SKADS: A sub-kilometer asteroid diameter survey.

Transcription

1 To be submitted to Icarus: May SKADS: A sub-kilometer asteroid diameter survey. Brett J. Gladman Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1 Canada Donald R. Davis and Carol Neese Planetary Sciences Institute, 1700 East Fort Lowell, Suite 106, Tucson, AZ, USA Gareth Williams IAU Minor Planet Center, Havard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge MA, USA Robert Jedicke Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI, 96822, USA JJ Kavelaars National Research Councial Hertzberg Institute for Astrophysics, 5071 West Saanich Road, Victoria BC, V9E 2E7, Canada Jean-Marc Petit Observatoire de Besançon, B.P. 1615, Besançon Cedex, France Hans Scholl Observatoire de la Côte d Azur, B.P. 4299, Nice Cedex 04, France

2 2 Matthew Holman Havard-Smithsonian Center for Astrophysics, MS-18, 60 Garden Street, Cambridge MA, USA Ben Warrington Joint Astronomy Center, STREET ADDRESS, Hilo, Hawaii, USA ABSTRACT We have acquired a sample of main-belt asteroids for which we have good-quality orbital and absolute H-magnitude determinations in order to study the orbital and size distribution of small main-belt asteroids. Based on six observing nights over a 11-night baseline we have detected, measured photometry for, and linked observations of 1087 asteroids which have one-week time baselines or more. The linkages allows the computation of full heliocentric orbits (as opposed to only statistical distances as some surveys have had in the past). Judged against known asteroids in the field, the typical uncertainty in the a/e/i orbital elements is less than 0.03 AU/0.03/0.5. The distances to the objects are sufficiently-well known that photometric uncertainties dominate most of their H-magnitude error budget, which are generally known to 0.3 magnitudes or better. The detected asteroids range from H R =12 22 and thus provide a set of objects to study down to sizes well below 1 kilometer in diameter. We find an onsky surface density of 210 asteroids in the ecliptic with opposition magnitudes brighter than m R = 23, with a slope of the cumulative number of asteroids per magnitude of 0.27 down to the limit of our survey. THE FOLLOWING SENTENCE PRELIMINARY. We find a preliminary estimate for the slope of the cumulative H-magnitude distribution to be 1.9/magnitude in the H R =14 17 range. We provide the information necessary such that anyone wishing to model the main asteroid belt can compare a detailed model to our detected sample.

3 3 1. Introduction The number of main belt asteroids rises steeply with decreasing diameter due to the power-law nature of the size distribution. Although collisions crater the largest asteroids and may occasionally break up a big one to create an asteroid family, objects larger than about 100 kilometers have not had their size distribution significantly modified since primordial times. However, asteroids a few kilometers and smaller in size are recent collisional shards of larger objects since their lifetime against collisional destruction is much less than the age of the Solar System. As such, these bodies preserve information about collisional breakup processes, the dominant physical process that has shaped the present asteroid belt. While our knowledge of the asteroid size distribution has dramatically increased in recent years, it is still poorly known at diameters D<10 km. Various models and extrapolations yield very different estimates of the number of km-sized and smaller main-belt asteroids (see Tedesco et al and references therein). The Palomar-Leiden Survey (PLS, Van Houten et al. 1970) was the best comprehensive orbital survey of faint main-belt asteroids, which photographically covered a selected area of the sky down to apparent magnitude V 20, corresponding to absolute magnitude H 17 and D 3 km. This survey covered approximately 200 square degrees of sky in a non-targeted search; that is, individual asteroids were seen to move across the contiguous patch of sky covered repeatedly over a time scale of about one month. Orbit solutions were then constructed which allowed calculation of the distances and thus absolute magnitudes. The orbital distribution of the approximately 500 well-determined orbits thus constructed showed little if any difference from the previously known population. However, at D<30 km, the PLS determined a shallower slope (about -2) for the incremental magnitude distribution than the collisional equilibrium value (-2.5) predicted by Dohnanyi s (1969) scale-independent theory, although many observational selection effects had to be accounted for in an approximate way to derive this result. On occasion scientists attempted to use data obtained for other projects to estimate the small

