An anisotropic distribution of spin vectors in asteroid families

An anisotropic distribution of spin vectors in asteroid families

J. Hanuš    M. Brož Astronomical Institute, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovičkách 2, 18000 Prague, Czech Republic
hanus.home@gmail.com
   J. Ďurech Astronomical Institute, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovičkách 2, 18000 Prague, Czech Republic
hanus.home@gmail.com
   B. D. Warner Palmer Divide Observatory, 17995 Bakers Farm Rd., Colorado Springs, CO 80908, USA    J. Brinsfield Via Capote Observatory, Thousand Oaks, CA 91320, USA    R. Durkee Shed of Science Observatory, 5213 Washburn Ave. S, Minneapolis, MN 55410, USA    D. Higgins Hunters Hill Observatory, 7 Mawalan Street, Ngunnawal ACT 2913, Australia    R. A. Koff 980 Antelope Drive West, Bennett, CO 80102, USA    J. Oey Kingsgrove, NSW, Australia    F. Pilcher 4438 Organ Mesa Loop, Las Cruces, NM 88011, USA    R. Stephens Center for Solar System Studies, 9302 Pittsburgh Ave, Suite 105, Rancho Cucamonga, CA 91730, USA    L. P. Strabla Observatory of Bassano Bresciano, via San Michele 4, Bassano Bresciano (BS), Italy    Q. Ulisse Observatory of Bassano Bresciano, via San Michele 4, Bassano Bresciano (BS), Italy    R. Girelli Observatory of Bassano Bresciano, via San Michele 4, Bassano Bresciano (BS), Italy
Received x-x-2013 / Accepted x-x-2013
Key Words.:
minor planets, asteroids: general, technique: photometric, methods: numerical
Abstract

Context:Current amount of 500 asteroid models derived from the disk-integrated photometry by the lightcurve inversion method allows us to study not only the spin-vector properties of the whole population of MBAs, but also of several individual collisional families.

Aims:We create a data set of 152 asteroids that were identified by the HCM method as members of ten collisional families, among them are 31 newly derived unique models and 24 new models with well-constrained pole-ecliptic latitudes of the spin axes. The remaining models are adopted from the DAMIT database or the literature.

Methods:We revise the preliminary family membership identification by the HCM method according to several additional criteria – taxonomic type, color, albedo, maximum Yarkovsky semi-major axis drift and the consistency with the size-frequency distribution of each family, and consequently we remove interlopers. We then present the spin-vector distributions for asteroidal families Flora, Koronis, Eos, Eunomia, Phocaea, Themis, Maria and Alauda. We use a combined orbital- and spin-evolution model to explain the observed spin-vector properties of objects among collisional families.

Results:In general, we observe for studied families similar trends in the (, ) space (proper semi-major axis vs. ecliptic latitude of the spin axis): (i) larger asteroids are situated in the proximity of the center of the family; (ii) asteroids with are usually found to the right from the family center; (iii) on the other hand, asteroids with to the left from the center; (iv) majority of asteroids have large pole-ecliptic latitudes (); and finally (v) some families have a statistically significant excess of asteroids with or . Our numerical simulation of the long-term evolution of a collisional family is capable of reproducing well the observed spin-vector properties. Using this simulation, we also independently constrain the age of families Flora (1.00.5 Gyr) and Koronis (2.5–4 Gyr).

Conclusions:

1 Introduction

An analysis of rotational state solutions for main belt asteroids was performed by many authors. All the authors observed the deficiency of poles close to the ecliptic plane (e.g., Magnusson 1986; Drummond et al. 1988; Pravec et al. 2002; Skoglöv & Erikson 2002; Kryszczyńska et al. 2007). Hanuš et al. (2011) showed that this depopulation of spin vectors concerns mainly smaller asteroids ( km), while the larger asteroids ( 130–150 km, Kryszczyńska et al. 2007; Paolicchi & Kryszczyńska 2012) have a statistically significant excess of prograde rotators, but no evident lack of poles close to the ecliptic plane. The observed anisotropy of pole vectors of smaller asteroids is now believed to be a result of YORP thermal torques111Yarkovsky–O’Keefe–Radzievskii–Paddack effect, a torque caused by the recoil force due to anisotropic thermal emission, which can alter both rotational periods and orientation of spin axes, see e.g., Rubincam (2000) and also collisions that systematically evolve the spin axes away from the ecliptic plane, and the prograde excess of larger asteroids as a primordial preference that is in agreement with the theoretical work of Johansen & Lacerda (2010). While the number of asteroids with known rotational states grows, we can study the spin vector distribution not only in the whole MBAs or NEAs populations, but we can also focus on individual groups of asteroids within these populations, particularly on collisional families (i.e., clusters of asteroids with similar proper orbital elements and often spectra that were formed by catastrophic break-ups of parent bodies or cratering events).

The theory of dynamical evolution of asteroid families (e.g., Bottke et al. 2006) suggests that the Yarkovsky222a thermal recoil force affecting rotating asteroids/YORP effects change orbital parameters of smaller asteroids (30–50 km) – the semi-major axis of prograde rotators is slowly growing in the course of time, contrary to retrograde rotators which semi-major axis is decreasing. This phenomenon is particularly visible when we plot the dependence of the absolute magnitude on the proper semi-major axis (see an example of such plot for Themis family in Figure 1, left panel). In addition, various resonances (e.g., mean-motion resonances with Jupiter or Mars, or secular resonances) can intersect the family and cause a decrease of the number of asteroids in the family by inducing moderate oscillations to their orbital elements(Bottke et al. 2001) as can be seen in Figure 1 for the Flora family, where the secular resonance with Saturn almost completely eliminated objects to the left from the center of the family (the resonance has its center at 2.13 AU for objects with , which is typical for Flora family members, it evolves objects that come to the proximity of the resonance). Some resonances can, for example, capture some asteroids on particular semi-major axes (Nesvorný & Morbidelli 1998).

Laboratory experiments strongly suggest that a collisionally-born cluster should initially have a rotational frequency distribution close to Maxwellian (Giblin et al. 1998) and an isotropic spin vector distribution.

Figure 1: Dependence of the absolute magnitude on the proper semi-major axis for the Themis family (left) and for the Flora family (right) with the likely positions of the family centers (vertical lines). We also plot three (, ) borders of the family for different parameters (different values correspond to a different initial extent of the family or different age and magnitude of the Yarkovsky semi-major axis drift) by gray lines, the optimal border corresponds to the middle line. The vertical dotted line represents the approximate position of the secular resonance for the inclination typical for Flora family members and the horizontal arrow its approximate range.

For several families, we already know their age estimates (e.g.,  Gyr for Koronis family, Bottke et al. 2001), and so we have a constraint on the time, for which the family was evolving towards its current state. As was shown in Bottke et al. (2001), the family evolution is dominated by Yarkovsky and YORP effects, and also collisions and spin-orbital resonances. The knowledge of the age should constrain some free parameters in various evolutionary models.

The spin-vector properties in an asteroid family were first studied by Slivan (2002) and Slivan et al. (2003), who revealed an anisotropy of spin vectors for ten members of the Koronis family. This was an unexpected result because collisionally-born population should have an isotropic spin-vector distribution. The peculiar spin-vector alignment in the Koronis family was explained by Vokrouhlický et al. (2003) as a result of the YORP torques and spin-orbital resonances that modified the spin states over the time span of 2–3 Gyr. The secular spin-orbital resonance with Saturn may affect the Koronis family members, according to the numerical simulations, it can (i) capture some objects and create a population of prograde rotators with periods  h, similar obliquities ( to ) and also with similar ecliptic longitudes in the ranges of ( to ) and ( to ); or (ii) create a group of low-obliquity retrograde rotators with rotational periods  h or  h. The prograde rotators trapped in the spin-orbital resonance were referred by Vokrouhlický et al. (2003) as being in Slivan states. Most members of the Koronis family with known rotational states (determined by the lightcurve inversion by Slivan et al. 2003, 2009; Hanuš et al. 2011, 2013) had the expected properties except the periods of observed prograde rotators were shifted to higher values of 7–10 h. Rotational states of asteroids that did not match the properties of the two groups were probably reorientated by recent collisions, which are statistically plausible during the family existence for at least a few Koronis members (e.g., asteroid (832) Karin was affected by a collision when a small and young collisional family within the Koronis family was born Slivan & Molnar 2012).

Another study of rotational states in an asteroid family was performed by Kryszczyńska (2013), who focused on the Flora family. She distinguished prograde and retrograde groups of asteroids and reported an excess of prograde rotators. This splitting into two groups is likely caused by the Yarkovsky effect, while the prograde excess by the secular resonance that significantly depopulates the retrograde part of the family (see Figure 1b, only retrograde rotators can drift via the Yarkovsky/YORP effects towards the resonance).

