The Flux Distribution and Sky Density of 25th Magnitude Main Belt Asteroids
aa r X i v : . [ a s t r o - ph . E P ] O c t Draft version October 30, 2019
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The Flux Distribution and Sky Density of 25th Magnitude Main Belt Asteroids
A. N. Heinze, Joseph Trollo, and Stanimir Metchev Institute for Astronomy, University of Hawaii, 2680 Woodlawn, Honolulu, HI, 96822, USA; [email protected] Asana, 1550 Bryant St Department of Physics and Astronomy, University of Western Ontario, 1151 Richmond St, London, ON N6A 3K7, Canada;[email protected] ∗ ABSTRACTDigital tracking enables telescopes to detect asteroids several times fainter than conventional tech-niques. We describe our optimized methodology to acquire, process, and interpret digital trackingobservations, and we apply it to probe the apparent magnitude distribution of main belt asteroidsfainter than any previously detected from the ground. All-night integrations with the Dark EnergyCamera (DECam) yield 95% completeness at R magnitude 25.0, and useful sensitivity to R = 25 . ±
15 asteroids per square degreebrighter than R = 25 . ±
23 brighter than R = 25 . R = 23mag. For a power law defined by dN/dR ∝ αR , we find marginally acceptable fits with a constantslope α = 0 . ± .
02 from R = 20 to 25.6 mag. Better fits are obtained for a broken power law with α = 0 . ± .
026 for R = 20 to 23.5 mag, steepening to α = 0 . ± .
025 for R = 23 . R = 23 . Keywords: minor planets, asteroids: general; astrometry; techniques: image processing; methods: dataanalysis INTRODUCTIONThe magnitude and size distribution of main-belt as-teroids (MBAs) holds clues to their physical propertiesand collisional evolution (e.g. Cheng 2004; Bottke et al.2005b; Durda et al. 2007). Digital tracking enables thedetection of asteroids up to ten times fainter than con-ventional methods (e.g. Zhai et al. 2014; Heinze et al.2015b), allowing us to probe the main belt populationdown to sizes as small as 100 meters with 4-meter classtelescopes.A significant change in asteroid properties is believedto occur in this size range: the transition from largeobjects whose cohesive strength comes mainly from self-gravity to small ones held together by material tensilestrength. Laboratory experiments and numerical hy-
Corresponding author: A. N. Heinze ∗ Department of Physics and Astronomy, Stony Brook Univer-sity, Stony Brook, NY 11794-3800, USA drocode modeling indicate this transition occurs at a sizesomewhat larger than 100 meters (Bottke et al. 2005a,and references therein). These theoretical results areindependently supported by measurements of asteroidrotation periods. Asteroids larger than about 300 me-ters observe a ‘spin barrier’, almost never having peri-ods shorter than two hours, while smaller objects canrotate much faster (see, e.g. Hergenrother & Whiteley2011). The two-hour spin barrier corresponds to the ro-tation period at which centrifugal breakup occurs for astrengthless, gravitationally bound object with typicalasteroidal density . Hence, asteroids larger than a fewhundred meters probably are strengthless ‘rubble piles’,while smaller asteroids have nonzero internal strength Typical density is about 2 g cm − (Carry 2012). The rotationperiod for centrifugal breakup of a strengthless gravitationallybound object depends only on density, not size. This is becausefor a given density and rotation period, surface gravity and cen-trifugal force both increase linearly with the object’s radius. Heinze et al. (Hergenrother & Whiteley 2011). Based on the ex-pected greater strength of the smallest asteroids, modelspredict that very small MBAs will be more abundantthan naive extrapolation of power-law fits to the dis-tribution of larger objects would indicate (Bottke et al.2005b; de El´ıa & Brunini 2007). While near-Earth ob-jects (NEOs) are routinely observed in this size range(hence the discovery that they violate the spin barrier),the observations we report herein may be the first toprobe extremely small MBAs with statistical power suf-ficient to test model predictions of an upturn in theirsize-frequency distribution.Very small MBAs have short collisional lifetimes inspite of their greater tensile strength (Bottke et al.2005b; Henych et al. 2018). Therefore they are youngerthan their larger counterparts, having been producedrelatively recently by the breakup of larger bodies dueto collision or (more rarely) to YORP-induced fission(e.g. Agarwal et al. 2013). Their abundance and dynam-ical distribution can tell us about the recent collisionalhistory of the main asteroid belt. Small MBAs also fre-quently impact larger objects. These impacts producecraters, change the spin axis of the target body, and insome cases even disrupt it. Hence, we must know theabundance of small MBAs to understand the collisionalenvironment inhabited by larger asteroids.Studying small MBAs can enhance our understand-ing of NEOs, since most (about 94%; see Bottke et al.2002) NEOs originate from the main belt (e.g.Morbidelli & Nesvorn´y 1999; O’Brien & Greenberg2005; de El´ıa & Brunini 2007; Minton & Malhotra2010). Relative to large objects, small MBAs are moresensitive to the Yarkovsky effect (Farinella et al. 1998;Nesvorn´y & Bottke 2004), which is believed to be themain process that transports MBAs into the unsta-ble orbital resonances from whence they evolve intoEarth-crossing orbits (Bottke et al. 2006). Since near-Earth objects (NEOs) approach Earth more closely thanMBAs, they can be detected down to small sizes: in fact,the vast majority of known NEOs are smaller than 1 km.Studying the same size cohort in the main belt will elu-cidate the selection effects involved in the main-belt toNEO transition and help us understand both popula-tions in more detail.Selection effects would be expected to include pref-erential transfer of the smallest objects, becausethey are most sensitive to the Yarkovsky effect (e.g.Nesvorn´y & Bottke 2004). This would suggest thatthe size distribution of small MBAs should be shal-lower than that of NEOs, since NEO dynamical life-times are far too short for them to reach a new colli-sional equilibrium after migrating from the main belt (Bottke et al. 2002). Another expected selection effecthas to do with location in the main belt: NEOs shouldpreferentially come from relatively small regions close tounstable orbital resonances. Bottke et al. (2002) predictthat 61% of NEOs are delivered from the main belt byjust two strong resonances: the ν secular resonance that sculpts the inner edge of the main belt, and the3:2 mean-motion resonance with Jupiter that carves outthe deepest Kirkwood Gap at a = 2 . ν secular resonance as a major source of NEOs, be-cause it has the unique property that only MBAs withretrograde rotation can be transported into it by theYarkovsky effect.Herein, we describe the observational methodology fordetecting extremely faint asteroids and probe the statis-tical distribution of their apparent magnitudes, therebylaying the foundation required in order to address thescientific questions raised above. Compared to our pi-lot project described in Heinze et al. (2015b), the cur-rent work reaches much fainter magnitudes, uses morefar more mature methodology and statistical analysis,and most importantly includes an incompleteness cor-rection to determine the apparent magnitude distribu-tion of main belt asteroids fainter than have ever pre-viously been systematically probed. Our current dataare also sufficient to determine the absolute magnitudedistribution of very small MBAs, which can be directlycompared with that of NEOs to probe size-dependentselection effects in the MBA to NEO transition. Wedefer our analysis of these absolute magnitude distribu-tions to our companion paper (Heinze et al., in prep.),which will rely heavily on the results presented herein.More detailed analysis of dynamical selection effects re-quires new data sets spanning more than two nights.The current work lays methodological foundations forsuch future research.Previous studies of small MBAs includeGladman et al. (2009), who used the 4m Mayall tele-scope at Kitt Peak to probe asteroids as faint as R magnitude 23.5; Parker et al. (2008), who measured thesize distributions of asteroid collisional families usingdata from the Sloan Digital Sky Survey; Yoshida et al. An asteroid in the ν secular resonance is dynamically unstablebecause its perihelion precesses at the same rate as that of Saturn,the 6th planet. aint Asteroids with DECam r magnitude ∼ . g and r colors of asteroids down to magnitude 23 using CFHT.The observations we present herein are 95% completeat R = 25 . R = 25 .
3. Hence, they constitute the most sensitiveground-based asteroid survey to date. Since we observedthe same field on two consecutive nights, we are ableto confirm the reality of most of our asteroids by de-tecting the same objects in two independent data sets.Two-night detection will also enable us to obtain dis-tances with ∼ .
5% accuracy (and hence absolute mag-nitudes) using the RRV method of Heinze & Metchev(2015a) and Lin et al. (2016). We reserve our analysis ofthe distance determinations and the absolute magnitudeand size distributions for a companion paper (Heinze etal., in prep.), which will rely heavily on the analyses ofcompleteness and flux overestimation bias we developherein.We describe our observations and image processingin Sections 2 and 3. In Sections 4 and 5 we presentthe digital tracking methodology that allows us to findand measure extremely faint asteroids. Section 6 quan-tifies the expected number of false detections amongour confirmed asteroids, demonstrating that it is neg-ligible. In Section 7 we analyze our detection complete-ness as a function of magnitude using a sophisticatedfake-asteroid simulation. Section 8 describes how welink detections of the same asteroid from one night tothe next. Our main results are in Sections 9 and 10: thelatter describes how we model and correct flux overes-timation bias, and presents the first-ever determinationof the sky density and apparent magnitude distributionof MBAs down to R magnitude 25.6. We offer our con-clusions in Section 11. OBSERVATIONSWe observed using the Dark Energy Camera (DE-Cam) on the 4 meter Blanco Telescope at Cerro TololoInter-American Observatory (CTIO) on UT 2014 March30, March 31, April 07, and April 08. Our primary ob-jective was to detect and characterize extremely faintMBAs using digital tracking.The DECam field of view is 2 degrees in diame-ter and comprises 60 active science CCDs with dimen-sions of 4096 × ). A single exposure captures2.7 square degrees of sky onto active science CCDs, whilea simple three-exposure dither pattern will fill in thegaps between detectors and deliver contiguous coverageover 3 square degrees.2.1. Choice of Target Fields
We planned our observations to achieve two primaryrequirements: to detect the smallest asteroids possiblein the main belt, and to recover as many as possible ofthem on multiple nights to constrain their distances andorbits. The first two nights (March 30-31) constituteda ‘discovery run’ in which thousands of new asteroidswould be found and measured over two nights. Mea-surements on two consecutive nights confirm the realityof faint objects and enable the calculation of accuratedistances using the RRV method (Heinze & Metchev2015a; Lin et al. 2016). We intended the second pairof nights (April 07-08) as a ‘recovery run’ in which mostof the asteroids would be recovered and approximate or-bits could be calculated from the resulting 8-9 day arcs.The requirement of detecting the smallest MBAs dic-tates targeting observations near the antisolar point,where asteroids are at their brightest because of theirlow phase angles. The requirement of recovering thesame asteroids on additional nights dictates offsettingthe target field from night to night in order to followthe mean sky motion of MBAs. Since the antisolar pointmoves eastward while MBAs near opposition are movingwestward in their retrograde loop, our target field couldbe centered accurately on the anti-solar point for onlyone night.We chose this to be the second night of our discoveryrun because the antisolar field on that night had fewerbright ( . The NOAO Data Handbook is available athttp://ast.noao.edu/sites/default/files/NOAO DHB v2.2.pdf
Heinze et al.
March 30 (approximately local midnight at CTIO). Theaverage sky motion of MBAs in this field on this date is − .
215 deg/day in RA and 0 .
092 deg/day in Dec. Ap-plying these offsets to our March 30 field gives the tar-get for our March 31 observations: RA 12:37:20.84, Dec-04:02:08.7. This location is within one arcminute of theantisolar point at local midnight on March 31: hence,our targeting was almost perfectly optimized. The tar-get fields for the two nights overlap heavily since theDECam field of view is much larger than the offset be-tween them, but the offset was important because itminimized the loss of asteroids off the edges of the fieldfrom one night to the next.To detect the faintest possible asteroids we targetedjust a single DECam field on each night. During briefintervals near the beginning and end of the night whenour target field was at airmass greater than 2.0, we ob-served photometric calibration fields and targets of op-portunity.2.2.