4 4 asteroid population. Shallow slopes for the main-belt size distribution down to sub-km sizes were found by Evans et al. (1998) from an analysis of serendipitous asteroid trails in WFPC2 HST images, but these slopes are not consistent with an extrapolation of the observed steep distribution of some populous asteroid families (Tanga et al. 1999), nor with the steep distribution of impactors (slope of about -4) inferred from Gaspra s cratering record. Jedicke and Metcalfe (1998) analyzed Spacewatch observations (acquired for near-earth object searches) which went as faint as V 21; the interpretation of this data set is very complicated due to biases inherent in its acquisition. It should be noted that many surveys for near-earth asteroids have thrown away in some empirical manner objects which appeared to have motions typical of main-belt asteroids. Analysis of the scans of the Sloan Digital Sky Survey (SDSS) by Ivezić et al. (2001) yielded an abundant population of main-belt asteroids down to apparent magnitude m R =21.5; although lacking enough observational arc to determine the orbits (and thus measure precise distances and therefore H-magnitudes of the asteroids), they statistically detected a flattening of the H-magnitude distribution at H Yoshida et al. (2003) conducted a similar experiment to m R 24.4 using the Subaru telescope and reported a similar flattening of the magnitude distribution. Here we report the first survey since PLS to discover and track a large sample ( 1000) of main-belt asteroids in order to determine their orbits, which in conjunction with their measured H values and V and R colors to allow modelling of the diameter distribution down to values below that which was possible for PLS. The lack of progress since PLS is due largely to the fact that the photographic limit had been reached and hence it was impossible to reach significantly fainter magnitudes. When CCDs on large-aperture telescopes became available (easily allowing a gain of 3 or more magnitudes compared to photographic Schmidts), they imaged such tiny fields that doing a wide field survey was essentially impossible for practical purposes; the shear of their differential motion meant that after a period of one week recovery would have to be done one by one, which was far too inefficient. We have exploited the field of view of CCD mosaic cameras to detect and track a sample of main-belt asteroids with diameters down to below a kilometer, and hence refer to our survey as the (S)ub-(K)ilometer (A)steroid (D)iameter (S)urvey, or SKADS.

5 5 2. Scientific Motivation Knowing the asteroid size distribution down to sub-kilometer sizes and resolving the question of the range of sizes over which the strength transition takes place is important for a number of reasons: All evaluations of the small size distribution of asteroids have assumed a single value of the albedo to convert the observed H-magnitude distribution into a diameter distribution. However, as noted by Cellino et al. (1991), the derived diameter distribution depends upon the assumption of albedos: picking a fixed albedo preserves the H-magnitude distribution, while varying the albedo distribution produces different diameter distributions. This point is illustrated in Fig 6 in Tedesco et al. (2005) which compares the diameter distribution derived by assuming a fixed albedo with that found assuming a more realistic albedo distribution. A difference by a factor of up to about three results at sizes less than 10 km diameter. Current hydrocode-based scaling theories for the collisional energy needed to disrupt asteroids of different sizes (Q* vs. D) predict a transition from the strength to gravity regime in the size range 0.1 to 30 km, but there are still strong discrepancies between different models. The different scaling laws predict very different population abundances at small sizes; Davis et al. (1999) and O Brien and Greenberg (2005) point out the different results for the present asteroid belt based on the different models for Q* as a function of size. Another major question regarding the asteroid size distribution is whether or not there is a second bump in the sub-10-km size range. This bump was interpreted by Durda et al. (1998) as being produced by the rapid variation in the strength of bodies in the sub-10-km size range and that the much more pronounced bump at larger sizes is a collisionally induced wave generated by the small size bump. Our survey was designed to sample this transition region and therefore provide a test for the scaling theories. Finding the asteroid abundance in the range D = km would yield data on a population of bodies which is much younger than the age of the solar system, having collisional lifetimes

6 6 of only 100 Myr, providing an important observational constraint on all models of the collisional evolution of the overall asteroid population, with the purpose of reconstructing the size distribution of the original population of asteroidal planetesimals (Davis et al. 1989, 1994). The ages of spacecraft-encountered asteroids such as 951 Gaspra, 243 Ida and 253 Mathilde are estimated from their crater populations. This requires knowing the projectile flux at sizes in the range from 0.01 to a few km (currently uncertain). This surveys provides a robust estimate at least at the upper end of this range of bodies. Analysis of the size distributions of some asteroid families shows steeply rising populations for decreasing sizes, much steeper 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 continues down to the 1-km size range, the family-derived fragments equal or even outnumber the entire background population at 1 km. The signature of this effect would be a dramatic increase in the slope of the size-frequency relation for D<5km. The delivery from the main asteroid belt of near-earth asteroids (Bottke et al. 2002) and meteorites (Morbidelli and Glaman 1998) depends in a critical way on the size distribution of small asteroids, both because the fragmentation rate depends on the impactor population (Farinella et al. 1993) and because the effectiveness of transport mechanisms such as the Yarkovsky effect (Farinella et al. 1998) is a sensitive function of the collision rate and of the abundance of small asteroids. The production of dust in the asteroid belt, such as the zodiacal dust cloud and the IRAS dust bands associated to asteroid families, depends in a critical way upon the number of existing small asteroids, both within families and in the background asteroid population (Nesvorny et al. 2006).

7 7 3. Experimental Design We wished to image as large portion of sky as possible to at least several magnitudes fainter than the PLS survey; our target was an R-band magnitude of m R = Observations would occur at opposition to allow as large a contiguous patch of sky to be imaged as possible (Fig. 1). Moving asteroids would be detected the first night and on subsequent nights the patch would be displaced westward at the average retrograde motion of the main-belt to keep the largest number of asteroids within the patch (the night-to-night shear of asteroids in and out of the moving patch can be modeled). The directly-measurable quantity is the H-magnitude of the detected asteroids. Determination of H requires knowledge of the distance to the asteroid, the geometry of the observation, and photometric measurement of the asteroid brightness. At opposition the R-band absolute magnitude H R of the asteroids can be estimated using: H R = m R 5log 10 (d ) P (φ) (1) where d and are the heliocentric and geocentric distances of the asteroid and P (φ) is the phase function. As usual (Bowell and Lumme, 1979) the phase function involves the slope parameter G, for which we take the standard value of Since we expect a large fraction of our sample to be within one magnitude of the detection limit and thus our photometry in many cases to be only accurate to 10%, distance determinations to better than 5% were targeted so that the bulk of the uncertainty in the H determination would be due to the unavoidable photmetric uncertainty. After experimentation using astrometric measurements of a previously-known sample of asteroids, we determined that a time baseline of roughly one week with sub-arcsecond astrometry would meet this goal, and thus all observations could be obtained within a single dark run. We thus aimed for 2 or 3 nights of observation at the beginning of a dark run, and then another 2 or 3 nights at the end of a dark run. Multi-night observations of the same asteroid could then be linked and an orbit calculated.

