© ESO, 2013

1. Introduction

An analysis of rotational state solutions for main belt asteroids has been performed by many authors. All these 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 mainly concerns smaller asteroids (D ≲ 40 km), while the larger asteroids (60 ≲ D ≲ 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 torques¹, and of collisions that systematically evolve the spin axes away from the ecliptic plane. The prograde excess of larger asteroids is probably caused by a primordial preference that agrees with the theoretical work of Johansen & Lacerda (2010). While the number of asteroids with known rotational states grows, we can not only study the spin vector distribution in the whole main-belt asteroids (MBAs) or near-Earth asteroids (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).

Fig. 1 Dependence of the absolute magnitude H on the proper semi-major axis ap 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 (ap, H) borders of the family for different parameters C (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 ν6 resonance for the inclination typical for Flora family members and the horizontal arrow its approximate range.

The theory of dynamical evolution of asteroid families (e.g., Bottke et al. 2006) suggests that the Yarkovsky²/YORP effects change orbital parameters of smaller asteroids (≲30–50 km) – the semi-major axis of prograde rotators slowly grows the course of time, unlike retrograde rotators whose semi-major axis decreases. This phenomenon is particularly visible when we plot the dependence of the absolute magnitude H on the proper semi-major axis a_p (see an example of such a plot for Themis family in Fig. 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 in the number of asteroids in the family by inducing moderate oscillations to their orbital elements (Bottke et al. 2001) as can be seen in Fig. 1 for the Flora family, where the secular ν₆ resonance with Saturn almost completely eliminated objects to the left of the center of the family. The ν₆ resonance has its center at 2.13 AU for objects with sinI ~ 0.09, which is typical of Flora family members. It develops objects which then approach 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 collisionallyborn cluster should initially have a rotational frequency distribution close to Maxwellian (Giblin et al. 1998) and an isotropic spin vector distribution. For several families, we already know their age estimates (e.g., 2.5  ±  1.0 Gyr for Koronis family, Bottke et al. 2001), and so we have a constraint on the time at which the family was evolving towards its current state. As shown in Bottke et al. (2001), the family evolution is dominated by Yarkovsky and YORP effects, as well as by 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 reveal an anisotropy of spin vectors for ten members of the Koronis family. This was an unexpected result because collisionallyborn 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 timespan of 2–3 Gyr. The secular s₆ 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 P ∈ (4,7) h, similar obliquities (42° to 51°) and also with similar ecliptic longitudes in the ranges of (24° to 73°) and (204° to 259°); or (ii) create a group of low-obliquity retrograde rotators with rotational periods P < 5 h or P > 13 h. The prograde rotators trapped in the s₆ spin-orbital resonance were referred to 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 for 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 made by Kryszczyńska (2013), who focuses on the Flora family. She distinguishes prograde and retrograde groups of asteroids and reports an excess of prograde rotators. This splitting into two groups is most likely caused by the Yarkovsky effect, while the prograde excess by the secular ν₆ resonance that significantly depopulates the retrograde part of the family. See Fig. 1b, only retrograde rotators can drift via the Yarkovsky/YORP effects towards the resonance.

Future 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 a_p’s), and the YORP effect should eliminate the spin vectors close to the ecliptic plane.

Disk-integrated photometric observations of asteroids contain information about an 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 (DAMIT³, 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 large 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 Sect. 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 Sect. 3.1. A combined spin-orbital model for the long-term evolution of a collisional family is described in Sect. 4, where we also compare the synthetic and observed spin-vector properties and constrain the 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 online catalog published by Nesvorný (2012), who used the hierarchical clustering method⁴ (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. The majority of asteroids are 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 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 collisionallyborn asteroid family. A fraction of objects have 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 those 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 a^⋆ and i − z, which usually define the core of the family well (see examples for Themis and Eunomia families in Fig. 2), and for each asteroid with available color indexes, we compared values a^⋆ and i − z to those that define the family;

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

• construction of a diagram of the proper semi-major axis vs. the absolute magnitude (see Fig. 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 the Eos family). We refer here to the (a_p, H) border of the family as the border of the V-shape; or by

• construction of 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).

These methods for determining family membership 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.

