker et al. (2008) it is known that large families are not limited to what found by the HCM, that seems to be biased to find compact, relatively young clusters, but there exists an extended population of objects with similar taxonomy and geometric albedo, that can extend to much larger regions in proper elements and frequencies domains: the family halo. Recently Brož & Morbidelli (2013) used numerical simulation to obtain an estimate of the Eos family and halo age, based on a determination of the family halo obtained using SDSS-MOC 4 data. Determining a good estimate of the possible orbital extension of the Hygiea family halo is therefore quite important, if one is interested in determining its age and, possibly, the original ejection velocity field. In this work we will try to make best use of all the new data on taxonomy (SDSS-MOC 4) and geometric albedo (WISE and NEOWISE) that is currently available to try to find the most possibly accurate determination of the Hygiea family halo. For this purpose we first selected all asteroids in the Hygiea region and we determined what number of these objects also have SDSS- MOC 4 and WISE albedo data. We found a total of 2188 asteroids that satisfied this criterion, 465 of which were members of the HCM halo. Because of the fact that most of the secular resonances in the region have a vertical structure in the (a, sin (i)) plane, that crosses the Hygiea, the Themis and the Veritas families, frequency family haloes in this region of the asteroid belt tend to very quickly merge with the local background. We thus decided to concentrate our attention on haloes in proper element domains, and, to include the information on SDSS-MOC 4 taxonomy and geometric albedo p V. We defined a distance metrics in a multidomain space as d md = d 2 + C PCV [( P C 1 ) 2 + ( P C 2 ) 2 + ( p V ) 2 ], (6) where, following the approach of Bus & Binzel (2002a,b) for a similar distance metric of proper elements and SDSS-MOC principal components (see also Carruba & Michtchenko 2007), C PCV Figure 12. An (a, sin(i)) projection of asteroids families haloes obtained with the new multidomain hierarchical clustering method in the Hygiea family orbital region. is a weighting factor equal to 10 6 (other choices in a range between 10 4 and 10 8 have been tested without significantly changing the robustness of the results), and d is the standard distance metrics in proper element domain given by equation (3). As first halo members, we selected asteroids that belong to the asteroids family, whose spectral type is compatible with that of the other members according to Mothé-Diniz et al. (2005), and that, of course, also have SDSS-MOC 4 and WISE/NEOWISE data. We then obtained dynamical groups using equation (6), for a value of cut-off d md just less than the value for which the family halo merges with the local background (and other families in the region). An advantage of this method is that it should automatically select asteroids close in proper element, SDSS-MOC 4 and WISE albedos, so reducing the number of interlopers usually found in dynamical group encountered in proper elements (or frequencies) domains only. Fig. 12 displays an (a, sin (i)) projection of the family haloes obtained with this new method. One can notice the separation between the halo of the Themis and Hygiea families, which, despite being of similar taxonomic type, are discerned as separated by this procedure. The limited number of objects in the Veritas and Eos haloes is caused by the fact that our sample of proper elements did not reach to the core regions of these two families (our goal in this paper was, after all, to study the Hygiea family). To check for the possible presence of interlopers in the families haloes obtained with equation (6), we plotted the values of (PC 1, PC 2 ) for the families halo of Hygiea (results are similar for other C-complex families such as Themis and Veritas) and Eos in Fig. 13, panel A. We find that only 14 out of 444 Hygiea halo members, i.e. about 3.15 per cent of the total, are possible S-complex asteroids. This is less than the 10 per cent of interlopers expected in dynamical families, according to statistical considerations (Migliorini et al. 1995), and much less than the 15 per cent of likely interlopers found with traditional HCM for the Hygiea halo. This method also produces somewhat robust results in albedo distribution. Fig. 13, panel B, displays histograms of number frequency values n i /N Tot as a function of geometric albedo p V for the members of the Hygiea and Eos haloes. We obtained mean values of albedos for the Hygiea halo of 0.068, and of for the Eos halo, compatible with C-complex and S-complex families, respectively. Our results are summarized in Table 4. In the next section we will investigate the cumulative size distributions of groups in the Hygiea area.

11 An analysis of the Hygiea family region 3567 Figure 13. Panel A: a projection in the (PC 1, PC 2 ) plane of members of the haloes of the Hygiea (blue crosses) and Eos families found with equation (6). Panel B: a histogram of number frequency values n i /N Tot as a function of geometric albedo p V for the members of the Hygiea (blue line) and Eos (red line) haloes. Table 4. Asteroids families haloes in the orbital region of the Hygiea group. We report the identification of the first body used to determine the family halo, the value of the cut-off d md (in m s 1 ) used to obtain the halo, the number of members of the halo and the number of possible interlopers, according to SDSS-MOC 4 taxonomy, that were found with this approach. First halo d md cut-off value Number of Number of likely member (m s 1 ) members interlopers Hygiea (2436) Themis (981) Veritas (5592) Eos (28916) CUMULATIVE SIZE DISTRIBUTIONS The size distribution of asteroids is one of the most significant observational constraints on their history, and it is also one of the hardest quantities to determine because of strong selection effects (Parker et al. 2008). As a next step of our preliminary analysis of the Hygiea region, we compute the cumulative H distribution N(<H) (i.e. the number of objects in a group with an absolute magnitude less than a given value) for families cores and haloes in the region. Here we limit our analysis only to families (and family haloes) in order to have statistically significant samples for the cumulative distribution. This analysis is also limited to the numbered asteroids only, since the multi-opposition ones do not have absolute magnitude data yet. As it is the case for several other families studied by Parker et al. (2008), family cumulative distributions for the groups in the region seem to be best approximated by a broken power law, for the two intervals in H between 12 and 14 and between 14 and 15. To obtain information on the collisional evolution of the newly found groups, we computed the exponents γ 1 and γ 2 that best fits the cumulative distributions in these two intervals, defined as the angular coefficients of the lines that best fitted the cumulative distribution in the (H,log 10 (N)) plane, in the two H intervals. Since in this work we are interested in the Hygiea family and its halo, we also computed the exponents γ i (i = 1, 2) for the Hygiea (and other families in the region) haloes, as obtained with the standard Figure 14. The cumulative distribution N(>H) of members of the Hygiea family core, dynamical halo (d) and SDSS-MOC 4 and WISE albedo corrected halo (dm). distance metric given by equation (3), and as obtained with our newly introduced method in multidomain spaces given by equation (6). Fig. 14 shows the cumulative H distribution N(<H) for members of the Hygiea family core (blue line), of the Hygiea family halo obtained in the space of proper element only (green line), and of the Hygiea family halo obtained in the multidomain space (magenta line). The vertical blue lines displays the limits of absolute magnitude for obtaining the γ i exponents as in Parker et al. (2008), and the inclined red lines are the lines that best fitted the cumulative distributions in the two H intervals. Our results are summarized in Table 5. We found a value of γ 1 for the Hygiea family core of that may indicate a relatively young age for the Hygiea family (background asteroids usually have values of γ 1 of 0.61). The Hygiea halo obtained in proper elements domain only has a value of γ 1 of 0.568, quite close to the value of background asteroids, which may