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r t i c l e i n f o a b s t r a c t

ticle history:ceived 26 April 2008vised 30 June 2008ailable online 3 August 2008

ywords:teroidsteroids, dynamicsotometry

Asteroid families, traditionally dened as clusters of objects in orbital parameter space, often havedistinctive optical colors. We show that the separation of family members from background interloperscan be improved with the aid of SDSS colors as a qualier for family membership. Based on an 88,000object subset of the Sloan Digital Sky Survey Moving Object Catalog 4 with available proper orbitalelements, we dene 37 statistically robust asteroid families with at least 100 members (12 familieshave over 1000 members) using a simple Gaussian distribution model in both orbital and color space.The interloper rejection rate based on colors is typically 10% for a given orbital family denition,with four families that can be reliably isolated only with the aid of colors. About 50% of all objectsin this data set belong to families, and this fraction varies from about 35% for objects brighter thanan H magnitude of 13 and rises to 60% for objects fainter than this. The fraction of C-type objects infamilies decreases with increasing H magnitude for H > 13, while the fraction of S-type objects abovethis limit remains effectively constant. This suggests that S-type objects require a shorter timescale forequilibrating the background and family size distributions via collisional processing. The size distributionvaries signicantly among families, and is typically different from size distributions for backgroundpopulations. The size distributions for 15 families display a well-dened change of slope and can bemodeled as a broken double power-law. Such broken size distributions are twice as likely for S-type familes than for C-type families (73% vs. 36%), and are dominated by dynamically old families. Theremaining families with size distributions that can be modeled as a single power law are dominated byyoung families (

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s of the families parent bodies (e.g., Marzari et al., 1999; Tangaal., 1999; Campo Bagatin and Petit, 2001; Michel et al., 2002;Ela and Brunini, 2007; Durda et al., 2007; and references

erein). Motivated by this rich information content, as well as theailability of new massive datasets, here we address the followingestions:

) What is the fraction of objects associated with families?) Do objects that are not associated with families show any he-liocentric color gradient?

) Do objects that are not associated with families have uniformsize distribution independent of heliocentric distance?

) Do objects associated with families have a different size distri-bution than those that are not in families?

) Do different families have similar size distributions?) Is the size distribution related to family color and age?

These questions have already been addressed numerous times.g., Mikami and Ishida, 1990; Cellino et al., 1991; Marzari et al.,95; Zappal et al., 1995; Morbidelli et al., 2003; Nesvorn et al.,05). The main advantages of the size distribution analysis pre-nted here, when compared to previous work, are

The large sample size: we use a set of 88,000 objects forwhich both SDSS colors and proper orbital elements computedby Milani and Kneevic (1994) are available;

Simple and well-understood selection effects: the SDSS sampleis >90% complete without a strong dependence on magnitude(Juric et al., 2002);

Improved faint limit: the sample of known objects listed inthe latest ASTORB le from January 2008 to which SDSS ob-servations are matched is now essentially complete to r 19.5(corresponding to H 17 in the inner belt, and to H 15 inthe outer belt);

Improved family denitions due to color constraints (rejectionof interlopers and separation of families overlapping in orbitalspace);

Improved accuracy of absolute magnitudes derived using SDSSphotometry, as described below.

The SDSS asteroid data are described in Section 2, and in Sec-n 3 we describe a novel method for dening asteroid familiesing both orbital parameters and colors. Analysis of the size dis-ibution for families and background objects is presented in Sec-n 4, and we summarize our results in Section 5.

SDSS observations of moving objects

1. An overview of SDSS

The SDSS is a digital photometric and spectroscopic surveyhich will cover about one quarter of the Celestial Sphere in theorth Galactic cap, and produce a smaller area (300) but mucheper survey in the Southern Galactic hemisphere (Stoughton et., 2002; Abazajian et al., 2003, 2004, 2005; Adelman-McCarthyal., 2006). SDSS is using a dedicated 2.5 m telescope (Gunn et

., 2006) to provide homogeneous and deep (r < 22.5) photom-ry in ve bandpasses (Fukugita et al., 1996; Gunn et al., 1998;ith et al., 2002; Hogg et al., 2001; Tucker et al., 2006) repeatable0.02 mag (root-mean-square scatter, hereafter rms, for sourcest limited by photon statistics; Ivezic et al., 2003) and with a ze-point uncertainty of 0.020.03 mag (Ivezic et al., 2004). Thex densities of detected objects are measured almost simultane-sly in ve bands (u, g , r, i, and z) with effective wavelengths

3540, 4760, 6280, 7690, and 9250 . The large survey sky cov-age will result in photometric measurements for well over 100

ob2SDSS MOC 4 139

illion stars and a similar number of galaxies.1 The complete-ss of SDSS catalogs for point sources is 99.3% at the brightd and drops to 95% at magnitudes of 22.1, 22.4, 22.1, 21.2, and.3 in u, g , r, i and z, respectively. Astrometric positions are accu-te to better than 0.1 arcsec per coordinate (rms) for sources with< 20.5 (Pier et al., 2003), and the morphological informationom the images allows reliable stargalaxy separation to r 21.5upton et al., 2002; Scranton et al., 2002). A compendium of otherchnical details about SDSS can be found on the SDSS web sitettp://www.sdss.org), which also provides interface for the publicta access.

2. SDSS moving object catalog

The SDSS, although primarily designed for observations of ex-agalactic objects, is signicantly contributing to studies of thelar System objects because asteroids in the imaging survey mustexplicitly detected and measured to avoid contamination of the

mples of extragalactic objects selected for spectroscopy. Prelim-ary analysis of SDSS commissioning data by Ivezic et al. (2001)owed that SDSS will increase the number of asteroids with accu-te ve-color photometry by more than two orders of magnitude,d to a limit about ve magnitudes fainter (seven magnitudeshen the completeness limits are compared) than previous multi-lor surveys (e.g. The Eight Color Asteroid Survey; Zellner et al.,85). For example, a comparison of SDSS sample with the Smallain-Belt Asteroid Spectroscopic Survey (Xu et al., 1995; Bus andnzel, 2002a, 2002b) is discussed in detail by Nesvorn et al.005).SDSS Moving Object Catalog2 (SDSS MOC) is a public, value-ded catalog of SDSS asteroid observations (Ivezic et al., 2002a).includes all unresolved objects brighter than r = 21.5 and withserved angular velocity in the 0.050.5/day interval. In additionproviding SDSS astrometric and photometric measurements, allservations are matched to known objects listed in the ASTORBe (Bowell, 2001), and to a database of proper orbital elementsilani, 1999; Milani and Kneevic, 1994), as described in detail byric et al. (2002). Juric et al. (2002) determined that the catalogmpleteness (number of moving objects detected by the softwareat are included in the catalog, divided by the total number ofoving objects recorded in the images) is about 95%, and its con-mination rate is about 6% (the number of entries that are notoving objects, but rather instrumental artifacts). The most recentSS MOC 4th data release contains measurements for 471,000oving objects. A subset of 220,000 observations were matched104,000 unique objects listed in the ASTORB le (Bowell, 2001).e large sample size increase between the rst and fourth releaseSDSS MOC is summarized in Fig. 1. The object counts in bothleases are well described by the following function (Ivezic et al.,01)

N

r= n(r) = no 10

ax

10bx + 10bx , (1)

here x = r rC , a = (k1 + k2)/2, b = (k1 k2)/2, with k1 andthe asymptotic slopes of log(n) vs. r relations. This functionoothly changes its slope around r rC , and we nd best-t val-s rC = 18.5, k1 = 0.6 and k2 = 0.2. The normalization constant,, is 7.1 times larger for SDSS MOC 4 than for the rst release. Indition to this sample size increase, the faint completeness limitr objects listed in ASTORB also improved by a about a magnitude,r 19.5 (the number of unique ASTORB objects increased fromThe recent Data Release 6 lists photometric data for 287 million unique objectsserved in 95832 of sky; see http://www.sdss.org/dr6/.http://www.sdss.org/dr6/products/value_added/index.html.