Further studies of rotational properties of collisional families should reveal the influence of the Yarkovsky and YORP effects, and possibly a capture of asteroids in spin-orbital resonances similar to the case of the Koronis family. The Yarkovsky effect should be responsible for spreading the family in a semi-major axis (retrograde rotators drift from their original positions towards the Sun, on the other hand, prograde rotators drift away from the Sun, i.e. towards larger ’s), and the YORP effect should eliminate the spin vectors close to the ecliptic plane.

Disk-integrated photometric observations of asteroids contain information about object’s physical parameters, such as the shape, the sidereal rotational period and the orientation of the spin axis. Photometry acquired at different viewing geometries and apparitions can be used in many cases in a lightcurve inversion method (e.g., Kaasalainen & Torppa 2001; Kaasalainen et al. 2001) and a convex 3D shape model including its rotational state can be derived. This inverse method uses all available photometric data, both the classical dense-in-time lightcurves or the sparse-in-time data from astrometric surveys. Most of the asteroid models derived by this technique are publicly available in the Database of Asteroid Models from Inversion Techniques (DAMIT333http://astro.troja.mff.cuni.cz/projects/asteroids3D, Ďurech et al. 2010). In February 2013, models of 347 asteroids were included there. About a third of them can be identified as members of various asteroid families. This high number of models of asteroids that belong to asteroid families allows us to investigate the spin-vector properties in at least several families with the largest amount of identified members. Comparison between the observed and synthetic (according to a combined orbital- and spin-evolution model) spin-vector properties could even lead to independent family age estimates.

The paper is organized as follows: in Section 2, we investigate the family membership of all asteroids for which we have their models derived by the lightcurve inversion method and present 31 new asteroid models that belong to ten asteroid families. An analysis of spin states within these asteroid families with at least three identified members with known shape models is presented in Section 3.1. A combined spin-orbital model for the long-term evolution of a collisional family is described in Section 4, where we also compare the synthetic and observed spin-vector properties and constrain ages of families Flora and Koronis.

2 Determination of family members

2.1 Methods for family membership determination

For a preliminary family membership determination, we adopted an on-line catalog published by Nesvorný (2012) who used the Hierarchical Clustering Method444In this method, mutual distances in proper semi-major axis (), proper eccentricity (), and proper inclination () space are computed. The members of the family are then separated in the proper element space by less than a selected distance (usually, it has a unit of velocity), a free parameter often denoted as “cutoff velocity“. (HCM, Zappalà et al. 1990, 1994). Nesvorný (2012) used two different types of proper elements for the family membership identification: semi-analytic and synthetic. The more reliable dataset is the one derived from synthetic proper elements, which were computed numerically using a more complete dynamical model. Majority of asteroids is present in both datasets. A few asteroids that are only in one of the datasets are included in the study as well (e.g., asteroids (390) Alma in the Eunomia family or (19848) Yeungchuchiu in the Eos family), because at this stage it is not necessary to remove objects that still could be real family members.

The HCM method selects a group of objects that are separated in the proper element space by less than a selected distance. However, not all of these objects are actually real members of the collisionally-born asteroid family. A fraction of objects has orbital elements similar to typical elements of the asteroid family members only by a coincidence, the so-called interlopers. Interlopers can be identified (and removed), for example, by:

  • inspection of reflectance spectra, because they are usually of different taxonomic types than that of the family members, we use the SMASSII (Bus & Binzel 2002) or Tholen taxonomy (Tholen 1984, 1989);

  • inspection of colors based on the Sloan Digital Sky Survey Moving Object Catalog 4 (SDSS MOC4, Parker et al. 2008), we used the color indexes and , which usually well define the core of the family (see examples for Themis and Eunomia families in Figure 2), for each asteroid with available color indexes, we compared values and to those that define the family;

  • inspection of albedos based on the WISE data (Masiero et al. 2011);

  • constructing a diagram of the proper semi-major axis vs. the absolute magnitude (see Figure 1), estimating the V-shape defined by the Yarkovsky semi-major axis drift and excluding outliers, i.e. relatively large asteroids outside the V-shape (see Vokrouhlický et al. 2006b, for the case of Eos family). We refer here the (, ) border of the family as the border of the V-shape;

  • constructing a size-frequency distribution (SFD) of the cluster, some asteroids can be too large to be created within the family and thus are believed to be interlopers (see, e.g., numerical simulations by Michel et al. 2011, who excluded the asteroid (490) Veritas from the Veritas family).

Figure 2: Dependence of the color indexes and (from the Sloan Digital Sky Survey Moving Object Catalog 4) for a C-type family Themis and S-type family Eunomia. The family corresponds to a compact structure in this parameter space marked by a rectangle. Note a qualitative difference between C- and S-types asteroids.

These methods for family membership determination have one common characteristic – we have to determine or choose a range for a quantity that defines the family members (range of spectra, sizes, or distance from the family center), which affects the number of objects we include in the family. Our criteria correspond to the fact that usually 99% of the objects are within the ranges.

2.2 New asteroid models

From the DAMIT database, we adopt 96 models of asteroids that are, according to the HCM method, members of collisional families.

Currently, we have about 100 new asteroid models that have not yet been published. Here, we present new physical models of 31 asteroids from this sample that are identified as members of asteroid families by the HCM method (we choose only asteroids that belong to ten specific families for which we expect a reasonable amount of members, i.e. at least three). These convex shape models are derived by the lightcurve inversion method from combined dense and sparse photometry. The derivation process is similar to the one used in Hanuš et al. (2013). The dense photometry was from two main sources: (i) the Uppsala Asteroid Photometric Catalogue (UAPC555http://asteroid.astro.helsinki.fi/, Lagerkvist et al. 1987; Piironen et al. 2001), where lightcurves for about 1 000 asteroids are stored, and (ii) the data from a group of individual observers provided by the Minor Planet Center in the Asteroid Lightcurve Data Exchange Format (ALCDEF666http://www.minorplanet.info/alcdef.html, Warner et al. 2009). The sparse-in-time photometry is downloaded from the AstDyS site (Asteroids – Dynamic Site777http://hamilton.dm.unipi.it/). We use data from the three most accurate observatories: USNO–Flagstaff station (IAU code 689), Roque de los Muchachos Observatory, La Palma (IAU code 950), and Catalina Sky Survey Observatory (CSS for short, IAU code 703, Larson et al. 2003).

To increase the number of asteroid models for our study of asteroid families, we perform additional analysis of our previous results of the lightcurve inversion. For many asteroids, we are able to determine a unique rotational period, but get multiple pole solutions (typically 3–5) with similar ecliptic latitudes , which is an important parameter. In Hanuš et al. (2011), we presented a reliability test, where we checked the physicality of derived solutions by the lightcurve inversion (i.e., if the shape model rotated around its axis with a maximum momentum of inertia). By computing models for all possible pole solutions and by checking their physicality, we remove the pole ambiguity for several asteroids, and thus determine their unique solutions (listed in Table 1). For other asteroids, the pole ambiguity remain and the models give us accurate period values and also rough estimates of ecliptic latitudes (if the biggest difference in latitudes of the models is ). We call these models partial and present them in Table 2. For the ecliptic latitude , we use the mean value of all different models. We define parameter as being the estimated uncertainty of , where and are the extremal values within all . The threshold for partial models is . We present 31 new models and 24 partial models. References to the dense lightcurves used for the model determination are listed in Table 3. In Section 4, we compare the numbers of asteroids in four quadrants of the (, ) diagram (defined by the center of the family and the value ) with the same quantities based on the synthetic family population. The uncertainties in are rarely larger than 20, and the assignment to a specific quadrant is usually not questionable (only in 4 cases out of 136 the uncertainty interval lies in both quadrants, most of the asteroids have latitudes ), and thus give us useful information about the rotational properties in asteroid families. Partial models represent about 20% of our sample of asteroid models.

The typical error for the orientation of the pole is (5–10)/ in longitude and 5–20 in latitude (both uncertainties depend on the amount, timespan and quality of used photometry). Models based purely on dense photometry are typically derived from a large number (30–50) of individual dense lightcurves observed during 5–10 apparitions, and thus the uncertainties of parameters of the rotational state correspond to lower values of the aforementioned range. On the other hand, models based on combined sparse-in-time data have, due to the poor photometric quality of the sparse data, the uncertainties larger (corresponding to the upper bound of the aforementioned range).

Models of asteroids (281) Lucretia and (1188) Gothlandia published by Hanuš et al. (2013) were recently determined also by Kryszczyńska (2013) from partly different photometric data sets. Parameters of the rotational state for both models agree within their uncertainties.

The spin vector solution of asteroid (951) Gaspra based on Galileo images obtained during the October 1991 flyby was already published by Davies et al. (1994b). Similarly, the solution of a Koronis-family member (243) Ida based on Galileo images and photometric data was previously derived by Davies et al. (1994a) and Binzel et al. (1993). Here we present convex shape models for both these asteroids. Our derived pole orientations agree within only a few degrees with the previously published values (see Table 5), which again demonstrates the reliability of the lightcurve inversion method.

