Icarus 202 (2009) 104–118

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Icarus

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On the asteroid belt’s orbital and size distribution

Brett J. Gladman a,∗, Donald R. Davis b, Carol Neese b, Robert Jedicke c, Gareth Williams d, J.J. Kavelaars e,Jean-Marc Petit f, Hans Scholl g, Matthew Holman h, Ben Warrington i, Gil Esquerdo b, Pasquale Tricarico b

a Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1, Canadab Planetary Science Institute, 1700 East Fort Lowell Road, Suite 106, Tucson, AZ 85719, USAc Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USAd IAU Minor Planet Center, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USAe Hertzberg Institute for Astrophysics, 5071 West Saanich Road, Victoria BC, V9E 2E7, Canadaf Observatoire de Besançon, B.P. 1615, 25010 Besançon Cedex, Franceg Observatoire de la Côte d’Azur, B.P. 4299, 06304 Nice Cedex 4, Franceh Harvard-Smithsonian Center for Astrophysics, MS-18, 60 Garden Street, Cambridge, MA 02138, USAi Joint Astronomy Center, 660 N. A’ohoku Place, Hilo, HI 96720, USA

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

Article history:Received 6 September 2006Revised 10 February 2009Accepted 11 February 2009Available online 26 February 2009

Keywords:AsteroidsCollisional physicsOrbit determination

For absolute magnitudes greater than the current completeness limit of H-magnitude ∼15 the mainasteroid belt’s size distribution is imperfectly known. We have acquired good-quality orbital and absoluteH-magnitude determinations for a sample of small main-belt asteroids in order to study the orbitaland size distribution beyond H = 15, down to sub-kilometer sizes (H > 18). Based on six observingnights over a 11-night baseline we have detected, measured photometry for, and linked observationsof 1087 asteroids which have one-week time baselines or more. The linkages allow the computation offull heliocentric orbits (as opposed to statistical distances determined by some past surveys). Judgedby known asteroids in the field the typical uncertainty in the (a/e/i) orbital elements is less than0.03 AU/0.03/0.5◦. The distances to the objects are sufficiently well known that photometric uncertainties(of 0.3 magnitudes or better) dominate the error budget of their derived H-magnitudes. The detectedasteroids range from H R = 12–22 and provide a set of objects down to sizes below 1 km in diameter.We find an on-sky surface density of 210 asteroids per square degree in the ecliptic with oppositionmagnitudes brighter than mR = 23, with the cumulative number of asteroids increasing by a factor of100.27/mag from mR = 18 down to the mR � 23.5 limit of our survey. In terms of absolute H magnitudes,we find that beyond H = 15 the belt exhibits a constant power-law slope with the number increasingproportional to 100.30H from H � 15 to 18, after which incompleteness begins in the survey. Examiningonly the subset of detections inside 2.5 AU, we find weak evidence for a mildly shallower slope forH = 15–19.5. We provide the information necessary such that anyone wishing to model the main asteroidbelt can compare a detailed model to our detected sample.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

The number of main belt asteroids rises steeply with decreas-ing diameter, with a size distribution roughly following a powerlaw. Although collisions crater the largest asteroids and may oc-casionally break up a big one to create an asteroid family, thesize distribution of objects larger than about 100 km has not beensignificantly modified since primordial times. However, asteroids afew kilometers and smaller in size are recent collisional shards oflarger objects since their lifetime against collisional destruction ismuch less than the age of the Solar System. As such, these bod-

* Corresponding author.E-mail address: gladman@astro.ubc.ca (B.J. Gladman).

0019-1035/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2009.02.012

ies preserve information about collisional breakup, the dominantphysical process that has shaped the present asteroid belt.

While our knowledge of the asteroid size distribution has dra-matically increased in recent years, it is still poorly known at di-ameters D < 10 km. Various models and extrapolations yield verydifferent estimates of the number of km-sized and smaller main-belt asteroids (see Tedesco et al., 2005 and references therein).

The Palomar–Leiden Survey (PLS, van Houten et al., 1970) wasthe best comprehensive orbital survey of faint main-belt asteroids,which photographically covered a selected area of the sky downto apparent magnitude V ∼ 20, corresponding to absolute magni-tude H ∼ 17 and D ∼ 3 km. This survey covered approximately200 square degrees of sky in a “non-targeted” search; that is, indi-vidual asteroids were seen to move across the contiguous patch ofsky covered repeatedly over a time scale of about one month. Orbit

The SKADS survey 105

solutions were then constructed which allowed calculation of thedistances and thus absolute magnitudes. The orbital distributionof the approximately 500 well-determined orbits thus constructedshowed little if any difference from the previously known popu-lation. However, at D < 30 km, the PLS determined a shallowerslope (about −2) for the incremental diameter distribution thanthe collisional equilibrium value (−2.5) predicted by the Dohnanyi(1969) scale-independent theory, although many observational se-lection effects had to be accounted for in an approximate way toderive this result.

While the PLS was designed as a main-belt survey, scien-tists have attempted to use data obtained for other projects toestimate the small asteroid population. Shallow slopes for themain-belt size distribution down to sub-km sizes were foundby Evans et al. (1998) from an analysis of serendipitous aster-oid trails in WFPC2 HST images, but these slopes are not con-sistent with an extrapolation of the observed steep distributionof some populous asteroid families (Tanga et al., 1999), nor withthe steep distribution of impactors (slope of about −4) inferredfrom Gaspra’s cratering record. Jedicke and Metcalfe (1998) an-alyzed Spacewatch observations (acquired for near-Earth objectsearches) which went as faint as V ∼ 21; the interpretation ofthis data set is very complicated due to biases inherent in itsacquisition. It should be noted that many surveys for near-Earthasteroids have “thrown away” in some empirical manner objectswhich appeared to have motions typical of main-belt asteroids.Analysis of the scans of the Sloan Digital Sky Survey (SDSS) byIvezic et al. (2001) yielded an abundant population of main-belt asteroids down to apparent magnitude mR = 21.5; althoughlacking enough observational arc to determine the orbits (andthus measure precise distances and therefore H-magnitudes ofthe asteroids), they concluded there was a flattening of the H-magnitude distribution to a shallower constant slope beginning atH � 15.5, which they fit by a rolling power law. Yoshida et al.(2003) conducted a similar experiment to mR � 24.4 using theSubaru telescope and reported a flattening of the H-magnitudedistribution, but which was a continuous drop of slope with in-creasing H rather than a transition to a new slope of singlevalue. Wiegert et al. (2007) analyzed imaging from the Canada–France Hawaii telescope intended to detect and track transnep-tunian objects (Kavelaars et al., 2009) to obtain short-arc orbitsfor 1525 asteroids and also found a flattening of the slope, al-beit to a steeper value than the previous surveys. Yoshida andNakamura (2007) conducted an improved version of their 2003 ex-periment.

Here we report the first survey since PLS to discover and tracka large sample (∼1000) of main-belt asteroids in order to deter-mine their orbits which, in conjunction with their measured Hvalues and V − R colors, allows modeling of the diameter distri-bution down to values below that which was possible for PLS.The lack of progress since PLS is due largely to the fact that thephotographic limit had been reached and hence it was impossibleto reach significantly fainter magnitudes. When CCDs on large-aperture telescopes became available (easily allowing a gain of 3or more magnitudes compared to photographic Schmidts), theyimaged such tiny fields that a wide field survey was essentiallyimpossible for practical purposes; the shear of their differentialmotion meant that after a period of one week recovery would haveto be done one by one, which was far too inefficient. We have ex-ploited the field of view of CCD mosaic cameras to detect and tracka sample of main-belt asteroids with diameters down to below akilometer, and hence refer to our survey as the (S)ub-(K)ilometer(A)steroid (D)iameter (S)urvey, or SKADS.