14 198

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g. 1. An illustration of the improvements in sample size between the rst ande fourth release of SDSS Moving Object Catalog. Symbols with (statistical) errorrs show differential counts for moving objects listed in the rst release from02 (dots: all objects detected by SDSS; circles: identied in ASTORB le) and theurth release from 2008 (triangles: all SDSS; squares: in ASTORB). Note that, in ad-tion to a sample size increase of about a factor of 7, the faint completeness limitr objects listed in ASTORB also improved by a about a magnitude (the numberunique ASTORB objects increased from 11,000 to 100,000). The dashed linesow a double-power law t described in text, with the d log(N)/dr slope changingm 0.60 at the bright end to 0.20 at the faint end. For illustration, the dot-dashedes shows a single power-law with a slope of 0.60.

11,000 to 100,000). Above this completeness limit, the SDSSOC lists color information for 33% of objects listed in ASTORB.The quality of SDSS MOC data was discussed in detail by Ivezical. (2001) and Juric et al. (2002), including a determination ofe size and color distributions for main-belt asteroids. An anal-is of the strong correlation between colors and the main-beltteroid dynamical families was presented by Ivezic et al. (2002b).dicke et al. (2004) reported a correlation between the family dy-mical age and its mean color for S-type families, and proposedat it is due to space weathering effects. This correlation was fur-er discussed and extended to C-type families by Nesvorn et al.005). Multiple SDSS observations of objects with known orbitalrameters can be accurately linked, and thus SDSS MOC also con-ins rich information about asteroid color variability, discussed intail by Szab et al. (2004) and Szab and Kiss (2008).

3. Errors in H magnitudes listed in the ASTORB le

As pointed out by Juric et al. (2002), there is a large systematicscrepancy between the absolute magnitudes listed in ASTORB led values implied by SDSS measurements. The latter are com-ted as

corr = HASTORB + V cV , (2)here HASTORB is the ASTORB value, cV is the apparent magnitudeJohnson system computed from information listed in ASTORB asscribed in Juric et al. (2002), and V is the observed magnitudenthesized from SDSS g and r magnitudes (SDSS MOC entries 47,, and 32, respectively).This discrepancy persists in the 4th release of SDSS MOC, as

ustrated in Fig. 2. The mean difference between H measured

SDSS and the values from ASTORB is 0.23 mag, and the root-

ean-scatter is 0.30 mag. The best-t shown in Fig. 2 implies that

4

wsu(2008) 138155

g. 2. A comparison of asteroid absolute magnitude, H , inferred from SDSS mea-rements, and the value listed in ASTORB le for 133,000 observations of about,000 unique objects observed at phase angles between 3 and 15 . The histogramows the data distribution [H = HSDSS HASTORB = Hcorr H , see Eq. (2)] ande dot-dashed line is a best t. The best t is a linear combination of three Gaus-ns: two (which simulate asteroid variability) are centered on 0.02, have widths0.08 and 0.20 mag, and have relative normalizations of 13% and 18%, respectively.eir sum is shown by the dashed line centered on H = 0.02. The third Gaus-n (which accounts for a large number of objects with bad photometry) has aidth of 0.28 mag and is shown by the dashed line centered on H = 0.33. Thatabout 69% of H measurements listed in ASTORB le are systematically too bright0.33 mag.

certainty of Hcorr is about 0.16 mag, with a negligible system-ic error (the latter is expected to be about 0.020.03 mag due tocertainties in absolute photometric calibration of SDSS imagingta; see Section 2.1). It is likely that this uncertainty is dominatedmagnitude variation due to rotation. The magnitude offset of

33 mag for 70% of measurements implied by the best t coulddue to measurements reported by LINEAR. A similar magnitude

fset at the faint end is a known problem in LINEAR calibration,d is currently being addressed with the aid of new calibrationtalogs (J.S. Stuart, private communication).Since the random error in H is twice as large as for Hcorr, weopt Hcorr in the remainder of this work. For a detailed analysisthis magnitude offset problem,3 we refer the reader to Juric et. (2002).

The asteroid families in SDSS MOC

The contrast between dynamical asteroid families and the back-ound population is especially strong in the space dened byoper orbital elements. These elements are nearly invariants ofotion and are thus well suited4 for discovering objects with com-on dynamical history (Valsecchi et al., 1989; Milani and Kneevic,92).The value of SDSS photometric data becomes particularly ev-

ent when exploring the correlation between colors and orbitalrameters for main-belt asteroids. Ivezic et al. (2002b) demon-rated that asteroid dynamical families, dened as clusters in or-tal element space, also strongly segregate in color space. We use

IAU Commission 15 has formed a Task Group on Asteroid Magnitudes to addressis problem, see http://www.casleo.gov.ar/c15-wg/index-tgh.html.

The current asteroid motion is usually described by osculating orbital elements

hich vary with time due to perturbations caused by planets, and are thus lessitable for studying dynamical families.

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g. 3. A plot of the color distribution in a and i z of 45,087 unique objects listedboth the SDSS MOC 4 and ASTORB le, and that have Hcorr < 16. The approximateundaries of three spectral classes are marked, and used in labeling family type.e color-coding scheme dened here is used in Figs. 46.

g. 4. A plot of the proper a vs. sin(i) for the same objects as shown in Fig. 3. Thelor of each dot is representative of the objects color measured by SDSS, accordingthe color scheme dened in Fig. 3. The three main regions of the belt, dened byong Kirkwood gaps, are marked.

e technique developed by Ivezic et al. (2002b) to visualize thisrrelation for 45,000 unique main-belt asteroids with Hcorr < 16ted in SDSS MOC 4 (Figs. 3 and 4). The asteroid color distributionSDSS bands shown in Fig. 3, and its comparison to traditionalxonomic classications, is quantitatively discussed by Ivezic et. (2001) and Nesvorn et al. (2005).A striking feature of Fig. 4 is the color homogeneity and dis-ctiveness displayed by asteroid families. In particular, the threeajor asteroid families (Eos, Koronis, and Themis), together with

e Vesta family, correspond to taxonomic classes K, S, C, and V,spectively (following Burbine et al., 2001, we assume that the 5SDSS MOC 4 141

s family is associated with the K class). Their distinctive opticallors indicate that the variations in surface chemical compositionithin a family are much smaller than the compositional differ-ces between families, and vividly demonstrate that asteroids be-nging to a particular family have a common origin.

1. A method for dening families using orbits and colors

Traditionally, the asteroid families are dened as clusters ofjects in orbital element space. The most popular methods foruster denition are the hierarchical clustering and the waveletalysis (Zappal et al., 1995; Nesvorn et al., 2005). Given therong color segregation of families, it is plausible that SDSS colorsn be used to improve the orbital family denitions and minimizee mixing of candidate family members and background popula-n.The SDSS colors used to construct Figs. 3 and 4 are the i zlor and the so-called a color, dened in Ivezic et al. (2001) as