Fig. 2 Dependence of the color indexes a⋆ and i − z (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. There is a qualitative difference between C- and S-types asteroids.

2.2. New asteroid models

From the DAMIT database, we adopt 96 models of asteroids that are, according to the HCM, 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. 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 (UAPC⁵, Lagerkvist et al. 1987; Piironen et al. 2001), where lightcurves for about 1000 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 (ALCDEF⁶, Warner et al. 2009). The sparse-in-time photometry is downloaded from the AstDyS site (Asteroids – Dynamic Site⁷). 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).

Fig. 3 Dependence of the absolute magnitude H on the proper semi-major axis ap for the eight 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 (ap, H) borders (two thick lines) of each family for values of the parameter C 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). Borderline cases and interlopers are identified by several methods including the position in the (ap, H) diagram, and thus could also lie close to the center of the family (e.g., in the case of the Flora family).

To increase the number of asteroid models for our study of asteroid families, we performed 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 β, 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 removed the pole ambiguity for several asteroids, and thus determined 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 β (if the biggest difference in latitudes of the models is <50°). We call these models partial and present them in Table 2. For the ecliptic latitude β, we use the mean value of all different models. We define parameter Δ ≡ |β_max − β_min|/2 as being the estimated uncertainty of β, where β_max and β_min are the extremal values within all β. The threshold for partial models is Δ < 25°.

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 Sect. 4, we compare the numbers of asteroids in four quadrants of the (a_p, β) diagram (defined by the center of the family and the value β = 0°) with the same quantities based on the synthetic family population. The uncertainties in β are rarely greater than 20°, and the assignment to a specific quadrant is usually not questionable (only in 4 cases out of 136 does the uncertainty interval lie in both quadrants, and most of the asteroids have latitudes |β| ≳ 30°), 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°)/cosβ in longitude λ and 5–20° in latitude β. 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 about five to ten 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 larger uncertainties, owing to the poor photometric quality of the sparse data (corresponds 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 according to the criteria described above for interlopers or borderline cases. Interlopers are asteroids that do not clearly belong to the family; for example, they have different taxonomic types or incompatible albedos or are far from the (a_p, H) border. On the other hand, borderline cases cannot be directly excluded from the family, since their physical or orbital properties are just not typical in the context of other members (higher/lower albedos, close to the (a_p, H) border). These asteroids are possible family members, but can just as easily be interlopers. In the penultimate column of Table 5, we show our revised membership classification for 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 λ and β and the period P), the semi-major axis a, the diameter D, and the albedo p_V from WISE (Masiero et al. 2011), the SMASS II (Bus & Binzel 2002), and Tholen (Tholen 1984, 1989) taxonomic types, and the reference to the model).

Although we got several members by the HCM for Vesta and Nysa/Polana families, we excluded these two families from further study of spin states. The Vesta family was created by a cratering event, and thus a majority of the fragments are rather small and beyond the capabilities of the model determination. Most of the models we currently have (recognized by the HCM) 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), hence should be treated individually. Additionally, we only have 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 suggested a membership in 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 Fig. 3, we show the (a_p, H) diagrams for all eight studied families. We plotted the adopted (a_p, H) border (from Brož et al. 2013) and labeled the members, borderline cases, and interlopers by different colors.

Several asteroids in our sample belong to smaller and younger subclusters 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 subclusters were probably created by secondary collisions. As a result, the spin states of asteroids in these subclusters were randomly reoriented. Because our combined orbital- and spin-evolution model (see Sect. 4) includes secondary collisions (reorientations), using asteroids from subclusters in the study of the spin-vector distribution is thus essential: asteroids from subclusters correspond to reoriented asteroids in our synthetic population.

Fig. 4 Dependence of the pole latitude β on the proper semi-major axis ap 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, whose 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, whose uncertainties are usually < 0.01 AU. The Eos family has an asymmetric V-shape (the (ap, H) border is asymmetric), which makes the center determination harder, so we marked two possible positions. One corresponds to the right (ap, H) border, the second to the left border. The uncertainties in β are usually 5–20°. In most cases, the value of |β| ≳ 30°, hence the quadrant to which the asteroid belongs (defined by the center of the family and the value β = 0°), is not changed.