2. Scientific motivation

Knowing the asteroid size distribution down to sub-kilometersizes and resolving the question of the range of sizes over whichthe strength transition takes place is important for a number ofreasons:

• Almost all evaluations of the small size distribution of aster-oids have assumed a single value of the albedo to convert theobserved H-magnitude distribution into a diameter distribu-tion. However, as noted by Cellino et al. (1991), the derived di-ameter distribution depends upon the assumption of albedos:picking a fixed albedo preserves the H-magnitude distribution,while varying the albedo distribution produces different diam-eter distributions. This point is illustrated in Fig. 6 in Tedescoet al. (2005) which compares the diameter distribution derivedby assuming a fixed albedo with that found assuming a morerealistic albedo distribution. A difference by a factor of up toabout three results at sizes less than 10 km diameter.

• Current hydrocode-based scaling theories for the specific colli-sional energy needed to disrupt asteroids of different sizes (Q*vs. D) predict a transition from the strength to gravity regimein the size range 0.1 to 30 km, but different scaling lawspredict very different population abundances at small sizes(Davis et al., 1999; O’Brien and Greenberg, 2005). Another ma-jor question regarding the asteroid size distribution is whetheror not there is a second “bump” relative to a power-law slopein the sub-10-km size range. This “bump” was interpreted byDurda et al. (1998) as being produced by the rapid variationin the strength of bodies in the sub-10-km size range and thatthe much more pronounced “bump” at larger sizes is a col-lisionally induced wave generated by the small size “bump.”Our survey was designed to sample the size range of the pur-ported small-size “bump” and reliably determine whether ornot such a feature exists in the size distribution.

• Finding the asteroid abundance in the range D = 0.5–2 kmwould yield data on a population of bodies which is muchyounger than the age of the Solar System, having collisionallifetimes of only ∼100 Myr. These data would provide animportant observational constraint on all models of the col-lisional evolution of the overall asteroid population, with thepurpose of reconstructing the size distribution of the originalpopulation of asteroidal planetesimals (Davis et al., 1989, 1994;Bottke et al., 2005).

• The ages of spacecraft-encountered asteroids such as 951Gaspra, 243 Ida and 253 Mathilde are estimated from theircrater populations. This requires knowing the projectile flux atcurrently uncertain sizes in the range from 0.01 to a few kmdiameter. This survey provides a robust estimate at least at theupper end of this range of bodies.

• Analysis of the size distributions of some asteroid familiesshows steeply rising populations for decreasing sizes, muchsteeper than the background non-family asteroid population(Tanga et al., 1999). It has been suggested (Zappalà and Cellino,1996; Tedesco et al., 2005) that, if this rate of increase contin-ues down to the 1-km size range, the family-derived fragmentsequal or even outnumber the entire background population at1 km. The signature of this effect would be a dramatic increasein the slope of the size-frequency relation for D < 5 km.

• The delivery from the main asteroid belt of near-Earth aster-oids (Bottke et al., 2002) and meteorites (Morbidelli and Glad-man, 1998) depends critically on the size distribution of smallasteroids, both because the fragmentation rate depends on theimpactor population (Farinella et al., 1993) and because theeffectiveness of transport mechanisms such as the Yarkovsky

106 B.J. Gladman et al. / Icarus 202 (2009) 104–118

effect (Farinella et al., 1998) is a sensitive function of the col-lision rate and of the abundance of small asteroids.

• The production of dust in the asteroid belt, such as the zo-diacal dust cloud and the IRAS dust bands associated withasteroid families, depends in a critical way upon the num-ber of existing small asteroids, both within families and in the“background” asteroid population (Nesvorny et al., 2006).

3. Experimental design

We wished to image as large a portion of sky as possible toat least several magnitudes fainter than the PLS survey; our targetwas an apparent R-band magnitude of mR = 23.0–23.5. Observa-tions would occur at opposition to allow as large a contiguous‘patch’ of sky to be imaged as possible (Fig. 1). Moving aster-oids would be detected the first night and on subsequent nightsthe ‘patch’ would be displaced westward at the average retrogrademotion of the main-belt to keep the largest number of asteroidswithin the patch (the night-to-night shear of asteroids in and outof the moving patch can be modeled).

Determination of the absolute magnitude H requires knowledgeof the distance to the asteroid, the geometry of the observation,and photometric measurement of the asteroid brightness. At oppo-sition the R-band absolute magnitude H R of the asteroids1 can beestimated using:

H R = mR − 5 log10(dΔ) − P (φ), (1)

where d and Δ are the heliocentric and geocentric distances of theasteroid and P (φ) is the phase function. Observing at opposition(where d = Δ + 1) simplifies many aspects of the data reduction(all asteroids dominantly move retrograde along the ecliptic) andsubsequent orbit linkage (since the retrograde rate is determinedby Δ to first order). As usual the phase function involves the slopeparameter G , for which we take the common value of 0.15 and usethe standard formalism (Bowell and Lumme, 1979). Since we ex-pect a large fraction of our sample to be within one magnitudeof the detection limit, and thus our photometry in many cases tobe only accurate to 10%, distance determinations to better than5% were targeted so that the bulk of the uncertainty in the Hdetermination would be due to the unavoidable photometric un-certainty. After experimentation using astrometric measurementsof a previously-known sample of asteroids, we determined that atime baseline of roughly one week with sub-arcsecond astrometrywould meet this goal, and thus all observations could be obtainedwithin a single dark run. We thus aimed for 2 or 3 nights of ob-servation at the beginning of a dark run, and then another 2 or3 nights at the end of a dark run. Multi-night observations of thesame asteroid could then be linked and an orbit calculated.

4. Observations

Six nights were allocated for this program at the 3.8-m May-all telescope at Kitt Peak National Observatory: March 21, 22, and23 UT 2001 at the beginning of the dark run, and March 29, 30,and 31 UT at the end of the dark run. An asteroid detected in atleast one night in both blocks would thus have an observed arclength of between 7 and 10 days if multiple observations could besuccessfully linked.

The KPNO mosaic CCD camera gives a 36 × 36 arcmin field ofview on the Mayall. We created a grid of contiguous sky cover-age along the ecliptic plane (Fig. 1) that was within a few degrees

1 In what follows we refer to H magnitudes when discussing general conceptsrelated to H magnitude, but when discussing our data specifically we refer to H R

magnitudes to make it clear that our precise absolute magnitudes are calibrated toreflectance in the R band.

Fig. 1. SKADS pointings for the night of March 21, 2001 UT. Horizontal dashed lineis the equator while the slanted solid line is the ecliptic plane; note that on thisdate the ecliptic opposition point is at the intersection of the ecliptic and equator.Image triplets on fields 121–124 were acquired first, before moving on to fields 117–120, etc. Boxes show the 36′ × 36′ field of view of the KPNO mosaic camera on theMayall telescope.

of the opposition point on the first night, covering 8.4 square de-grees of sky (which accounts for gaps between the mosaic’s CCDs).The fields were translated 13.5 and 5.4 arcmin to the west andnorth (respectively) per day to keep as many main-belt asteroids aspossible in the imaged region. These translation rates were deter-mined using the average rates of known main-belt asteroids withina few degrees of the initial pointings.

Filter choice was governed by a balance between obtainingmaximum magnitude depth (discovering as many objects as pos-sible) and the desire to acquire some color information on theasteroids. Obtaining V − R colors allows us to estimate (even ifonly on a probabilistic basis) the taxonomic type (Section 10) ofeach detected asteroid. On photometric nights we thus acquiredtriplets of 120-s images for object detection in R-band with aninter-exposure spacing of roughly 15 min. Immediately before orafter one of these three R-band exposures on any given night, weacquired a 120-s V -band exposure; the temporal proximity mini-mizes any possible light curve variations in the color measurement.

However, in nights of below-median seeing we judged color in-formation less important than retaining maximum depth in orderto not lose the faintest asteroids. In such cases we acquired 150-sec exposures in a wide-bandpass VR filter. This strategy workedwell in allowing detection of faint asteroids on worse than me-dian nights; for example, we lost few objects in 1.7′′ seeing withthe VR filter when compared to adjacent-night R-band imaging in1.2′′ seeing.