0.89(g r) + 0.45(r i) 0.57. (3)e a color is the rst principal component of the asteroid colorstribution in the SDSS r i vs. g r colorcolor diagram (foransformations between the SDSS and Johnson system see Ivezical., 2007). Similar principal component analysis was also per-rmed by Roig and Gil-Hutton (2006), who considered the distri-tion of taxonomic classes (especially V-type asteroids) in SDSSinciple components by comparing directly to spectroscopic data,d by Nesvorn et al. (2005), whose two principal componentse well correlated with the a and i z colors (we nd that= 0.49PC1 0.16 reproduces the measured a values with ans of 0.026 mag for objects with r < 18). The principal colors de-

ved by Nesvorn et al. (2005) include the u band, which becomesisy at the faint end. Given that the completeness of the knownject catalog (ASTORB) reaches a faint limit where this noise be-mes important, we use the a and i z colors to parametrizee asteroid color distribution. Therefore, the family search is per-rmed in a ve-dimensional space dened by these two colorsd the proper semi-major axis, sine of the inclination angle andcentricity.There are numerous techniques that could be used to search for

ustering in a multi-dimensional space (e.g. Zappal et al., 1995;esvorn et al., 2005; Carruba and Michtchenko, 2007). They differthe level of supervision and assumptions about underlying datastribution. Critical assumptions are the distribution shape forch coordinate, their correlations, and the number of independentmponents. We utilize three different methods, one supervisedd two fully automatic. The automatic unsupervised methods aresed on the publicly available code FASTMIX5 by A. Moore andcustom-written code based on Bayesian non-parameteric tech-ques (Ferguson, 1973; Antoniak, 1974).In the supervised method (1) families are manually identiedd modeled as orthogonal (i.e. aligned with the coordinate axes)aussian distributions in orbital and color space. The two unsu-rvised methods (2 and 3) also assume Gaussian distributions,t the orientation of individual Gaussians is arbitrary, and thetimal number of families is determined by the code itself. Allree methods produce fairly similar results and here we describely the supervised method (1), and use its results in subsequentalysis. The two unsupervised methods produce generally similarsults for the objects associated with families, but tend to over-assify the background into numerous (5060) small families, andeir details and results are not presented thoroughly in this paper.See http://www.cs.cmu.edu/~psand.

14 198

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demesisj 2.733 0.018 0.085 0.003 0.087 0.003 0.080 0.037 0.040 0.089 2.50 2.50 2.50 2.82dia 2.737 0.048 0.105 0.005 0.029 0.019 0.160 0.052 0.010 0.096 2.50 2.50 2.50 2.82duaj 2.740 0.035 0.090 0.005 0.042 0.011 0.050 0.044 0.030 0.082 2.70 3.50 2.50 2.82lorisj 2.744 0.040 0.152 0.005 0.254 0.008 0.060 0.030 0.050 0.052 3.00 3.00 2.50 2.82erxiaj 2.751 0.036 0.087 0.002 0.136 0.006 0.090 0.037 0.070 0.096 3.00 3.00 2.50 2.82onj 2.768 0.029 0.159 0.004 0.134 0.020 0.100 0.044 0.020 0.067 3.00 3.00 2.50 2.82niaj 2.761 0.046 0.063 0.013 0.070 0.012 0.090 0.059 0.000 0.104 2.20 2.60 2.50 2.82raj 2.787 0.019 0.136 0.002 0.195 0.003 0.120 0.037 0.050 0.052 3.50 2.80 2.50 2.82asiliaj 2.851 0.009 0.259 0.002 0.123 0.002 0.050 0.037 0.050 0.059 3.50 3.20 2.82 3.05ronisj 2.904 0.046 0.037 0.002 0.053 0.019 0.090 0.044 0.020 0.059 3.00 3.00 2.82 3.05sj 3.021 0.059 0.175 0.012 0.073 0.015 0.050 0.044 0.030 0.052 3.50 3.00 2.91 3.60relaj 3.122 0.032 0.285 0.007 0.195 0.016 0.150 0.052 0.080 0.067 3.00 2.50 2.91 3.60emisj 3.126 0.071 0.024 0.008 0.153 0.019 0.120 0.037 0.010 0.059 2.50 2.50 2.91 3.60gieaj 3.144 0.066 0.090 0.008 0.137 0.019 0.110 0.037 0.010 0.082 2.50 2.50 2.91 3.60iaohuaj 3.149 0.015 0.177 0.004 0.197 0.005 0.070 0.037 0.080 0.067 3.00 2.50 2.91 3.60sula 3.156 0.060 0.279 0.012 0.090 0.028 0.070 0.044 0.060 0.067 2.50 3.00 2.91 3.60ritasj 3.168 0.007 0.160 0.003 0.063 0.004 0.080 0.030 0.050 0.052 3.00 3.50 2.91 3.60eobalda 3.170 0.014 0.247 0.006 0.251 0.011 0.160 0.037 0.040 0.089 2.75 3.50 3.00 3.50The family name (the lowest-numbered member), sorted by semi-major axis.The mean proper semi-major axis (AU) and its Gaussian dispersion ( ) adopted for this family.The mean sine of proper inclination and its Gaussian dispersion adopted for this family.The mean proper orbital eccentricity and its Gaussian dispersion adopted for this family.The mean a color (see text for denition) and its Gaussian dispersion adopted for this family.

f The mean i z color and its Gaussian dispersion adopted for this family.The maximum deviation in orbital space from the adopted mean values, in units of adopted dispersions (see text).The maximum deviation in color space from the adopted mean values, in units of adopted dispersions (see text).

i The minimum and maximum semi-major axis adopted for this family.j Family matched to Nesvorn et al. (2005).

We select from the SDSS MOC 4 the rst observation of all ob-cts identied in ASTORB, and for which proper orbital elementse also available, resulting in 87,610 objects. Among these, theree 45,502 objects with Hcorr < 16. We split the main sample intoree subsets using semi-major axis ranges dened by the majorrkwood gaps (see Fig. 4): inner (a < 2.50), middle (2.50 < a 2.82) belt. For each subset, we produce thevs. sin(i) diagrams color-coded analogously to Fig. 4, and useem to obtain preliminary identication of asteroid families inth orbital and color space. Approximate rectangular bounds aresigned to these visually identied families, from which medianentroid) and standard deviation, , for the three orbital elementse estimated. Using these estimates, for each asteroid we computestance in orbital space from a given family centroid as

orbit =d2a + d2e + d2i , (4)

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Histograms in Dorbit were used to determine a preliminaryvalue of Orb, the maximum orbital distance from a family cen-troid for an object to be ascribed family membership. These ini-tial Orb are determined from the differential Dorbit distribution asthe position on the rst local minimum (the further rise of countswith increasing Dorbit is due to the background objects and otherfamilies). The object distribution in the Dorbit vs. a and Dorbit vs.(i z) diagrams was used to rst dene approximate rectangularbounds for each family, and then to compute the color centroidand standard deviation in a and (i z) for each candidate family.In order to use color as a family discriminator, we dene analo-gously to orbital elements

Dcolor =d21 + d22, (8)

where2 A. Parker et al. / Icarus

ble 1opted family denitions in the orbital and color space

milya apb ab ip c i c epd ed ae

raj 2.272 0.065 0.084 0.023 0.133 0.027ptistina 2.277 0.035 0.096 0.006 0.143 0.006 staj 2.350 0.072 0.114 0.009 0.100 0.015igonej 2.371 0.028 0.087 0.006 0.207 0.007 cCuskey 2.374 0.065 0.049 0.008 0.151 0.015 terpe 2.374 0.076 0.017 0.007 0.185 0.011sa-Polanaj 2.388 0.044 0.042 0.005 0.184 0.019assaliaj 2.405 0.029 0.025 0.002 0.163 0.006dree 2.405 0.034 0.084 0.005 0.170 0.013utonia 2.574 0.039 0.070 0.021 0.169 0.049taj 2.584 0.024 0.131 0.004 0.173 0.005ariaj 2.612 0.053 0.254 0.008 0.090 0.021itidika 2.618 0.067 0.216 0.013 0.244 0.023 nomiaj 2.632 0.073 0.227 0.011 0.150 0.018isaj 2.650 0.035 0.040 0.004 0.178 0.007 eonaj 2.660 0.038 0.202 0.004 0.168 0.007 noj 2.664 0.026 0.230 0.004 0.234 0.005oleaj 2.671 0.033 0.066 0.004 0.190 0.005 nan 2.694 0.069 0.045 0.011 0.057 0.013a

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ae i zf iz f Orbg Colh amini amaxi