2.3 Family members and interlopers

We revise the family membership assignment by the HCM method according to the above-described criteria for interlopers or borderline cases. Interlopers are asteroids which do not clearly belong to the family, for example, they have different taxonomic types, incompatible albedos or are far from the (, ) border. On the other hand, borderline cases cannot be directly excluded from the family, their physical or orbital properties are just not typical in the context of other members (higher/lower albedos, close to the (, ) border). These asteroids are possible family members, but can be easily interlopers as well. In Table 5, last but one column, we show our revised membership classification of each object (M is a member, I an interloper and B a borderline case), the table also gives the rotational state of the asteroid (the ecliptic coordinates of the pole orientation and and the period ), the semi-major axis , the diameter and the albedo from WISE (Masiero et al. 2011), the SMASS II (Bus & Binzel 2002) and Tholen taxonomic types (Tholen 1984, 1989), and the reference to the model).

Although we got for Vesta and Nysa/Polana families several members by the HCM method, we excluded these two families from our further study of spin states. Vesta family was created by a cratering event, and thus majority of fragments are rather small and beyond the capabilities of the model determination. Most of the models we currently have (recognized by the HCM method) are not compatible with the SFD of the Vesta family and thus are interlopers. On the other hand, Nysa/Polana family is a complex of two families (of different age and composition), thus should be treated individually. Additionally, we have only five member candidates for the whole complex, so even if we assign them to the subfamilies, the numbers would be too low to make any valid conclusions.

In Table 4, we list asteroids for which the HCM method suggested a membership to families Flora, Koronis, Eos, Eunomia, Phocaea and Alauda, but using the additional methods for the family membership determination described above, we identified them as interlopers or borderline cases.

In Figure 3, we show the (, ) diagrams for all eight studied families. We plot the adopted (, ) border (from Brož et al. 2013) and label the members, borderline cases and interlopers by different colors.

Several asteroids in our sample belong to smaller and younger sub-clusters within the studied families (e.g, (832) Karin in the Koronis family, (1270) Datura in the Flora family or (2384) Schulhof in the Eunomia family). These sub-clusters were likely created by secondary collisions. As a result, the spin states of asteroids in these sub-clusters were randomly reoriented. Because our combined orbital- and spin-evolution model (see Section 4) includes secondary collisions (reorientations), using asteroids from sub-clusters in the study of the spin-vector distribution is thus essential: asteroids from sub-clusters correspond to reoriented asteroids in our synthetic population.

Figure 3: Dependence of the absolute magnitude on the proper semi-major axis for the eight studies families: Flora, Koronis, Eos, Eunomia, Phocaea, Themis, Maria and Alauda with the likely positions of the family centers (vertical lines). We also plot the possible range of the (, ) borders (two thick lines) of each family for values of the parameter from Brož et al. (2013) (different values correspond to a different initial extent of the family or different age and magnitude of the Yarkovsky semi-major axis drift.). The pink triangles represent the members from our sample (M), green circles borderline cases (B) and blue circles interlopers (I). Note that borderline cases and interlopers are identified by several methods including the position in the (, ) diagram, and thus could also lie close to the center of the family (e.g., in the case of the Flora family).

3 Observed spin vectors in families

Figure 4: Dependence of the pole latitude on the proper semi-major axis for eight studied asteroid families: Flora, Koronis, Eos, Eunomia, Phocaea, Themis, Maria and Alauda. Family members are marked by circles and borderline cases by squares, which sizes are scaled proportionally to diameters (only the scale for (15) Eunomia was decreased by half to fit the figure). The vertical lines correspond to the likely centers of the asteroid families, which uncertainties are usually AU. The Eos family has an asymmetric V-shape (the (, ) border is asymmetric) which makes the center determination harder, so we marked two possible positions (one corresponds to the right (, ) border, the second to the left border). The uncertainties in are usually 5–20. In most cases, the value of and thus the quadrant to which the asteroid belongs (defined by the center of the family and the value ) is not changed.

There are eight asteroid families for which we find at least three members (together with borderline cases) in our data set of asteroid models (after the family membership revision, labeled by M or B in the last column of Table 5) – Flora (38 members), Koronis (23), Eos (16), Eunomia (14), Phocaea (11), Themis (9), Maria (9), and Alauda (3) families. Having the models and membership, we now can proceed to the discussion of the spin states in families in general (Section 3.1), and for families Flora and Koronis (Sections 3.23.3).

3.1 Spin-vector orientations in individual families

In Figure 4, we show the dependence of asteroid’s pole latitudes in ecliptic coordinates (if there are two possible pole solutions for an asteroid, we take the first one in Table 1, because it corresponds to a formally better solution, additionally, latitudes for both ambiguous models are usually similar) on the semi-major axes. We mark the family members by circles and borderline cases by squares, which sizes are scaled proportionally to diameters to show also the dependence on the diameter. Vertical lines in Figure 4 correspond to the likely centers of the asteroid families, which we determine by constructing the V-shaped envelope of each family (we use all members of each family assigned by the HCM method, see Figure 1 and Figure 3). The Eos family has an asymmetric V-shape (the (, ) diagram), we compute centers for both wings of the V-shape individually. For the Flora family, we use only the right wing of the V-shape to derive the center, while the left one is strongly affected by the secular resonance.

In the study of spin-vector properties in families, we simply use the ecliptic coordinates for the pole orientation: ecliptic longitude and latitude . A formally better approach would be to use the coordinates bound to the orbital plane of the asteroid: orbital longitude and latitude . The orbital latitude can be then easily transformed to obliquity, which directly tells us, if the asteroid rotates in a prograde or retrograde sense. However, due to several reasons, we prefer the ecliptic coordinates: (i) most of the asteroids have low inclinations and thus the difference between their ecliptic and orbital latitudes are only few degrees, the maximum differences for the families with higher inclination (Eos, Eunomia, Phocaea, Maria) are 20–30; (ii) the orbital coordinates of the pole direction cannot be computed for partial models, because we do not know the ecliptic longitude, these models represent about 20% of our studied sample; (iii) the positions of the asteroids in the (, ) diagrams (i.e., to which quadrant they belong), namely if they have or are the sufficient information. Because most of the asteroids have latitudes larger than 30, their positions in the (, ) are similar (this is not true only for three asteroids out of 136); and (iv) we compare the (, ) diagrams (numbers of objects in the quadrants) between the observed and synthetic populations for ecliptic latitudes, so the consistency is assured.

In general, we observe for all studied families similar trends: (i) larger asteroids are situated in the proximity of the center of the family; (ii) asteroids with are usually found to the right from the family center; (iii) on the other hand, asteroids with to the left from the center; (iv) majority of asteroids have large pole-ecliptic latitudes (); and finally (v) some families have a statistically significant excess of asteroids with or .

Case (i) is evident for families Flora, Eunomia, Phocaea, Themis or Maria. We have no large asteroids in the samples for the remaining families.

Cases (ii) and (iii) are present among all families with an exception of Eos, where all the asteroids are close to the (badly constrained) center. This phenomenon can be easily explained by the Yarkovsky drift, which can change asteroid’s semi-major axes , namely it can increase of prograde rotators, and decrease of retrograde once. The magnitude of the Yarkovsky drift is dependent on the asteroid size, is negligible for asteroids with diameters 50 km (the case of Eos), and increases with decreasing diameter. For Flora, Eunomia, Phocaea or Maria family, we can see that the smallest asteroids in the sample ( 5–10 km) can be situated far from the family center, and we can also notice a trend of decreasing size with increasing distance from the center that probably corresponds to the magnitude of the Yarkovsky effect and the initial velocities the objects gained after the break-up.

Observation (iv) is a result of the dynamical evolution of the asteroid’s spin vector orientations dominated by the YORP effect, which increases the absolute value of the pole-ecliptic latitude (see papers Hanuš et al. 2011, 2013, where this effect is numerically investigated and compared with the observed anisotropic spin vector distribution of the sample of 300 MBAs).

Case (v) concerns families Flora, Eunomia, Phocaea, Themis and Maria. The different number of asteroids with and among these families is statistically significant and cannot be coincidental. The obvious choice for an explanation are mean-motion or secular resonances. Indeed, the secular resonance removed many objects with from the Flora family (see Section 3.2 for a more thorough discussion), the 8:3 resonance with Jupiter truncated the Eunomia family, which resulted into the fact that there are no objects with  AU, and similarly, the 3:1 resonance with Jupiter affected the Maria family, for which we do not observe objects with smaller than 2.52 AU. Near the Phocaea family at  AU, the 3:1 resonance with Jupiter is situated. Due to the high inclination of objects in the Phocaea family (), the resonance affects asteroids with  AU, which corresponds to the probable center of the family. The resonance removed a significant number of objects between 2.40 AU and 2.45 AU, and all objects with larger.