8 8 4. Observations Six nights were allocated for the observations for this program at the 3.8-m Mayall telescope at Kitt Peak National Observatory: March 21, 22, and 23 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 at least one night in both blocks would thus have an observed arc length of between 8 and 10 days if multiple observations of it could be successfully linked. The KPNO mosaic CCD camera gives a arcminute field of view on the Mayall. We created a grid of contiguous sky coverage along the ecliptic plane (Fig. 1) that was within a few degrees of the opposition point on the first night, covering 8.4 square degrees of sky (which accounts for gaps between the mosaic s CCDs). The fields were translated 13.5 and 5.4 arcmin to the west and north (respectively) per day to keep as many main-belt asteroids as possible in the coverage, these rates being determined by the average rates of known main-belt asteroids within a few degrees of the initial pointings. Filter choice was governed by balance between obtaining maximum magnitude depth (and thus discovery as many objects as possible) and the desire to acquire some color information on the asteroids. Obtaining V-R colors allows us to estimate (even if only on a probablistic basis) the spectral type (Sec. 9) of each detected asteroid. On photometric nights we thus acquired triplets of 120-sec images for object detection in R-band with an inter-exposure spacing of roughly 15 minutes. Immediately before or after one of these three R-band exposures on any given night, we acquired a 120-sec V-band exposure; the temporal proximity minimizes any possible light curve variations in the color measurement. However, in nights of below-median seeing we judged color information less important than retaining maximum depth in order to not lose the faintest asteroids. In such cases we acquired 150-sec exposures in a wide-bandpass VR filter. This strategy worked well in allowing detection of faint asteroids on worse than median nights; for example, we lost few objects in 1.7 seeing with the VR filter when compared to adjacent-night R-band imaging in 1.2 seeing.

9 9 We were fortunate to acquire data on all 24 of the detection fields in all six of our allocated nights. Although the majority of the observing time was photometric, on only about 2.5 nights was the seeing sufficiently good (<1.3 ) that we acquired V and R data on the fields for color determination. 5. Reduction to object catalogues The final image set consisted of about 25 Gbytes of data per observing night. Using standard procedures, the instrumental signature was removed using a 2-D bias image and flattened with a superflat made from the data frames themselves (flattening was good to 1%). Each of the mosaic cameras eight CCDs for each pointing were then passed through a movingobject detection pipeline (Petit et al. 2004). Briefly, the pipeline software takes the triplet of images, finds common stars, aligns the three frames to a common pixel coordinate system, catalogues all detectable objects, eliminates all objects that are present in the same location on all three images, and finally identifies linearly-moving objects consistent with main-belt asteroids. These candidate asteroids are shown to a human operator, and for confirmed real detections an astrometric solution of the CCD was computed and used to measure sub-acrsecond astrometric positions. A plate solution of fourth order based on USNO catalog stars was used for the astrometric measurements of most of the SKADS asteroids; when the asteroid is far from the available USNO stars systematic astrometric errors on that asteroid could reach several arcseconds. When asteroids were linked later these cases became clear and the asteroids were often re-measured using a plate solution based on hand-selected stars near the asteroid s position. Photometric zeropoints were determined nightly from multiple observations of Landolt standard stars; these were acquired at a variety of airmasses from 1 to 2 and the extinction terms were measured and included in our photometric measruements. Knowing the airmass of observation, R and V brightnesses were measured when the asteroid was not confused; photometric errors based on aperture correction photometry are provided if there were photometric conditions. On frames