0 0.044 0.040 0.059 2.60 2.75 1.50 2.500 0.030 0.050 0.067 2.20 3.00 1.50 2.500 0.052 0.270 0.089 2.75 3.50 1.50 2.500 0.037 0.060 0.082 2.50 3.10 1.50 2.500 0.052 0.010 0.074 2.50 2.20 1.50 2.500 0.052 0.040 0.104 2.00 3.00 1.50 2.500 0.052 0.040 0.067 3.00 3.50 1.50 2.500 0.052 0.040 0.082 3.50 2.50 1.50 2.500 0.052 0.030 0.089 3.00 3.00 1.80 2.500 0.052 0.030 0.089 2.20 2.60 2.50 2.820 0.037 0.040 0.067 2.50 3.00 2.50 2.820 0.037 0.010 0.059 2.50 3.00 2.50 2.820 0.044 0.030 0.067 2.50 3.00 2.50 2.820 0.044 0.020 0.052 2.50 3.00 2.50 2.820 0.052 0.050 0.067 3.00 3.00 2.50 2.820 0.037 0.060 0.059 2.60 2.80 2.50 2.820 0.052 0.030 0.082 3.00 3.00 2.50 2.820 0.044 0.040 0.096 4.00 3.00 2.50 2.820 0.044 0.010 0.111 2.00 2.60 2.50 2.82= (acentroid aobject)

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= (izcentroid izobject)iz

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e use histograms in Dcolor to dene Col, the maximum colorstance from a family centroid for an object to be ascribed fam-membership. Fig. 8 illustrates the Dorbit and Dcolor histograms

r the Vesta and Baptistina families, and the bottom panels showe distributions of family and background objects in the Dorbit vs.color plane.In cases of families which formed from the disruption of a dif-

rentiated parent body, the color distribution might not providewell-dened morphology. However, we did not nd any casehere a subset of objects selected using Dorbit did not result in onetwo well dened color distributions. Nevertheless, it is possibleat a small fraction of objects could be rejected from a family duedifferent colors than the majority of other members.All objects that have both Dorbit Orb and Dcolor Col are

en considered to be a family member. With a given estimate ofmily populations, this procedure is iterated and all parameterse rened. It typically takes one to two iterations to converge. Allonverged families are removed from the sample and the pro-ss was repeated until there were no family candidates with morean 100 members. This condition is the result of the require-ent that the statistical errors for the slope of absolute magnitudestribution of the families are smaller (typically 0.010.02) thanausible systematic errors (0.030.04), as discussed below.Using this procedure, we found 37 families which account for% of all objects. Their dening parameters are listed in Table 1.ditional three candidate families that had fewer than 100 mem-rs in the last iteration were discarded (see last three entriesTable 2). The family names were determined by comparison

ith Zappal et al. (1995) and Nesvorn et al. (2005), and, whencorresponding family was found, by searching for the lowest-mbered asteroid in the Milani and Kneevic (1994) catalog ofoper orbital elements. In a small number of cases, it is possibleat the name-giving object has a color that is inconsistent withe majority of objects in the family. We ignored such cases andtained nomenclature from Zappal et al. (1995) and Nesvorn et. (2005) in order to ease the comparison. That is, the deningoperties of families are the orbital and color parameters listed inble 1, rather than their names.The separation of the main-belt asteroids into families and theckground objects is illustrated in Figs. 5 and 6. While there isme residual structure in the background, both in color and or-tal space, it is much less prominent than the structure displayedidentied families.

2. The comparison of resulting families with previous work

Important questions about the quality of family associations de-ved here, that are relevant for subsequent analysis, are

The contamination: are all families robust? The completeness: are any families missed? What is the impact of variations in adopted family deni-

tions on the resulting family properties such as the numberof members and their size distribution?

In this section we address the rst two questions and discusse third one below (Section 4.3).It is unlikely that the derived sample of 37 families containsy spurious family because of the conservative requirement thatfamily must include at least 100 members. Indeed, we have re-cted three familes that are likely real because they had fewerndidate members (5, 90 and 46). The very narrow color distri-

tion of all selected families provides another argument for theirbustness.

lissiSDSS MOC 4 143

ble 2e median orbital parameters and SDSS colors for detected families and rejectedmps (last three)

milya Nb ap c ipd epe af i zgra 6164 2.28 0.08 0.13 0.13 0.05ptistina 310 2.28 0.10 0.14 0.04 0.03sta 3793 2.35 0.11 0.10 0.12 0.32cCuskey 1043 2.36 0.05 0.15 0.12 0.01igone 307 2.37 0.09 0.21 0.09 0.05terpe 387 2.38 0.02 0.18 0.09 0.03sa-Polana 2928 2.39 0.04 0.18 0.13 0.04dree 649 2.40 0.08 0.17 0.13 0.02assalia 730 2.41 0.03 0.16 0.07 0.04utonia 3405 2.57 0.07 0.16 0.12 0.04ta 225 2.59 0.13 0.17 0.08 0.04aria 1315 2.61 0.25 0.09 0.12 0.02itidika 698 2.61 0.22 0.24 0.11 0.03nomia 2995 2.63 0.23 0.15 0.13 0.03isa 185 2.65 0.04 0.18 0.08 0.05eona 428 2.66 0.20 0.17 0.11 0.05no 354 2.66 0.23 0.23 0.07 0.03olea 172 2.66 0.07 0.19 0.09 0.04nan 624 2.67 0.04 0.06 0.11 0.02mesis 129 2.73 0.09 0.09 0.09 0.02dia 598 2.74 0.11 0.03 0.16 0.01dua 442 2.75 0.09 0.04 0.05 0.05erxia 252 2.75 0.09 0.14 0.08 0.07on 914 2.76 0.16 0.13 0.10 0.03loris 121 2.76 0.15 0.25 0.06 0.05nia 1106 2.76 0.06 0.07 0.08 0.01ra 248 2.79 0.14 0.20 0.12 0.04asilia 127 2.85 0.26 0.12 0.05 0.03ronis 1267 2.90 0.04 0.05 0.09 0.02s 4367 3.04 0.18 0.07 0.04 0.03rela 411 3.12 0.29 0.20 0.15 0.08emis 1073 3.13 0.02 0.15 0.11 0.01giea 1076 3.15 0.09 0.13 0.11 0.01iaohua 150 3.15 0.18 0.20 0.07 0.05sula 644 3.15 0.28 0.09 0.06 0.06ritas 250 3.17 0.16 0.06 0.08 0.05eobalda 100 3.17 0.25 0.25 0.15 0.01h 90 2.87 0.11 0.05 0.17 0.04h 5 2.77 0.08 0.09 0.08 0.14h 46 2.78 0.06 0.07 0.00 0.25The family name (the lowest-numbered member).The number of objects in SDSS MOC 4 associated with this family.The median proper semi-major axis (AU).The median sin of proper inclination.The median proper orbital eccentricity.

f The median a color (see text for denition).The median i z color.Rejected clumps with less than 100 members.