The asymmetry of asteroids with and in the Themis family is caused by a selection effect: in the family, there are no objects with absolute magnitude  mag (i.e., large asteroids) and  AU, on the other hand, with  AU, there are more than a hundred of such asteroids (see Figure 1a). Our sample of asteroid models derived by the lightcurve inversion method is dominated by larger asteroids, and it is thus not surprising that we did not derive models for the Themis family asteroids with  AU.

The Flora and Koronis families are interesting also from other aspects, and thus are discussed in more detail in Sections 3.2 and 3.3.

3.2 The Flora family

The Flora cluster is situated in the inner part of the main belt between 2.17–2.40 AU, its left part (with respect to the (, ) diagram) is strongly affected by the secular resonance with Saturn, which is demonstrated in Figure 1b. The probable center of the family matches the position of asteroid (8) Flora at AU. Because of the relative proximity to the Earth, more photometric measurements of smaller asteroids are available than for more distant families, and thus more models were derived. So far, we identified 38 models of asteroids that belong to the Flora family (together with borderline cases).

The majority of asteroids within this family have (68%, due to small inclinations of the family members, majority of the objects with are definitely prograde rotators, because their obliquities are between 0 and 90) and lie to the right from the center of the family, confirming the presence of the Yarkovsky drift. Nine out of twelve asteroids with can be found in Figure 4 near or to the left from the center of the family. The exceptions are the borderline asteroids (1703) Barry and (7360) Moberg, and asteroid (7169) Linda with close to 2.25 AU (see Figure 4). The borderline category already suggests that the two asteroids could be possible interlopers and their rotational state seems to support this statement. However, it is also possible that these asteroids have been reoriented by a non-catastrophic collisions. Rotational state of another borderline asteroid (800) Kressmannia is also not in agreement with the Yarkovsky/YORP predictions, and thus it could be an interloper (or reoriented). The asteroid (7169) Linda classified as member could still be an interloper, which was not detected by our methods for interloper removal, or could be recently reoriented by a non-catastrophic collision (the typical timescale for a reorientation (Farinella et al. 1998, see Eq. 5) of this 4km-sized asteroid with rotational period h is  Myr, which is comparable with the age of the family). The depopulation of poles close to the ecliptic plane is also clearly visible.

The resonance to the left from the center of the family creates an excess of retrograde rotators not only among the family, but also among the whole main belt population if we use the currently available sample of asteroid models (there are 300 asteroid models in DAMIT database, in the Flora family, there are 14 more asteroids with than with (corresponds to the prograde excess), which corresponds to about 6% of the whole sample, this bias needs to be taken into consideration, for example, in the study of rotational properties among MBAs).

The missing asteroids with were delivered by this resonance to the orbits crossing the orbits of terrestrial planets and are responsible, for example, for the retrograde excess of the NEAs (La Spina et al. 2004): the resonance contributes to the NEA population only by retrograde rotators, other major mean-motion resonances, such as the 3:1 resonance with Jupiter, deliver both prograde and retrograde rotators in a similar amount.

We did not observe a prograde group of asteroids with similar pole-ecliptic longitudes in the Flora family (i.e., a direct analog of the Slivan state in the Koronis family) that was proposed by Kryszczyńska (2013). Although Kryszczyńska (2013) claims that Slivan states are likely observed among the Flora family, no corresponding clustering of poles of the prograde rotators is shown, particularly of ecliptic longitudes. We believe that the term Slivan state was used incorrectly there.

3.3 The Koronis family

The Koronis family is located in the middle main belt between 2.83–2.95 AU with the center at  AU. We identified 23 members (together with borderline cases) with determined shape models.

The concept given by the Yarkovsky and YORP predictions work also among the Koronis family (asteroids with lie to the left from the family center, asteroids with to the right, see Figure 4). In addition to that, Slivan (2002) and Slivan et al. (2003) noticed that prograde rotators have also clustered pole longitudes. These asteroids were trapped in a secular spin-orbital resonance and are referred as being in Slivan states (Vokrouhlický et al. 2003). Several asteroids were later recognized as being incompatible with the Slivan states, such as (832) Karin and (263) Dresda by Slivan & Molnar (2012). Asteroid (832) Karin is the largest member of a young (5.8 Myr, Nesvorný & Bottke 2004) collisional family that is confined within the larger Koronis family. The spin state of (832) Karin was thus likely affected during this catastrophic event and changed to a random state. Asteroid (263) Dresda could be randomly reoriented by a non-catastrophic collision that is likely to happen for at least a few of 27 asteroids in the Koronis cluster with known spin state solutions, or its initial rotational state and shape did not allow a capture in the resonance. All four borderline asteroids have rotational states in agreement with the Yarkovsky/YORP concept which may support their membership to the Koronis cluster. On the other hand, rotational states of asteroids (277) Elvira and (321) Florentina do not match the expected values, and thus could be again interlopers or affected by reorientations.

Being trapped in the spin-orbital resonance does not necessarily mean that the asteroid is a member of the Koronis family, it rather indicates that its initial orbital position, the rotational state and the shape were favorable for being trapped in the resonance. For example, asteroids (311) Claudia, (720) Bohlinia, (1835) Gajdariya and (3170) Dzhanibekov have expected rotational states but are either rejected from the Koronis family or classified as borderline cases by our membership revision.

4 Long-term evolution of spin vectors in asteroid families

Here we present a comparison of the observed spin-vector orientations in several asteroid families with a numerical model of the temporal spin-vector evolutions. We use a combined orbital- and spin-evolution model, which was described in detail in Brož et al. (2011). We need to account for the fact that the Yarkovsky semi-major axis drift sensitively depends on the orientation of the spin axis, which is in turn affected by the YORP effect and non-disruptive collisions. This model includes the following processes, which are briefly described in the text: (i) impact disruption; (ii) gravitational perturbations of planets; (iii) the Yarkovsky effect; (iv) the YORP effect; (v) collisions and spin-axis reorientations; and (vi) mass shedding.

Impact disruption

To obtain initial conditions for the family just after the breakup event we use a very simple model of an isotropic ejection of fragments from the work of Farinella et al. (1994). The distribution of velocities "at infinity" follows the function

(1)

with the exponent being a free parameter, a normalization constant and the escape velocity from the parent body, which is determined by its size  and mean density  as The distribution is usually cut at a selected maximum allowed velocity to prevent outliers. The initial velocities of individual bodies are generated by a straightforward Monte–Carlo code and the orientations of the velocity vectors in space are assigned randomly. We also assume that the velocity of fragments is independent of their size.

We must also select initial osculating eccentricity  of the parent body, initial inclination , as well as true anomaly  and argument of perihelion  at the time of impact disruption, which determine the initial shape of the synthetic family just after the disruption of the parent body.

Gravitational perturbations of planets

Orbital integrations are performed using the SWIFT package (Levison & Duncan 1994), slightly modified to include necessary online digital filters and a second-order symplectic integrator (Laskar & Robutel 2001). The second-order symplectic scheme allows us to use a time-step up to .

Our simulations include perturbations by four outer planets, with their masses, initial positions and velocities taken from the JPL DE405 ephemeris (Standish et al. 1997). We modify the initial conditions of the planets and asteroids by a barycentric correction to partially account for the influence of the terrestrial planets. The absence of the terrestrial planets as perturbers is a reasonable approximation in the middle and outer part of the main belt (for orbits with and ).888For the Flora family located in the inner belt we should account for terrestrial planets directly, because of mean-motion resonances with Mars, but we decided not do so, to speed-up the computation. Anyway, the major perturbation we need to account for is the  secular resonance, which is indeed present in our model.

Synthetic proper elements are computed as follows. We first apply a Fourier filter to the (non-singular) orbital elements in a moving window of 0.7 Myr (with steps of 0.1 Myr) to eliminate all periods smaller than some threshold (1.5 kyr in our case); we use a sequence of Kaiser windows as in Quinn et al. (1991).

The filtered signal, mean orbital elements, is then passed through a frequency analysis code adapted from Šidlichovský & Nesvorný (1996) to obtain (planetary) forced and free terms in Fourier representation of the orbital elements. The isolated free terms are what we use as the proper orbital elements.

Yarkovsky effect

Both diurnal and seasonal components of the Yarkovsky accelerations are computed directly in the N-body integrator. We use a theory of Vokrouhlický (1998) and Vokrouhlický & Farinella (1999) for spherical objects (but the magnitude of the acceleration does not differ substantially for non-spherical shapes Vokrouhlický & Farinella 1998). The implementation within the SWIFT integrator was described in detail by Brož (2006).