Observing Conditions and Acquired Data
On March 30, we acquired a total of 147 images of thetarget field. All were 90-second exposures: 131 in thewide VR filter for maximum sensitivity, and 16 morein the r filter for calibration purposes. On March 31we acquired 225 images of our primary field, each onea 90-second exposure. Of these, 210 were in the VRfilter and 5 each were in the g, r, and i filters. The rea-son for the smaller number of images on March 30 wasthe failure of one of the DECam instrument computers,which took about three hours to recover. On both nightswe dithered the pointing after every exposure, followinga quasi-random pattern to ensure maximum cancella-tion of instrumental systematics and artifacts. We useda large number of DECam observing scripts that eachtook a few images dithered on a hexagon or linear se-quence, and we built up the quasi-random pattern bycontinually changing the scale and orientation of theseregular sequences. Our dither amplitudes were chosento fill in the gaps between detectors seamlessly, whilekeeping to the same field as much as possible.As described above, the nights of March 30-31 con-stituted our discovery run. Based on extrapolating thepower laws of Gladman et al. (2009), we expected tofind about three thousand asteroids down to a limitingmagnitude fainter than 25.0, and to confirm their realityby detecting them on both nights. On April 07-08, weintended to recover most of the newly discovered aster-oids and calculate approximate orbits from the resulting8-9 day arcs.We were blessed with mostly clear weather on all fournights, but the seeing during the recovery run was ex- tremely poor, in the 3-4 arcsecond range. Background-limited sensitivity to faint objects has the same depen-dence on seeing as it has on telescope aperture: thus,a 4-meter telescope in 4 arcsecond seeing is no betterthan a 1-meter telescope in 1 arcsecond seeing. Oursensitivity on April 07 and 08 is at least one magni-tude worse than during our discovery run, and approxi-mate orbits will be calculable only for a minority of thenew asteroids found in our discovery run. Fortunately,we can use the rotational reflex velocity (RRV) method(Heinze & Metchev 2015a; Lin et al. 2016) to calculatethe distances and absolute magnitudes of all the newlydiscovered asteroids using only the data from March 30and 31, when the seeing was 1-1.5 arcsec (Figure 2).Our 50% completeness limit on these nights was fainterthan 25th magnitude, representing a regime of flux andabsolute magnitude that has never before been system-atically analyzed in the main belt. Hence, we focus ourcurrent analysis on these nights, deferring to a futurepaper the analysis of orbital statistics for the brightersubset of asteroids recovered on April 07-08.2.3.
Image Selection for Digital Tracking Analysis
We obtained 147 images of our target field on March30 and 225 on March 31. To ensure uniform imagestacks, we applied our digital tracking analysis only toimages acquired in the wide VR filter, restricting us to131 images on March 30 and 210 on March 31.To identify any additional images that should be re-jected from the final stack, we define a noise parameter ξ that is inversely proportional to the SNR (or, equiva-lently, directly proportional to the fractional uncertaintyon the measured flux) of a faint source whose detectionis entirely background-limited. The background noisegoes as the square root of the number of sky photonsthat overlap with the point spread function (PSF) ofthe source: hence, ξ is proportional to the square rootof the sky brightness B times the effective radius r ofthe PSF, which in turn is proportional to the measuredseeing on the image. We take the seeing measurement s i on a given image to be the full width at half maxi-mum (FWHM) of stars on that image (Figure 2). Themeasured flux from the source varies directly with theatmospheric transparency, which we have called ǫ andplotted in Figure 3. Hence, our noise parameter for im-age i is given by: ξ i = s i √ B i ǫ i (1)For optimal detection of faint, non-moving objects, wewould create a stack of images weighted by 1 /ξ . To sim-plify the analysis, and in particular the determination of aint Asteroids with DECam Figure 1.
Left:
Diagram of the DECam focal plane, taken from the NOAO Data Handbook. CCDs labeled with F or G(colored magenta or green) are for focus or guiding, respectively, while those whose labels begin with N (north) or S (south) arescience CCDs which get their own extensions in the multi-extension FITS images that are DECam’s raw output. The colors ofthe science CCDs (orange, pink, blue, yellow) indicate which of four sets of readout electronics controls them.
Right:
A single90-second exposure from our data set, with the images from all detectors repixellated onto a master astrometric grid. Note thatdetectors N30 and S30 from the diagram are dead, but all other science detectors are fully functional.
Figure 2.
Left:
Seeing as a function of time for all VR images of our antisolar field obtained on March 30. The large gapbetween UT 4 and UT 7 is due to the failure of two of the control computers for DECam. The instrument was restored tofull operation before dawn.
Right:
The same plot for our March 31 observations. Red points indicate poor quality images notused in our final digital tracking stacks. Bad images on March 31 were mostly due to poor seeing. Note that seeing is directlyproportional to the PSF radius r used to calculate the noise parameter ξ . precise angular velocities, we elect not to apply such avariable weighting to our digital tracking stacks. Weare therefore interested in using ξ not to weight images,but rather to identify any that are of such poor qualitythat including them in an unweighted stack would re-duce the ultimate sensitivity. By interpreting ξ as thefractional uncertainty on the measured flux of a hypo-thetical point source, we can identify a threshold valueof ξ that separates useful images from those that wouldonly contribute noise. We sort the images in order ofincreasing ξ , and then explore the predicted fractional uncertainty on a uniformly weighted stack of these im-ages, truncated at image m before the end of the list: ξ stack = pP mi =1 ξ i m (2)Under Equation 2, the images at the beginning ofthe ordered list (that is, those with the lowest ξ i ) con-tribute the greatest reduction in the value of ξ stack , butpoorer quality images continue to make some contribu-tion through the bulk of the list. However, we find asmall number of images at the end of the list actuallydo cause ξ stack to increase when they are included. The Heinze et al.
Figure 3.
Left:
Atmospheric transmission ǫ relative to the best image for all VR images of our antisolar field obtained onMarch 30. Right:
The same plot for our March 31 observations. Red points indicate poor quality images not used in our finaldigital tracking stacks. The overall curvature on both nights is the effect of airmass. The sky was clear all night on March 31,but on March 30 some of our latest images were affected by clouds and rejected for this reason. threshold values for ξ are 66.0 and 81.0 for March 30 and31, respectively, where the higher threshold for March31 is due to the poorer seeing on that night. We re-ject images with ξ i exceeding these thresholds from ourdigital tracking analysis. Of a total 131 target-field im-ages taken in the VR filter on March 30, we reject 8and retain 123. Atmospheric transparency is the domi-nant cause of image rejection on this night, due to lightclouds blowing across our target field late in the datasequence (see Figure 3). For March 31, of a total 210images taken in the VR filter, we reject 8 and retain 202.The dominant cause of rejection is bad seeing near thebeginning of the data sequence (see Figure 2).Our digital tracking analysis proceeds with 123 imagesfor March 30 and 202 for March 31. As each image is a90 second exposure, the cumulative integrations are 3.08hr and 5.05 hr, respectively. The temporal span from thefirst image to the last is 7.67 hr on March 30 and 7.30hr on March 31. The mean seeing over accepted imageswas 1.11 arcsec on March 30 and 1.46 arcsec on March31. Figure 4 illustrates the clean, deep image obtainedby stacking the 202 images used for asteroid detectionon March 31. DATA PROCESSING3.1.
Basic Processing
We begin our processing of DECam data by read-ing the compressed, multi-extension fits.fz file for eachimage using CFITSIO, and converting it into multi-ple, single-extension, uncompressed fits files, each cor-responding to a single detector. Our processing thenproceeds for several steps almost as if each individualdetector were a separate CCD imager. A standard se-quence of calibration images, including biases and flatsin all relevant bands, was obtained on each night of our observations. For each night, we create a master biasframe for each detector using a median stack. We createflat frames in the same manner, after bias-subtractionand normalization. However, the normalization is notperformed independently for each detector. Instead, allthe subframes corresponding to a given flat image arenormalized by the same factor: the factor required toobtain a unit mean on detector N04, which we chooseas our reference detector because of its central position(see Figure 1). By this procedure, we allow the flat-field itself to remove detector-to-detector variations inquantum efficiency.We proceed to bias-subtract and flatfield all of ourscience frames independently. We then subtract thesky backgrounds using a polynomial model of the skyemission on each detector. A two-dimensional quadraticmodel produces excellent results, after the stars are it-eratively rejected from the fit. As the DECam detectorsshow no noticeable fringing in our wavelengths of inter-est, more advanced methods of sky subtraction are notnecessary. 3.2.
Astrometric Re-pixellation
We begin the task of re-pixellating our images on aconsistent astrometric grid by by fitting RA and DECas quadratic functions of x and y pixel position on eachdetector, using Gaia DR1 as our astrometric referencecatalog. We use the resulting astrometric fit to placethe data from each subframe of a given image into a sin-gle large, astrometrically registered array, which has di-mensions of 32,000 × α and δ , and a reference pixel loca-tion x and y (both corresponding to the exact center aint Asteroids with DECam Figure 4.
A star-registered all-night integration from March 31, created by median-stacking the 202 images we ultimatelyused for asteroid detection for that night. Sensitivity to stars and galaxies extends well past R = 25 mag, and digital trackingenabled us to achieve comparable sensitivity for unknown asteroids. Even the inset only hints at the tens of thousands of faintgalaxies visible in the full-size image. of the repixellated image), the x, y pixel coordinates ofa given α, δ position are given by: x − x = ( α − α ) cos ( δ ) /s pix y − y = ( δ − δ ) /s pix (3)Where s pix is the invariant pixel scale. More sophisti-cated projections would also be compatible with digitaltracking. We select this relatively crude option becausein Heinze et al. (2015b) we intensively analyzed its be-havior with respect to digital tracking, demonstratingthat it would produce good results as long as the tar-get declination is less than 60 degrees from the celestial equator. As the current data span a declination rangeof only -5.1 to -3.0 degrees, the distortion is very small.In constructing the repixellated image, we trim offand discard the outermost 10 pixels of each subframeto avoid edge anomalies present on the detectors. Dueto variations in the pixel scale of raw DECam images,our re-pixelization creates systematic photometric vari-ations from center to edge across the full DECam field.As the full extent of these photometric effects is less than1%, they are negligible for our purposes.Repixelation using Gaia delivered systematic astro-metric error so small it is difficult to measure. We have Heinze et al. determined that the median systematic error is less than0.007 arcsec. The maximum systematic error occursnear the image edges, and appears to be about 0.013arcsec. These errors are negligible compared to randomerrors for our target asteroids, and will make no ap-preciable contribution to errors on distances computedusing the RRV method.The effects of saturated bleed-streaks from bright starsmust be handled during repixelation in a way that en-ables us to remove them without leaving artifacts. Wefind that pixels immediately adjacent to bleeds typicallyhave anomalous values, so we mark all such pixels as sat-urated. Where a bleed-streak reaches the edge of DE-Cam detector, it broadens out as the electrons ‘pool’ atthe detector edge, and induces a variable negative offsetin the pixel values over a wide region surrounding thepool. Since this offset affects only a tiny fraction of ourdata, we elect simply to mask the affected pixels.Our repixelation uses bilinear interpolation, but re-verts to nearest-neighbor sampling whenever any of thefour pixels being interpolated is masked or saturated.This preserves hard edges in the resampled images, andenables us to remove all saturated bleeds at the imagesubtraction stage.3.3.