Our method of iterative removal of identied families and themultaneous use of colors and orbital parameters to search formaining families appears very robust when compared to two au-mated methods also employed. Nevertheless, given that all threeethods from this work assume Gaussian distributions of colorsd orbital parameters, it is prudent to compare our family listith families obtained by other means, such as the hierarchicalustering method. We have cross-correlated the list of 37 familiestermined here with the list of 41 families obtained by Nesvornal. (2005) using the hierarchical clustering method. Nesvorn et. (2005) based their study on a larger sample of objects withoper orbital elements (106,000 vs. 88,000 analyzed here;te that the latter sample extends to 1.5 mag fainter ux limitt is smaller because it includes only objects observed by SDSS),d did not place a requirement on the minimum number of ob-cts per family. Therefore, it is plausible that families missed byr selected method may be present in their list.Out of 41 families from the Nesvorn et al. (2005) list, 27 are

ted in Table 2. This is encouraging level of agreement given thegnicant difference in applied methodology. We examined in de-

14

Fipapa4 A. Parker et al. / Icarus 198 (2008) 138155g. 5. Illustration of the decomposition of the main-belt asteroid population into families and background objects in proper a vs. sin(i) (left panels) and proper a vs. e (rightnels). The top panels show all (background and family) objects in the data subset. The two middle panels show objects from 37 identied families, and the bottom twonels show the background population.

the

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inonrere(BMfahanuloobg. 6. Analogous to Fig. 5, except that the top three panels show the e vs. sin(i) distribution for the three main regions dened by strong Kirkwood gaps (a < 2.5 left,5 < a < 2.82 middle, 2.82 < a < 3.5 right; see Fig. 4). The middle row shows family members (with several families of note labeled), and the bottom row shows theckground population. For a high-resolution version of this gure with complete labeling, see http://www.astro.washington.edu/ivezic/sdssmoc/sdssmoc.html.

il each of the fourteen Nesvorn et al. (2005) families missingom our list and searched for them in the sample of backgroundjects. We did not nd any candidate family that included morean 100 members, though most appear to be real clusters.Among the ten families from our list that we could not identifythe Nesvorn et al. (2005) list, three were detected by at leaste method discussed by Zappal et al. (1995), and thus are likelyal (Euterpe, Teutonia, and Henan). Of the remaining seven, thecognition of four families was greatly aided by color informationaptistina from Flora, Mitidika from Juno, Lydia from Padua, andcCuskey from Nysa-Polana). It is likely that the remaining threemilies were not detected by Nesvorn et al. (2005) because theyve steep absolute magnitude distributions and thus only a smallmber of members were present in the (older) version of cata-

We further compared our list of families to those presented inMoth-Diniz et al. (2005), who used spectroscopic measurementsto probe asteroid family structure. Of their 21 nominal families, allbut three (Renate, Hoffmeister, and Meliboea) can be matched tofamilies detected here. Of the numerous smaller (many with fewerthan 100 members) clumps they identify, 10 correspond to fam-ilies listed here, while eight families (Teutonia, Mitidika, Euterpe,Andree, Lydia, Ursula, Lyxaohua and Theobalda) present in our listdo not appear in Moth-Diniz et al. (2005). They resolved the re-maining family in our data set, Flora, into a number of smallerclumps that merged into a single family at high cutoff velocities.

A good example of the separation of dynamically mixed fami-lies using SDSS colors is provided by the small family Baptistinawhich is buried within the Flora family. Figs. 7 and 8 (rightAsteroid families ing used by Nesvorn et al. (2005). For example, among the 3405jects in the Teutonia family, only 37 have Hcorr < 14.

paidSDSS MOC 4 145nels) illustrate how different a color distributions enable theentication of 5% of the objects nominally assigned Flora fam-

14 198

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g. 7. An illustration of color differences for families with practically identical or-tal parameter distributions. The dashed histogram shows the a color distributionr 6164 candidate members of the Flora family. The solid histogram shows the alor distribution for 310 candidate members of the Baptistina family, which is eas-separated from the Flora family thanks to the SDSS color information.

(a = 0.13) membership by their orbital parameters as beingembers of the Baptistina family (a = 0.04). While initiallyzzled why a similar color-aided search by Nesvorn et al. (2005)d not yield any additional families, we have found that all color-parated families extracted here are dominated by faint objectsd thus may not have been present in suciently large numbersthe older catalog. We conclude that all the families discussedre are robustly detected, and that it is very unlikely that weissed any family with more than 100 hundred members. It is,wever, possible that the background is composed of numerousmilies dominated by small objects that are not discernible withe presently available catalog. Hence, the fraction of 50% of ob-cts associated with families is only a lower limit (this fraction isfunction of object size, as discussed below).Fig. 11 shows the color dependence of the family and back-

ound populations on semi-major axis. We note that the mediancolor for the background population becomes bluer as the semi-ajor axis increases in the same way as the median color for themily population.

The size distribution of asteroid populations

The known object catalog is complete to r 19.5; above thisit an ASTORB entry is found for practically every SDSS movingject. Depending on the distance and orientation of the observedject, this apparent magnitude limit corresponds to a complete-ss limit ranging from H 17 in the inner belt to H 15 in theter belt. Brighter than these limits, selection function is essen-ally equal to 1 for the purposes of this work (SDSS managed toserve only about 1/3 of all ASTORB objects, but this is essen-ally a random selection without an impact on derived absoluteagnitude distribution of individual families). These simple selec-on effects allow us to derive robust family size distributions tory small size limits. The only other study of family size distribu-ons that approached the same size limit is that of Morbidelli et. (2003), who had to introduce an ad hoc indirect correction for

lection effects in the known object catalog (which is supportedour analysis, as discussed below).

Thtu(2008) 138155

The transformation between the asteroid absolute magnitude,, and its effective diameter, D , requires the knowledge of thesolute visual albedo pV ,

= 18.1 2.5 log(pV0.1

) 5 log(D/1 km). (11)

e absolute albedo is not known for the overwhelming major-y of objects in our sample. However, the albedo is known to berongly correlated with colors (Zellner, 1979; Shoemaker et al.,79; Delbo, 2004); for example, the C-like asteroids (a < 0) havemedian albedo of 0.04 and the S-like asteroids have a medianbedo of 0.14. Given that the color variations within a family areall, it seems plausible that the albedo variation within a familyalso small (this is supported by the data compiled by Tedesco et., 2005; see their Table 7). With this assumption, the shapes ofe absolute magnitude distribution and the distribution of log(D)e the same. Hereafter, we will interchangeably use the absoluteagnitude distribution and size distribution, where the latterplies the distribution of log(D). For simplicity, in the remainderanalysis we only use the differential distributions.If the differential absolute magnitude distribution, n(H) =

N/H , can be described by

g(n) = Const.+ H, (12)d the albedos of objects within a given family or population aremilar, then it follows from Eq. (11) that the differential size distri-tion can be described as n(D) Dq , with the size distributiondex

= 5 + 1. (13)hile the absolute magnitude distributions derived here oftennnot be described by a single power-law, Eq. (13) is still usefulr locally relating the slope of the H distribution to the slope ofplied (differential) D distribution. For example, a model basedan equilibrium cascade in self-similar collisions developed by

ohnanyi (1969) predicts q = 3.5 and = 0.5.

1. The comparison of size distributions for families and background

We compare the size distributions for the family populationd for the background, separately for the three regions dened bymi-major axis. The differential absolute magnitude distributionse shown in Fig. 9. To aid the comparison of different panels, weot for reference the differential distribution derived from the cu-ulative distribution reported by Ivezic et al. (2001)

r) = no 10ax

10bx + 10bx , (14)here x = Hcorr HC , a = (k1 + k2)/2, b = (k1 k2)/2, with HC =.5, k1 = 0.65 and k2 = 0.25 (Table 4 in Ivezic et al., 2001).Ivezic et al. (2001) were able to t this functional form because

eir sample extended to a 1.5 mag fainter H limit (Hcorr 17.5)an the sample discussed here. Given this sample difference, forch H distribution shown in Fig. 9 we instead t a brokenwer law: a separate power-law t for the bright and faint end.hile this procedure is expected to yield a shallower slope at theint end than the above Ivezic et al. (2001) t, it is preferred herecause it decouples the bright and faint ends. The separation ofe bright and faint ends was attempted in H steps of 0.5 mag, ande value that minimizes the resulting 2 was adopted as the best. The statistical errors for the best-t slopes are typically 0.0102, but it is likely that their uncertainty is perhaps a factor ofo or so larger due to systematic effects (see Section 4.3 below).a few cases, the best t is consistent with a single power law.

e best-t power-law parameters for differential absolute magni-de distributions shown in Fig. 9 are listed in Table 3.

the

FidybesigfamobDa

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(tg. 8. Top left: Dorbit histogram for the Vesta family and surrounding objects (dened from the Vesta family centroid). Top right: Equivalent to top left plot, but for thenamically buried Baptistina family. The dotted line represents the Dorbit histogram without any color constraints, which climbs smoothly to very high numbers of objectscause of the inclusion of Flora family objects. The solid line represents the Dorbit histogram with the color constraints applied, which displays a stronger clusteringnature. Vertical dashed line represents Orb cutoff values selected for these families. Middle left: Dcolor histogram for objects that met the Dorbit criteria for the Vestaily. Middle right: Dcolor histogram for objects in a preliminary orbital denition of the Baptistina family, showing strong color distinction from the background Flora

jects. Vertical dashed line represents Col cutoff values selected for these families. Bottom left and right: Dorbit vs. Dcolor for Vesta and Baptistina families, respectively.shed box denes Orb and Col boundaries (objects inside are assigned family membership).