YORP effect

The evolution of the orientation of the spin axis and of the angular velocity is given by:

(2)
(3)

where - and -functions describing the YORP effect for a set of 200 shapes were calculated numerically by Čapek & Vokrouhlický (2004) with the effective radius , the bulk density , located on a circular orbit with the semi-major axis . We assigned one of the artificial shapes (denoted by the index ) to each individual asteroid from our sample. The - and -functions were then scaled by the factor

(4)

where , , denote the semi-major axis, the radius, and the density of the simulated body, respectively, and is a free scaling parameter reflecting our uncertainty in the shape models and the magnitude of the YORP torque, which dependents on small-sized surface features (even boulders, Statler 2009) and other simplifications in the modeling of the YORP torque. In Hanuš et al. (2013), we constrained this parameter and find to be the optimal value when comparing the results of the simulation with the observed latitude distribution of main belt asteroids. In our simulation, we used this value for .

The differential equations (2), (3) are integrated numerically by a simple Euler integrator. The usual time step is .

Collisions and spin-axis reorientations

We neglect the effect of disruptive collisions because we do not want to loose objects during the simulation, but we include spin axis reorientations caused by collisions. We use an estimate of the time scale by Farinella et al. (1998).

(5)

where , , , and corresponds to period  hours. These values are characteristic for the main belt.

Mass shedding

If the angular velocity approaches a critical value

(6)

we assume a mass shedding event, so we keep the orientation of the spin axis and the sense of rotation, but we reset the orbital period  to a random value from the interval  hours. We also change the assigned shape to a different one, since any change of shape may result in a different YORP effect.

Synthetic Flora, Koronis and Eos families

Figure 5: A simulation of the long-term evolution of the synthetic Flora (top), Koronis (middle) and Eos (bottom) families in the proper semi-major axis  vs. the pole latitude  plane. Left: objects larger than , which almost do not evolve in . Right: objects with , with the initial conditions denoted by empty circles and an evolved state at 1 Gyr denoted by full circles. The sizes of symbols correspond to the actual diameters . The initial conditions for Flora correspond to an isotropic size-independent velocity field with and , and a uniform distribution of poles (i.e. ). We increase the number of objects 10 times compared to the observed members of the Flora (Koronis and Eos as well) family in order to improve statistics. We retain their size distribution, of course. The objects in Flora family are discarded from these plots when they left the family region (eccentricity , inclination ), because they are affected by strong mean-motion or secular resonances ( in this case). Thermal parameters were set as follows: the bulk density , the surface density , the thermal conductivity , the thermal capacity , the Bond albedo and the infrared emissivity . The time step for the orbital integration is and  for the (parallel) spin integration. The parameters for Koronis and Eos are chosen similarly, only for Koronis, we use , and for Eos and .

In Figure 5 (top panel), we show a long-term evolution of the synthetic Flora family in the proper semi-major axis  vs. the pole latitude  plane for objects larger and smaller than . The values of the model parameters are listed in the figure caption. Larger asteroids do not evolve significantly and remain close to their initial positions. On the other hand, smaller asteroids () are strongly affected by the Yarkovsky and YORP effects: They drift in the semi-major axis, differently for prograde and retrograde rotators, and their pole orientations become mostly perpendicular to their orbits (corresponds to the proximity of the ecliptic plane for small inclinations). After the simulation at  Gyr, we observe a deficiency of asteroids with to the left from the family center and a deficiency of asteroids with to the right from the family center.

The asymmetry of the synthetic Flora family with respect to its center (red vertical line in the Figure 5) caused by the secular resonance is obvious. The down-right quadrant (,  AU) still contains for  Gyr many objects, because for some of them the evolution in and is rather small, and other were delivered to this quadrant by collisional reorientations.

The appearance of the evolved proper semi-major axis  vs. the pole latitude  diagrams for Koronis and Eos families are qualitatively similar to the one of the Flora family. Because the asteroid samples for Koronis and Eos families are dominated by intermediate-sized asteroids ( km), the evolution in and is on average slower than in the Flora family. We show the state of the simulation for Koronis family in 4 Gyr and for Eos in 1.5 Gyr (based on the expected ages). The Eos family thus seems less evolved than Koronis family.

We also check the distributions of the proper eccentricities and inclinations of the synthetic Flora/Koronis/Eos objects if they (at least roughly) correspond to the observed family. However, the number of objects to compare is rather low, and seems insufficient for a detailed comparison of distributions in 3D space of proper elements (, , ).

Ages of Flora, Koronis and Eos families

Figure 6: Time evolution of the metric , where correspond to the numbers of synthetic objects in quadrants () that are defined by the center of the family and value , for synthetic Flora, Koronis and Eos families (red lines). The spread corresponds to 100 different selections of objects (we simulate 10 times more objects to reach a better statistics), the upper curve denotes the 90% quantile and the bottom 10%. Thick horizontal line is the observed ratio with the uncertainty interval.

To quantitatively compare the simulation of the long-term evolution of the synthetic families in the proper semi-major axis  vs. the pole latitude  plane with the observation, we construct the following metric: we divide the (, ) plane into four quadrants defined by the center of the family and value and compute the ratio , where correspond to the numbers of synthetic objects in quadrants (). In Figure 6, we show the evolution of the metric during the simulation of families Flora, Koronis and Eos for all synthetic objects with  km, and the value of the same metric for the observed population for comparison.

For the Koronis family (middle panel), the synthetic ratio reaches the observed one after  Gyr and remains similar until the end of the simulation at  Gyr. Bottke et al. (2001) published the age  Gyr for the Koronis family. Unfortunately, we cannot constrain the age of the Eos family from this simulation due to objects with the relatively small evolution in and . The fit for the Flora family is not ideal, the reason could be differences in the initial velocity field or the true anomaly of the impact. The best agreement is for the age  Gyr, which is approximately in agreement with the dynamical age in Nesvorný et al. (2005):   Gyr.

5 Conclusions

We identify 152 asteroids, for which we have convex shape models and simultaneously, the HCM method identifies them as members of ten collisional families. Due to a large number of expected interlopers in families Vesta and Nysa/Polana, we exclude these families from the study of the rotational properties. In the remaining sample of asteroids from eight families, we identify % of objects that are interlopers or borderline cases (see Table 4). We use several methods, described in Section 2.1, for their identification. The borderline cases are still possible members of the families and thus are included in our study of the spin-vector distribution.

From the dependence of the asteroid’s pole latitudes on the semi-major axes, plotted in Figure 4, we can see fingerprints of families spreading in and spin axis evolution due to Yarkovsky and YORP effects: Asteroids with lie on the left side from the center of the family, and on the other hand, asteroids with on the right side. The asymmetry with respect to the family centers is in most cases caused by various resonances that cut the families, in the case of Themis family, a selection effect is responsible.

However, we do not observe a perfect agreement with the Yarkovsky and YORP effects predictions. A few (eight) individual objects that have incompatible rotational states could: (i) be incorrectly determined; (ii) be interlopers; (iii) have initial rotational states that cause only a small evolution in the (, ) space (i.e., they are close to their initial positions after the break-up); or (iv) be recently reoriented by collisional events.

In the case of the Flora family, significantly less asteroids with () than with () are present. The secular resonance is responsible for this strong deficit, because objects with are drifting towards this resonance and are subsequently removed from the family (they become part of the NEAs population where they create an excess of retrograde rotators).

We do not find an analog of the Slivan states (observed in the Koronis family) among any other of the studied families.

We simulate a long-term evolution of the synthetic Flora, Koronis and Eos families (Figure 5) in the proper semi-major axis  vs the pole latitude  plane and compare the results with the properties of observed asteroid families. We obtain a good qualitative agreement between the observed and synthetic spin-vector distributions. For all three families, we compute evolution of the number of objects in the four quadrants of the families in the (, ) diagram, and we estimate ages for families Flora  Gyr and Koronis (2.5 to 4 Gyr) that are in agreement with previously published values. However, we do not estimate the age of the Eos family due to a small evolution of the objects in the (,) diagram.

The uncertainties seem to be dominated by the observed quadrant ratios. We expect that increasing the sample size by a factor of 10 would decrease the relative uncertainty by a factor of about 3, which is a good motivation for further work on this subject.

Acknowledgements.
The work of JH and JD has been supported by grants GACR P209/10/0537 and P209/12/0229 of the Czech Science Foundation, and the work of JD and MB by the Research Program MSM0021620860 of the Czech Ministry of Education. The work of MB has been also supported by grant GACR 13-013085 of the Czech Science Foundation.