Photometric Calibration
Photometric calibration of our data is complicated bythe use of the nonstandard, wide VR filter. Our objec-tive is to report magnitudes that correspond as closely aspossible to the magnitudes that the same objects wouldhave in the Johnson R filter. We choose this filter be-cause it overlaps essentially completely with the VR fil-ter; and because it was the filter used by Gladman et al.(2009), to which, as the most rigorous survey of faintMBAs prior to this work, we are interested in compar-ing our results.For photometric calibration purposes, we obtainedDECam images of a field centered at 14:42:00, -00:05:00which is a standard calibration field for DECam becauseit contains many thousands of well-measured stars withSDSS photometry. We observed this field in the g , r , i , and VR filters, but elected in the end to extract ourcalibration from the VR observations only.We used photometric transformations given at theSDSS website to convert from cataloged magnitudesin the SDSS filters to R magnitudes. We experimentedwith two sets of transformation equations. One fromJester et al. (2005), derived for stars with Rc − Ic < .
15, is given in Equation 4. The other, derived by http://classic.sdss.org/dr4/algorithms/sdssUBVRITransform.html Robert Lupton in 2005, is unpublished but derived froma catalog query documented on the SDSS website. Itresults in Equation 5. V = g − . ∗ ( g − r ) − . V − R = 1 . ∗ ( r − i ) + 0 .
22 (4) R = r − . ∗ ( r − i ) − . R band magnitudes derived from SDSS gri photometryto fluxes measured on our VR images, and hence to cal-culate the approximate R magnitudes of other objectsbased on our VR images.We do this using aperture photometry with a largeaperture of radius 20 pixels (5.25 arcsec), deriving mag-nitude zeropoints of 25.798 at airmass 1.21 on March 30,and 25.771 and 25.756 at airmass 1.16 and 1.35, respec-tively, for March 31. Since these zeropoints are derivedfor repixelated images processed through the customizedmethodology described above, they would not be ex-pected to correspond exactly to those determined forother pipelines. The difference of 0.027 magnitudes inthe low-airmass zeropoints for March 30 and 31 appearsto represent a genuine difference in atmospheric trans-parency during the measurements of the SDSS calibra-tion field on the respective nights, in the sense that theatmosphere was slightly less transparent for the March31 images. This was not true of March 31 in general, aswe discuss below.For final calibration of asteroid magnitudes, we wishto use stars measured on the same images as the aster-oids. Hence, we use the SDSS-derived calibrations tomeasure the magnitudes of stars in the science fields,based on averaged results from subsets of 7 science im-ages taken at approximately the same airmass as theSDSS calibration images. Final magnitudes of the as-teroids are then based on these stars, without the needfor further inter-image comparison. We select the setof stars to be measured on the science images accord-ing to strict criteria that will also make them appropri-ate templates for generating fake asteroids (see § aint Asteroids with DECam R magnitudes to calculate magnitude zeropointsfor each of our science images. We fit these photomet-ric zeropoints as a function of airmass to find slopes of0 . ± . . ± . ∼ .
008 mag) better average atmospheric transparency.We discuss the issue of relative photometric calibrationfor the two nights further in § r filter, we also per-formed a calibration mapping VR fluxes directly toSDSS r magnitudes. We found a difference of +0.26 magrelative to our approximate R magnitudes. Our 50%sensitivity limit of R = 25 . r = 25 .
56 mag. However, since both Gladman et al.(2009) and Yoshida et al. (2003) used the R band (andderiving either R or r mags from VR fluxes is an ap-proximation), we focus our analysis on R magnitudes.Interested readers can convert to r band by adding 0.26mag.3.4. Subtracting Stars and Other Stationary Objects
We subtract stars, galaxies, and other stationary ob-jects from our science images prior to the digital trackinganalysis, using the method of Alard & Lupton (1998).In this method, a template image of the same targetfield is subtracted from each science frame after beingconvolved with a kernel designed to make the PSF ofthe two images identical. In our case, the template im-age is a stack of other images taken far enough awayin time (i.e. 10 minutes) that the slowest-moving as-teroids don’t self-subtract. The convolution kernal isconstructed from a sum of basis functions that have theform of radial Gaussians multiplied by polynomials in x and y . Alard & Lupton (1998) used three Gaussianswith σ = 1, 3, and 9 pixels, respectively, and multipliedthem by polynomials of order 6, 4, and 2. This pro- duces a total of 49 basis functions, with each of whichthe template image must be independently convolved— a procedure that is very computationally demand-ing. Fortunately, we find that such a large number ofpolynomials is not necessary for our DECam data.For our March 31 data, we adopt two Gaussians, with σ = 1 pixel and σ = 2 pixels respectively, each mul-tiplied by a 2nd order polynomial for a total of only12 basis functions. Subtracting the 210 images of ourMarch 31 data took about 2 weeks using a high-enddesktop workstation with 32 cores. For our March 30data, which was taken in somewhat better seeing, weadopt three Gaussians with σ = 0 .
6, 1 .
2, and 2 . × n best-matching images where n is de-fined by the minimum number required to meet twoconditions. These conditions impose minimum accept-able values for the minimum coverage ( mincov ) and themean coverage ( meancov ) of the stack, where coverageis defined as the number of images that contribute dataat a given pixel location. The thresholds mincov and meancov are designed to ensure that sufficient images gointo the template stack to render its background noisemuch lower than that of the science frame.We initially attempted the subtraction with mincov = 5 and meancov = 12, drawing images for the tem-plate stack from the pool of all science images separatedby at least 0.15 hr in time from the one being processed.This was very effective, but too slow, as the level of PSFsimilarity between different images had to be evaluatedso many times. We decided instead to draw the sub-traction images from a smaller set of 21 images, eachmade by stacking an independent subset of 10 scienceimages. Further, we restricted the set of images avail-0 Heinze et al. able for subtracting any given science image to seven,rather than the full 21. In this context, we set mincov = 2 and meancov = 2: such small values do not incurmore background noise than the previous settings, sincenow each subtraction image is the stack of ten scienceimages.We address sky background variations independentlyafter the star subtraction, by fitting and subtracting asecond order polynomial model of the residual sky back-ground in each 4000 × subtracted images and find the RMS through the stackat each pixel. Large values in the RMS map accuratelyindicate stars too bright to subtract cleanly. We maskpixels on the subtracted science images if their valueon the RMS map is greater than 15 ADU, scaled byan adjustment factor calculated independently for each4000 × DIGITAL TRACKING ANALYSIS4.1.
Introduction to Digital Tracking
In Heinze et al. (2015b) we have given a detailed de-scription of digital tracking (also called synthetic track-ing; Shao et al. (2013); Zhai et al. (2014)) and a reviewof past applications. We provide a much briefer intro-duction here.Digital tracking is the technique of shifting and stack-ing a large number of images to discover moving objectsmuch fainter than could be seen in a single image. Sincethe angular velocities of the objects to be discoveredare initially unknown, it is necessary to probe a rangeof trial velocities, which we refer to as ‘trial vectors’,and produce a separate ‘trial stack’ for each trial vec-tor. The trial vectors, each of which corresponds to atwo-dimensional angular velocity in the plane of the sky,must be chosen to span the range of ‘angular velocityphase space’ inhabited by the objects of interest. Thedensity of trial vectors must be chosen so that for anyobject in the region of angular velocity phase space be-ing explored, there will be at least one trial stack thataccurately registers all its images and renders it as asharp point source.As in Heinze et al. (2015b), here we search a rectan-gular region of angular velocity parameter space definedas specific ranges in eastward and northward angular ve-locity, chosen to include most MBAs in our field. Thisrectangular region is illustrated in Figure 5.The optimal spacing of trial vectors in angular velocityphase space is given by Equation 6 (Heinze et al. 2015b).∆ m = √ b max t int (6)Where b max is the maximum permissible blur in arc-seconds, and t int is the temporal span of the digitaltracking integration. For March 30 and 31 we have t int = 7.67 hr and t int = 7.30 hr, respectively. We mayreasonably set b max equal to the seeing, which for thetwo nights was 1.11 arcsec and 1.46 arcsec, respectively.This produces ∆ m = 0.205 arcsec/hr for March 30 and∆ m = 0.283 arcsec/hr for March 31. We conservativelyadopt a spacing of 0.20 arcsec/hr for March 30 and 0.25arcsec/hr for March 31.We search a rectangular region in angular velocityphase space that extends from -54 to -15 arcsec/hr to-ward the east, and -10 to +32 arcsec/hr toward thenorth. This was chosen to include the vast majorityof known MBAs in the vicinity of our field at the timeof our observations. As shown in Figure 5, the actualdistribution is elliptical rather than rectangular, and aconsiderably smaller area in angular velocity phase spacecould have spanned all the real objects. We could have aint Asteroids with DECam −50 −40 −30 −20 − Angular Velocities of Asteroids Detected March 31