The data and best ts shown in the top three panels in Fig. 9monstrate that the absolute magnitude distributions are notentical: the outer main-belt shows a atter distribution, and thener belt shows a steeper distribution than the middle belt regionr objects with Hcorr < 14. This is in conict with the Ivezic et al.001) nding that the size distribution appears universal through-t the belt. However, here we analyze a sample about seven timesrger; the statistical errors at the bright end for the Ivezic et al.001) sample were too large to detect this effect (see their Figs. 21d 22). Nevertheless, the Ivezic et al. (2001) size distribution re-ains valid when the whole belt is treated together because theunts underprediction of their t in the outer belt is compensatedits overprediction in the inner belt.

ing of Hcorr distribution as the semi-major axis increases is validfor each subpopulation separately. Objects associated with familiesalways show the attening at the faint end, while the backgroundpopulations admit a single power-law t in the middle and outerbelt.

Due to different Hcorr distributions for family and backgroundpopulations, the fraction of objects associated with families is afunction of Hcorr. The top left panel in Fig. 10 shows this depen-dence separately for blue (dominated by the C taxonomic type)and red (dominated by the S type) subsets. For both subsets, thefraction of objects in families signicantly increases from 20% to50% between Hcorr = 9 and Hcorr = 11. The two color-selectedsubsets show different behavior for Hcorr > 11: for blue subset theAsteroid families inThe separation of populations into families and backgroundhe middle and bottom rows in Fig. 9) shows that the atten-

fritSDSS MOC 4 147action of objects in families decreases from 50% to 30%, whilestays constant at the 60% level for red families. Since blue fam-

14 198

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telofatithofg. 9. The differential absolute magnitude distributions corresponding to panels in Fig. 6 are shown as symbols with (Poisson) error bars. The solid line shows arbitrarilynormalized best-t distribution from Ivezic et al. (2001). The two dashed lines show the best-t broken power law: a separate power-law t for the bright and faint end.some cases, the two lines are indistinguishable. The best-t parameters are listed in Table 3. The two arrows show the best-t break magnitude (left) and the adoptedmpleteness limit (right).

es typically have larger semi-major axis than red families, it isssible that this decrease in family membership is due to in-easing color rejection at the faint end. However, the remainingo panels in Fig. 10 demonstrate that this is not the case becausee color rejection rate is both fairly independent of Hcorr, and tooall to account for the observed decrease of blue family member-ip.The dominance of background objects for Hcorr < 13 is consis-

nt with the background population having a signicantly shal-wer size distribution for large objects than the families. Thelloff of blue family fraction and the slow climb in color rejec-on rate toward large values of H in Fig. 10 conrms that

to our detection limit it appears that the red family and back-ground populations have identical size distributions for objectswith Hcorr > 13. Morbidelli et al. (2003) suggest that the back-ground population is composed of many small families whichformed from small-diameter objects and as such should havesteep size distributions for objects larger than 1 km, producinga background population with an initial size distribution (for ob-jects >1 km) steeper than that for families which formed fromthe breakup of larger objects. Differences between the size dis-tributions of these two populations should eventually disappearthrough collisional processing. Because the blue family and back-ground populations appear to have signicantly different size dis-8 A. Parker et al. / Icaruscorre size distributions for blue families are shallower for valuesHcorr > 13. Because the red family fraction is effectively at

trid(2008) 138155ibutions while the red populations size distributions appearentical (for Hcorr > 13), we infer that the time for equilibrating

TaBe

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tal population as a function of Hcorr magnitude, with the solid histogram representing redp right panel compares the fraction of objects in families to the total population as a funlid histogram) and when they are not (dotted histogram). The bottom panel shows the rpopulations. The top left panel shows the fraction of objects in families to the Asteroid families in the SDSS MOC 4 149

ble 3st-t parameters for counts shown in Fig. 9

rangea Nb Hminc Hmaxc HBd 1e 2f

02.50 30,702 11.0 16.5 14.0 0.76 0.46mily 16,309 11.0 16.0 13.5 0.79 0.53ckground 14,393 11.0 16.0 13.5 0.69 0.46

02.82 32,500 11.0 16.0 13.5 0.73 0.42mily 14,261 11.0 16.0 13.5 0.67 0.44ckground 18,239 11.0 15.5 13.5 0.54 0.57

23.60 24,367 12.0 15.5 13.5 0.56 0.40mily 9547 12.0 15.0 13.5 0.57 0.37ckground 14,820 12.0 15.0 13.5 0.56 0.52

The range of proper semi-major axis for dening the inner, middle and outerain belt (AU).The number of objects in each subsample; the rst line corresponds to the full

mple, and the following two to subsamples classied as families and background,spectively. The total number of objects is 87,569.The minimum and maximum H magnitude used in tting the H distribution.The best-t break H magnitude (see text).The bright H distribution slope.

f The faint H distribution slope.

e background and family size distributions is longer for blue ob-cts than red. This difference in the equilibration time may bee either to differences in asteroid internal structure and mate-al properties between taxonomic classes or due to environmentalriations in collisional processing rates, as red objects are moreevalent in the inner belt and blue objects in the outer belt.We note that the background population in the outer beltows a curious excess of large objects (Hcorr < 11.5) compared

to best power-law t (Fig. 9, bottom right). We have inspectedthe orbital parameter and color distributions for 58 objects with10 < Hcorr < 11 and found that they are not associated with anyidentied family, nor generally clustered.

4.2. The comparison of size distributions for individual families

The inspection of differential Hcorr distributions for the 37 fam-ilies identied here shows that many, but not all, display a clearchange of slope such as seen for family populations in Fig. 9. Wehave attempted a broken power-law t for all families. When thetwo best-t slopes differ by less than 0.05, we enforce a singlepower-law t. This procedure yields 22 families described by a sin-gle power law and 15 families with a robust detection of the slopechange. Their best-t parameters are listed in Tables 4 and 5, re-spectively, and a few examples of measured H distributions andbest ts are shown in Fig. 12.

For families whose absolute magnitude distributions are de-scribed by a single power law, the median best-t power-law slopeis 0.56, with a standard deviation of 0.16 (determined form inter-quartile range). This scatter is signicantly larger than the mea-surement errors and indicate that families do not have a universal sizedistribution. Similarly, for families with a best-t broken powerlaw, the medians and standard deviations for the bright andfaint slopes are (0.66, 0.24) and (0.32, 0.15), respectively (againnote the signicant scatter relative to the measurement errors),with the median Hcorr where the slope changes of 14.2 (D 6 kmobjects (a > 0) and the dotted histogram representing blue objects (a < 0). Thection of magnitude when colors are used as a constraint on family membershipejected fraction due to color constraints as a function of magnitude.