We were fortunate to acquire data on all 24 of the detectionfields in all six of our allocated nights. Although the majority ofthe observing time was photometric, on only about 2.5 nights wasthe seeing sufficiently good (<1.3′′) that we acquired V and R dataon the fields for color determination.

5. Reduction to object catalogs

The final image set consisted of about 25 Gbytes of data perobserving night. Using standard procedures, the instrumental sig-

The SKADS survey 107

nature was removed using a 2D bias image and flattened with asuperflat made from the data frames themselves (flattening wasgood to ∼1%).

Each of the mosaic cameras eight CCDs for each pointing werethen passed through a moving-object detection pipeline (Petit etal., 2004). Briefly, the pipeline software takes the triplet of im-ages, finds common stars, aligns the three frames to a commonpixel coordinate system, catalogs all detectable objects, eliminatesall objects that are present in the same location on all three im-ages, and finally identifies linearly-moving objects consistent withmain-belt asteroids. These candidate asteroids are shown to a hu-man operator who eliminates instrumental artifacts which havealigned to produce a false detection, and for confirmed real de-tections an astrometric solution of the CCD was computed andused to measure sub-arcsecond astrometric positions. A plate solu-tion of fourth order based on USNO catalog stars was used for theastrometric measurements of most of the SKADS asteroids; whenthe asteroid was far from the available USNO stars systematic as-trometric errors on that asteroid could reach several arcseconds.These cases became clear when the asteroids were subsequentlylinked and the poor nights were re-measured using a plate solu-tion based on hand-selected stars near the asteroid’s position.

Photometric zeropoints were determined nightly from multi-ple observations of Landolt standard star fields (Landolt, 1992);these were acquired at a variety of airmasses from 1 to 2 andthe extinction terms were measured and included in our photo-metric measurements. Knowing the airmass of observation, R andV brightnesses were measured when the asteroid was not con-fused with background objects; photometric errors based on aper-ture correction photometry are provided if there were photometricconditions. On frames where the VR filter was used, an R-bandbrightness estimate was obtained assuming an average color; whenthese photometric measurements are quoted in our database aphotometric error will not be listed (and these photometric re-sults were not and should not be used for any quantitative work).An internal designation for an asteroid detected in a given imagetriplet is of the form pxffcn, where “p” is simply a leading letterto be explained below, x is the night of observation (1–6), ff is thefield number (1–24), c is the CCD number on the mosaic (1–8),and n uniquely identifies the detected asteroid on the CCD with acharacter from the set (1,2, . . . ,9,0,a,b, . . . ,h). No more than 18asteroids were ever detected on a single CCD.

The number of detected asteroids on each of six nights were1034, 1298, 1274, 1130, 1005, and 1072, where the reader shouldrecall the first three nights were sequential, and then there was a5-night gap before the final three sequential nights. The vast ma-jority of the detected asteroids are expected to be the same asthose that were detected on nearby nights (Fig. 2). Between thetwo 3-night blocks there will be more substantial shear of objectsinto and out of the 8.4◦ patch but this effect can be accuratelymodeled as a function of (a, e, i) in our characterized survey. Itwas then necessary to determine how many and which of the de-tections were multiple apparitions of the same asteroid, and usethe linked observations to calculate heliocentric orbits.

6. Linkage and orbit determination

Starting from the six nightly catalogs of asteroid triplets, pre-liminary orbits were fit to the available observations, with thegoal of first identifying multiple observations of the same aster-oids, and once these linkages were made, standard methods wereused to calculate the heliocentric orbital elements of the asteroids.For our science goal, a highly-accurate set of orbital elements wasless important than determining the instantaneous distance to theasteroid, since it is only the distance which is needed to obtain theH magnitude. We quickly confirmed that the quality of the avail-

Fig. 2. Locations of all 1277 multi-night detections from SKADS on March 21.35 UT2001. The horizontal solid line is the celestial equator, while the diagonal line showsthe position of the ecliptic; at the time given here the opposition point was veryclose to their intersection (observations occurred close to the time of the vernalequinox).

able astrometric data was more than good enough for the purposeof obtaining distances, and that accurate measurements of all sixorbital elements could be obtained as long as at least a week ofobservational baseline was available.

The linking of observations of the same asteroid was accom-plished using a multi-pass procedure. Because the field coveragewas contiguous there was a small chance that an object be de-tected in one field and then be re-detected an hour later in animmediately adjacent field to the west. The same object wouldthus appear twice in a single night’s observations under two dif-ferent designations. Searching for these links was accomplished byfitting Väisälä orbits between all possible combinations of objectsand rejecting those linkages that produced unacceptable residualsin the observations. Only seven cases (of 6813 triplets) of the sameobject being observed twice on the same night were found.

The next stage was to link the detections to observations on asecond night, via an iterative procedure. This was accomplished byfitting Väisälä orbits as above or by matching motion vectors. Thefirst pass used observations from adjacent nights. The second passused observations from nights separated by two days. Subsequentpasses used observations from nights separated by more than twonights.

The third stage was to link the two-night links to third nights.Using the Väisälä orbits generated in the second stage, predictedpositions were generated for each object for all the times of ob-servation. Using these predicted positions, a search for possiblematches was undertaken. If an object was within 20′′ of a pre-dicted position and had consistent motion with the predicted mo-tion, a Gaussian orbit determination was attempted. If this wassuccessful, the object was considered to be a three-nighter. Afterthe first pass using a 20′′ search aperture, the search was repeatedon unlinked objects using 40′′ , 60′′ and 120′′ apertures.

The final stage was to find additional nights for the three-nighters. Using the Gaussian orbits determined in stage three, pre-dicted positions were generated for each object for all the times ofobservation. The search for matches was done as above, using sev-

108 B.J. Gladman et al. / Icarus 202 (2009) 104–118

Table 1Example of available SKADS data, giving information on two of the 1277 SKADS detections seen on 3 or more nights. All four angular elements are given in degrees, and themean anomaly M is given for March 21, 2001. See Figs. 4 and 5 for typical orbital element uncertainties. Each of these two asteroids was not detected on one of the 6 nightsof observation. The lack of photometric information on two of the 21013 exposure indicates that the asteroid was too close to a star for reliable photometry. When error barsare not provided on the photometry this indicates the conditions were either not photometric or that the seeing was poor and the VR filter was used (in which case theR-band magnitude is an estimate only accurate to about a tenth of a magnitude). The ‘Near’ field indicates which of the three R-band exposures the V -band exposure wastaken adjacent to (either ∼3 min before or after). The complete table is provided electronically (see text).

Master a (AU) e I Ω ω M H R

s10124 2.37 0.187 0.50 148.1 251.5 127 17.8

Nightly Rmag1 Rmag2 Rmag3 Vmag Near V − R10124 21.2 21.2 21.3 – – –20124 21.0 21.0 21.1 – – –30126 21.34 ± 0.06 21.24 ± 0.08 21.26 ± 0.08 21.65 ± 0.08 1 0.30 ± 0.10not seen – – – – – –50114 21.4 21.5 21.5 – – –60114 21.6 21.6 21.6 – – –

s21013 2.81 0.245 9.00 5.0 1.9 170 17.2

Nightly Rmag1 Rmag2 Rmag3 Vmag Near V − R11012 22.1 21.8 21.9 – – –21013 21.9 – — – – –31014 22.12 ± 0.08 22.15 ± 0.09 21.98 ± 0.10 22.60 ± 0.13 3 0.61 ± 0.1641034 22.28 ± 0.10 22.36 ± 0.09 23.34 ± 0.09 22.80 ± 0.13 1 0.52 ± 0.16Not seen – – – – – –61033 22.17 ± 0.14 22.00 ± 0.13 22.19 ± 0.14 23.07 ± 0.22 3 0.8 ± 0.3

Table continued in the NASA PDS on-line system 1

eral search apertures. When a match was found, the Gaussian orbitwas recomputed and used to generate new predicted positions andthe search was repeated.