3. Observed spin vectors in families

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. Now that we have the models and membership, we can proceed to the discussion of the spin states in families in general (Sect. 3.1), and for families Flora and Koronis (Sects. 3.2, 3.3).

3.1. Spin-vector orientations in individual families

In Fig. 4, we show the dependence of asteroid’s pole latitudes in ecliptic coordinates on the semi-major axes. 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. To determine the centers of families, we use all members of each family assigned by the HCM, see Figs. 1 and 3. The Eos family has an asymmetric V-shape (the (a_p, H) diagram), so 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 ν₆ secular resonance.

In the study of spin-vector properties in families, we simply use the ecliptic coordinates for the pole orientation: ecliptic longitude λ and latitude β. A formally better approach would be to use the coordinates bound to the orbital plane of the asteroid: orbital longitude λ_orb and latitude β_orb. The orbital latitude can then be easily transformed to obliquity, which directly tells us whether the asteroid rotates in a prograde or retrograde sense. However, for several reasons, we prefer the ecliptic coordinates: (i) most of the asteroids have low inclinations and thus the differences between their ecliptic and orbital latitudes are only a few degrees, and 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 (a_p, β) diagrams (i.e., to which quadrant they belong), namely if they have β > 0° or β < 0°, are sufficient information. Because most of the asteroids have latitudes larger than 30°, their positions in the (a_p, β_orb) are similar (not true only for three asteroids out of 136); and (iv) we compare the (a_p, β) 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 similar trends for all studied families: (i) larger asteroids are situated in the proximity of the center of the family; (ii) asteroids with β > 0° are usually found to the right of the family center; (iii) asteroids with β < 0° are to the left of the center; (iv) the majority of asteroids have large pole-ecliptic latitudes (|β| ≳ 30°); and finally (v) some families have a statistically significant excess of asteroids with β > 0° or β < 0°.

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 the 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 a; that is, it can increase a of prograde rotators, and decrease a of retrograde once. The magnitude of the Yarkovsky drift is dependent on the asteroid size, is negligible for asteroids with diameters D ≳ 50 km (the case of Eos), and increases with decreasing diameter. For the Flora, Eunomia, Phocaea, or Maria families, we can see that the smallest asteroids in the sample (D ~ 5–10 km) can be situated far from the family center, and we can also notice a trend toward decreasing size with increasing distance from the center that probably corresponds to the magnitude of the Yarkovsky effect and the initial velocities v_ini(D) that 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 β > 0° and β < 0° among these families is statistically significant and cannot be coincidental. The obvious choice for an explanation are mean-motion or secular resonances. Indeed, the ν₆ secular resonance removed many objects with β > 0° from the Flora family (see Sect. 3.2 for a more thorough discussion). The 8:3 resonance with Jupiter truncated the Eunomia family, which resulted in there being no objects with a_p > 2.70 AU; similarly, the 3:1 resonance with Jupiter affected the Maria family, for which we do not observe objects with smaller a_p than 2.52 AU. The 3:1 resonance with Jupiter is situated near the Phocaea family at a = 2.50 AU. Due to the high inclination of objects in the Phocaea family (I ~ 24°), the resonance affects asteroids with a_p > 2.40 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 a_p.

The asymmetry of asteroids with β > 0° and β < 0° in the Themis family is caused by a selection effect: in the family, there are no objects with absolute magnitude H < 12 mag (i.e., large asteroids) and a_p < 3.10 AU. On the other hand, with a_p > 3.10 AU, there are more than a hundred such asteroids (see Fig. 1a). Our sample of asteroid models derived by the lightcurve inversion method is dominated by larger asteroids, so it is not surprising that we did not derive models for the Themis family asteroids with a_p < 3.10 AU. The Flora and Koronis families are also interesting for other aspects, and thus are discussed in more detail in Sects. 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, and its left part (with respect to the (a_p, H) diagram) is strongly affected by the secular ν₆ resonance with Saturn, which is demonstrated in Fig. 1b. The probable center of the family matches the position of asteroid (8) Flora at a = 2.202 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 β > 0° (~68%; due to small inclinations of the family members, the majority of the objects with β > 0° are definitely prograde rotators, because their obliquities are between 0° and 90°) and lie to the right of the center of the family, confirming the presence of the Yarkovsky drift. Nine out of twelve asteroids with β < 0° can be found in Fig. 4 near to or to the left of the center of the family. The exceptions are the borderline asteroids (1703) Barry and (7360) Moberg, and asteroid (7169) Linda with a_p close to 2.25 AU (see Fig. 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 noncatastrophic collisions. The rotational state of another borderline asteroid (800) Kressmannia also does not agree with the Yarkovsky/YORP predictions, so it could be an interloper (or reoriented). The asteroid (7169) Linda classified as a member could still be an interloper, which was not detected by our methods for interloper removal, or could have recently been reoriented by a noncatastrophic collision. The typical timescale for a reorientation (Farinella et al. 1998, see Eq. (5)) of this 4km-sized asteroid with rotational period P = 27.9 h is τ_reor ~ 500 Myr, which is comparable to the age of the family. The depopulation of poles close to the ecliptic plane is also clearly visible.