15 198

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ble 4st-t H distribution parameters for families described by a single power law

milya Nb Hbc H f d S e Agef

ptistina 310 14.0 16.0 0.49 cCuskey 1043 12.5 16.0 0.49 igone 307 14.0 16.0 0.59 terpe 387 13.5 16.0 0.52 dree 649 14.0 16.0 0.70 assalia 730 14.5 16.0 0.97 0.3 0.1ta 225 14.0 16.0 0.35 1.5 0.5itidika 698 12.5 15.5 0.61 isa 185 13.0 15.5 0.47 0.5 0.2no 354 14.5 16.0 0.63 olea 172 14.0 15.5 0.85 mesis 129 14.0 16.0 0.70 0.2 0.1dia 598 14.0 16.0 0.81 dua 442 14.0 16.0 0.50 erxia 252 14.5 15.5 0.71 0.5 0.2loris 121 13.0 15.5 0.35 0.7 0.4nia 1106 13.0 15.5 0.52 0.2 0.1asilia 127 13.5 15.5 0.71 0.05 0.04iaohua 150 13.0 15.0 0.56 0.3 0.2sula 644 11.0 15.0 0.47 eobalda 100 13.0 15.5 0.44 ritas 250 12.0 15.0 0.50 8.3 0.5 MyrThe family name (the lowest-numbered member).The number of objects in SDSS MOC 4 associated with this family.Bright H magnitude limit used for tting.Faint H magnitude limit used for tting.The best-t power-law index for H distribution.

f The family age in Gyr (except for Veritas in Myr), taken from Nesvorn et al.005), when available.

ble 5st-t H distribution parameters for families described by a broken power law

milya Nb Hbc H f d HB e 1f 2g Ageh

ra 6164 11.0 16.0 14.0 0.66 0.40 1.1 0.5sta 3793 12.5 16.0 14.8 0.89 0.50 sa-Polana 2928 13.5 16.5 15.0 0.80 0.39 utonia 3405 12.5 15.5 14.0 0.94 0.59 aria 1315 11.0 15.5 14.5 0.53 0.38 3.0 1.0nomia 2995 11.0 15.5 13.5 0.76 0.28 2.5 0.5eona 428 12.0 15.5 14.2 0.57 0.29 0.7 0.5nan 624 14.0 16.0 15.1 0.79 0.46 on 914 12.0 15.5 14.2 0.86 0.26 1.2 0.4ra 248 13.0 16.0 14.8 0.37 0.18 0.5 0.2ronis 1267 12.0 15.0 13.5 0.55 0.26 2.5 1.0s 4367 10.5 15.0 13.9 0.52 0.32 2.0 0.5rela 411 13.2 15.0 14.0 1.04 0.62 emis 1073 11.0 15.0 13.4 0.46 0.10 2.5 1.0giea 1076 12.7 15.0 14.4 0.62 0.18 2.0 1.0The family name (the lowest-numbered member).The number of objects in SDSS MOC 4 associated with this family.Bright H magnitude limit used for tting.Faint H magnitude limit used for tting.The break H magnitude used for tting.

f The best-t power-law index for H distribution at the bright end.The best-t power-law index for H distribution at the faint end.The family age in Gyr, taken from Nesvorn et al. (2005), when available.

r pV = 0.1). We discuss correlations of these best-t parametersith the family color and age in Section 4.4.

3. Systematic deviations in size distribution due to variations inmily denitions

Before proceeding with the analysis of correlations betweenze distributions and other family properties such as color ande, we analyze the systematic deviations in size distribution duevariations in family denitions. For example, the color con-

raints may result in a size-dependent incompleteness because

the increased photometric noise at the faint end. Similarly, thesumption of Gaussian distributions for orbital parameters and

nath(2008) 138155

g. 11. Median a color as a function of semi-major axis for several populations.lid histogram represents the family population, dashed line represents the back-ound population, and the dotted line represents the median of the family a colorntroids (unweighted by family size) for each Main Belt region (inner, mid, andter).

lors may result in incomplete families due to extended halos,pointed out by Nesvorn et al. (2005). This effect may also

duce size-dependant systematics because small objects are scat-red over a larger region of orbital space, as shown below.The Vesta family offers a good test case because of its uniquelor distribution (which is due to the inuence of 1 m ab-rption feature on the measured i z color). The top panel ing. 13 compares the Hcorr distributions for the adopted Vestamily and for a less constraining orbital cut dened simply by06 < sin(i) < 0.16 and e < 0.16, that yields 30% more candidateembers. Apart from this overall shift in the normalization, the re-lting distributions have statistically indistinguishable shapes. Theiddle panel compares the adopted family and a much less con-raining color cut, a > 0 (i.e. no constraint on the i z color), thatelds 50% more objects. Again, the slope of the two distributionse indistinguishable.We detect a signicant difference, however, when we split theopted family in the core and outskirt parts using Dorbit < 1d 1.75 < Dorbit < 2.75 [see Eq. (4)]. As the bottom panel ing. 13 shows, the outskirt subsample has a steeper Hcorr dis-ibution than the core subsample. The best-t power-law slopesthe 14 < Hcorr < 16 region are 0.45 and 0.59 for the core andutskirt subsample, respectively. We note that despite this slopefference, the change of slope between the bright (0.89) and faintd is robustly detected.Another method to see the same size sorting effect is to in-ect the dependence of Dorbit on Hcorr. We nd that the medianorbit for objects in the Vesta family increases from 1.0 to 1.5 ascorr increases from 14 to 17.This size sorting effect is not a peculiar property of the Vesta

mily as it is seen for a large fraction of families. It is caused by ancreased scatter in all three orbital parameters as Hcorr increases.is is not surprising as the velocity eld of the fragments pro-ced in the disruption of an asteroid familys parent body mayve been size-dependent. For most families the sorting is domi-

ted by the increased dispersion in the semi-major axis. One ofe most striking examples, the Eos family, is shown in Fig. 14.

the

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foto14g. 12. Analogous to Fig. 9, except that the absolute magnitude distributions for selected asteroid families are shown. The rst three panels (from top left to bottom right)ow examples of families that follow a single power-law magnitude distribution, and the remaining six panels show magnitude distributions for families that require auble power-law t. The best-t parameters are listed in Tables 4 and 5, respectively.

discussed by Nesvorn et al. (2005), this increase of dispersionsize decreases can be also be explained as the drift induced bye Yarkovsky effect (see also Vokrouhlick, 1999 and Bottke et al.,01).

4. Correlations between size distributions and family color and age

We analyze the correlations between the best-t size distri-tion parameters listed in Tables 4 and 5, and family color ande. The age, when available, is taken from the compilation byesvorn et al. (2005).The dependence of the power-law index on the mean a color

r families described by a single power law is shown in the

0.19). These differences are not statistically signicant. Within eachcolor-selected subsample (blue vs. red, i.e. a < 0 vs. a > 0), thereis no discernible correlation between the slope and color.

Families that require a broken power law t are twice aslikely for red families (a > 0) dominated by S-type asteroids) thanfor blue families dominated by C-type asteroids (73% vs. 36%).As illustrated in Fig. 15, the size distributions are systematicallysteeper for S-type families, and the bright and faint end slopesappear to be correlated. The median values of the bright andfaint end slopes are (0.57, 0.18) for blue families, and (0.79, 0.39)for red families.

For a subset of families that have available age estimates, wend that families with broken power law size distributions areAsteroid families inp left panel in Fig. 15. The mean and standard deviation forblue families are (0.55, 0.13), and for 8 red families are (0.65,

dolaSDSS MOC 4 151minated by old families, while those that admit a single powerw are dominated by young families, with the age separation

152 A. Parker et al. / Icarus 198 (2008) 138155

Fig. 13. (continued) The dependence of absolute magnitude distribution on familydenition for Vesta family. The top panel compares the magnitude distribution ob-tained for adopted family denition (open circles) to that obtained for a much lessconstraining cut on orbital elements (closed squares). The middle panel comparesthe adopted distribution (open circles) to that obtained for a much less constrainingcolor cut (closed squares). In both cases the shapes of the magnitude distributionsare similar, with only signicant change in the number of selected candidate mem-bers. The bottom panel separates the adopted population into the orbital coreregion (open circles) and outskirt region (closed squares). Note that the latter dis-tribution is steeper (i.e. the small members are more prevalent in the outskirtregion).