At the end of this process we obtained 1277 orbits of main-beltasteroids that were observed on three or more nights of the SKADSobservations. Of these, 255, 366, 286, and 370 were observed on6, 5, 4, and 3 nights (respectively). Because of the spacing of thenights (3 nights of observation, 5 nights off, and then another 3nights), any asteroid observed on 4 or more nights (907 objects)automatically has an arc baseline of at least 7 days. Of the 370 3-night linkages, 190 have observations from only one 3-night block;such “1-block” asteroids have orbits (and distances) known withconsiderably poorer precision. A linked orbit is identified by its‘master’ designation which is simply the internal designation onthe first night of observation from the first pair-linkage above, withthe “p” replaced by an “s”. For example, the SKADS master desig-nation s21013 is the linked astrometry of p11012, p21013, p31014,p41034, and p61033; this asteroid was not detected on night 5. Ta-ble 1 gives example entries for the type of data available for eachasteroid detected in SKADS; the entire table is available on-line aspart of the Planetary Data System’s Asteroid archive web site, inthe Small Bodies Node data services at http://www.psi.edu/pds.

Fig. 3 shows heliocentric orbital elements for all 1087 main-beltSKADS asteroids with observational arcs of 8 nights or more. Manywell-known features of the main-belt’ orbital distribution are seen,such as the 3:1 resonance, the Flora and Koronis families, and theinner edge of the belt in the a/i distribution caused by the ν16secular resonance. We will discuss the color information presentedin Fig. 3 in Section 10 when we discuss the two-color photometry.

In contrast to previous short-arc surveys, SKADS orbital ele-ments are sufficiently accurate to permit (after de-biasing) thecalculation of the absolute population of many sub-groups of themain asteroid belt. We observed two multi-block jovian trojans,with observed arc lengths varying from 7 to 10 days. Jovian trojansare not present in their intrinsic (but flux-limited) proportions inthe SKADS multi-block list because our shifting of the fields at anaverage main belt rate makes them less likely to be present in bothblocks; our fields were retrograded night to night along the eclip-tic faster than the average rate of the trojans. This will also be true(to a lesser extent) for the Hildas, for which 6 are present in theSKADS multi-block orbits. A simulation of the survey, that respects

the phase relationship of the resonant objects with Jupiter, wouldallow one to estimate the intrinsic Hilda fraction brighter than anH magnitude of roughly 17. At the time of observation, jovian Tro-jans would have been roughly 120◦ from Jupiter and thus we areonly sensitive to the very highest libration-amplitude objects.

None of our 1277 3-night detections nor any of the one-nightor two-night detections (thus a total of 2240 detections) appearto be Hungaria-group asteroids (a < 2 AU, at high orbital incli-nation), which also suffer from shear problems to some extent.However, extracting all Hungarias with H < 15 from the MPC or-bital database, we calculate that at any instant 0.1% of the asteroidswithin ±0.5◦ of the ecliptic are Hungarias; thus our non-detectionof a single Hungaria is not statistically alarming as we only expectof order one such detection. We note that this 0.1% fraction is astrong function of the H threshold used, rising to about 0.3% byH = 18. We believe that the Hungarias are over-represented in theMPC database due to systematically better recovery (even at diam-eters of only a few km) in automated surveys due to the proximityof these asteroids. We do not believe that Hungarias would be verydifficult to recognize even with short arcs.

One goal of SKADS was to be sensitive to asteroids below 1 kmin diameter (about H = 18) throughout most of the main belt (toroughly 3.2 AU heliocentric distance). For such an asteroid near op-position, our mR � 23 limiting magnitude (see below) correspondsto an absolute magnitude of H R = 18.8, and thus even at the outeredge of the main belt we have some sensitivity to sub-kilometerasteroids, with greatly increasing sensitivity as we approach theinner edge. In the direction that SKADS was looking nearly all as-teroids with diameters larger than 1 km were detected. At 2 AUheliocentric distance our mR � 23 limit yields detections as faintas H R = 21.5, corresponding to a diameter of only several hundredmeters.

The H-magnitude measurement incorporated all available R-band photometry to compute an average H R magnitude (that is,our H is tied to the R band). Each R-band photometric measurewas converted to an H R magnitude and these values were aver-aged. The H-magnitudes of the SKADS detections ranges from H =12.1 (numbered asteroid 6301, re-detected by SKADS) to the small-est asteroid with H = 22.3 (SKADS internal designation s1061a),corresponding to a diameter ratio of about two orders of magni-tude.

The SKADS survey 109

Fig. 3. Heliocentric orbital elements for all asteroids detected by SKADS having orbital arc lengths of 8 or more days. Two Trojan asteroids (with e < 0.1 and i < 10◦) arenot shown. This is a biased sample of the main-belt orbital distribution since not all orbits are equally likely to be present in both blocks of sky coverage 8 days apart.For asteroids with average mR > 22.5 points are open squares, but brighter objects are coded red triangles if V − R > 0.38 or blue solid squares if V − R < 0.38. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6.1. Orbit accuracy

The accuracy of the derived orbital elements was judged bysearching for links with previously-known asteroids in the MinorPlanet Center database. The much-longer available arc from thearchived Minor Planet Center observations yielded the ‘true’ or-bit for the asteroid and the discrepancy between the true orbital

elements and those derived from our 7–10 day arc provided anestimate of the accuracy of the elements when based on SKADS or-bits alone. (Note that a minimum arc of 7 days results for a SKADSmulti-block asteroid linked on 3 or more nights.)

In total, 135 previously-known asteroids were re-discovered inSKADS. We found that the orbits with �7-day arcs provided dis-tances and orbital elements accurate to better than five percent.

110 B.J. Gladman et al. / Icarus 202 (2009) 104–118

Fig. 4. Discrepancy of SKADS orbits against the ‘true’ orbit as determined by thepreviously-known orbit in the MPC, for SKADS asteroids having orbital arc lengthsof 7 or more days. Horizontal lines bound ±1-sigma variations from the mean error.These 1-sigma variations in a, e, heliocentric distance r and H are 0.03 AU, 0.03, 0.11AU, and 0.36 magnitudes, respectively. See text for discussion of the discrepanciesin H .

Figs. 4 and 5 present the difference between the SKADS-derived el-ements and the ‘true’ MPC elements for the previously-known as-teroids in our sky coverage. The orbits are sufficiently accurate thatthe dominant source of uncertainty in the derived H-magnitudescomes from the photometric uncertainty. In fact, since all of theSKADS asteroids that have been identified with previous discov-eries have H < 18, one might be surprised that the discrepancywith the MPC H values are so large. Because these H < 18 aster-oids are detected with magnitudes far above our roughly mR ∼ 23limit, we believe that this reflects real systematic errors in theH magnitudes of the MPC asteroids (see Jedicke et al., 2002;Juric et al., 2002, for discussion).

After the PLS survey, other studies which have probed largesamples with apparent magnitudes fainter than 20 (Ivezic et al.,2001; Yoshida et al., 2003, 2007; Wiegert et al., 2007) have hadonly very short observational arcs available (of order hours) andwere forced to apply statistical ranging techniques to estimatedistances and therefore H magnitudes of the detections (whichcould have errors up to a magnitude). The SKADS sample is thelargest available sample of asteroids with good orbits detectedin a uniform and well-characterized way. The high-precision H-magnitude measurements allow one to more accurately estimateasteroid diameters, and it is the diameter distribution that al-lows one, when coupled to a well-characterized understanding ofthe detection conditions, to study the main-belt orbital and sizedistribution and furnish constraints for collisional evolution mod-els.