10 10 where the VR filter was used, an R-band brightness estimate was obtained assuming an average color; when these photometric measurements are quoted in our database a photometric error will not be listed. An internal designation for an asteroid detected in a given image triplet is of the form pxffcn was given, where p is simply a leading letter, x is the night of observation (1 6), ff is the field number (1 24), c is the CCD number on the mosaic (1 8), and n uniquely identifies the detected asteroid on the CCD and for which a letter starting a was then used if the digits 1 0 were exhausted due to exceeding 10 asteroids per CCD (no more than 18 asteroids were ever detected on a single CCD). The number of detected asteroids on each of six nights were 1034, 1298, 1274, 1130, 1005, and 1072, where the reader should recall the first three nights were sequential, and then there was a 5-night gap before the final three sequential nights. Thus the vast majority of the detected asteroids are expected to be the same as those that were detected on the nearby nights (Fig. 2). Between the two 3-night blocks there is expected to be some more substantial shear of objects into and out of the 8.4 patch, but this effect can be accurately modelled as a function of (a, e, i) inour characterized survey. It was then now necessary to determine how many and which of the detections were multiple apparitions of the same asteroid, and use the linked observations to calculate the heliocentric orbits. 6. Linkage and Orbit determination Starting from the six nightly catalogues of asteroid triplets, preliminary orbits were fit to the available obseravations, with the goal of first identifying multiple obervations of the same asteroids, and once these linkages were made, standard methods were used to calculate the heliocentric orbital elements of the asteroids. For our science goal, a highly-accurate set of orbital elements was less important than determining the instantaneous distance to the asteroid, since it is only the distance which is needed to obtain the H magnitude. We quickly confirmed that the quality of the available astrometric data was more than good enough for the purpose of obtaining distances, and that accurate measurements of all six orbital elements could be obtained as long as at least a week of

11 11 baseline was available. The linking of observations of the same asteroid was accomplished using a multi-pass procedure. Because the field coverage was contiguous there was a small chance that an object be detected in one field and then be re-detected an hour later in an immediately adjacent field to the west. The same object would thus appear twice in a single night s observations under two different designations. Searching for these links was accomplished by fitting Väisälä orbits between all possible combinations of objects and rejecting those linkages that produced unacceptable residuals in the observations. Only seven cases (of 6813 triplets) of the same object being observed twice on the same night were found. The next stage was to link the detections to observations on a second night, via an iterative procedure. This was accomplished by fitting Väisälä orbits as above or by matching motion vectors. The first pass used observations from adjacent nights. The second pass used observations from nights separated by two days. Subsequent passes used observations from nights separated by more than two nights. The third stage was to link the two-night links to third nights. Using the Väisälä orbits generated in the second stage, predicted positions were generated for each object for all the times of observation. Using these predicted positions, a search for possible matches was undertaken. If an object was with 20 of a predicted position and had consistent motion with the predicted motion, a Gaussian orbit determination was attempted. If this was successful, the object was considered to be a three-nighter. After the first pass using a 20 search apeture, the search was repeated 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, predicted positions were generated for each object for all the times of observation. The search for matches was done as above, using several search apertures. When a match was found, the Gaussian orbit was recomputed and used to generate new predicted positions and the search was repeated.

12 12 At the end of this process we obtained 1277 orbits of main-belt asteroids that were observed on three or more nights of the SKADS observations. Of these, 255, 366, 286, and 370 were observed on 6, 5, 4, and 3 nights (respectively). Because of the spacing of the nights (3 nights of observation, 5 nights off, and then another 3 nights), any asteroid observed on 4 or more nights (907 objects) automatically has an arc baseline of at least 8 days. Of the night linkages, 190 have observations from only one 3-night block; such 1-block asteroids have orbits (and distances) known with considerably poorer precision. A linked orbit is identified by its master designation which is simply the internal designation on the first night of observation from the first pair-linkage above, with the p replaced by an s. For example, the SKADS master designation s21013 is the linked astrometry of p11012, p21013, p31014, p41034, and p61033; this asteroid was not detected on night 5. Table. 1 gives example entries for the type of data available for each asteroid detected in SKADS; the entire table is available on-line as part of the Planetary Data System s Asteroid archive, in the Small Bodies Node data services at Fig. 3 shows heliocentric orbital elements for all 1087 main-belt SKADS asteroids with observational arcs of 8 nights or more. Many well-known features of the main-belt orbital distribution are seen, such as the 3:1 resonance, the Flora and Koronis families, and the inner edge of the belt in the a/i distribution caused by the ν 16 secular resonance. Unlike previous short-arc surveys, SKADS elements are sufficiently accurate to permit (after de-biasing) the calculation of the absolute population of many sub-groups of the main asteroid belt. We observed XX jovian Trojans, with observed arc lengths varying from 40 minutes (in which case it is only the heliocentric distance which is indicating the likely orbit) to all 6 nights. Jovian trojans are not present in their intrinsic (but flux-limited) proportions in the SKADS multi-block list because our shifting of the fields at an average main belt rate makes them less likely to be present in both blocks; our fields were retrograded night to night along the ecliptic faster than the average rate of the trojans. This will also be somewhat true for the Hildas, at least 4 of which are present in the multi-block orbits. A simulation of the survey would allow one to estimate the instrinsic Hilda fraction brighter than an H magnitude of roughly 17.