The ν₆ resonance to the left of 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 the DAMIT database). In the Flora family, there are 14 more asteroids with β > 0° than with β < 0° (i.e, we have a 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 β < 0° 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 ν₆ resonance contributes to the NEA population only by retrograde rotators, and 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 to be observed in the Flora family, no corresponding clustering of poles of the prograde rotators is shown, particularly for 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 a = 2.874 AU. We identified 23 members (together with borderline cases) with determined shape models. The concept given by the Yarkovsky and YORP predictions also work among the Koronis family (asteroids with β < 0° lie to the left from the family center, asteroids with β > 0° to the right, see Fig. 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 s₆ and are referred to 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 probably affected during this catastrophic event and changed to a random state. Asteroid (263) Dresda could be randomly reoriented by a noncatastrophic 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 that agree with the Yarkovsky/YORP concept, which may support their membership in 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 again be interlopers or be 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 to 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 used 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 is sensitive to the orientation of the spin axis, which is in turn affected by the YORP effect and nondisruptive 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 used 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 $\mathrm{d}\mathit{N}\mathrm{\left(}\mathit{v}\mathrm{\right)}\mathrm{d}\mathit{v}\mathrm{=}{\mathit{C}}^{\mathrm{\prime }}\mathit{v}{\mathrm{(}{\mathit{v}}^{\mathrm{2}}\mathrm{+}{{\mathit{v}}_{\mathrm{esc}}^{\mathrm{2}}}^{\mathrm{)}}}^{\mathrm{-}\mathrm{(}\mathit{\alpha }\mathrm{+}\mathrm{1}\mathrm{)}\mathit{/}\mathrm{2}}\mathrm{d}\mathit{v,}$(1)with the exponent α as a free parameter, C′ a normalization constant and v_esc the escape velocity from the parent body, which is determined by its size D_PB and mean density ρ_PB as ${\mathit{v}}_{\mathrm{esc}}\mathrm{=}\sqrt{\mathrm{(}\mathrm{2}\mathit{/}\mathrm{3}\mathrm{)}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{PB}}}\hspace{0.17em}{\mathit{D}}_{\mathrm{PB}}\mathit{.}$ The distribution is usually cut at a selected maximum-allowed velocity v_max to prevent outliers. The initial velocities | v | of individual bodies are generated by a straightforward Monte-Carlo code, and the orientations of the velocity vectors v 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 e_i of the parent body, initial inclination i_i, as well as true anomaly f_imp and argument of perihelion ω_imp 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 were 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 timestep up to Δt = 91 d.

Our simulations included perturbations by four outer planets, with their masses, initial positions and velocities taken from the JPL DE405 ephemeris (Standish et al. 1997). We modified 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 parts of the main belt (for orbits with a > 2.5 AU and e < 0.6)⁸.