Fig. 14. The correlation between the proper semi-major axis and absolute magni-tude for the Eos family. The increased dispersion of the semi-major axis for faint(small) objects is probably caused by the Yarkovsky effect. Note that this family isintersected by several mean motion resonances with Jupiter (e.g., 7:3 at ap 2.95).

boundary at 1 Gyr. We note that the size distribution was usedfor some of age estimates compiled by Nesvorn et al. (2005), sothis conclusion may be a bit of circular reasoning, though the ma-jority of age estimates are derived independently of the observedsize distribution.

The correlations between the mean color and family age re-ported by Jedicke et al. (2004) and Nesvorn et al. (2005) arereproduced when using the a color for families discussed here.Fig. 16 illustrates a good agreement with the analytic ts to the ob-served correlations obtained by Nesvorn et al. (2005), and furtherdemonstrates the correlation between the observed size distribu-tions and age.

5. Discussion and conclusions

We have used a large sample of asteroids (88,000) for whichboth orbital elements and SDSS colors are available to derive im-proved membership for 37 asteroid families. The addition of colorstypically rejects about 10% of all dynamically identied candidatemembers due to mismatched colors. Four families can be reliablyisolated only with the aid of colors. About 50% of objects in thisdata set belong to families, with this fraction representing a lowerlimit due to a conservative requirement that a candidate familymust include at least 100 members. The resulting family de-nitions are in good agreement with previous work (e.g. Zappalet al., 1995; Nesvorn et al., 2005) and all the discrepancies arewell understood. Although York et al. (2000) have observed onlyabout 1/3 of all known asteroids, it is remarkable that the samplediscussed here provides color information for more than an orderof magnitude more objects associated with families than analyzedin the published literature.

This data set enables the determination of absolute magnitudeFig. 13.(sou

ize) distributions for individual families to a very faint limit with-t a need to account for complex selection effects. We verify that

the

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chlaniseefg. 15. A summary of relationships between the family median color and parameters of the best-t power-law magnitude distributions. The top left panel shows the slope ofe best-t power-law as a function of color a (see text for denition) for 25 families that follow a single power-law magnitude distribution. The top right panel shows theright slope of the best-t double power-law as a function of color a for 12 families that follow a double power-law magnitude distribution, and the bottom left panel isalogous plot for the faint slope. Note that blue (C type) families have much shallower magnitude distributions than redder families. The bottom right panel demonstratese strong correlation between the bright and faint slopes, that seems independent of color. The errors of alpha are dominated by systematics. As per discussion inction 4.3, we estimate these systematic errors to be about 0.050.1.

ze distribution varies signicantly among families, and is typi-lly different from size distributions for background populations.nsequently, the asteroid size distribution cannot be describeda universal function that is valid throughout the main belt

.g. Jedicke and Metcalfe, 1998; Ivezic et al., 2001, and referenceerein). This nding will have an inuence on conclusions de-ved from modeling the size distribution under this assumption.g. Bottke et al., 2005, and references therein). In particular, itnot clear how to interpret a detailed dependence of the criti-l specic energy (energy per unit mass required to fragment anteroid and disperse the fragments to innity) on asteroid size de-ved from such models, when the starting observational constraintsize distribution is an average over multiple families with sig-

cantly varying size distributions.We show that for objects with Hcorr < 13, the background pop-

ation dominates (family fraction decreases toward lower Hcorr,dicating a shallower size distribution for large objects), while forjects with Hcorr > 13, the red family fraction remains effectivelynstant to our completeness limit while the blue family fractionlls off. This indicates that the time to collisionally equilibratee family and background populations (see, e.g., Morbidelli et al.,03) is shorter for red objects than blue.The size distributions for 15 families display a well-denedange of slope and can be modeled as a broken double power-w. The rst evidence for this effect and a discussion of its sig-cance are presented by Morbidelli et al. (2003). Using a data

We also nd such broken size distributions are twice as likely forS-type familes than for C-type families (73% vs. 36%), and are dom-inated by dynamically old families. The remaining families withsize distributions that can be modeled as a single power law aredominated by young families (

15 198

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g. 16. The correlation between the mean a color for families dened here andeir age taken from Nesvorn et al. (2005). Families whose absolute magnitudestribution can be described by a single power law are shown by circles, andose that require a broken power law as squares. Note that the former are dom-ated by young families (1 Gyr), and the latter by old families. The two linese the best-t color-age relation from Nesvorn et al. (2005), converted using= 0.49PC1 0.16, where PC1 is the rst principal color component derived bysvorn et al. (2005) [note that Nesvorn et al. (2005) used several very youngmiles not discussed here to constrain the slopes of plotted relations].

rveys, such as Pan-STARRS (Kaiser et al., 2002) and LSST (Tyson,02), will obtain even more impressive samples, both in size, di-rsity of measurements and their accuracy. For example, LSST willan the whole observable sky every three nights in two bands to5 depth equivalent to V = 24.7. These data will enable muchproved analysis due to several factors:

Due to hundreds of observations, the orbits will be determineddirectly, instead of relying on external data, resulting in a sam-ple about 3040 times larger than discussed here;

The effective faint limit will be extended by about 5 mag,corresponding to ten times smaller size limit (diameters ofseveral hundred meters);

Due to many photometric observations obtained with thesame well-calibrated system, the uncertainties in absolutemagnitudes will be an order of magnitude smaller;

The addition of the y band (at 1 m) will improve the colorclassication due to better sensitivity to the 1 m absorptionfeature present in spectra of many asteroids.

These new data will undoubtedly reinvigorate both observa-onal and theoretical studies of main-belt asteroids.

knowledgments

We are grateful to E. Bowell for making his ASTORB le publiclyailable, and to A. Milani, Z. Kneevic and their collaborators fornerating and distributing proper orbital elements. M.J. gratefullyknowledges support from the Taplin Fellowship and from NSFant PHY-0503584. Funding for the SDSS and SDSS-II has beenovided by the Alfred P. Sloan Foundation, the Participating In-itutions, the National Science Foundation, the U.S. DepartmentEnergy, the National Aeronautics and Space Administration, thepanese Monbukagakusho, the Max Planck Society, and the Higherucation Funding Council for England. The SDSS Web site istp://www.sdss.org/. The SDSS is managed by the Astrophysicalsearch Consortium for the Participating Institutions. The Partic-ating Institutions are the American Museum of Natural History,trophysical Institute Potsdam, University of Basel, University of

mbridge, Case Western Reserve University, University of Chicago,rexel University, Fermilab, the Institute for Advanced Study, the

Iv(2008) 138155

pan Participation Group, Johns Hopkins University, the Joint In-itute for Nuclear Astrophysics, the Kavli Institute for Particle As-ophysics and Cosmology, the Korean Scientist Group, the Chineseademy of Sciences (LAMOST), Los Alamos National Laboratory,e Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-stitute for Astrophysics (MPA), New Mexico State University, Ohioate University, University of Pittsburgh, University of Portsmouth,inceton University, the United States Naval Observatory, and theniversity of Washington.

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The size distributions of asteroid families in the SDSS Moving Object Catalog 4IntroductionSDSS observations of moving objectsAn overview of SDSSSDSS moving object catalogErrors in H magnitudes listed in the ASTORB file

The asteroid families in SDSS MOCA method for defining families using orbits and colorsThe comparison of resulting families with previous work

The size distribution of asteroid populationsThe comparison of size distributions for families and backgroundThe comparison of size distributions for individual familiesSystematic deviations in size distribution due to variations in family definitionsCorrelations between size distributions and family color and age

Discussion and conclusionsAcknowledgmentsReferences