13 13 None of our night detections nor the one-night or two-night detections (thus a total of XX detections) BG BRETT: COUNT THAT NUMBER! appear to be Hungaria-group asteroids (a 2 AU, at high orbital inclination). Extracting all Hungarias with H<15 from the MPC orbital database, we calculate that 0.1% of them are instantaneously within 0.5 of the ecliptic, and thus our non-detection of a single Hungaria is not statistically alarming. We note that this 0.1% fraction is a strong function of the H threshold used, rising to about 0.3% by H=18; we believe that the Hungarias are over-represented in the MPC database due to systematically better recovery (even at diameters of only a few km) in automated survey due to the proximity of these asteroids. We do not believe that Hungarias would be very difficult to recognize even with short arcs. A main goal of SKADS was to be sensitive to asteroids below 1 km in diameter (about H=18) out to the outer edge of the main belt (roughly 3.2 AU heliocentric distance). For such an asteroid near opposition, our m R 23 limiting magnitude (see below) corresponds to an absolute magnitude of H R =18.8, and thus even at the outer edge of the main belt we are sensitive to sub-kilometer asteroids, with increasing sensitive as we approach the inner edge. At 2 AU heliocentric distance this corresponds to Thus, in the direction that SKADS was looking essentially all objects with diameters larger than 1 km were deteced. 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 measure was converted to an H R magnitude and these values were averaged. The H-magnitudes of the SKADS detections ranges from H=11.8 (numbered asteroid 6301, re-detected) to the smallest asteroid with H=21.9 (SKADS internal designation s1061a), corresponding to a diameter ratio of two orders of magnitude Orbit Accuracy The accuracy of the derived orbital elements was judged by then searching for links with previously-known asteroids in the Minor Planet Center database. The much-longer available arc from the archived Minor Planet Center observations yielded the true orbit for the asteroid and the discrepancy between the true orbital elements and those derived from our 8 10 day arc provided

14 14 an estimate of the accuracy of the elements when based on SKADS orbits alone. In total, 135 previously-known asteroids were identified as being re-discovered in SKADS. We found that the orbits with 8-day arcs provided distances and orbital elements accurate to better than five percent. Figs. 4 and 5 present the difference between the derived elements and the true elements for the previously-known asteroids in our sky coverage. The orbits are sufficiently accurate that the dominant source of uncertainty in the derived H-magnitudes comes from the photometric uncertainty. In fact, since all of the SKADS asteroids that have been identified with previous discoveries that have H<18, one might be surprised that the discrepancy with the MPC H values are so large. Because these H<18 asteroids are detected with magntidues far above our roughly m R 23 limit, we believe that this reflects real systematic errors in the H magnitudes of the MPC asteroids (see Jedicke et al. 2002, Juric et al for discussion). The SKADS sample is the largest available sample of asteroids with good orbits detected in a uniform and well-characterized way. After the PLS survey, other studies which have probed large samples with apparent magnitudes fainter than 20 (Ivezić et al. 2001, Yosihda et al. 2003) have had only very short observational arcs available (of order hours) and were forced to apply statistical ranging techniques to estimate distances and therefore H magnitudes of the detections (which could have errors up to a magnitude). But it is the diameter distribution (measured via H) that allows one, when coupled to a well-characterized understanding of the detection conditions, to study the main-belt orbital and size distribution and furnish constraints for collisional evolution models. 7. Characterization The SKADS survey was carefully designed so that precise characterization of the survey (thorough knowledge of sky coverage, sensitivity, and field timing) would allow us to de-bias the survey. Thus, great effort was expended to determine the detection efficiency on each night as a function of asteroid apparent magnitude and rate. For each CCD triplet of each night we implanted 250 artificial asteroids in the image, using

15 15 a model PSF created from a set of bright stars on each individual CCD frame. Thus, changes in seeing (and the concomitant loss of sensitivity) are correctly modelled. Since at opposition some of the closest asteroids were moving at rates that (in the best seeing) could induce some mild trailing losses (with the objects moving by of order the seeing during the 2-minute exposure), the implanted flux for each objects was actually added in 10 bundles of 10% of the flux, that was implanted at positions corresponding to the randomly-chosen rate and direction of the artificial object. Artificial objects were implanted at rates of arcsec/hour, moving retrograde with a dispersion of rates covering all possible main-belt orbital inclinations. Artificial object magnitudes were chosen from m R =17 to 24, with a square-root weighting to place more artificial objects in the fainter end of the distribution (where knowledge of the declining detection efficiency is more important). These frames were then passed through the same processing steps as the original data, except that a human operator did not view and confirm the artificial objects. Comparing the implanted list with those found by the pipeline software, the fraction of detected objects as a function of magnitude and rate were determined. This precise characterization is needed in order to calculate a detection bias. For each night the detection efficiency is determined as a function of object apparent magnitude and rate of motion. The fraction η of artificial asteroids detected as a function of these variables is fit by smooth functions. The crucial efficiency versus magnitude behavior is fit to the formulation η(m R )= η o c (m R 17) 2 1+exp ( m R m L ) (2) w where η o 98% is the efficiency at m R =17, c 0.5% measures the strength of a quadratic drop, which changes to an exponential falloff over a width w near the magnitude limit m L. Table 2 tabulates these parameters for the six nights of observation and Fig. 6 shows an example. Brighter than m R =17 the efficiency is taken to be η o. The slow dropoff from m R =17to22isduetothe gradually increasing confusion engendered by the growing number counts of background stars and