Eastward angular velocity (arcsec/hr) N o r t h w a r d angu l a r v e l o c i t y ( a r cs e c / h r) ● ● ● ●●● ●● ●● ● ● ●●● ● ●● ●●● ●●●●● ● ●● ●● ● ●● ●● ● ●● ●● ● ●● ●● ●●● ●● ●●● ●●● ●● ●● ●● ●●● ● ●● ●● ●● ● ●●● ●●●● ● ●● ●● ●● ●● ●● ● ●●● ●● ●● ●● ● ●●● ●● ●●● ● ●● ● ●●● ●●● ●● ●●● ●● ●●●● ●● ●●● ● ● ● ●●● ● ●● ●● ●● ●● ●●● ●● ● ● ●●● ●● ● ●● ●●● ● ●●●● ●●● ●● ●● ●● ●● ●● ●● ● ●●● ●●●● ●● ●●● ● ●● ● ●● ●● ●● ●● ●● ●● ●● ●● ● ●● ●●● ●●● ● ●●● ●● ●● ●● ●●● ● ●● ●● ● ●● ●● ● ●●● ● ● ●●●● ● ●● ● ●● ● ● ●● ●● ●●● ●● ●●● ● ●● ●● ● ● ●● ●●●●● ●● ●●● ●●●●● ●●● ●● ●●●● ●● ● ●● ●●● ●● ●●● ●● ● ●● ●● ● ●●● ●● ●● ●● ● ●● ●●● ●● ● ●● ●● ●● ● ●● ● ●●● ●● ● ●●● ●● ●●●●● ● ●●●● ● ●● ●●● ● ●●● ●●● ●● ●●●● ● ●●● ●●●●● ● ●● ●● ●● ● ● ●● ● ●● ●● ● ●● ●●● ●●● ● ●●● ● ● ●●● ●●● ●● ● ● ●●● ● ●●●● ●●● ● ●●●●● ● ● ●● ●●●● ●● ●● ●●● ●● ● ● ●● ●● ●● ●●● ● ●●● ● ● ●●● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ● ●●● ●●●● ● ●●● ●●● ● ● ●● ● ●● ● ● ● ●● ● ●●● ●● ●●● ●●● ●● ●● ● ●●●●●● ●●● ●● ●● ● ●● ●● ●● ●● ●● ●● ●● ●●● ● ● ●●●● ● ●● ●●● ● ●●● ●● ●●●● ●● ●● ●● ●● ● ●● ●●●● ● ●● ●●● ●● ●● ●● ●●●● ●●● ●●● ●●● ●●● ●●● ●●● ●● ● ●● ● ● ●● ●● ●● ●● ●●●● ● ●● ● ● ●● ●●●● ● ● ●● ●● ●● ●●● ●●●●● ●●● ●●●● ●● ●●●● ● ●●● ● ●● ●●● ●● ● ●● ● ●●●● ●●● ● ●● ● ● ● ●●● ●● ●● ●●● ●● ● ● ●● ● ●● ●●● ●●●● ●● ●●● ●● ● ● ●● ●● ●● ●●● ●● ●● ●● ● ●● ●● ●● ● ●● ●● ● ●● ●●● ● ●● ●●●● ● ●●● ● ●●● ● ● ● ●●●●● ●● ● ●●● ● ●● ●● ●●● ●● ● ● ●●●● ●●● ● ●● ●● ●● ●● ●●● ●●●●● ●● ●● ●● ●● ● ●● ● ●● ●● ●● ●●● ● ● ● ●●● ●● ●● ●● ● ●● ●● ●●● ● ●● ●● ●● ● ● ●● ●● ●● ●● ● ●● ●● ●● ● ●● ●●●● ●● ● ●● ●● ●● ● ●●● ●● ●●● ● ●● ● ●● ●● ●●● ● ● ● ● ●●● ●● ● ●● ●● ●●● ●●● ● ● ●● ●● ●● ●●● ● ●● ●● ● ●●● ●●● ●● ● ● ●● ●● ● ●● ● ●●● ●●● ●● ● ●● ● ●● ●● ●● ●● ● ●●● ●● ●●●● ●●● ● ●● ● ●● ●● ●●● ●● ●●●● ● ●● ●●●● ●● ●● ● ● ●● ●● ● ●●● ●●● ●● ●● ●● ●● ●● ● ●● ● ● ●●●● ●● ● ●● ●●● ●● ● ● ●●●● ●●● ● ●●● ●● ● ●●● ● ● ●● ●● ● ● ●● ● ●●● ●●● ● ● ● ● ●● ●●● ● ● ●● ●●●● ●●● ● ● ●● ●● ●●● ●● ●●● ● ● ●● ● ● ● ●● ●● ●● ●● ● ●●● ● ●●● ●● ●● ● ●● ●●● ● ● ● ●●● ●●● ●●● ●● ●●● ●● ● ●●●● ● ● ● ●● ●●●● ● ●●● ●●● ● ●● ● ●● ● ●● ●● ●● ● ●● ●● ●● ● ● ● ●●● ●● ● ●●●● ● ●●● ● ●●●● ●● ●●●● ● ●● ●● ●●● ● ●● ●●● ●●● ●●● ●● ● ●● ●● ●● ● ●●● ● ●● ●●●●● ● ● ● ●● ● ●● ● ●● ● ●●● ● ● ●●● ●● ●● ● ●● ● ●● ●●● ● ●● ●●● ● ●● ●● ●● ●● ●● ● ●●● ● ●● ● ●● ● ● ●● ● ●●●● ● ●●●● ●● ● ●● ●● ● ● ●● ●●● ●● ●●● ●● ●●●● ●● ●● ● ●● ●● ● ●●● ●● ●●● ●●● ● ●●●● ● ● ● ●● ●●● ●● ● ●● ●● ●●● ●●●●● ●● ● ● ● ●● ●●● ●●● ●●● ●● ●●●● ●●● ●● ●● ●● ● ●● ● ●●● ●●●● ● ● ●●● ● ●● ● ● ●● ●●●● ● ● ● ● ●●●● ● ●●●● ●● ● ●● ●● ●● ● ●●● ● ●● ●●● ● ●● ●●● ● ●● ●● ●●●●● ●● ●● ●● ●● ●● ●●● ●● ●●● ●● ●● ●●● ● ●● ●● ● ●●●● ●●● ●● ●● ●● ●●● ●● ● ●●● ●● ●●●● ●● ●●● ●● ● ●● ● ● ●● ●● ●● ●● ●● ● ●● ●● ● ●● ●● ●● ●● ●● ● ● ●● ●● ●●● ● ●● ●●● ●● ●● ●●●● ● ●● ●●● ● ● ●● ●●● ●●● ●●● ●●● ●● ●● ● ●● ●● ●● ●●● ●●● ● ●●●● ●● ● ●● ●● ●● ●● ●● ●● ●● ● ●● ● ● ●● ●●● ●● ● ● ●● ●●● ●●● ● ● ●● ● ●● ●●● ● ● ● ● ● ●● ●● ●● ●●●● ● ● ●● ● ●● ●● ● ●● ● ●● ● ●● ● ●●● ●● ●●● ●●● ● ●●●●● ● ● ●●● ●● ● ● ●● ●● ●● ●● ●● ● ●● ● ● ●● ● ●● ● ● ●●● ● ● ● ● ●●●●● ● ●● ●●● ●● ● ●●● ● ●● ● ●● ● ●● ●● ●● ●● ●● ● ● ●● ●● ● ●●●● ●● ● ● ●● ●●● ●● ●● ●●● ●● ●●● ●●●●● ● ●● ● ●●● ●●●● ●●●●● ● ●● ●● ●● ● ●● ●● ●● ●● ●●● ●● ●● ●●● ● ●●● ●●● ●●●● ●● ●●●● ●● ●● ●● ●● ● ●● ● ●● ●● ●● ● ●● ● ● ● ●● ●● ●● ●●●● ● ● ●● ●●● ●● ●● ●●● ● ●● ●● ●● ●● ●● ● ●●● ●● ●● ●● ●● ●●● ●●● ● ●● ●● ●● ●● ●● ●●●● ● ●●● ●●● ● ● ●●●● ●●● ● ● ●● ●● ● ●● ●●● ●● ●● ●●●●● ● ●● ●●● ●●● ●● ● ●● ●●●● ●● ● ●● ●● ● ●●● ● ●●●● ● ● ●●● ● ●● ●●● ● ●●● ●● ● ● ●● ● ●●●● ●● ● ●●● ●● ● ●● ● ●● ●● ●● ●● ●●● ●● ●● ●●● ●● ●● ●●● ●●● ● ●● ●● ● ●● ● ●● ● ● ●●●● ●● ● ● ●●●●●● ● ●●● ●● ● ●● ●● ●●● ● ● ●● ● ●● ●● ● ●●● ● ●● ●● ●●●● ●● ●● ● ●● ● ● ●●● ●●●●● ●● ●●● ● ●● ●●●● ● ● ● ●● ●● ● ●● ●●● ●●● ●●●● ●●●● ●● ●●● ●●● ●● ● ●●●● ● ●● ●● ● ●● ●● ● ● ●● ●● ● ●● ● ●● ●●● ● ●● ●●● ●●● ●● ● ●● ● ● ●●● ●● ● ● ●● ●●● ● ●● ● ●● ●● ● ●● ●●● ●●● ●● ● ●● ●● ●● ●●● ● ● ●●●● ●●● ●● ●● ●● ●● ●● ● ●●● ●●●● ●● ● ●● ● ●●● ●●● ●●●● ● ●● ●● ●● ●●● ●● ● ●● ●● ● ●● ●●● ●●● ● ● ●● ●●● ●●●●●● ● ● ●●● ● ● ●●● ●● ●●●● ●● ● ●● ●● ● ●● ●● ●●● ●● ●● ●●● ● ●● ● ●●● ● ● ●● ●● ●● ●● ●● ●● ●● ● ●● ●● ●●● ●● ●●● ●● ●● ● ●●● ●●● ●● ●●● ●● ●● ●● ● ●● ● ●●● ● ●● ● ●● ● ●●● ●●● ● ●●● ●● ●● ●● ●● ● ●● ●●●● ● ●● ● ● ●●● ●●● ●● ● ●● ●● ●● ●● ●●● ●● ●● ●●●● ●●● ● ●●● ● ●● ● ●●●● ●● ●● ●● ●● ●● ●● ●● ● ●● ● ●●● ● ● ● ●●● ● ●●●● ●● ●● ●● ●● ●●● ●●● ●●● ● ●●● ●● ● ●●● ●●● ●● ●●●● ●●● ● ●● ● ●●● ●●●● ●● ● ●● ●● ●● ●● ●● ● ● ● ●● ●● ●●● ●●● ●●● ● ●● ●●● ●●●●● ● ●● ●●● ● ●● ●●●● ●● ●● ●● ● ●●● ● ●●● ●●● ● ●●● ●● ● ● ●● ●●● ●● ● ●●● ●● ●● ● ●● ●●●● ●● ●● ●● ●● ● ●● Boundary of digital tracking region J up i t e r T r o j an s H il da s Main BeltAsteroids E c li p t i c Figure 5.
The angular velocity phase space searched in our digital tracking analysis, with the phase space positions of asteroidsdetected in our March 31 data plotted. The dark red dashed line indicates motion parallel to the ecliptic. The green lineindicates the boundary in angular velocity space that we use in § predicted this region based on known objects and re-duced the digital processing runtime by 30-60% by tar-geting an optimized elliptical region or regions. How-ever, we consider the regions of ‘white space’ in the an-gular velocity plot to be a useful check on false positivesin our analysis: a spurious asteroid would be equallylikely to occur at any angular velocity, so the absenceof detections in the corner regions of the plot suggests there are very few false positives. We address this ques-tion more rigorously in §
6, but the blank areas of theFigure 5 are a good initial indication. We note also thatthe generous rectangle we searched in angular velocityphase space enabled us to detect a considerable numberof Jupiter Trojan asteroids, which we had not antici-pated.2
Heinze et al.
Computational Requirements
We performed our digital tracking analysis primarilyusing idle time on two processing nodes of the computingcluster belonging to the Asteroid Terrestrial-impact LastAlert System (ATLAS, Tonry et al. 2018). Each nodehas 24 cores and 128 GB of memory.4.2.1.
Memory
As mentioned above, each of the input imagesto our digital tracking analysis has dimensions of32,000 × × σ clipped median, and the pixel values are converted backto full floating-point precision prior to stacking. Thisdoes not use excessive memory since it can be done onepixel at a time. With half-precision storage, we can loadall of the images from March 31 using only about 70%of our computers’ 128 GB of memory, and hence canperform the digital tracking analysis efficiently.4.2.2. Processing
The angular velocity phase space region and step sizesdetermined in § × × . × vector-pixels forour March 30 data and 1 . × for March 31. Onthe ordinary, 24-core nodes of the ATLAS computingcluster, we obtained typical processing rates of about1 . × vector-pixels per hour per node. Thus, wecould complete the digital tracking analysis in about 80node-days. However, as described below we actually an-alyzed the entire data set three times: once to detectreal asteroids, once to probe false positives, and oncewith fake asteroids inserted to probe our completenessrate. The overall analysis thus required the equivalentof 240 node-days, although some of it was performedon an experimental, 1024-core supercomputer, which wefound to be equivalent (for purposes of digital tracking)to about 7.5 ordinary cluster nodes.4.3. Detection of Candidate Asteroids
As in Heinze et al. (2015b), our digital tracking anal-ysis does not save the trial stacks, which would requireprohibitively large volumes of storage. Instead, our codeautomatically detects sources on each trial stack andwrites a detection log. This detection log is the primaryoutput of the digital tracking analysis, and must then befurther analyzed to identify and precisely measure thereal asteroids, as described below.Our digital tracking code operates on images binned2 × × = 361 aint Asteroids with DECam B is then taken to be the mean ofthe surviving pixels, and the noise κ is their standarddeviation. We note that the rejection of deviant pointsby the trim mean causes κ to underestimate the truenoise: in the case of Gaussian statistics with 10% trimmean rejection, κ = σ/ . σ by1 . κ would be like taking a robust standard deviation,which is very similar conceptually to what we have done.We detect asteroids by finding pixels in the smoothedtrial stack whose brightness exceeds the sky brightness B by at least 10 κ , which is equivalent to 7 . σ in Gaus-sian terms. Bright asteroid images create many pixelsbrighter than our detection threshold, so we reduce re-dundant detections by deleting all candidates within a‘redundancy radius’ of a more significant detection. Toguard against the loss of actual distinct asteroids, weconservatively set the redundancy radius to just 5 pixelson the trial stacks: that is, 2.625 arcsec. The redundantdetections that survive this cull are dealt with in post-processing of the digital tracking detection log, as wedescribe below in § POST-PROCESSING OF DIGITAL TRACKINGDETECTIONSThe output of the digital tracking analysis describedin § κ (see § § × Spurious Detections at the Edge of the Image
Our detection logs have large numbers of spurious de-tections near the edge of the region of valid data. Thesedo not correspond to the edges of individual DECamdetectors (those vanish completely, thanks to our well-optimized dither), but rather to the boundary beyondwhich we acquired no data with any detector. Thisboundary is rendered approximately circular by the lay-out of the DECam detectors, and has nonzero widthsince the actual edge of the data does not occur at thesame pixel coordinates for all trial stacks.We do not know in detail the cause of the spuriousdetections at the edge of our data region. They appeardespite several aspects of our code that are designedto prevent noise at the edges of a stack from being er-roneously recorded as a significant detection. The largenumbers of spurious edge detections first appeared whenwe began storing the digital tracking images in memoryat half precision, but we have not been able to identifythe aspect of half-precision storage that causes them.As described above, we have extensively validated theoutput of our half-precision code relative to previousfull-precision results for actual asteroid candidates: theonly significant difference is the appearance of this vastnumber of false positives at the images edges.Since real asteroids could not be usefully detected atthe extreme edge of the images anyway, we have electedsimply to cull the detection logs using a map of the edgeregions where false detections occur. We construct thismap from the detection logs themselves, first plotting4
Heinze et al. the pixel coordinates of all the detections, and then us-ing a combination of blurring and boundary-detection toidentify the full extent of the false-detection zone whileencroaching as little as possible on the valid data inte-rior to it. We find additional use for this map later inour analysis: it provides a measurement of the area onthe sky over which we had good sensitivity.Our detection logs from March 30 and 31 originallycontained 7.5 million and 6.4 million candidates, respec-tively. After removing the spurious edge-detections, thenumbers dropped to 3.3 million and 2.4 million, the vastmajority of which are redundant detections of bright as-teroids, as we describe below.5.2.