Synthetic proper elements are computed as follows. We first apply a Fourier filter to the (nonsingular) 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, which are 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 used a theory of Vokrouhlický (1998) and Vokrouhlický & Farinella (1999) for spherical objects (but the magnitude of the acceleration does not differ substantially for nonspherical shapes Vokrouhlický & Farinella 1998). The implementation within the SWIFT integrator is 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 $\begin{array}{ccc}\frac{\mathrm{d}\mathit{\omega }}{\mathrm{d}\mathit{t}}& \mathrm{=}& \mathit{c}{\mathit{f}}_{\mathit{i}}\mathrm{\left(}\mathit{ϵ}\mathrm{\right)}\mathit{,} \mathit{i}\mathrm{=}\mathrm{1}\mathit{...}\mathrm{200}\mathit{,}\\ \frac{\mathrm{d}\mathit{ϵ}}{\mathrm{d}\mathit{t}}& \mathrm{=}& \mathit{c}\frac{{\mathit{g}}_{\mathit{i}}\mathrm{\left(}\mathit{ϵ}\mathrm{\right)}}{\mathit{\omega }}\mathit{,}\end{array}$where f- and g-functions describing the YORP effect for a set of 200 shapes were calculated numerically by Čapek & Vokrouhlický (2004) with the effective radius R₀ = 1 km and the bulk density ρ₀ = 2500 kg/m³, located on a circular orbit with the semi-major axis a₀ = 2.5 AU. We assigned one of the artificial shapes (denoted by the index i) to each individual asteroid from our sample. The f- and g-functions were then scaled by the factor $\mathit{c}\mathrm{=}{\mathit{c}}_{\mathrm{YORP}}{\left(\frac{\mathit{a}}{{\mathit{a}}_{\mathrm{0}}}\right)}^{-2}{\left(\frac{\mathit{R}}{{\mathit{R}}_{\mathrm{0}}}\right)}^{-2}{\left(\frac{{\mathit{\rho }}_{\mathrm{bulk}}}{{\mathit{\rho }}_{\mathrm{0}}}\right)}^{-1}\mathit{,}$(4)where a, R, and ρ_bulk denote the semi-major axis, the radius, and the density of the simulated body, respectively, and c_YORP is a free scaling parameter reflecting our uncertainty in the shape models and the magnitude of the YORP torque, which depends 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 c_YORP = 0.2 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 c_YORP.

The differential Eqs. (2) and (3) are integrated numerically by a simple Euler integrator. The usual time step is Δt = 1000 yr.

Collisions and spin-axis reorientations.

We neglected the effect of disruptive collisions because we do not want to lose objects during the simulation, but we included spin axis reorientations caused by collisions. We use an estimate of the timescale by Farinella et al. (1998). ${\mathit{\tau }}_{\mathrm{reor}}\mathrm{=}\mathit{B}{\left(\frac{\mathit{\omega }}{{\mathit{\omega }}_{\mathrm{0}}}\right)}^{{\mathit{\beta }}_{\mathrm{1}}}{\left(\frac{\mathit{D}}{{\mathit{D}}_{\mathrm{0}}}\right)}^{{\mathit{\beta }}_{\mathrm{2}}}\mathit{,}$(5)where B = 84.5 kyr, β₁ = 5/6, β₂ = 4/3, D₀ = 2 m, and ω₀ corresponds to period P = 5 h. These values are characteristic of the main belt.

Mass shedding.

If the angular velocity approaches a critical value ${\mathit{\omega }}_{\mathrm{crit}}\mathrm{=}\sqrt{\frac{\mathrm{4}}{\mathrm{3}}\mathit{\pi G}{\mathit{\rho }}_{\mathrm{bulk}}}\mathit{,}$(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 P = 2π/ω to a random value from the interval (2.5,9) h. We also change the assigned shape to a different one, since any change in shape may result in a different YORP effect.

Fig. 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 ap vs. the pole latitude β plane. Left: objects larger than D > 30 km, which almost do not evolve in β. Right: objects with D ≤ 30 km, 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 D. The initial conditions for Flora correspond to an isotropic size-independent velocity field with α = 3.25 and vesc = 95 m s-1, and a uniform distribution of poles (i.e. sinβ). We increase the number of objects 10 times compared to the observed members of the Flora (Koronis and Eos as well) family 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 ep = 0.1to0.18, inclination sinIp = 0.05to0.13), because they are affected by strong mean-motion or secular resonances (ν6 in this case). Thermal parameters were set as follows: the bulk density ρbulk = 2500 kg m-3, the surface density ρsurf = 1500 kg m-3, the thermal conductivity K = 0.001 W m-1 K-1, the thermal capacity Ct = 680 J kg-1, the Bond albedo A = 0.1, and the infrared emissivity ϵ = 0.9. The time step for the orbital integration is dt = 91 days and dtspin = 103 yr for the (parallel) spin integration. The parameters for Koronis and Eos are chosen similarly, only for Koronis do we use vesc = 100 m s-1, and vesc = 225 m s-1 and ρsurf = 2500 kg m-3 for Eos.