16 16 galaxies. The value of w is mostly controlled by the constancy of the observing conditions since a variety of seeing conditions softens the roll-over to zero efficiency near the limit. The limiting magnitude m L varies from m R =22.6 to 23.2 due to the variation in average seeing from night to night. 8. Debiased apparent magnitude Having detemined the detection efficiencies, it is possible to de-bias the apparent magnitude distribution into an intrinsic luminosity function ; that is, the number of asteroids as a function of apparent magnitude. This kind of estimate has been previously by Ivezić et al. (2001) and Yosihda et al. (2003). Since almost all the detections in our sky coverage over the a block of three sequential nights will be the same asteroids, the bias-corrected number as a function of magnitude should be virtually identical. This night-to-night consistency is thus one test of the quality of our characterization. Fig. 7 shows that down to magintude m R =23 we find a consistent number of asteroids over the entire magnitude range. This figure also shows that our characterization appears to have a systematic problem past 23rd magnitude, as the differential number of counts ceases to rise. Although this is in principle physically possible if there are collisional waves in the size distribution (Campo Bagatin et al. 1994, Durda et al. 1998), we feel it more likely to be due to an overestimation of our detection efficiency at the faintest end. Petit et al. (2004) illustrate how this roll-over effect can occur in surveys for which the detection efficiency is determined by implanting artificial objects and detecting them with an automated pipeline; if the artificial objects are not subject to the same operator verification process as the real asteroids then one counts very low signal-to-noise objects discovered by the pipeline as found when a human operator might reject them as not being sure detections. This results in a systematic overestimation of the detection efficiency at the very faintest end of the survey (starting roughly at the magnitude of 50% detection efficiency). Removing this effect would require a human operator to examine a significant number of the tens of thousands of artificially-implanted objects.

17 17 Fig. 8 presents a cumulative version of the apparent magnitude distribution, which clearly shows a change starting at m R 19 where the power-law slope drops from 0.6 to 0.27, the same values as Ivezić et al. (2001) reported down to the Sloan limit of m R =21.5. Our deeper data shows that this shallow slope continues down to an apparent magnitude of at least 23rd magnitude. Using our de-biased magnitude distribution, we estimate Σ 210 asteroids per square degree brighter than m R =23, where Σ(< m R ) = (m R 23), 20 <m R < 23 (3) is the cumlative density of asteroids on the sky (at opposition within 1 degree of the ecliptic). If the 0.27 slope continued to magnitude m R =24.4, this would imply an on-sky density of Σ 500 per square degree, in contrast with Yoshida et al. (2003) who estimate 290/square degree at that depth. Since SKADS already detected Σ = 210 asteroids per square degree brighter than m R =23, this would imply that the slope of the cumulative apparent magnitude distribution would have to precipitously drop to 0.10 or less just past our magnitude limit. We doubt that this is the case. SO, WHAT TO DO HERE; OUR DEBIASED NUMBER COUNTS * DO * SHOW THAT THE DIFFERENTIAL DENSITY DROPS, BUT I DON T THINK WE CAN BELIEVE IT...IT D BE A REAL SHAME IF IT WAS TRUE. WE RE REALLY SAYING THAT WE PREFER TO THINK OUR CHARACTERIZATION BREAKS DOWN RIGHT AT THE LIMIT *AND* THAT YOSHIDA ET AL HAVE A WEEK CHARACTERIZATION PAST 23rd (AND BOTH OF THOSE SEEM PLAUSIBLE TO ME RATHER THAN THE DROP IN THE SLOPE IS PRECISELY AT THE SKADS MAG LIMIT). OPINIONS? Assuming that the size distribution is the same everywhere in the asteroid belt, the average albedo does not vary with heliocentric distance, and that the size distribution is a single power law over the belt s entire radial range, a 0.27 slope in the luminosity functions implies a differential asteroid size distribution with index 2.4. Since all of these are likely untrue at some level, a more complex modelling of SKADS is needed, based on the H-magnitude distribution. In a future paper we will compute a full (a, e, i, H) de-biasing of the main belt to address these issues. For our purposes here we show only an uncorrected cumulative H-magnitude distribution (Fig. 9); this is

18 18 only an approximation because asteroids with large H magnitudes are not equally likely to have been detected on at least one night in both SKADS blocks (since faint asteroids are less likely to be detected multiple times). The slope α of the cumulative number versus H R distribution is related to the differential dize distrbiution q of the underlying population by q =5α +1 if a power law of slope q holds at all radial distances to which the survey is sensitive at that H range. As previously shown by Ivezić et al. (2001) using an uncorrected survey, the main belt shows an initially steeper slope at bright H magnitudes that drops to a shallower one; Ivezić et al. (2001) estimate the transition H magnitude at However, the SKADS detections show the transition to a shallower slope of α 0.38 already established at H R 13.5, and this roughly-constant slope continuing down to at least H = 17. This slope is considerably steeper than the Sloan estimate of 0.25±0.01, but consistent with the differential slope of 2.9 estimated by PLS and the Spacewatch (Jedicke et al. 1998) surveys. We know that the SKADS detection efficiency to drop beginning around H=17, so it may be that the α =0.38 slope of the H-magnitude distribution continues past this point; confirmation of this will require a detailed analysis of the SKADS detection efficiency as a function of H. But brighter than H R =17 we expect to have seen essentially all asteroids in the SKADS search volume, and thus this slope to be accurately deteremined. ANY OTHER COMMENTS HERE ON THE PREVIOUS PARAGRAPH? 9. V-R colors of SKADS detections Each linked SKADS asteroid with an orbit has observations on 3, 4, 5, or 6 nights. Of the 1277 multi-night asteroids detected, all but 44 had at least one V R color measurement, and may have had as many as three. During the periods of photometric conditions in which a 120-sec V -band exposure was acquired, this image was acquired within 3 minutes of a 120-sec R-band exposure, so that rotational variation is unlikely to have had any effect on the color. The photometric accuracy of the data is primarily limited by signal to noise. Fig. 10 shows the color quality of the data. Most of the asteroids have V R colors accurate to 0.15 magnitudes. After