Removing Duplicates
Millions of duplicate detections of real asteroids sur-vive our edge-culling and the simple redundancy cut de-scribed in § Images for Asteroid Analysis
Given the four dimensional location of an asteroid can-didate in the detection log, we can calculate its loca-tion on each of the input images, extract small ‘postagestamps’ centered on this location from each image, andstack these stamps to produce an image of the asteroidfor manual examination. This image is superior to thetrial stack evaluated internally by the digital trackingcode because it is made at the full resolution of the inputimages; because the images are registered at sub-pixelprecision using bilinear rather than nearest-neighbor in-terpolation; and because the image data have not hadto be loaded into memory at half precision. We use suchstacks in several ways for manually vetting the asteroidcandidates identified by the clustering process describedin § aint Asteroids with DECam . The new angularvelocity measurement is simply the point in angular ve-locity space where the quadratic fit to the flux reachesits maximum.We apply a second type of verification by making mul-tiple stamp-stacks at the same angular velocities, butwith different subsets of the data, created by even split-ting of the temporally ordered set of images. For manualexamination, we arrange these in a pyramid shape: thestack of all the images is at the top of the pyramid; be-low come two stacks each made of half the images; belowthat three stacks each made with a third of the data, etc.Real asteroids should be consistently detected in multi-ple layers of the pyramid, until the stacks become tooshallow and they fade into the noise. In some cases anasteroid may change significantly in brightness due torotation (or move off the image or into a masked regionnear a bright star), and hence will be absent on severalpyramid tiles. Our general criterion for confirming anasteroid as real is that it must appear fairly consistentlyon at least two tiles in some row of the pyramid image:equivalently, it must be detectable in two independentsubsets of the data.Figure 7 shows examples of two pyramid images, prob-ing much fainter asteroids than Figure 6. The asteroidin the second panel of Figure 7 gets passed by another,much brighter object at a distance of less than 6 arcsecon the sky and 2.9 arcsec/hr in angular velocity. Bothasteroids were independently recovered by our asteroidclustering analysis. The fact that the fainter one was notconsumed by the brighter object’s cluster despite theirproximity in all four dimensions is a testimony to the ac-curate tuning of our clustering thresholds, as describedin § ×
10 tilings of stamp-stacksat the nominal parameters. This allowed us to rapidlyscreen 100 asteroids at a time by eye, and aided in catch-ing classification mistakes or glitches in our fits for re-fining the pixel and angular velocity coordinates of eachobject. 5.4.
Creation of Final Asteroid Lists
Using tiled arrays of stamp-stacks; check images; andpyramid images, we visually screened the candidate listsoutput by our clustering analysis. Of the original lists of We use this type of flux calculation only in the current context,not for actual photometry, which we discuss later. on thescale of the PSF (and hence properly accounting for cor-relations between pixels), by performing identical forcedphotometry in a large number of non-overlapping skyapertures near the asteroid. These apertures are con-strained to be centered within a 100 pixel (26.25 arcsec)box centered on the asteroid, and yet must not overlapa disk of 12 pixel (3.15 arcsec) radius around the object.The sky noise is evaluated as the STDEV of the forcedphotometry after iterative rejection of the 10% most de-viant points, and is converted to Gaussian σ using thefactor of 1.267 discussed in § σ value of the sky noise.The effective FWHM of our asteroid images can varyfor several reasons. The fastest-moving asteroids willbe slightly ( . Heinze et al.
Figure 6.
Examples of the ‘check images’ described in the text, which we use to check the reality of candidate asteroids andrefine our angular velocity measurements. The tiles in each panel cover a grid in angular velocity space with a spacing of 0.5arcsec/hr: optimal for illustrative purposes but coarser than the sampling we ultimately used for precise velocity determination.Each tile has a width of 151 pixels (40 arcsec). The data shown are from March 30 (left) and March 31 (right). These arerelatively bright asteroids in our context: each is the 600th entry in the significance-ranked list output by the cluster analysisfor its respective night. The trails in the March 30 images have time-gaps due to the DECam computer failure. The March 31images are fuzzier but have a less noisy background relative to March 30: the seeing was better on March 30, but more goodimages were available to be stacked on March 31. measured depends on the aperture radius being probed(at larger radii not as many apertures can be fit in theallowed region), but is typically a few hundred. The me-dian optimal radius is 3.1 pixels (0.81 arcsec) for March30 and 3.7 pixels (0.97 arcsec) for March 31, consistentwith the generally better seeing on March 30.If the internal significance calculation in our digitaltracking code were perfect, all of the detections it out-put would have signifance at least 7.89 σ . In fact, ourmore sophisticated analysis finds that the vast majorityof detections passed by our manual screening do haveat least this significance: only 2.2% and 7.7% fall be-low it for March 30 and 31, respectively. Whether theserelatively low-significance detections correspond to realasteroids is a question we address further below.5.5. Magnitude Measurements
We extract optimized photometry for each asteroidfrom its stamp-stack using a method similar to our sig-nificance calculation: by testing a large number of aper-tures and selecting the optimal one. The backgroundnoise is also calculated in the same way. However, theoptimization is different because we now seek to min-imize the fractional uncertainty on the flux, and this includes an additional source of uncertainty beyond thesky background noise: the uncertainty on the aperturecorrection. The aperture corrections and their uncer-tainties were determined by measuring 100 bright aster-oids at both the large (20 pixel radius) aperture usedfor the stellar photometric calibrations and the range ofsmaller apertures referred to above. Since the aperturecorrection has a larger fractional uncertainty for smallapertures, the optimal aperture for photometry is typ-ically larger than for the significance calculation. Themedian optimal photometric aperture has a radius of 3.7pixels (0.97 arcsec) for March 30 and 4.3 pixels (1.13 arc-sec) on March 31. These values can be compared withthe optimal radii of 0.81 arcsec and 0.97 arcsec for thesignificance calculation on the respective nights. FALSE POSITIVE ANALYSISIn order to extract accurate information on asteroidpopulations from our data, it is essential to analyze boththe rate of false positives and that of false negatives (sur-vey incompleteness). We describe in this section the in-tensive analyses we have carried out to determine detec-tion thresholds for a negligible false-positive rate. In the aint Asteroids with DECam Figure 7.
Examples of the pyramid images described in the text, which we use to confirm the reality of asteroid candidates.Each panel represents a distinct asteroid from March 31, while successive rows show the division of the data into multipleindependent, temporally ordered subsets. Tiles are 40 arcsec on a side. The left panel shows asteroid candidate 3,561 fromthe significance-ranked list output by our clustering analysis: this asteroid is definitively real (confirmed in the March 30 data)but is near the faint limit of our sensitivity. The right panel shows another real asteroid, ranked 2,189 in significance, whichentertainingly gets passed at a distance of less than six arcseconds by a much brighter asteroid whose angular velocity differsby only 2.9 arcsec/hr. The brighter asteroid is independently detected as number 517 in the ranked list. next section we will quantify our magnitude-dependentcompleteness.Digital tracking enables a very realistic probe ofthe false positive rate. By randomly re-assigning (i.e.scrambling) the time-stamps of our images, we can re-peat the digital tracking analysis using identical dataand methodology and yet with the certainty that anydetections will be false. This is because scrambling theimage times makes it impossible for real asteroids to beregistered in any of the trial stacks, and hence they van-ish in the clipped median combine.We have performed a full digital tracking analysis ofour images with scrambled timestamps, down to the cre-ation of check images and the sophisticated significancecalculation described above. We find that the most sig-nificant candidate detection in these scrambled data setsis 7 . σ for March 30 and 7 . σ for March 31. Undermanual examination, these detections appear indistin-guishable from real asteroids with the same significance.To determine if the detection of spurious sources atthese significance levels is surprising, we estimate thenumber of noise realizations of the PSF that our analysishas probed. As described in § × . × pixels. Our digital tracking analysis assumed thatthe effective area of a PSF is 9 binned pixels, in whichcase the valid data on each trial stack covers the size of20 million PSFs. Multiplying this by the number of trialstacks, we find that we have effectively sampled about7 . × and 4 . × realizations of the PSF-scalenoise on our two nights, respectively.The expected number of spurious detections from pureGaussian noise equals unity for a threshold x lim beyondwhich the one-sided tail of a normalized Gaussian inte-grates to the inverse of these numbers of realizations.Thus, for N real = 7 . × and 4 . × on the twonights respectively, we seek x lim such that: Z ∞ x lim e − x / σ σ √ π = 1 N real (7)Solving Equation 7 yields x lim = 7 . σ and 7 . σ forMarch 30 and 31, respectively. These numbers are closeto the values of 7 . σ and 7 . σ for the most significantdetections actually found in the time-scrambled digital8 Heinze et al. tracking analysis, so we conclude that such detectionsare not surprising — and that the methods we have em-ployed for rejecting spurious sources leave behind some-thing quite close to pure Gaussian noise. This quantita-tive conclusion aligns with the visual impression that ouroptimized dithering, star subtraction, masking, shifting,and clipped-median stacking do indeed produce imagesmore free of artifacts than is usually possible with as-tronomical data.We desire an extremely low false-positive rate in orderto ensure that the scientific conclusions we ultimatelydraw about the population of small MBAs will not beinvalidated by false positives. Hence, we adopt the maxi-mum significance values seen in the scrambled-time anal-ysis (7 . σ and 7 . σ for March 30 and 31, respectively)as our minimum thresholds for the independent detec-tion of real objects. A few of the real detections manu-ally classified as ‘a’ (and many of those classified as ‘b’)fall below this threshold. Class ‘a’ candidates falling be-low the thresholds are reclassifed as ‘subsig’, and are notconsidered as definitive, independent detections (eventhough their much greater abundance relative to sourcesof similar significance in the time-scrambled data indi-cates most of them are real). Only the ‘a’ class objectswith significance above the threshold are considered tobe definitively, independently detected on a single night.The totals in various categories are given in Table 1.With the thresholds we have adopted, we expect no morethan one false positive per night — a rate far too lowto meaningfully affect scientific conclusions drawn fromour analysis. COMPLETENESS ANALYSISBecause the main science result of our survey is thestatistical distribution of asteroids as a function of flux,we put a great deal of effort into a rigorous evaluationof our fractional detection rate as a function of appar-ent magnitude. To accomplish this, we repeated ourentire digital tracking analysis on images to which wehad added fake asteroids. We took pains to make surethese fake asteroids would be subject to all aspects of ouranalysis that could affect our sensitivity to real ones. Weplaced them in the images prior to the subtraction of sta-tionary sources, so that if (contrary to our expectations)any real asteroids were dimmed by self-subtraction, somight be the fake ones. We placed them at locationsbased on realistic dynamical orbits so that if (again, con-trary to expectations) nonlinearity in the sky motion ofreal asteroids reduced our sensitivity, the fake asteroidswould experience the same effect. Finally, we calculatedthe pixel locations where our fake asteroids should beplaced not by taking our astrometric mapping (Equa-
Table 1.