Fig. 5. Discrepancy of SKADS angular orbital elements against the ‘true’ orbit asmeasured by the previously-known orbit in the MPC, for SKADS asteroids havingorbital arc lengths of 7 or more days. Horizontal lines bound ±1-sigma variationsfrom the mean error. These 1-sigma variations in I , Ω , ω, and M are 0.7, 1.0, 27,and 28 degrees, respectively. The ω and M errors are strongly anti-correlated sincethe longitude of the particles at the time of observation are known; the 1-sigmaerror in the longitude of pericenter (ω + M) is only 5 degrees.

7. Characterization

The SKADS survey was carefully designed so that precise char-acterization of the survey (thorough knowledge of sky coverage,sensitivity, and field timing) would allow us to ‘de-bias’ the sur-vey. Subtle systematic effects can easily affect the slope that oneconcludes for the H-magnitude distribution. Thus, great effort wasexpended to determine the detection efficiency on each night as afunction of asteroid apparent magnitude and rate.

For each CCD triplet of each night we implanted 250 artificialasteroids in the image, using a model PSF created from a set ofbright stars on each individual CCD frame. Thus, changes in see-ing (and the concomitant loss of sensitivity) are correctly modeled.Since at opposition some of the closest asteroids were moving atrates that (in the best seeing) could induce some mild trailinglosses (with the objects moving by of order the seeing during the2-min exposure), the implanted flux for each objects was actuallyadded in 10 bundles of 10% of the flux, that was implanted at po-sitions corresponding to the randomly-chosen rate and direction ofthe artificial object.

Artificial-object magnitudes were chosen from mR = 17 to 24,with a square-root weighting to place more artificial objects in thefainter end of the distribution (where knowledge of the decliningdetection efficiency is more important). Artificial objects were im-planted at rates of 12–49 arcsec/h, moving retrograde with a dis-persion of rates covering all possible main-belt orbital inclinations.It is important to note that because this was done at random lo-cations (and not on a grid), this realistically measures the fractionof objects lost due to the background confusion which increases

The SKADS survey 111

Table 2Parameterization of the efficiency functions for each night, using variables definedin Eq. (2).

Night ηo c mL w

21 0.961 0.0056 22.59 0.2222 0.984 0.0061 23.23 0.1923 0.989 0.0051 22.99 0.1529 0.979 0.0046 23.13 0.1730 0.984 0.0057 22.83 0.2031 0.974 0.0043 22.94 0.15

Fig. 6. Detection efficiency versus R-band magnitude for the SKADS observationsof March 21 2001 UT. Parameters for the fit are given in Table 2. Horizontal errorbars just indicate the bin width, whereas the vertical error bars are due to countingstatistics.

steadily as fainter magnitudes are approached, and also measuresthe fraction of asteroids which ‘leave’ the CCDs during an imagetriplet as a function of rate of motion.

These frames were then passed through the same processingsteps as the original data, except that a human operator did notview and confirm the artificial objects. Comparing the implantedlist with those found by the pipeline software, the fraction ofdetected objects as a function of magnitude and rate were deter-mined. This precise characterization is needed in order to calculatea detection bias.

For each night the detection efficiency is determined as a func-tion of object apparent magnitude and rate of motion. The fractionη of artificial asteroids detected as a function of these variablesis fit by smooth functions. The crucial efficiency versus magnitudebehavior is fit to the formulation

η(mR) = ηo − c(mR − 17)2

1 + exp(mR −mLw )

, (2)

where ηo � 98% is the efficiency at mR = 17, c ∼ 0.5% measuresthe strength of a quadratic drop, which changes to an exponentialfalloff over a width w near the magnitude limit mL . Table 2 tabu-lates these parameters for the six nights of observation and Fig. 6shows the efficiency for the first night as an example (this happensto be the shallowest night of SKADS). Brighter than mR = 17 the ef-ficiency is taken to be ηo . The value of w is mostly controlled bythe constancy of the observing conditions since a variety of seeingconditions softens the roll-over to zero efficiency near the limit.

The limiting magnitude mL varies from mR = 22.6 to 23.2 due tothe variation in average seeing from night to night.

Generic features of Fig. 6 are important to understand for quan-titative interpretation of the resulting de-biased distributions. Theslow drop-off from mR = 17 to 22 is due to the gradually in-creasing confusion engendered by the growing number counts ofbackground stars and galaxies. This slow decline over ≈5 magni-tudes will be present in all CCD data sets which use just a fewimages to detect moving targets. Even a gradual drop at the levelof ten percent (or more) has a systematic effect on the debiasedsize distribution; thus surveys which have taken their detectionefficiency to be constant above some fixed magnitude limit willunderestimate the true slope of the H-magnitude distribution, aswe quantify below.

8. Debiased apparent magnitude

Having determined the detection efficiencies, it is possible tode-bias the apparent magnitude distribution into an intrinsic ‘lu-minosity function’; that is, the number of asteroids as a function ofapparent magnitude. One can simply divide the differential num-ber counts in a given magnitude bin by the estimated detectionefficiency in that bin to produce the actual number of asteroidsin the survey region as a function of magnitude; these are usuallythen compiled into a cumulative number of asteroids brighter thana given mR . Such distributions have been previously measured byIvezic et al. (2001) and Yoshida et al. (2003).

Since almost all the detections in our sky coverage over ablock of three sequential nights will be the same asteroids, thebias-corrected number as a function of magnitude should be vir-tually identical. This night-to-night consistency is thus one testof the quality of our characterization. Fig. 7 shows that downto magnitude mR = 23 we find a consistent number of asteroidsover the entire magnitude range. This figure also shows that ourcharacterization appears to have a systematic problem past 23rdmagnitude, as the differential number of counts ceases to rise.Although this is in principle physically possible if there are ‘col-lisional waves’ in the size distribution (Campo Bagatin et al., 1994;Durda et al., 1998), we feel it more likely to be due to an over-estimation of our detection efficiency at the faintest end. Petit etal. (2004) illustrate how this roll-over effect can occur in surveysfor which the detection efficiency is determined by implanting ar-tificial objects and detecting them with an automated pipeline;if the artificial objects are not subject to the same operator ver-ification process as the real asteroids then one counts very lowsignal-to-noise objects discovered by the pipeline as found whena human operator might reject them as not being sure detections.This results in a systematic overestimation of the detection effi-ciency at the very faintest end of the survey (starting roughly atthe magnitude of 50% detection efficiency). Removing this effectwould require a human operator to examine a significant numberof the tens of thousands of artificially-implanted objects.

Fig. 8 presents a cumulative version of the apparent magni-tude distribution, which clearly shows a change starting at mR �19where the power-law slope drops from �0.6 to �0.27, the samevalues as Ivezic et al. (2001) reported down to the SDSS limit ofmR = 21.5. (We will always quote slopes as the multiplicative coef-ficient in the cumulative distribution; thus a slope of 0.27 indicatesthat the number of asteroids brighter than a given magnitude in-creases by the factor 100.27 per magnitude.) Our deeper data showsthat this shallow 0.27 slope continues down to an apparent mag-nitude of at least mR = 23. Using our de-biased magnitude distri-bution, we estimate Σ ∼ 210 asteroids per square degree brighterthan mR = 23, where

Σ(< mR) = 210 × 100.27∗(mR−23), 20 < mR < 23 (3)

112 B.J. Gladman et al. / Icarus 202 (2009) 104–118

Fig. 7. The de-biased number of SKADS asteroids in each of the first three nightsof the survey. Each differential bin is de-biased by the detection efficiency for thatnight (March 21 as black squares connect by solid lines, March 22 as red trian-gles connected by dotted lines, and March 23 as blue circles connect by dashedlines). Error bars reflect both Poisson statistical errors in the number of detectedobjects and the uncertainty in the detection efficiency in that bin. The decrease inthe differential number of counts past 23rd magnitude is almost certainly due to anoverestimate of the true detection efficiency at the very limit of the survey ratherthan a true change in the size distribution. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. The cumulative number of SKADS asteroids as a function of apparent R-bandmagnitude, for two nights of the survey. The field area is 8.4 square degrees, allow-ing conversion to sky density Σ . Power laws have been fit to the detections brighterthan 19th magnitude, and for the magnitude range 20.5–22.5. March 22nd data isin black, while March 23rd is in grey (or red in the color version of this figure).The consistency on the two nights shows the uniformity of the survey characteriza-tion.

is the cumulative density of asteroids on the sky (at oppositionwithin 1 degree of the ecliptic). If the 0.27 slope continued to mag-nitude mR = 24.4, this would imply an on-sky density of Σ � 500per square degree, in contrast with Yoshida et al. (2003) who es-timate ∼290/square degree at that depth. Since SKADS alreadydetected Σ = 210/square degree brighter than mR = 23, this wouldimply that the slope of the cumulative apparent magnitude distri-bution would have to drop precipitously to 0.10 or less just pastour magnitude limit. We see no physical reason to expect just adramatic change; it is plausible that a slight over-estimation ofthe Yoshida et al. detection efficiency as their survey limit is ap-proached is responsible for an under-estimation of the true surfacedensity in this magnitude range.