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Asteroid
[deg] [deg] [deg] [deg] [hours]
243 Ida 259 66 74 61 53 6 134 122 25
364 Isara 282 44 86 42 4 1 98 104
540 Rosamunde 301 81 127 62 3 1 135 83
550 Senta 63 40 258 58 9 1 151 85
553 Kundry 197 73 1 64 5 1 61 80
621 Werdandi 247 86 66 77 12 2 146 71
936 Kunigunde 47 57 234 50 154 88
951 Gaspra 20 23 198 15 71 4 117 89
1286 Banachiewicza 214 62 64 60 81 51
1353 Maartje 266 73 92 57 154 139
1378 Leonce 210 67 46 77 89 113
1423 Jose 78 82 121 134
1446 Sillanpaa 129 76 288 63 2 1 76 73
1464 Armisticia 194 54 35 69 2 1 231 67
1503 Kuopio 170 86 27 61 116 68
1527 Malmquista 274 80 49 107
1618 Dawn 39 60 215 51 93 91
1633 Chimay 322 77 116 81 2 1 127 83
1691 Oort 45 68 223 58 86 60
1703 Barry 46 76 221 71 89 138
1805 Dirikis 364 48 188 61 117 91
1835 Gajdariya 34 74 204 69 66 86
1987 Kaplan 356 58 233 89 8 2 81 28
2430 Bruce Helin 177 68 15 1 112
3279 Solon 268 70 3 1 137
3492 Petra-Pepi 9 57 202 16 15 1 25 111
4399 Ashizuri 266 48 45 61 4 1 20 84
4606 Saheki 44 59 222 68 6 1 123
6159 1991 YH 266 67 62 67 3 1 102
6262 Javid 93 76 275 66 3 1 106
6403 Steverin 246 77 109 73 2 1 74
7043 Godart 73 62 235 80 4 1 121
7169 Linda 11 60 198 61 5 1 95
Table 1: continued.
999 For each asteroid, the table gives the ecliptic coordinates and of the pole solution with the lowest , the corresponding mirror solution and , the sidereal rotational period , the number of dense lightcurves observed during apparitions, and the number of sparse data points for the corresponding observatory: , and . The uncertainty of the sidereal rotational period corresponds to the last decimal place of and of the pole direction to 5–10 if we have multi-apparition dense data or 10–20 if the model is based mainly on sparse data (i.e., only few dense lightcurves from 1–2 apparitions).
Asteroid P N N N N
[deg] [deg] [hours]
391 Ingeborg 60 7 24 2 141 96
502 Sigune 44 3 9 2 157 52
616 Elly 67 23 4 1 101 133
1003 Lilofee 65 10 107 83
1160 Illyria 47 23 96 100
1192 Prisma 65 14 5 1 44 43
1276 Ucclia 49 22 114 45
1307 Cimmeria 63 9 2 1 91 54
1339 Desagneauxa 65 17 78 120
1396 Outeniqua 62 7 2 1 112 68
1493 Sigrid 78 7 78 103
1619 Ueta 39 6 5 1 122 51
1623 Vivian 75 8 77 58
1738 Oosterhoff 72 8 109 105
1838 Ursa 47 17 102 91
2086 Newell 60 12 10 1 24 84
3017 Petrovic 73 8 3 1 114
3786 Yamada 56 2 3 1 71
3896 Pordenone 32 9 3 1 22 71
4209 Briggs 56 25 2 1 64
4467 Kaidanovskij 54 13 20 107
6179 Brett 42 20 6 1 93
7055 1989 KB 61 11 7 1 117
7360 Moberg 18 18 3 1 103
Table 2: continued.
101010For each asteroid, there is the mean ecliptic latitude of the pole direction and its dispersion , the other parameters have the same meaning as in Table 1. The uncertainty of the sidereal rotational period corresponds to the last decimal place of .
Asteroid Date Observer Observatory (MPC code)
364 Isara 2009 5 – 2009 05 Warner (2009) Palmer Divide Observatory (716)
391 Ingeborg 2000 8 – 2000 12 Koff et al. (2001) Antelope Hills Observatory, Bennett (H09)
502 Sigune 2007 6 – 2007 6 Stephens (2007b) Goat Mountain Astronomical Research Station (G79)
553 Kundry 2004 12 – 2005 1 Stephens (2005) Goat Mountain Astronomical Research Station (G79)
616 Elly 2010 1 – 2010 1 Warner (2010) Palmer Divide Observatory (716)
2010 2 – 2010 2 Durkee (2010) Shed of Science Observatory, USA (H39)
621 Werdandi 2012 1 22.9 Strabla et al. (2012) Bassano Bresciano Observatory (565)
2012 1 – 2012 2 Strabla et al. (2012) Organ Mesa Observatory (G50)
1307 Chimmeria 2004 9 – 2004 9 Warner (2005) Palmer Divide Observatory (716)
1396 Outeniqua 2006 3 – 2006 3 Warner (2006) Palmer Divide Observatory (716)
1446 Sillanpaa 2009 3 – 2009 3 HigginsOn line at http://www.david-higgins.com/Astronomy/asteroid/lightcurves.htm Hunters Hill Observatory, Ngunnawal (E14)
1464 Armisticia 2008 1 – 2008 1 Brinsfield (2008b) Via Capote Sky Observatory, Thousand Oaks (G69)
1619 Ueta 2010 9 – 2010 10 Higgins (2011) Hunters Hill Observatory, Ngunnawal (E14)
2010 9 – 2010 9 Stephens (2011b) Goat Mountain Astronomical Research Station (G79)
1633 Chimay 2008 4 – 2008 4 Brinsfield (2008a) Via Capote Sky Observatory, Thousand Oaks (G69)
1987 Kaplan 2000 10 – 2000 10 Warner (2001, 2011) Palmer Divide Observatory (716)
2011 12 – 2011 12 Warner Palmer Divide Observatory (716)
2086 Newell 2007 1 – 2007 2 Stephens (2007c) Goat Mountain Astronomical Research Station (G79)
2403 Bruce Helin 2006 9 – 2006 9 HigginsOn line at http://www.david-higgins.com/Astronomy/asteroid/lightcurves.htm Hunters Hill Observatory, Ngunnawal (E14)
3279 Solon 2006 11 – 2006 11 Stephens (2007a) Goat Mountain Astronomical Research Station (G79)
3492 Petra-Pepi 2011 6 – 2011 7 Stephens (2011a) Goat Mountain Astronomical Research Station (G79)
3786 Yamada 2002 7 – 2002 8 Stephens (2003) Goat Mountain Astronomical Research Station (G79)
3896 Pordenone 2007 10 – 2007 10 HigginsOn line at http://www.david-higgins.com/Astronomy/asteroid/lightcurves.htm Hunters Hill Observatory, Ngunnawal (E14)
4209 Briggs 2003 9 – 2003 9 Warner (2004) Palmer Divide Observatory (716)
4399 Ashizuri 2008 6 – 2008 6 Brinsfield (2008a) Via Capote Sky Observatory, Thousand Oaks (G69)
4606 Saheki 2009 1 – 2009 3 Brinsfield (2009) Via Capote Sky Observatory, Thousand Oaks (G69)
6159 1991 YH 2006 3 – 2006 3 Warner (2006) Palmer Divide Observatory (716)
6179 Brett 2009 4 – 2009 4 Warner & Pray (2009) Palmer Divide Observatory (716)
6262 Javid 2010 2 – 2010 2 PTFPalomar Transient Factory survey (Rau et al. 2009), data taken from Polishook et al. (2012).
6403 Steverin 2004 9 – 2004 9 Warner (2005) Palmer Divide Observatory (716)
7043 Godart 2008 8 – 2008 8 Durkee Shed of Science Observatory, USA (H39)
2008 8 – 2008 9 Pravec et al. (2012) Goat Mountain Astronomical Research Station (G79)
7055 1989 KB 2007 5 – 2007 5 Stephens (2007b) Goat Mountain Astronomical Research Station (G79)
2007 5 – 2007 6 HigginsOn line at http://www.david-higgins.com/Astronomy/asteroid/lightcurves.htm Hunters Hill Observatory, Ngunnawal (E14)
7169 Linda 2006 8 – 2006 8 Higgins & Goncalves (2007) Hunters Hill Observatory, Ngunnawal (E14)
7360 Moberg 2006 4 – 2006 4 Oey (2006) Leura (E17)
Table 3: continued.
111111
Asteroid Status Reason
Flora
9 Metis Interloper Far from the (, ) border, peculiar SFD
43 Ariadne Interloper Associated at m/s, peculiar SFD