Asteroid Candidates Detected with DigitalTracking Number MatchDate Category Number matched a rateMarch 30 ‘a’ 2760 2720 98.6%March 30 ‘subsig’ 16 11 68.8%March 30 ‘b’ 22 0 0March 30 total 2798 2738 97.9%March 31 ‘a’ 2973 2902 97.6%March 31 ‘subsig’ 37 14 37.8%March 31 ‘b’ 81 1 1.2%March 31 total 3091 2917 94.4% a ‘Number matched’ is the count of asteroids in each cat-egory that were detected and unambiguously matchedin data from the other night. It includes objects thatwere automatically detected on both nights as well asthose that were initially detected in only one night’sdata but were then successfully recovered in data fromthe other night at a sky position and angular velocitypredicted based on the original detection (see § tion 3) at face value, but by obtaining an entirely new,5th order global astrometric fit to the repixellated im-ages. Thus, in the unlikely event that errors in our astro-metric repixelation using Equation 3 were large enoughto cause blurring and loss of sensitivity in our stackedimages of real asteroids, the fake asteroids should expe-rience similar degradation.7.1. Orbital Population of Fake Asteroids
As we developed our strategy for the fake asteroidtest, we realized that by judicious choice of the inputpopulation of asteroids, we could link the statistics ofour actual detected asteroids to the total population ofasteroids in the whole asteroid belt that had the samecharacteristics. Our first step in doing this was to simu-late the entire population of small MBAs. We based thissimulation on the known orbits of MBAs with orbitalsemimajor axis between 1.7 and 4.1 AU and absolutemagnitude H < .
0. We chose this fairly bright abso-lute magnitude threshold to ensure a sample of knownasteroids that would be complete for all main-belt orbits,avoiding a statistical bias against orbits in the outer beltwhere small objects are harder to see. We chose thesemimajor axis range to include the entire main belt,but to reject Jupiter Trojans and NEOs (i.e., all ob-jects with perihelia inside of 1.3 AU). Mars-crossing andJupiter-crossing asteroids were retained, since their or- aint Asteroids with DECam . The final list, afterculling as described above, comprised 25409 orbits. Wewished to use these orbits of well known MBAs as a basisfor generating a much larger, but statistically identical,set of simulated MBA orbits. We found that althoughtwo of the Keplerian orbital elements, argument of per-ihelion and mean anomaly at the epoch, are uniformlydistributed, the other four are not and have substantialcorrelations with one another. Rather than attempt toarrive at analytical approximations of the very complexdistributions of these four parameters, we elected to baseeach simulated asteroid on one of the real asteroids. Thetwo elements mentioned above would be chosen from auniform distribution, but the other four elements wouldbe adopted from the real asteroid’s orbit and modifiedby a Gaussian fuzz with amplitude chosen to random-ize the orbits as much as possible without erasing thestructure and correlations of the original distributions.We simulated asteroid orbits according to this pro-tocol, setting the epoch for each orbit to JD 2456746.5:that is, 00:00 UT on 2014 March 30, near the start of ourobservations. For each asteroid orbit we calculated thecorresponding RA and Dec at 00:00 UT on March 30.We terminated the simulation when a sufficient numberof simulated asteroids had been found to be within 1.7degrees of our field center at this time. This is signif-icantly larger than the 1.1 degree radius of DECam’sfield of view, ensuring that all simulated MBAs thatcould appear on our DECam images on either March 30or March 31 would be included.We aimed to simulate 20000 asteroids within the 1.7-degree target region. Since our input orbital distribu-tion is carefully chosen not to be biased against objectsin distant orbits, such objects will be better-representedin the simulation than in our actual survey, which, be-ing flux-limited, has reduced sensitivity to more distantobjects in the outer main belt. This bias would producea considerably faster average angular velocity for realasteroids detected in our images relative to our simu-lated asteroids. Since detection efficiency might dependon angular velocity, we wished to ‘re-bias’ the simulatedasteroids to more closely imitate the real population.Hence, for asteroids more than a threshold geocentricdistance d thresh of 1.5 AU (chosen as a representativedistance for the inner half of the main belt), we assigneda probability of being visible given by P vis = 10 − α dM , http://ssd.jpl.nasa.gov/dat/ELEMENTS.NUMBR where α is the slope of the MBA apparent magnitudedistribution found by Gladman et al. (2009), and dM is the difference between the asteroid’s actual appar-ent magnitude and the magnitude it would have at thethreshold distance (note that, as defined, dM is positivefor all the relevant asteroids). For asteroids near opposi-tion, which are the only ones relevant here, dM is givento sufficient accuracy by: dM = 2 . (cid:18)(cid:18) ∆ d thresh (cid:19) · (cid:18) r ( d thresh + 1) (cid:19)(cid:19) (8)Where ∆ and r are the geocentric and heliocentric dis-tances of the asteroid, respectively, in AU. For asteroidswith ∆ > d thresh , we randomly assign a status of visibleor invisible according to probability P vis . We allowedthe simulation to proceed until it generated 20009 vis-ible asteroids in the target area, at which point it hadgenerated 35275 total asteroids in the target region and5 . × total asteroids anywhere in the Solar System.For the purpose of inserting fake asteroids into ourdata, only the 20009 visible asteroids should be consid-ered. The statistical value of the remaining simulatedasteroids, including those flagged as invisible, will beconsidered in our companion paper (Heinze et al., inprep.) on the absolute magnitude distribution and totalpopulation of the main belt.7.2. Placing Fake Asteroids in the Images
To insert the simulated asteroids into the actual im-ages, we use the catalog of bright, isolated, and unsatu-rated field stars described in § ×
40 pixel(10.5 × Heinze et al.
Table 2.
Magnitude Distribution of Simu-lated AsteroidsMagnitude Range Fraction of Asteroids20.0–21.0 0.0221.0–22.0 0.0222.0–23.0 0.0223.0–24.0 0.0924.0–24.5 0.1024.5–25.0 0.2025.0–25.5 0.3025.5–26.0 0.25 position at 10 instants spanning the 90-second exposuretime of the image, interpolating from the 5-minute sam-pling of the simulated ephemeris. For each of these tenpositions, we identified the nearest object from our list ofisolated stars that could be translated to the asteroid’sposition (within a tolerance of 0.1 pixels) by shifting aninteger number of pixels. Thus, we were able to buildup the asteroid images without blurring due to to inter-polation. Since, on average, less than 100 stars wouldhave to be examined to find one that could be movedto the asteroid’s location within the specified tolerance,we expect that the stars used to build up each asteroid’simage will generally come from near the asteroid in theDECam field. We have not attempted to set any max-imum distance, since the PSF is remarkably consistentacross the DECam field.Once a suitable star is identified to construct part ofa simulated asteroid’s image, a region 40x40 pixels cen-tered on the star is copied, scaled by the magnitudedifference, further scaled to account for the fact thatit will only contribute one tenth of the asteroid’s finalflux, shifted, and added back in to the original image.The star positions are measured individually on eachimage. As mentioned above, however, the asteroid po-sitions are converted from the celestial coordinates ofthe ephemeris file to pixel positions using a 5th order fitbased on Gaia astrometry, which is entirely independentof the astrometric fit originally used to repixellate theimage. Thus, any flaws in the original astrometric solu-tion, which could blur the stacked images of real aster-oids, can affect the simulated asteroids in a similar way.The simulated asteroids, like real objects, follow coher-ent orbits between the two nights. Hence, the same fakeasteroid can be found on both nights and its distancecan even be calculated using the RRV method and com-pared with the actual distance, which in the case of thefake asteroids is exactly known by construction. Comparison of fake asteroid detections across the twonights made us aware of a photometric inconsistency inthe star catalogs used as templates for the fake asteroids(and photometric calibration for real asteroids as well).To explore this, we identified 1491 stars in common be-tween the March 30 and March 31 catalogs. We foundthat on average, we had measured the same star to be0.053 magnitudes fainter on March 30 relative to March31. Three effects contribute to this surprisingly largeoffset. First, it is due in part to the difference in SDSS-derived zeropoints described in § § § Matching Detected Fake Asteroids to Input
We analyzed the fake asteroids using the same stepsas for the real asteroids, including manual screening bymeans of stamp-stacks, tiled images, check images, andpyramid images. As for the real asteroids, we createdlists of fake asteroids with classifications of ‘a’, ‘subsig’,and ‘b’. Up to this point, the fake asteroid test wasblind. Only when the classified lists were complete didwe compare them with the input asteroids. These listsare provided in Table 3, which is exactly analogous toTable 1 for the real objects.Comparing to the input catalog, we find that forMarch 30, 13 of the 23 detections classified as ‘subsig’and 3 of the 71 classified as ‘b’ actually correspondedto input asteroids. For March 31, 11 of the 29 ‘subsig’detections and 2 of the 59 ‘b’ detections correspondedto input asteroids. Thus we see that the ‘subsig’ classis much more likely than the ‘b’ class to correspond to aint Asteroids with DECam Table 3.
Detections in our Fake AsteroidAnalysisDate Category Number FoundMarch 30 ‘a’ 4424March 30 ‘subsig’ 23March 30 ‘b’ 71March 30 total 4518March 31 ‘a’ 4379March 31 ‘subsig’ 29March 31 ‘b’ 59March 31 total 4467 genuine objects, but both are full of false positives. Em-barassingly, we found that 5 of the 4424 class ‘a’ detec-tions for March 30 did not correspond to input asteroids.Examining them revealed four of them to be obviousmistakes in the form of motion-mismatched detectionsof other fake asteroids. We note that we would not ex-pect such mistakes to occur for the real asteroids, whichwere subjected to more intensive manual screening anddouble-checking. The final unmatched ‘a’ detection inMarch 30 looks plausibly genuine and is possibly a falsepositive arising from the noise distribution. However,it is almost 1 σ more significant than the most signifi-cant detection from the time-scrambled data set. Thisis especially odd since the fake asteroids were added tothe time-scrambled images, and hence the same falsepositives should exist in both analyses. Direct compar-ison of stamp-stacks of time-scrambled images withoutand with fake asteroids show the false positive source isfar more significant in the latter case: hence, somethingabout the addition of the fake asteroids seems to haveboosted its significance to values not representative ofthe actual false positive rate. There were no analogousmistakes or false positives in the March 31 fake aster-oid analysis: every one of the class ‘a’ detections wasmatched to an input fake asteroid.7.4. Survey Completeness from Fake Asteroids
We calculate our flux-dependent detection complete-ness for each night individually, using only fake asteroidsclassified as ‘a’ in Table 3, which totalled 4424 and 4379for March 30 and 31, respectively. For comparison, thenumbers of fake asteroids input on the respective nightswere 6595 and 6688, where these totals include only ob-jects input within sky area over which could detect as-teroids, as defined by the boundary map we describedin § ∼ ASTEROIDS DETECTED ON BOTH NIGHTS8.1.