9. H R magnitude distribution

Assuming that the size distribution is the same everywhere inthe asteroid belt, that the average albedo does not vary with he-liocentric distance, and that the size distribution is a single powerlaw over the belt’s entire radial range, a 0.27 slope in the apparentmagnitude distribution would imply a differential asteroid size dis-tribution with index 2.4 (dN/dD ∝ D−q with q = 2.4). Since all ofthese assumptions are likely untrue at some level, a more complexmodeling of SKADS is needed, based on the H-magnitude distribu-tion. In a future paper we will present a full (a, e, i, H) de-biasingof the SKADS orbital and H distribution to calculate calibrated dis-tributions of these parameters.

However, since our survey is relatively deep and the distancesare well known, we can make some preliminary conclusions aboutthe H-magnitude distribution down to the level where our obser-vational incompleteness becomes serious. Our measurements pro-vide a ‘raw’ H-magnitude distribution. Here ‘raw’ indicates thatalthough in Section 8 we corrected the mR distribution for ourcharacterized apparent magnitude bias, it is much more complexto do this for H . Thus the H-magnitude distributions shown heresuffer from incompleteness in our multi-block catalog not only be-cause of simple flux sensitivity on any one night, but also becauseasteroids with large H magnitudes are not equally likely to havebeen detected on at least one night in both SKADS blocks (sincefaint asteroids are less likely to be detected multiple times). Therequirement to be seen at least two nights in one block and onenight in another means that beginning around H = 17 (which cor-responds to mR � 22 in the outer belt) we cease to efficientlydetect small asteroids in the outer belt on at least 3 nights. Thiswill be modeled explicitly using the characterization informationfrom SKADS (Jedicke et al., 2009, in preparation).

The slope α of the cumulative number versus H R distribution

log10 N(< H R) = C + αH R (4)

is related to the differential size distribution q of the underlyingpopulation by q = 5α + 1 if a power law of slope q holds at allradial distances to which the survey is sensitive at that H range,and the albedo distribution is independent of distance. Because ofthe formulation of Eq. (4), note that the slope is α for both thedifferential and cumulative H distributions.

Fig. 9 shows the raw distribution in (r, H R) space for all themulti-block SKADS asteroids and Fig. 10 shows its differential num-ber counts. While a fit to the entire sample down to H R = 17.1yields a slope of α = 0.38 (corresponding to q = 2.9) and es-sentially the same as that determined by Jedicke and Metcalfe(1998), our statistics for H R <15 are poor since our areal cov-erage is not sufficient to have detected enough large asteroids.In fact, for H R � 15 the main-belt asteroid distribution is essen-tially completely known and is well represented by a power-lawof slope α = 0.51 in the range H = 11–15. SKADS was instead

The SKADS survey 113

Fig. 9. The SKADS multi-block detections in (r, H R ) space. Two jovian Trojan aster-oids with r � 5 AU are excluded. The curved line shows the approximate locus ofpoints with mR = 22.5, after which incompleteness begins to set in for the multi-block objects. Note that to magnitude H R � 17.5 we are above the incompletenesscurve at the outer edge of the belt (around r � 3.7 AU).

designed to study the H > 15 regime down to a diameter be-low one kilometer. Examining this end of the distribution, we seethe expected change in behavior at H R � 17 where our surveybegins to be incomplete. Surprisingly however, these data showa very stable slope in the range H = 15–17, in contrast withprevious studies (Jedicke and Metcalfe, 1998; Ivezic et al., 2001;Wiegert et al., 2007) which conclude that the power-law slope ischanging slope over this H-magnitude range. We have investedconsiderable effort trying to understand this difference.

We first explored how the accuracy of the heliocentric dis-tances could affect the results. SKADS is the first survey since PLSto have both accurate distances and H magnitudes. The four re-cent projects (Spacewatch, SDSS, SMBAS, and Wiegert et al., 2007)had only short observational arcs (2 h or less), and were forcedto adopt imprecise methods to estimate asteroid heliocentric dis-tances Rapp. Various techniques were used in these papers toestimate distances. Since the formulae (similar to Jedicke, 1996)used by Ivezic et al. (2001) for the SDSS study are explicitly givenin their Appendix A, along with the empirical correction Rbest =Rapp × (1.16 − 0.03Rapp), we have used this algorithm to studysystematic effects. We took a single night of astrometry (about40-minute arcs) for 786 SKADS asteroids with high-quality orbitsused their ecliptic-velocity components to compute distances us-ing the SDSS algorithm, and then compared the results with the‘true’ values as derived from the SKADS multi-block orbit. Fig. 11’supper panel duplicates the effect already discussed in Ivezic et al.(2001) and the lower panel shows the resulting H discrepancies(mostly confined to ±1 mag, but up to 4 mags in the inner belt).The ‘DC offset’ used to calculate Rbest still leaves a systematic dis-

Fig. 10. The H R magnitude distribution of the SKADS sample, uncorrected for eitherdetection efficiency above the flux limit or incompleteness past the detection limit.As the previous figure shows, SKADS is essentially complete to H R = 17.1. Since forH R < 15 we have poor statistics and the entire main belt catalogue is complete,we measure the slope from 14.8 < H < 17.4 (solid line) and determine a power-lawslope of 0.29 ± .03. Faintward of the completeness limit asteroids fall below theextrapolation due to our falling detection efficiency (Fig. 6), and then rapidly due tocomplete insensitivity to the outer belt (Fig. 9).

tance trend which is then magnified by the 1/r4 dependence ofreflected flux into a systematic H trend.

A second systematic effect that is present in the H distributionis the effect of incompleteness due to the fact that the apparentmagnitude limit causes asteroids with different H R magnitudes tohave variable detection efficiency at different distances. Examin-ing Fig. 6, for asteroids brighter than apparent magnitude mR � 22there is a gradual decline of detection efficiency (dominantly dueto the increasing number of background sources which cause con-fusion and thus asteroid non-detection); we call this the ‘shoulder’of the detection efficiency. Then over the next magnitude the de-tection efficiency drops rapidly to zero due to loss of flux. Lastly,objects beyond mR ∼ 23 have essentially no probability of beingdetected. Switching to absolute magnitude (and thus spreading agiven mR out over the belt due to differing distances), the effectsof the gradual ‘shoulder’ are difficult to see in Fig. 9, although themore rapid drop in the last magnitude is easily visible at eachheliocentric distance. Many surveys have dealt with their incom-pleteness by ‘cutting’ their sample using an apparent magnitudecut brighter than their apparent-magnitude detection limit (abovebut parallel to the curved line of Fig. 9), and then taking the de-tection efficiency above this limit to be constant. Because we havemeasured the SKADS detection efficiency as a function of magni-tude, we could in principle divide each object by the detectionefficiency of the apparent magnitude at which it was detected (thisis difficult to do accurately for surveys with short observationalarcs since the uncertainty in distance translates to an uncertainH magnitude). Although SKADS has excellent distance measure-ments, those asteroids with high-quality distance are necessarilythose with detections on at least three nights so it is unclear whichnight’s detection efficiency should be applied. This should be donevia simulating the detection efficiency for a full model of the aster-oid belt, and is beyond the scope of this paper. However, we canmake an approximate correction in the region where our detectionefficiency is non-negligible by dividing each magnitude with anH magnitude determined by its detection efficiency on the worstnight (Table 2) that it was actually detected to get a ‘partially fluxdebiased’ H-magnitude distribution. Fig. 12 shows the result of thisexercise, which confirms that the debiased H-magnitude distribu-tion continues to follow a power-law increase down to H R � 18;the method cannot correct past this point because Fig. 9 showsthat the efficiency has now reached zero in the outer portion of thebelt. The power-law slope of this partially-debiased distribution is