19 19 transforming the commonly-used ECAS filter set to match the V and R bandpasses for the KPNO mosaic camera filters, it was determined (Howell, 2002, private communication) that V R=0.4 was a separator between the dominant S and C spectral types of the main asteroid belt. Fig. 10 shows how our ability to assign asteroids to spectral types based solely on their V R color is largely only statistical. The larger spread in colors as R-band magnitude increases is of course just due to the mounting errors due to declining signal to noise. Fig. 10 illustrates that beyond 19th magnitude we cease to be able to be certain which side of the V R=0.4 line a given asteroid falls, although this could be assigned probablistically. Using this color information, modelers can create different prescriptions to map detected asteroids to spectral class and thus albedo. Coupled with detailed knowledge of the detection biases, this wll allow exploration of the size distribution in different dynamical regions of the belt. 10. Conclusion The SKADS survey provides the necessary information for detailed modelling of the main asteroid belt s size distribution, with sufficiently-precise orbital information that one could examine different orbital zones of the belt. Using colour as a proxy for spectral type will allow one to estimate albedos and compute true diameters for the asteroids, allowing theorists to model the main-belt size distribution. To measure the true size distribution it is necessary to know the detection efficiency as a function of orbital parameters and H magnitude. This is a complicated function of the orbital elements and needs full knowledge of the detection timing and field pointings, all of which are provided in a publicly-accessible form on the NASA Planetary Data System web site 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.

20 20 The full version of Table 1, containing linkages and, orbital elements for the linked objects, and full photometric information with estimated uncertainties. Lists of astrometric observations that are unlinked or have only two nights of observation. We confirm the roll-over in the apparent magnitude luminosity function at m R 18 to shallow cumulative power-law slope of This continues to at least magnitude m R =23, the 50% limit of our well-characterized survey. Using our accurately-determined distances to compute H R absolute magnitudes, we find a steeper H R magnitude slope than the Sloan survey seported, corresponding to a differential population power law of slope index 2.9 valid in the roughly 1 to 10 kilometer diameter range.

21 21 Acknowledgements This work was carried out with support from NASA s Planetary Astronomy program and a time allocation at Kitt Peak National Observatories. BG and BW acknowledge NSERC, CFI, and the Canada Research Chairs program for support. REFERENCES Bottke, W. F., Morbidelli, A., Jedicke, R., Petit, J. -M., Levison, H F., Michel, P., Metcalfe, T. S., Debiased orbital and absolute magnitude distribution of the near-earth objects. Icarus 156, Bowell, E., Lumme, K., Colorimetry and magnitudes of asteroids. In: Gehrels, T., Ed., Asteroids, Burns, J. A., The evolution of satellite orbits. In: Burns, J. A., Matthews, M. S., Eds., Satellites, A. Campo Bagatin, A. Cellino, D.R. Davis, P.Farinella, and P. Paolicchi, Wavy size distributions for collisional systems with a small-size cutoff. Planet. Space Sci. 42, Cellino, A., Zappala, V., and Farinella, P., The size distribution of main-belt asteroids from IRAS data. Mon. Not. R. astr. Soc xxx, yyy. Davis, D. R., Weidenschilling, S. J., Farinella, P., Paolicchi, P., Binzel, R. P., Asteroid collisional history - effects on sizes and spins. In: Binzel, R. P., Gehrels, T., Matthews, M. S., Eds., Asteroids II, Davis, D. R., Ryan, E. V., Farinella, P., Asteroid collisional evolution: results from current scaling algorithms. Planet. Space Sci. 42, Davis, D. R., Farinella, P., Marzari, F., The missing Psyche family: collisionally eroded or never formed? Icarus 137, Dohnanyi, J. W., Collisional models of asteroids and their debris. J. Geophys. Res. 74, Durda, D. D., Greenberg, R., Jedicke, R., Collisional models and scaling laws: a new interpretation of the shape of the main-belt asteroid size distribution. Icarus 135, Evans, R. W., and 23 colleagues, Asteroid trails in Hubble Space Telescope. Icarus 131,

22 22 Master a (AU) e i Ω ω M H R s Nightly Rmag1 Rmag2 Rmag3 Vmag Near V-R ± ± ± ± ±0.10 not seen s Nightly Rmag1 Rmag2 Rmag3 Vmag Near V-R ± ± ± ± ± ± ± ± ± ±0.16 not seen ± ± ± ± ± Table continued in the electronic supplement Table 1: Example 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 the mean anomaly M is given for March 21, See Figs. 5 and 6 for typical orbital element uncertainties. Each of these two asteroids was not detected on one of the 6 nights of observation. The lack of photometric information on two of the exposure indicates that the asteroid was too close to a star for reliable photometry. When error bars are not provided on the photometry this indicates the night was either not photometric or that the seeing was poor and the VR filter was used (in which case the R-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 was taken adjacent to (either 3 minutes before or after). The complete table is provided electronically (see text).