Matching Asteroids from Night-to-Night
By its nature, digital tracking analysis produces accu-rate angular velocity measurements for every asteroid.We can therefore extrapolate the motion of an asteroiddetected in one night’s data to predict where it shouldbe in the data from the other night. However, a sim-pleminded extrapolation will produce predictions offsetto the west of the asteroids’ actual locations if lookingforward in time (i.e., from March 30 detections to pre-dictions for March 31), and offset to the east if we lookback in time. The reason for this systematic error is therotational reflex velocity (RRV; see Heinze & Metchev2015a; Lin et al. 2016): that is, the part of the asteroid’sangular velocity that is due neither to the orbital veloc-ity of the asteroid nor that of the Earth, but rather to theobserver’s rotation about the geocenter. This rotationalvelocity averages to zero over a full day, but throughoutthe nightime hours it maintains a positive componentin the same direction as the Earth’s orbit. Asteroids atopposition are all moving westward on the sky becausethey are in their retrograde loop as the Earth overtakesthem due to its faster orbital motion around the Sun,and the RRV term simply increases this westward angu-lar velocity. If this too-fast westward motion is used to2
Heinze et al. . . . . . . March 30 Detection Completeness from Fake Asteroids
Input 'R' band magnitude of fake asteroids F r a c t i on de t e c t ed
50% completeat mag 25.36Fake asteroid results Analytical detection model
Chi−square function . . . . . . Two−Night Detection Completeness from Fake Asteroids
Input 'R' band magnitude of fake asteroids F r a c t i on de t e c t ed
50% completeat mag 25.47Fake asteroid results Analytical detection model
Chi−square function
Figure 8.
Detection completeness for faint asteroids as a function of magnitude. The black line and points with error barsshow the histogram-based completeness from the fake asteroid simulation. Results for March 30 and for the two-night analysis( § § extrapolate forward or backward by one day, it producesthe errors described above.To extrapolate asteroid positions more accuratelyfrom one night to the next, we wish to estimate andremove the RRV contribution to our measured angularvelocity. The RRV contribution is equal to the rota-tional velocity of the observer (projected on the planeof the sky and averaged over the time span of the obser-vations) divided by the distance to each asteroid. Sinceour observations were designed so that the target fieldtransited the meridian near the center of our observ-ing sequence, a simplified calculation yields an excellentapproximation for the projected, averaged velocity. If t obs is the time interval over which the digital track-ing observations were obtained (7.30 and 7.84 hr, re-spectively, for our two nights), the angle through whichthe Earth rotates between the start and midpoint (orequivalently midpoint and end) of the observations is θ obs = 2 π ( t obs ) / (1 day), and the velocity in question isgiven by: v rot = v eq cos( θ lat ) sin( θ obs ) θ obs (9)where v eq is the equatorial rotation velocity of the Earthand θ lat is the latitude of the observatory. For our March31 observations, Equation 9 gives v rot = 1386 km/hr. It remains to estimate ∆, the asteroid’s distance fromEarth, which can be done only very approximately froma single night’s data. Fortunately, a rough approxima-tion will suffice to remove the majority of the RRV offset.If all asteroids moved in circular orbits at zero inclina-tion, their distances could be calculated exactly fromtheir angular velocities, and we base our distance esti-mates on this simplified scenario . For known asteroidsin our data, this approximation systematically underes-timates ∆ by 10%, with an RMS scatter of 14%. Thoughbetter approximations could be devised, this one provedcompletely sufficient for our purposes, enabling us topredict second-night asteroid positions with an RMS er-ror of less than 4 arcsec and a mean systematic offset of0.5 arcsec (see below).Given v rot from Equation 9 and ∆ from our circular-orbit approximation, we take the RRV contribution tothe angular velocity to be v rot / ∆, entirely in the RAdirection. This is applied as a correction to remove the Note that the distance estimate described here is not equivalentto the RRV method of Heinze & Metchev (2015a) and Lin et al.(2016). RRV distances are far more accurate and have no intrinsicsystematic bias, but they require asteroids already to be linkedover two nights. The night-to-night linkage we describe here isa prerequisite for the accurate RRV distances we will present inour companion paper (Heinze et al., in prep). aint Asteroids with DECam . § × . σ , which we obtained by solving Equation 7 with N real = 9 × . The value of N real comes from the factthat each of the ‘check images’ used to match asteroidshas an area about 9 × times larger than the ∼ × σ and were ultimately rejected (even though mostwere probably real) to ensure a negligible rate of falsepositives. For the remaining matches, all with > . σ significance, successful detection at a predicted locationin an independent data set confirms the reality of the as-teroids beyond reasonable doubt — though most of themwere already known to be real based on single-night de-tections above the ∼ σ thresholds quoted above.8.2. Discussion of Detected Two-Night Asteroids
Table 1 gives the numbers of objects in various cate-gories that were detected, either automatically or manu-ally, on a second night. In category ‘a’ (morphologicallynormal-looking detections above the significance limit)less than 2.5% of all detections failed to be matched.Out of a total of 103 candidates in category ‘b’ (detec-tions that looked morphologically peculiar under man-ual screening) only one object was matched on a secondnight: the manual screener’s impression of somethingbeing wrong with a given detection was usually accu-rate. In the ‘subsig’ category (morphologically normalbut below the single-night significance limit), slightlyunder 50% of detections were confirmed on a secondnight. These candidates are thus confirmed as real aster-oids for purposes of our two-night analysis, though theydo not count as independent single-night detections.Among the small number of confidently-detectedsingle-night asteroids that could not be manuallymatched, the cause in many cases was obvious: the as-teroid had moved out of our target field. There werea few cases where the image quality seemed good butthe ‘check image’ showed apparently empty sky or onlya very low significance ( < σ ) detection. We believethat in most of these cases, the asteroid was real andpresent in the sky area covered by the image, but hadsimply become too faint for detection due to rotationalbrightness variations. Observations of Near-Earth aster-oids — in the same size range as the main belt objectswe have detected — show that they often vary by morethan 0.7 magnitudes in brightness as they rotate (e.g.Hergenrother & Whiteley 2011). Such variations couldeasily cause an object that was detected at good signifi-cance (e.g. 8 σ ) on one night to fade below our detectionlimit on the other.We find that 2525 asteroids were automatically, inde-pendently detected on both nights; 206 asteroids wereautomatically detected on March 30 and manually re-covered in the March 31 data; and 392 asteroids wereautomatically detected on March 31 and manually re-covered in the March 30 data. Thus, a total of 3123asteroids were each detected on both nights. Table 44 Heinze et al. gives the breakdown of these numbers in terms of man-ual classifications. In addition to the 3123 two-nightasteroids, 40 and 71 detections on March 30 and 31, re-spectively, could not be recovered on a second night butwere confidently real. Hence, we have detected a totalof 3234 distinct asteroids.8.3.
Survey Completeness for Two-Night Asteroids
Since our fake asteroids were placed using self-consistent orbits from night to night, fake asteroids canbe linked from one night to the next in the same wayas real asteroids. Hence, we can determine the frac-tion of fake asteroids that were detected on both nights,and create a new completeness curve that correspondsto two-night detections. We obtain the effective magni-tude of asteroids detected on both nights, whether realor fake, by simply averaging the measured magnitudesfor each individual night. We compare the fake asteroidsdetected on both nights to the input fake asteroids that could in principle have been detected on both nights,rather than to the total number of input fake asteroids,some of which moved into or out of our field of viewfrom one night to the next. The resulting completenesscurve is shown in the right panel of Figure 8. The factthat lower-significance asteroids could be confirmed bydetection on a second night contributes to higher com-pleteness at faint magnitudes relative to our single-nightresults. ABUNDANCE AND MAGNITUDEDISTRIBUTION OF FAINT ASTEROIDSWe have described in § § . In the The only exception is our power law fits to the cumulative mag-nitude distribution, which we analyze only for the single-nightdata sets, finding almost identical results for each night.
20 21 22 23 24 25
Magnitude Histograms for Real Asteroids 'R' band magnitude N u m be r pe r . m ag b i n Raw CountsCompleteness correctedBest fit power law
Magnitude limitof good correction
March 30March 31Two−Night Analysis
Figure 9.
Histograms of apparent R magnitude for realmain-belt asteroids showing both raw counts and counts cor-rected for survey incompleteness based on the completenesscurves from the fake asteroid test (see Figure 8). The cor-rection fails for magnitudes fainter than 25.3, for reasonsdescribed in §
10. Note that the greater sensitivity enabledby confirming asteroid detections on both nights ( §
8) pro-duces significantly higher completeness at faint magnitudesfor the two-night analysis. interests of space, we illustrate certain types of analysisusing example figures showing only one or two of oursets of asteroids, but we provide numerical results forall three data sets in tabular form.9.1.
On-Sky Number Density
We begin by simply calculating the number of aster-oids per square degree, down to various magnitude lim-its. 9.1.1.
Effective Sky Area Probed
The effective sky areas probed by our observations,calculated using the boundary maps described in § aint Asteroids with DECam Table 4.
Confirmed Asteroids Detected on Both NightsBreakdown of manual classifications d March 30 March 31Description Total ‘a’ ‘subsig’ ‘b’ ‘a’ ‘subsig’ ‘b’Automatically detected on both nights 2525 2518 7 0 2520 5 0Automatically detected only on March 30 e
206 202 4 0 NA NA NAAutomatically detected only on March 31 f
392 NA NA NA 382 9 1All two-night detections g d The ‘a’, ‘subsig’, and ‘b’ classifications are explained in § § . σ and 7 . σ derived in § e All of these detections were manually recovered in the March 31 data. f All of these detections were manually recovered in the March 30 data. g This is the total count of asteroids that were each detected on both nights, regardless of whether thedetections were automatic or manual. field. The effective sky area is the field overlap averagedover the motion arcs of all the asteroids, and is equalto 2.848 square degrees. Thanks to our well-optimizedchoice of target fields, which offset the field centers fromone another by an amount that accurately matched theaverage daily motion of our target asteroids, this is only3% smaller than the area covered on March 30. The 3%reduction represents the statistical fraction of detectablemain belt asteroids that, being near the edge of the fieldin one night, would not be in the field at all on the other.9.1.2.
Culling the Lists of Real Asteroids
We exclude the Hilda asteroids and the Jupiter Tro-jans from our final lists, for several reasons. First, wewish to probe the main-belt asteroids specifically, andthe Hildas and Trojans constitute distinct populationswith larger average distance and different size-frequencydistributions (Terai et al. 2018). Second, we wish to ob-tain results directly comparable to other surveys (e.g.Gladman et al. 2009) that targeted MBAs. Finally, wewant our measurements of the on-sky density of aster-oids at opposition to be substantially independent of thetime of year (i.e. ecliptic longitude) and the position ofJupiter. This is expected if we confine our analysis toMBAs, which are distributed fairly uniformly in eclip-tic longitude, but not if we include the Jupiter Trojans(which are strongly clustered around Jupiter’s L4 andL5 Lagrange points) or the Hilda asteroids (which are significantly concentrated at ecliptic longitudes 180 and ±
60 degrees from Jupiter).We identify Hilda asteroids and Jupiter Trojans usingboundaries in angular velocity phase space indicated bygreen and red lines in Figure 5. On March 30 and 31respectively, we reject 231 and 242 Jupiter Trojans, and28 and 29 Hilda asteroids, leaving 2501 and 2702 inde-pendently detected main belt asteroids.Prior to removing the Jupiter Trojans and Hilda aster-oids from our list of 3123 two-night asteroids, we performpreliminary culling to ensure the final list can be used forstatistical analysis herein and in our companion paper(Heinze et al., in prep.) on the absolute magnitude dis-tribution. We remove 170 two-night asteroids that weremanually recovered only near the very edge of our im-ages on one of the nights: they were outside the nominalboundary of valid data described in § Heinze et al.