114 B.J. Gladman et al. / Icarus 202 (2009) 104–118

Fig. 11. Comparison of heliocentric distances r and derived H magnitudes betweenSKADS and a calculation using only 40-minute arcs using the method described byIvezic et al. (2001). A systematic error (top panel) exists in the distance determina-tions where asteroids really in the inner belt (r ∼ 2 AU) are thought to be furtheraway and thus larger (smaller H) than in reality (bottom panel). In the outer beltthe effect is reversed and objects are thought to be up to 1 AU closer and system-atically smaller (larger H) than in reality.

α = 0.30 ± 02, the same value as before the rough de-biasing, butnow extending from H R = 15–18. We thus conclude that after thesteeper α = 0.5 slope exhibited by the H < 15 complete belt, atsmaller sizes the asteroid belt flattens to a slope of α = 0.3 andmaintains this until H R > 18.

9.1. Comparison with other work

While there can be little argument regarding the H-magnitudedistribution for H < 15 since there is now an essentially-completeinventory of the main belt, different publications have derived dif-ferent slopes for the H-magnitude distribution for H >15. Jedickeand Metcalfe (1998) estimated slopes in a variety of distance andH-magnitude regimes, mostly sensitive to H < 15 because of therelatively shallow depth of the Spacewatch survey; they found astrong slope change near H = 13 and varying slopes from H �15–17 to 0.22–0.26 ± 0.04 (adding systematic and random uncer-tainties). The Ivezic et al. (2001) analysis of the Sloan data yieldsan asymptotic slope for H 15.5 of 0.24–0.28 ± 0.01, althoughthe error does not account for systematic effects (Ivezic, 2007, pri-vate communication). Yoshida et al. (2003) report a continuousdecline in slope as fainter magnitudes are reaches, with values ofα = 0.27, 0.23, and 0.20 (all ±0.01) in the inner, middle, and outerbelt, respectively, for asteroids with 0.5 < D < 1 km. Yoshida etal. (2007) then reported a dramatically-different conclusion, witha constant slope of 0.26 or 0.27 (depending on whether it is overthe entire main belt for H > 17.6, or separated by asteroid color).Most recently, Wiegert et al. (2007) measured slopes of 0.31±0.01,0.36 ± 0.01, and 0.36 ± 0.02 in the inner, middle, and outer belt.

Fig. 12. A rough flux debias (see text) of the H-magnitude distribution. The solidline from H R = 15 to 18 is a linear regression fit to the debiased data, having slopeα = 0.30±0.02, and whose extrapolation is shown. Brighter than H R = 15 this fit ismarginal to the SKADS data, but the belt is known to obey α ∼ 0.5 in this region;fainter than H R = 18 we cannot correct for flux bias since our sensitivity in theouter belt becomes zero. The dashed curve shows the H-magnitude distributionconcluded by Sloan, matched to H R = 15.5 where they conclude the flattening inslope begins.

So although there is agreement that beyond H > 15 the slope hasbecome shallower, the value of the slope and whether there is avariation in different parts of the belt is unclear.

Our H R > 15 slope measurement, taking the entire belt intoconsideration, is α = 0.30 ± 0.02. Because we designed SKADSto provide accurate distances and a well-measured detection ef-ficiency, we have attempted to explore the effect of systematicassumptions if we had analyzed our data with a poorer knowledgeof the detection efficiency. More explicitly, if a survey is assumedto have a constant (and nearly 100%) completeness above a mag-nitude cut, what effect does a ‘shoulder’, as discussed above, haveon the results? To address this, we created a synthetic main-beltmodel with a single power-law exponent of α = 0.38, and whoseorbital distribution mimics that of the H < 15 main-belt popula-tion (although the orbit distribution turns out to be unimportantfor the conclusion below). We then simulated an observationalsurvey which had a 100% detection efficiency to apparent mag-nitude mR = 20 and then a linear decrease to 70% at mR = 22.4where we apply a magnitude ‘cut’. This generic kind of broadshoulder in detection efficiency is a good representation of an ob-servational survey with a gradual shoulder caused by increasingbackground confusion, even a survey whose mR cut is felt to bebrighter than the beginning of the abrupt drop (at mR ∼ 22 forSKADS in the night shown by Fig. 6). In this simulation the in-formation on H is essentially perfect since the model detectiondistances and resulting H magnitudes are assumed to have no er-rors. The raw H counts of the simulated detections (Fig. 13, lower

The SKADS survey 115

Fig. 13. An illustration of the systematic effect created by ignoring a shoulder in theapparent-magnitude detection efficiency. The lower points shows the differentialnumber counts of detections drawn from a main-belt model (see text) where thedetection efficiency declines linearly beginning at mR = 20.0 to 70% at mR = 22.4.The upper points show the resulting H R -magnitude distribution when the simu-lated detections are de-biased using the algorithm of Fan et al. (2001) assuming thatthe detection efficiency is constant The debiased distribution is shifted upwards forclarity. The input model slope (α = 0.38 over the entire asteroid belt) is shown asa dashed line for reference in both cases. The debiased distribution follows an in-correct power-law slope of ∼0.33 (dotted line) past H = 16 due to neglecting theefficiency drop-off in the last 2.4 magnitudes of the simulated survey.

data set) show a noticeable flattening beginning at the H magni-tude corresponding to mR = 20 at the outer edge of the belt, andthe slope continues to flatten as the ‘shoulder’ advances across theentire belt; beginning at H ∼18.5 the apparent magnitude cut be-gins to completely remove the population beginning at the outeredge of the belt and working in to the inner edge. The incomplete-ness caused in the last 2–3 magnitudes, by the fact that asteroidswith this H-magnitude may only be seen in the inner asteroidbelt (see Fig. 9), can only be corrected by a model. Wiegert etal. (2007) did not apply such a correction. Yoshida et al. (2003)performed a complex model-dependent bias correction, but do notappear to differentiate between the complete and the incompleteregime. Ivezic (2001) applied a correction, assuming that the H-distribution was independent of distance and that their detectionefficiency was constant above their magnitude cut, by using thealgorithm of Fan et al. (2001). Although tests show (Ivezic, 2007,private communication) that the Fan et al. algorithm does prop-erly correct a simulated belt if the survey’s detection efficiency isconstant to a magnitude cut; in the more realistic case where theefficiency drops as the limit is approached, the debiased distribu-tion exhibits a slope flatter than the input (Fig. 13, upper data set)if an efficiency variation is not included in the de-biasing. That is,a spurious slope results at the faint end of the distribution; thevalue of this incorrect slope is a function of the size of the shoul-der and systematic distance errors (Fig. 11), but always produces ashallower slope than the input model.