Synthetic Flora, Koronis, and Eos families.

In Fig. 5 (top panel), we show a long-term evolution of the synthetic Flora family in the proper semi-major axis a_p vs. the pole latitude β plane for objects larger and smaller than 30 km. 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 (D < 30 km) 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 (corresponding to the proximity of the ecliptic plane for small inclinations). After the simulation at t = 1 Gyr, we observe a deficiency of asteroids with β > 0° to the left of the family center and a deficiency of asteroids with β < 0° to the right of the family center.

The asymmetry of the synthetic Flora family with respect to its center (Fig. 5) caused by the secular ν₆ resonance is obvious. The own-right hand quadrant (β < 0°, a_p > 2.202 AU) still contains many objects for t = 1 Gyr, because for some of them the evolution in β and a_p is rather small, and others were delivered to this quadrant by collisional reorientations.

The appearance of the evolved proper semi-major axis a_p vs. the pole latitude β 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 (D ~ 20−50 km), the evolution in a_p and β 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 the Koronis family.

We also checked the distributions of the proper eccentricities and inclinations of the synthetic Flora/Koronis/Eos objects for whether they (at least roughly) correspond to the observed family. However, the number of objects to compare is fairly low and seems insufficient for any detailed comparison of distributions in 3D space of proper elements (a_p, e_p, sinI_p).

Fig. 6 Time evolution of the metric (k2 + k4)/(k1 + k3), where ki correspond to the numbers of synthetic objects in quadrants i (i = 1,2,3,4) that are defined by the center of the family and value β = 0°, 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 (k2 + k4)/(k1 + k3) with the uncertainty interval.

Ages of the Flora, Koronis, and Eos families.

To quantitatively compare the simulation of the long-term evolution of the synthetic families in the proper semi-major axis a_p vs. the pole latitude β plane with the observation, we constructed the following metric: we divide the (a_p, β) plane into four quadrants defined by the center of the family and value β = 0° and compute the ratio (k₂ + k₄)/(k₁ + k₃), where k_i correspond to the numbers of synthetic objects in quadrants i (i = 1,2,3,4). In Fig. 6, we show the evolution of the metric (k₂ + k₄)/(k₁ + k₃) during the simulation of families Flora, Koronis, and Eos for all synthetic objects with D < 30 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 t = 2.5 Gyr and remains similar until the end of the simulation at t = 4 Gyr. Bottke et al. (2001) published the age t = (2.5 ± 1.0) Gyr for the Koronis family. Unfortunately, we cannot constrain the age of the Eos family from this simulation owing to objects with the relatively small evolution in a_p and β. The fit for the Flora family is not ideal, and the reason could be differences in the initial velocity field or the true anomaly f_imp of the impact. The best agreement is for the age t = (1.0 ± 0.5) Gyr, which is approximately in agreement with the dynamical age in Nesvorný et al. (2005): (1.5 ± 0.5) Gyr.

5. Conclusions

We have identified 152 asteroids for which we have convex shape models, and simultaneously the HCM identifies them as members of ten collisional families. Owing to a large number of expected interlopers in families Vesta and Nysa/Polana, we excluded these families from the study of the rotational properties. In the remaining sample of asteroids from eight families, we identified ~20% of objects that are interlopers or borderline cases (see Table 4). We used several methods, described in Sect. 2.1, for their identification. The borderline cases are still possible members of the families and thus were 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 Fig. 4, we can see fingerprints of families spreading in a and spin axis evolution due to Yarkovsky and YORP effects: Asteroids with β < 0° lie on the left side of the center of the family, and asteroids with β > 0° on the right side. The asymmetry with respect to the family centers is in most cases caused by various resonances that cut the families, and in the case of Themis family, a selection effect is responsible.