352 Gisela Borderline Associated at m/s, big object
364 Isara Interloper Big, peculiar SFD, close to (, ) border
376 Geometria Interloper Far from the (, ) border, peculiar SFD
800 Kressmannia Borderline Associated at m/s, lower albedo
1188 Gothlandia Borderline Associated at m/s
1419 Danzing Interloper Far from the (, ) border
1703 Barry Borderline Associated at m/s
2839 Annette Interloper Associated at m/s, C type
7360 Moberg Borderline Redder (color from SDSS MOC4)
Koronis
167 Urda Borderline Close to the (, ) border
208 Lacrimosa Interloper Far from the (, ) border, peculiar SFD
311 Claudia Borderline Close to the (, ) border
720 Bohlinia Borderline Close to the (, ) border
1835 Gajdariya Interloper Close to the (, ) border, incompatible albedo
2953 Vysheslavia Borderline Close to the (, ) border
3170 Dzhanibekov Interloper Behind the (, ) border, incompatible albedo
Eos
423 Diotima Interloper Far from the (, ) border, big, C type
590 Tomyris Borderline Close to the (, ) border
Eunomia
85 Io Interloper Behind the (, ) border, peculiar SFD, incompatible albedo
390 Alma Borderline Borderline albedo, borderline in (, , ) space
4399 Ashizuri Borderline Close to the (, ) border
Phocaea
290 Bruna Borderline Close to the (, ) border
391 Ingeborg Interloper Clearly outside (, )
852 Wladilena Borderline Slightly outside (, )
1963 Bezovec Interloper C type, incompatible albedo (=0.04)
5647 1990 TZ Interloper Incompatible albedo (=0.64)
Themis
62 Erato Borderline Close to the (, ) border
1633 Chimay Borderline Close to the (, ) border
Maria
695 Bella Borderline Close to the (, ) border
714 Ulula Borderline Close to the (, ) border
Alauda
276 Adelheid Interloper Far from the (, ) border, big
Table 4: continued.
121212 In the table, we also give the name of the asteroid, the family membership according the HCM method, if it is an interloper or a borderline case and the reason. Peculiar SFD means a size frequency distribution that is incompatible with the SFD typically created by catastrophic collisions or cratering events (i.e., a large remnant, large fragment and steep slope). Quantity corresponds to the cutoff value of the HCM method for a particular family.
Asteroid Bus/DeMeo Tholen M/I/B Reference
[deg] [deg] [deg] [deg] [hours] [AU] [km]
Flora
8 Flora 335 5 155 6 2.2014 141.0 S 0.260.05 M Torppa et al. (2003)
9 Metis 180 22 2.3864 169.0 S 0.130.02 I Torppa et al. (2003)
43 Ariadne 253 15 2.2034 72.1 Sk S 0.230.05 I Kaasalainen et al. (2002)
281 Lucretia 128 49 309 61 2.1878 11.8 S SU 0.200.01 M Hanuš et al. (2013)/Kryszczyńska (2013)
352 Gisela 205 26 23 20 2.1941 26.7 Sl S 0.190.02 B Hanuš et al. (2013)
364 Isara 282 44 86 42 2.2208 35.2 S 0.160.03 I this work
376 Geometria 239 45 63 53 2.2886 39.0 Sl S 0.190.04 I Hanuš et al. (2011)
540 Rosamunde 301 81 127 62 2.2189 20.3 S 0.220.05 M this work
553 Kundry 197 73 1 64 2.2308 9.6 S 0.250.04 M this work
685 Hermia 197 87 29 79 2.2359 10.9 0.280.05 M Hanuš et al. (2011)
700 Auravictrix 67 46 267 51 2.2295 20.6 0.140.05 M Kryszczyńska (2013)
800 Kressmannia 345 37 172 34 2.1927 17.0 S 0.150.02 B Hanuš et al. (2011)
823 Sisigambis 86 74 2.2213 15.8 0.230.03 M Hanuš et al. (2011)
915 Cosette 350 56 189 61 2.2277 12.3 0.230.04 M Ďurech et al. (2009)
951 Gaspra 20 23 198 15 2.2097 12.2 S S 0.330.13 M this work
19 21 Davies et al. (1994b)11footnotemark: 1
1056 Azalea 242 61 49 48 2.2300 13.0 S 0.250.04 M Hanuš et al. (2013)
1088 Mitaka 280 71 2.2014 16.0 S S 0.160.02 M Hanuš et al. (2011)
1185 Nikko 359 34 2.2375 11.3 S S 0.20 M Hanuš et al. (2011)/Ďurech et al. (2009)
1188 Gothlandia 133 84 335 81 2.1907 12.7 S 0.250.02 B Hanuš et al. (2013)/Kryszczyńska (2013)
1249 Rutherfordia 204 72 31 74 2.2243 14.1 S 0.220.02 M Hanuš et al. (2013)
1270 Datura 60 76 2.2347 8.2 0.24 M Vokrouhlický et al. (2009)
1307 Cimmeria 63 2.2505 10.1 S 0.220.02 B this work
1396 Outeniqua 62 2.2480 11.7 0.210.01 M this work
1419 Danzig 22 76 193 62 2.2928 14.1 0.240.05 I Hanuš et al. (2011)
1446 Sillanpaa 129 76 288 63 2.2457 8.8 0.210.01 M this work
1514 Ricouxa 251 75 68 69 2.2404 8.1 0.180.04 M Hanuš et al. (2011)
1518 Rovaniemi 62 60 265 45 2.2255 9.0 0.260.04 M Hanuš et al. (2013)
1527 Malmquista 274 80 2.2274 10.3 0.220.02 M this work
1619 Ueta 39 2.2411 9.9 S 0.250.03 M this work
1675 Simonida 23 58 227 54 2.2332 11.1 0.250.03 M Kryszczyńska (2013)
1682 Karel 232 32 51 41 2.2388 7.1 0.24 M Hanuš et al. (2011)
1703 Barry 46 76 221 71 2.2148 9.4 0.220.03 B this work
1738 Oosterhoff 72 2.1835 8.7 S 0.280.04 M this work
1785 Wurm 11 57 192 47 2.2359 6.2 S 0.24 M Hanuš et al. (2013)
2017 Wesson 159 81 356 79 2.2521 7.2 0.200.05 M Kryszczyńska (2013)
2094 Magnitka 107 57 272 48 2.2323 12.1 0.130.01 M Hanuš et al. (2013)
2112 Ulyanov 151 61 331 61 2.2547 7.5 0.24 M Hanuš et al. (2013)
2510 Shandong 256 27 71 27 2.2531 9.0 S 0.20 M Hanuš et al. (2013)
2709 Sagan 308 8 124 16 2.1954 6.8 S 0.24 M Hanuš et al. (2013)
2839 Annette 341 49 154 36 2.2166 7.6 0.060.01 I Hanuš et al. (2013)
3279 Solon 268 70 2.2027 5.9 0.24 M this work
7043 Godart 73 62 235 80 2.2447 5.7 0.230.04 M this work
7169 Linda 11 60 198 61 2.2487 4.5 0.24 M this work
7360 Moberg 18 2.2510 7.7 0.220.04 B this work
31383 1998 XJ 110 74 279 63 2.1853 4.1 0.290.03 M Hanuš et al. (2013)
Koronis
158 Koronis 30 64 2.8687 47.7 S S 0.140.01 M Ďurech et al. (2011)
220 68 35 65 Slivan et al. (2003)
167 Urda 249 68 107 69 2.8535 44.0 Sk S 0.160.04 B Ďurech et al. (2011)
225 73 40 75 Slivan et al. (2003)
208 Lacrimosa 170 68 350 71 2.8929 45.0 Sk S 0.170.06 I Slivan et al. (2003)
243 Ida 259 66 74 61 2.8616 28.0 S S 0.240.07 M this work
263 67 Davies et al. (1994a); Binzel et al. (1993)22footnotemark: 2
263 Dresda 105 76 285 80 2.8865 25.5 S 0.180.02 M Slivan et al. (2009)
277 Elvira 121 84 2.8856 31.2 S 0.200.05 M Hanuš et al. (2011)
50 80 244 81 Slivan et al. (2009)
311 Claudia 214 43 30 40 2.8976 25.8 S 0.240.03 B Hanuš et al. (2011)
209 48 24 48 Slivan et al. (2003)
321 Florentina 264 63 91 60 2.8856 34.0 S S 0.140.01 M Slivan et al. (2003)
462 Eriphyla 108 35 294 34 2.8737 41.9 S S 0.170.02 M Slivan et al. (2009)
534 Nassovia 66 41 252 42 2.8842 38.6 Sq S 0.120.02 M Hanuš et al. (2011)
58 50 244 50 Slivan et al. (2003)
720 Bohlinia 230 41 40 43 2.8873 34.0 Sq S 0.200.02 B Slivan et al. (2003)