The SDSS asteroid data set lacked a direct measurement ofthe detection efficiency as a function of magnitude and assumeda constant detection efficiency down to their magnitude cut ofmR ∼ 21.5. The experiment shown in Fig. 13 used a detection ef-ficiency function (or selection function, in the terminology of Fanet al., 2001), where the detection efficiency at the cut was about70%, may be a reasonable approximation to the real Sloan selectionfunction, and the 0.05 drop they find in the slope for H R >15.5(right at the point where their mR cut begins to affect the outermain belt) is of comparable magnitude to the slope difference be-tween our result and that of Ivezic et al. (2001). Thus, since SKADS

Fig. 14. A rough flux debias (see text) of the H-magnitude distribution of inner-belt asteroids (heliocentric distances less than 2.5 AU). The solid line from H R =15–19.8 is a linear regression fit to the debiased data, having slope α = 0.23 ± 0.04.The slope is poorly measured since this region of the belt contains only about onequarter of the detections. Beyond H R = 20 a rough flux debias cannot correct the Hcounts.

has both high-quality distances and a well-measured magnitudeefficiency, our data set is much less influenced by systematic ef-fects. We thus believe our constant H R = 15–18 slope of α = 0.30to be accurate, with no strong evidence of any change of slopedown to the SKADS limit. However, as Fig. 12 shows, in this mag-nitude range the relation for the H-distribution given by Ivezic etal. (2001) gives an acceptable representation of the SKADS data (asjudged by a chi-squared test).

9.2. H distribution of the inner belt

Examining Fig. 9 shows that SKADS detected asteroids withH R >18 out to about 3.5 AU, which includes most of the mainasteroid belt except a small number of outer-belt asteroids thatare at their aphelia. Since the 3:1 mean-motion resonance withJupiter is often used as a dividing line between the inner and outerbelt, in this section we confine the analysis to SKADS detectionscloser than 2.5 AU, resulting in the survey maintaining good de-tection efficiency down to H R � 19.5, thus including asteroids wellbelow a kilometer in diameter. The penalty is a smaller samplesize; only 276 SKADS asteroids were detected at r < 2.5 AU. Ap-plying the same analysis as before, we find (Fig. 14) again thata simple power-law fit to the data in the range H R = 15–19.5 isacceptable, and that a roll-over to a shallower slope does not ap-pear to be needed. The slope for just the inner belt detectionsis lower than for the entire belt fit as a whole, but the value ofαinner = 0.23 ± 0.04 is less well constrained due to poorer statis-tics; this slope is α = 0.070±0.045 shallower than our measure-

116 B.J. Gladman et al. / Icarus 202 (2009) 104–118

Fig. 15. Photometric information related to SKADS. Upper left: A sample of main-belt asteroids of known taxonomic type transformed from the ECAS to the KPNO V − Rcolor system, relative to the proposed V − R = 0.38 for S/C separation. Upper right: All SKADS asteroid colors on March 23rd, as a function of R-band magnitude. Lower left:Uncertainty in our measured V − R color as a function of the color. 50th and 90th percentiles for the V − R uncertainty are shown. Uncertainties do not appear to dependon color, which is important to insure that no systematic bias exists against a given spectroscopic type. Lower right: Color uncertainty as a function of R-band magnitude.Asteroids that were brighter than mR � 21.5 in our sample have V − R colors accurate to 0.15 magnitudes or better.

ment of the slope for the belt as a whole, and not convincinglydifferent.

SKADS lacks enough detections to conclude whether or not theinner-belt size distribution differs significantly from the outer beltas a whole, providing only very weak evidence for a shallowerslope in the inner belt. Wiegert et al. (2007) concluded that theinner belt had a shallower slope than the outer, as did Ivezic etal. (2001) if one makes the reasonable assumption that the inner

belt is dominated by red (mostly S-type) asteroids and the outerbelt by blue (mostly C-type) asteroids. Yoshida et al. (2003) makethe opposite conclusion, finding the outer belt to be significantlyshallower than the inner belt.

10. V − R colors of SKADS detections

Each linked SKADS asteroid with an orbit has observations on3, 4, 5, or 6 nights. Of the 1277 multi-night asteroids detected, all

The SKADS survey 117

but 44 had at least one V − R color measurement, and some hadas many as three. During the periods of photometric conditions inwhich a 120-sec V -band exposure was acquired, this image wasacquired within 3 min of a 120-s R-band exposure, so that rota-tional variation is unlikely to have had any effect on the color.

The photometric accuracy of the data is primarily limited bysignal to noise. Fig. 15 shows the color quality of the data. Most ofthe asteroids have V − R colors accurate to 0.15 magnitudes. Af-ter transforming the commonly-used ECAS filter set to match theV and R bandpasses for the KPNO mosaic camera filters, it was de-termined (Howell, 2002, private communication) that V − R = 0.38was a separator between the dominant S and C taxonomic typesof the main asteroid belt. Fig. 15 shows how one may be able toassign asteroids to taxonomic types based solely on their V − Rcolor. The larger spread in colors as R-band magnitude increasesis due to the mounting errors due to declining signal to noise.Beyond 19th magnitude it will be difficult to securely determinewhich side of the V − R = 0.38 line a given asteroid falls, althoughthis could be assigned probabilistically.

We provide this information in the SKADS catalog for thosewho may wish to do color-based modeling. Using this color in-formation, modelers can create different prescriptions to map de-tected asteroids to taxonomic class and thus albedo. Coupled withdetailed knowledge of the detection biases, this will allow explo-ration of the size distribution in different dynamical regions of thebelt.

11. Conclusion

The SKADS survey provides the necessary information for de-tailed modeling of the main asteroid belt’s size distribution, withsufficiently-precise orbital information that one could examine dif-ferent orbital zones of the belt if a full 4-dimensional de-biasingin (a, e, i, H) space were performed. Using color as a proxy fortaxonomic type will allow one to estimate albedos and computetrue diameters for the asteroids, allowing theorists to model themain-belt size distribution. To measure the true size distribution itis necessary to know the detection efficiency as a function of or-bital parameters and H magnitude. This is a complicated functionof the orbital elements and needs full knowledge of the detec-tion timing and field pointings, all of which are provided in apublicly-accessible form on the NASA Planetary Data System website http://www.psi.edu/pds. The data release will consist of:

• Field pointings on each night.• Full astrometric information for all detections on each night,

in standard IAU Minor Planet Center format.• The full version of Table 1, containing linkages, orbital ele-

ments for the linked objects, and full photometric informationwith estimated uncertainties.

• Lists of astrometric observations that are unlinked or have onlytwo nights of observation.

We confirm the roll-over in the apparent magnitude luminosityfunction at mR � 18–19 to a shallow cumulative power-law slopeof 0.27. This continues to at least magnitude mR = 23, the 50%limit of our well-characterized survey.

SKADS confirms that the H R > 15 slope is shallower than theH R < 15 regime. Using our accurately-determined distances tocompute H R absolute magnitudes, we find that a magnitude slopeof α = 0.30 ± 0.02 holds in the belt from H R � 15–18, correspond-ing to an incremental power-law diameter distribution with indexof −2.5 (valid in the roughly 1 to 8 km diameter range). Althoughshallower than the Dohnanyi result of −3.5, this is steeper thanthe estimate from previous surveys, and will have implications for

size-strength scaling laws for the collisional processes of the mainasteroid belt.

In particular, the lack of any firm evidence for a roll-over toa shallower H-magnitude slope before at least H = 18 (and per-haps even deeper) calls into question modeling that attempts toeither (1) explain the claimed rollover via collisional physics (e.g.Bottke et al., 2005), or (2) use it to explain features seen in craterdistributions (e.g., Strom et al., 2005; O’Brien et al., 2006). If theα = 0.30 slope continues to diameters of 200 m, this results in apopulation ∼2–3 times larger than a distribution fit to the lunarhighland craters; thus if the size distribution does not dramaticallyflatten after H = 18 it may call into question the identification ofthe asteroid belt as the source of the impactors creating the lunarhighlands.

Acknowledgments

We acknowledge the role of Paolo Farinella in the conceptionof this research program, and dedicate this paper to his memory.Great thanks are extended to Zeljko Ivezic for his cooperation indiagnosing systematic effects when weak magnitude-dependent ef-ficiencies are present. This work was carried out with support fromNASA’s Planetary Astronomy program and a time allocation at KittPeak National Observatories. BG and BW acknowledge NSERC, CFI,and the Canada Research Chairs program for support.

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