However, we did not observe perfect agreement with the Yarkovsky and YORP effects predictions. A few individual objects (eight) that have incompatible rotational states could (i) be incorrectly determined; (ii) be interlopers; (iii) have initial rotational states that only cause a small evolution in the (a_p, β) 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 fewer asteroids with β < 0° (~32%) than with β > 0° (~68%) are present. The secular ν₆ resonance is responsible for this strong deficit, because objects with β < 0° 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 did not find any analog of the Slivan states (observed in the Koronis family) among any other of the studied families.

We simulated a long-term evolution of the synthetic Flora, Koronis, and Eos families (Fig. 5) in the proper semi-major axis a_p vs the pole latitude β plane and compared the results with the properties of observed asteroid families. We obtained a good qualitative agreement between the observed and synthetic spin-vector distributions. For all three families, we computed evolution of the number of objects in the four quadrants of the families in the (a_p, β) diagram, and we estimated ages for families Flora (1.0 ± 0.5) Gyr and Koronis (2.5 to 4 Gyr) that agree with previously published values. However, we did not estimate the age of the Eos family due to a small evolution of the objects in the (a_p,β) 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.

Online material

Acknowledgments

The work of J.H. and J.D. has been supported by grants GACR P209/10/0537 and P209/12/0229 of the Czech Science Foundation, and the work of J.D. and M.B. by the Research Program MSM0021620860 of the Czech Ministry of Education. The work of M.B. has been also supported by grant GACR 13-013085 of the Czech Science Foundation.

References

All Tables

All Figures

Fig. 1 Dependence of the absolute magnitude H on the proper semi-major axis ap 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 (ap, H) borders of the family for different parameters C (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 ν6 resonance for the inclination typical for Flora family members and the horizontal arrow its approximate range.

In the text

	Fig. 2 Dependence of the color indexes a⋆ and i − z (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. There is a qualitative difference between C- and S-types asteroids.
In the text

Fig. 3 Dependence of the absolute magnitude H on the proper semi-major axis ap for the eight 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 (ap, H) borders (two thick lines) of each family for values of the parameter C 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). Borderline cases and interlopers are identified by several methods including the position in the (ap, H) diagram, and thus could also lie close to the center of the family (e.g., in the case of the Flora family).

In the text

Fig. 4 Dependence of the pole latitude β on the proper semi-major axis ap 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, whose 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, whose uncertainties are usually < 0.01 AU. The Eos family has an asymmetric V-shape (the (ap, H) border is asymmetric), which makes the center determination harder, so we marked two possible positions. One corresponds to the right (ap, H) border, the second to the left border. The uncertainties in β are usually 5–20°. In most cases, the value of |β| ≳ 30°, hence the quadrant to which the asteroid belongs (defined by the center of the family and the value β = 0°), is not changed.

In the text

Fig. 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 ap vs. the pole latitude β plane. Left: objects larger than D > 30 km, which almost do not evolve in β. Right: objects with D ≤ 30 km, 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 D. The initial conditions for Flora correspond to an isotropic size-independent velocity field with α = 3.25 and vesc = 95 m s-1, and a uniform distribution of poles (i.e. sinβ). We increase the number of objects 10 times compared to the observed members of the Flora (Koronis and Eos as well) family 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 ep = 0.1to0.18, inclination sinIp = 0.05to0.13), because they are affected by strong mean-motion or secular resonances (ν6 in this case). Thermal parameters were set as follows: the bulk density ρbulk = 2500 kg m-3, the surface density ρsurf = 1500 kg m-3, the thermal conductivity K = 0.001 W m-1 K-1, the thermal capacity Ct = 680 J kg-1, the Bond albedo A = 0.1, and the infrared emissivity ϵ = 0.9. The time step for the orbital integration is dt = 91 days and dtspin = 103 yr for the (parallel) spin integration. The parameters for Koronis and Eos are chosen similarly, only for Koronis do we use vesc = 100 m s-1, and vesc = 225 m s-1 and ρsurf = 2500 kg m-3 for Eos.

In the text

Fig. 6 Time evolution of the metric (k2 + k4)/(k1 + k3), where ki correspond to the numbers of synthetic objects in quadrants i (i = 1,2,3,4) that are defined by the center of the family and value β = 0°, 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 (k2 + k4)/(k1 + k3) with the uncertainty interval.

In the text