23 23 Farinella, P., Gonczi, R., Froeschle, C., Froeschle, C., The injection of asteroid fragments into resonances. Icarus 101, Farinella, P., Vokrouhlicky, D., Hartmann, W. K., Meteorite delivery via Yarkovsky orbital drift. Icarus 132, Ivezic, Z., and 32 colleagues, Solar system objects observed in the Sloan Digital Sky Survey commissioning data. Astron. J. 122, Jedicke, R., Metcalfe, T. S., The orbital and absolute magnitude distributions of main belt asteroids. Icarus 131, Jedicke, R., Larsen, J. and Spahr, T. (2002). Observational Selection Effects in Asteroid Surveys. In Asteroids III, (W.F. Bottke, A. Cellino, P. Paolicchi and R. Binzel, Eds.) Univ. of Arizona Press, Tucson, pp Juric, M. and 15 co-authors. (2002). Comparison of Positions and Magnitudes of Asteroids Observed in the Sloan Digital Sky Survey with those Predicted for Known Asteroids. Astron. J., 124, Morbidelli, A., Gladman, B., Orbital and temporal distributions of meteorites originating in the asteroid belt. Meteor. & Planet. Sci. 33, Nesvorny,D., Vokrouhlicky, D., Bottke, W.F., Sykes, M., Physical properties of asteroid dust bands and their sources. Icarus 181, Petit, J. -M., Holman, M., Scholl, H., Kavelaars, J., Gladman, B., A highly automated moving object detection package. Mon. Not. Royal Astron. Soc. 347, Tanga, P., Cellino, A., Michel, P., Zappal, V., Paolicchi, P., dell Oro, A., On the size distribution of asteroid families: the role of geometry. Icarus 141, Tedesco, E., F., Cellino, A., Zappal, V., The statistical asteroid model. I. The main-belt population for diameters greater than 1 kilometer. Astron. J. 129, van Houten, C. J., van Houten-Groeneveld, I., Herget, P., Gehrels, T., The Palomar-Leiden survey of faint minor planets. Astron. Astrophys. Supp. 2, Yoshida, F., Nakamura, T., Watanabe, J. -I., Kinoshita, D., Yamamoto, N., Fuse, T., Size and spatial distributions of sub-km main-belt asteroids. Pub. Astron. Soc. Pacif. Japan 55, Zappala, V., Cellino, A., Reconstructing the original ejection velocity fields of asteroid families. Icarus 124,

24 24 Fig. 1. SKADS pointings for the night of March 21, 2001 UT. Horizontal dashed line is the equator while the slanted solid line is the ecliptic plane; note that on this date the ecliptic opposition point is at the intersection of the ecliptic and equator. Image triplets on fields were acquired first, before moving on to fields , etc. Boxes show the field of view of the KPNO mosaic camera on the Mayall telescope. This preprint was prepared with the AAS L A TEX macros v5.2.

25 Decl. (J2000.0) R.A. (J2000.0) Fig. 2. Locations of all 1277 multi-night detections from SKADS, on March UT The solid line is the celestial equator, while the diagonal dashed line shows the position of the ecliption; at the time given here the opposition point was very close to their intersection (observations occured close to the time of the vernal equinox). Asteroids near the fringes of the field coverage are those which were discovered roughly 1 week later and which had high-inclination orbits which caused them to enter the field coverage for the March 29th-31st block due to differential shear of the sample over the 10-day period.

26 26 Fig. 3. Heliocentric orbital elements for all asteroids detected by SKADS having orbital arc lengths of 8 or more days. A few Trojan asteroids are not not 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. THIS FIGURE MUST STILL BE REDONE WITH OUR FINAL MULTI-BLOCK ORBITS.

27 27 Fig. 4. Discrepancy of SKADS orbits against the true orbit as measured by the previously-known orbit in the MPC, for SKADS asteroids having orbital arc lengths of 8 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, 0.11 AU, and 0.36 magnitudes, respectively. See text for discussion of the H errors.

28 28 Fig. 5. Discrepancy of SKADS angular orbital elements against the true orbit as measured by the previously-known orbit in the MPC, for SKADS asteroids having orbital arc lengths of 8 or more days. Horizontal lines bound ±1-sigma variations from the mean error. These 1-sigma variations in i, Ω,ω, andm are 0.7, 1.0, 27, and 28 degrees, respectively. The ω and M errors are strongly anti-correlated since the longitude of the particles at the time of observation are known; the 1-sigma error in the longitude of pericenter is only 5 degrees.

29 29 Fig. 6. Detection efficiency versus R-band magnitude for the SKADS observations of March UT. Parameters for the fit are given in Table 2. Horizontal error bars just indicate the bin width, whereas the vertical error bars are due to counting statistics (the number of objects per bin is weighted towards the faint magnitudes).