832 Karin 242 46 59 44 2.8644 16.3 0.210.05 M Hanuš et al. (2011)
230 42 52 42 Slivan & Molnar (2012)
1223 Neckar 252 28 69 30 2.8695 25.7 S 0.150.03 M Hanuš et al. (2011)
259 41 73 40 Slivan et al. (2003)
1289 Kutaissi 158 79 338 74 2.8605 22.6 S 0.160.04 M Slivan et al. (2003)
1350 Rosselia 166 72 2.8580 21.1 Sa S 0.200.05 M Hanuš et al. (2011)
1389 Onnie 183 75 360 79 2.8661 14.7 0.170.04 M Hanuš et al. (2013)
1423 Jose 78 82 2.8602 20.0 S 0.280.04 M this work
1482 Sebastiana 262 68 91 67 2.8723 17.6 0.210.05 M Hanuš et al. (2011)
1618 Dawn 39 60 215 51 2.8688 17.5 S 0.150.04 M this work
1635 Bohrmann 5 38 185 36 2.8534 17.5 S 0.210.02 M Hanuš et al. (2011)
1742 Schaifers 56 52 247 68 2.8892 16.6 0.110.02 M Hanuš et al. (2011)
1835 Gajdariya 34 74 204 69 2.8331 12.8 0.270.04 I this work
2953 Vysheslavia 11 64 192 68 2.8282 12.8 S 0.250.07 B Vokrouhlický et al. (2006a)
3170 Dzhanibekov 216 62 30 63 2.9291 9.6 S 0.300.04 I Hanuš et al. (2013)
4507 1990 FV 137 50 307 51 2.8689 11.0 0.280.02 M Hanuš et al. (2013)
6262 Javid 93 76 275 66 2.9063 7.8 0.290.04 M this work
Eos
423 Diotima 351 4 3.0684 177.3 C C 0.070.00 I Marchis et al. (2006)
573 Recha 74 24 252 48 3.0138 44.4 0.130.02 M Hanuš et al. (2011)
590 Tomyris 273 47 120 46 3.0006 31.1 0.180.03 B Hanuš et al. (2011)
669 Kypria 31 40 190 50 3.0114 29.2 S 0.170.02 M Hanuš et al. (2013)
807 Ceraskia 325 23 132 26 3.0185 21.4 S 0.210.05 M Hanuš et al. (2013)
1087 Arabis 334 7 155 12 3.0150 45.6 S 0.100.01 M Hanuš et al. (2011)
1148 Rarahu 148 9 322 9 3.0161 26.3 K S 0.220.06 M Hanuš et al. (2011)
1207 Ostenia 310 77 124 51 3.0207 22.9 0.130.02 M Hanuš et al. (2011)
1286 Banachiewicza 214 62 64 60 3.0223 22.6 S 0.160.03 M this work
1291 Phryne 106 35 277 59 3.0130 22.4 0.190.04 M Hanuš et al. (2011)
1339 Desagneauxa 65 3.0211 26.1 S 0.120.02 M this work
1353 Maartje 266 73 92 57 3.0120 42.2 0.070.00 M this work
1464 Armisticia 194 54 35 69 3.0035 23.3 0.130.36 M this work
2957 Tatsuo 88 57 246 37 3.0221 22.9 K 0.290.02 M Hanuš et al. (2013)
3896 Pordenone 32 3.0057 20.0 0.130.01 M this work
5281 Lindstrom 238 72 84 81 3.0125 20.0 M Hanuš et al. (2013)
19848 Yeungchuchiu 66 70 190 67 3.0075 13.2 0.210.03 M Hanuš et al. (2013)
Eunomia
15 Eunomia 363 67 2.6437 259.0 S S 0.210.06 M Kaasalainen et al. (2002)
85 Io 95 65 2.6537 161.0 B FC 0.060.03 I Ďurech et al. (2011)
390 Alma 54 48 263 73 2.6517 31.2 DT 0.130.02 B Hanuš et al. (2013)
812 Adele 301 44 154 69 2.6594 13.6 0.240.03 M Hanuš et al. (2013)
1333 Cevenola 8 79 201 40 2.6336 17.1 0.170.04 M Hanuš et al. (2011)
1495 Helsinki 356 33 2.6392 13.3 0.230.02 M Hanuš et al. (2013)
1503 Kuopio 170 86 27 61 2.6263 18.4 0.300.06 M this work
1554 Yugoslavia 281 34 78 64 2.6194 17.2 0.100.01 M Hanuš et al. (2013)
1927 Suvanto 90 39 277 6 2.6497 12.5 0.260.04 M Hanuš et al. (2013)
2384 Schulhof 194 57 46 36 2.6099 11.7 0.270.02 M Hanuš et al. (2013)
3017 Petrovic 73 2.6074 12.7 0.210.02 M this work
3492 PetraPepi 9 57 202 16 2.6159 12.2 0.230.03 M this work
4399 Ashizuri 266 48 45 61 2.5759 8.8 0.280.06 B this work
4467 Kaidanovskij 54 2.6383 11.6 0.21 M this work
8132 Vitginzburg 33 66 193 48 2.6263 11.6 0.21 M Hanuš et al. (2013)
Phocaea
25 Phocaea 347 10 2.4002 75.1 S S 0.230.02 M Hanuš et al. (2013)
290 Bruna 286 80 37 74 2.3372 10.4 0.420.08 B Hanuš et al. (2013)
391 Ingeborg 60 2.3202 19.6 S S 0.20 I this work
502 Sigune 44 2.3831 19.5 S 0.230.02 M this work
852 Wladilena 218 41 57 16 2.3627 31.1 0.160.02 B Hanuš et al. (2013)
1192 Prisma 65 2.3660 7.2 0.23 M this work
1568 Aisleen 109 68 2.3520 12.0 0.180.03 M Hanuš et al. (2011)
1963 Bezovec 218 16 50 49 2.4231 45.0 C 0.040.01 I Hanuš et al. (2013)
1987 Kaplan 356 58 2.3822 14.6 0.210.04 M this work
2430 Bruce Helin 177 68 2.3627 12.7 Sl S 0.23 M this work
5647 1990 TZ 266 69 2.4241 9.3 S 0.640.07 I Hanuš et al. (2013)
6179 Brett 42 2.4278 5.8 0.23 M this work
7055 1989 KB 61 2.3496 6.7 0.330.15 M this work
10772 1990 YM 16 46 2.3901 6.2 0.380.06 M Hanuš et al. (2013)
Themis
62 Erato 87 22 269 23 3.1217 95.4 Ch BU 0.060.00 B Hanuš et al. (2011)
222 Lucia 107 54 290 51 3.1349 56.5 BU 0.120.02 M Hanuš et al. (2013)
621 Werdandi 247 86 66 77 3.1193 27.1 FCX 0.150.02 M this work
936 Kunigunde 47 57 234 50 3.1383 39.6 0.110.01 M this work
1003 Lilofee 65 3.1483 31.4 0.150.04 M this work
1623 Vivian 75 3.1347 29.6 0.08 M this work
1633 Chimay 322 77 116 81 3.1748 37.7 0.080.01 B this work
1691 Oort 45 68 223 58 3.1664 33.2 CU 0.070.01 M this work
1805 Dirikis 364 48 188 61 3.1333 28.1 0.090.01 M this work
Maria
616 Elly 67 2.5526 22.6 S 0.190.04 M this work
695 Bella 87 55 314 56 2.5391 41.2 S 0.240.03 B Hanuš et al. (2011)
714 Ulula 224 10 41 5 2.5352 39.2 S 0.270.04 B Hanuš et al. (2011)
787 Moskva 330 60 122 19 2.5396 40.3 0.120.02 M Hanuš et al. (2013)
875 Nymphe 42 31 196 42 2.5539 15.2 0.190.02 M Hanuš et al. (2013)
1160 Illyria 47 2.5604 14.8 0.220.04 M this work
1996 Adams 107 55 2.5587 13.5 0.140.01 M Hanuš et al. (2013)
3786 Yamada 56 2.5503 16.7 0.230.04 M this work
6403 Steverin 246 77 109 73 2.5945 6.9 0.490.05 M this work
Vesta
63 Ausonia 305 21 120 15 2.3952 90.0 Sa S 0.160.03 Torppa et al. (2003)
306 Unitas 79 35 2.3580 49.0 S S 0.170.06 Ďurech et al. (2007)
336 Lacadiera 194 39 37 54 2.2518 69.0 Xk D 0.050.01 Hanuš et al. (2011)
556 Phyllis 34 54 209 41 2.4654 38.5 S S 0.180.03 Marciniak et al. (2007)
1933 Tinchen 113 26 309 36 2.3530 6.5 0.290.06 Hanuš et al. (2013)
2086 Newell 60 2.4014 9.8 Xc 0.20 this work
6159 1991 YH 266 67 62 67 2.2914 5.4 0.460.13 this work
8359 1989 WD 121 68 274 68 2.3500 8.2 0.220.03 Hanuš et al. (2013)
Nysa/Polana
44 Nysa 99 58 2.4227 70.6 Xc E 0.550.07 Kaasalainen et al. (2002)
135 Hertha 272 52 2.4285 77.0 Xk M 0.150.05 Torppa et al. (2003)
1378 Leonce 210 67 46 77 2.3748 22.5 0.030.00 this work
1493 Sigrid 78 2.4297 22.1 Xc F 0.040.00 this work
4606 Saheki 44 59 222 68 2.2518 6.7 0.330.02 this work
Alauda
276 Adelheid 199 20 9 4 3.1162 125.0 X 0.060.01 I Marciniak et al. (2007)
1276 Ucclia 49 3.1698 40.0 0.050.01 M this work
1838 Ursa 47 3.2111 48.6 0.040.01 M this work
4209 Briggs 56 3.1564 30.9 0.090.03 M this work
Table 5: continued.
131313 For each asteroid, the table gives the spin state solution (i.e., ecliptic coordinates and of the spin axis and the sidereal rotational period