10% errors . Finally, we remove 241 Jupiter Trojans,and 27 Hildas, leaving 2668 two-night asteroids detectedunder conditions of well-quantified sensitivity — a listdirectly comparable to the similar one we constructedfor the two-night fake asteroids.We plot the distributions of these objects’ apparent R band magnitudes in Figure 9.9.1.3. Calculated On-Sky Number Densities
Interesting magnitude thresholds for our sky den-sity calculation include the value of 23.0 for whichGladman et al. (2009) found 210 asteroids per squaredegree; the value of 24.4 for which Yoshida et al. (2003)found ∼
290 while Gladman et al. (2009) extrapolatedtheir power law to estimate ∼ R mag 24.9. At mag 25.0 it is still 96.4%and 94.2% for March 30 and 31 respectively, and 99.3%for the two-night analysis. Hence, we will undercountthe true numbers of asteroids only slightly if we sim-ply sum our detected asteroids out to the respectivethresholds. Table 5 presents raw counts as well ascompleteness-corrected values, and the former are likelyto be as accurate as the latter out to magnitude 25.0.Fainter than 25.0, the incompleteness becomes signifi-cant and only the corrected numbers should be regardedas meaningful measurements. It is important to notethat all of these on-sky number densities apply only toasteroids at opposition on the ecliptic (i.e., near the an-tisolar point). If we move away from the antisolar pointbut stay on the ecliptic, the sky density would be ex-pected to drop quickly due to the rapid falloff in aster-oid brightness with increasing phase angle. Moving offthe ecliptic, the density should drop even faster becausethere are fewer asteriods in highly inclined orbits (e.g.Terai et al. 2007, 2013).Table 5 shows that our results are mutually consis-tent from night to night, and are in excellent agree-ment with those of Gladman et al. (2009) at R =23.0. At magnitude 24.4, our March 31 measurement This cull is needed because the current analysis will be founda-tional for our companion paper on absolute magnitudes. Manyof the culled objects may also have been Jupiter Trojans, whoseRRV signals are hard to measure due to their distance. Thecomparison with fake asteroids is not affected because none ofthe latter had such large distance uncertainties. of 463 ±
12 asteroids per square degree agrees muchmore closely with the extrapolated estimate of ∼ σ (except for R = 25 . . ◦ away from the antisolarpoint would be expected to produce a small but signifi-cant reduction in the brightness of the asteroids. Sincethe March 31 observations were centered almost exactlyon the antisolar point, they should give the most accu-rate measurement of the on-sky density of faint asteroidsat opposition. On this date we find 870 ±
18 asteroidsper square degree brighter than the R = 25 . Power Laws to Fit Asteroid Distributions
Published results on the magnitude and size distribu-tions of faint asteroids have typically been presented interms of a cumulative power law distribution over eithermagnitude or size. In terms of diameter D , the powerlaw is given by Equation 10 (e.g. Wiegert et al. 2007): N ( > D ) ∝ D − b c (10)In the approximation that the distribution of asteroidalbedos is independent of diameter D , it follows fromEquation 10 that the cumulative distribution of aster-oid absolute magnitudes is given by Equation 11 (e.g.Gladman et al. 2009): N ( < M ) ∝ α c M (11)where M is the absolute magnitude, and the power lawslopes b c and α c are related by b c = 5 α c .Equations 10 and 11 describe the cumulative distribu-tion. We can also explore the corresponding differentialpower laws: dNdD ∝ D − b d (12) dNdM ∝ α d M (13) aint Asteroids with DECam Table 5.
On-Sky Number Density of Asteroids
Raw Counts Completeness Corrected
Magnitude Total Asteroids Total AsteroidsDate threshold detected per deg detected per deg March 30 23.0 603 206 ± ± ± ± ± ± a · · · ± · · · ± ±
12 1326 452 ± ±
12 1384 463 ± ±
13 1296 455 ± a · · · ± · · · ± ±
15 1963 670 ± ±
15 2084 697 ± ±
15 1917 673 ± a · · · ± · · · ± ±
17 2431 829 ± ±
17 2599 870 ± ±
17 2398 842 ± a · · · ± · · · ± ±
17 2810 958 ± ±
17 3022 1011 ± ±
18 2883 1012 ± a Unweighted numerical average over the three data sets. Uncertainty does notgo down under the average since the individual uncertainties are mostly dueto Poisson noise of counting a heavily-overlapping set of asteroids. Averages oftotal counts are not meaningful since the effective sky areas are not the same.At mag 25.6 the completeness correction fails and all values are underestimates,so the average would not be interesting.
If the slope b d is constant over a large range in size,the cumulative and differential size slopes are related by b c = b d −
1. In this case, the differential and cumula-tive magnitude slopes are the same, α c = α d , due to theexponential form of Equations 11 and 13. If the sameslopes described all main belt asteroids, these magnitudeequations would refer equally to apparent magnitudes(e.g. R ) and to absolute magnitude H , and the slope ofthe diameter power law ( b d or b c ) could be calculatedfrom the apparent magnitude distribution without am-biguity or error.In fact, abundant evidence exists (e.g. Wiegert et al.2007; Yoshida & Nakamura 2007; Gladman et al. 2009)that the power law slopes are not constant. Hence, thecumulative slopes b c and α c measured over a given rangeof size or magnitude constitute a type of weighted av-erage over changing differential slopes b d and α d for allthe asteroids that contribute to the distribution — i.e., not only those in the range being fit, but also objectslarger/brighter that nevertheless add in to the cumu-lative total. This blending is a disadvantage of usingthe cumulative distributions. An advantage is the muchlarger number of asteroids per bin, and consequentlysmaller statistical errors. With over 2500 asteroids, wehave a large enough sample to produce meaningful fitsto the differential distributions, which we do in § § R magni-tude distribution, reserving our fits to the distributionsof absolute magnitude H to a companion paper (Heinzeet al., in prep.), in which we also describe how we ob-tained ∼ .
5% accurate distances to the asteroids we de-tected on both nights. Since we are fitting the apparentmagnitude distribution herein, the slopes we derive area weighted average over the absolute magnitude distri-8
Heinze et al. butions at various size ranges and distances in the mainbelt. For example, the range R = 24 −
25 correspondsto H = 22 . − . −
100 m for 15%albedo) near the inner edge of the main belt at 1.0 AUfrom the Earth, but a range of H = 19 . − . −
430 m) 2.4 AU from Earth in the outer mainbelt. This average nature of our results must be keptin mind when interpreting their implications for the sizedistributions.9.3.
Cumulative Power Laws
Following Gladman et al. (2009), we fit the cumula-tive distribution of R magnitudes for our real asteroidsusing Equation 11. This distribution is shown in Figure10. We use weighted least-squares to fit a line to thequantity log ( n ), where n is the number of detected as-teroids. We calculate the uncertainty on the logarithmusing Equation 14: σ log ( n ) = σ n n ln(10) (14)Where we use σ n = √ n for magnitudes brighter than R = 24 .
9, but include the uncertainty on the complete-ness correction for fainter magnitudes.Consistent with Gladman et al. (2009), we find a dra-matic transition in the slope α c at a relatively brightmagnitude between R = 19 and R = 20. Brighterthan R = 19, we find very steep values of α c =0 . ± .
02 and 0 . ± .
04 for March 30 and 31, re-spectively. In this regime, Gladman et al. (2009) found α c = 0 . ± .
16 and 0 . ± .
15 on two successivenights. Since both studies are plagued by small num-ber statistics for these relatively rare, bright asteroids,the fact that we find steeper slopes than Gladman et al.(2009) may not be significant. The change in the slope ofthe apparent magnitude distribution that both we andGladman et al. (2009) identify near R = 19 mag proba-bly corresponds to the dramatic change in the size dis-tribution that Yoshida et al. (2019) detect at a diameterof about 6km in an analysis of completely independentdata that includes space-based infrared measurements.More interesting for our current analysis is the fainterregime. From R = 20 . α c = 0 . ± .
014 and 0 . ± .
014 on two nights.Our values in the same regime are in full agreement: wefind 0 . ± .
002 and 0 . ± .
002 for March 30 and31. Translating to b c (subject to the caveats about av-eraging noted in § b c = 1 . ± .
01. Forcomparison, using observations in a similar magnituderange Wiegert et al. (2007) found b c = 1 . ± .
02 in the g ′ band but a steeper slope of b c = 1 . ± .
08 in r ′ . Wewould have expected their r ′ result to be more compa-rable to our own observations in R ; note, however, that they calculated approximate distances and sizes for alltheir asteroids, and hence attempted to fit the actual sizedistribution rather than merely the apparent magnitudedistribution as we have done. They also observed far-ther from opposition than we or Gladman et al. (2009),so if there is a systematic difference in phase functionsbetween large and small asteroids, this could explain thedifference in measured power law slopes.Extending our fit of the cumulative distribution from R = 20 . R = 25 . α c = 0 . ± .
003 and 0 . ± .
002 for March30 and 31. Since our weighted fits emphasize the faintbins containing the largest numbers of asteroids, thefact that these slopes agree with Gladman et al. (2009)down to two magnitudes fainter indicates that thereis no dropoff in the abundance of small asteroids atmagnitudes fainter than the R ∼ . R ∼ . α c to the size slope (again, subject to manycaveats), we find β c = 1 . ± .
01, which disagrees withthe value of ∼ . β c = 1 . ± .
02 foundby Yoshida & Nakamura (2007) based on observationsthat were sensitive down to R ∼ . R = 20 . R = 19, but the factthat the difference plot trends upward at the faintestmagnitudes suggests a second transition, this time toa steeper slope for the faintest asteroids. This couldbe the first detection of increasing abundance of smallMBAs below the strength/gravity transition, as pre-dicted by Bottke et al. (2005b) and de El´ıa & Brunini(2007). We use the higher resolution enabled by the dif-ferential rather than cumulative distributions to explorethis possibility further in § Differential Power Laws
The cumulative distribution of our asteroid magni-tudes (Figure 10) shows systematic deviations from thebest-fit power law. To explore this further, we fit differ-ential, rather than cumulative, power laws to the magni-tude histograms. The differential distributions are nois-ier than the cumulative ones due to small number statis-tics, but they have the advantage that successive binsare independent. Where sufficiently large numbers of aint Asteroids with DECam
16 17 18 19 20 21 22 23 24 25 − + + + + Cumulative Distribution of Asteroids' Apparent Magnitudes
Apparent R−band Magnitude C u m u l a t i v e N u m be r o f A s t e r o i d s S l ope = . S l ope = . S l o p e = . S l o p e = . March 30March 31
Difference from Power Law Fit (logarithmic, vertically offset and rescaled)
March 30March 31 −10%0.0+10%
Figure 10.
Cumulative distribution of asteroids based on the corrected histograms shown in Figure 9. Our sensitivity extendsabout one magnitude fainter than any previous survey. As discussed in § § Heinze et al. objects have been measured, the differential distributionoffers much better resolution for determining transitionpoints in the power law slope: by contrast, in the cu-mulative distribution the effect of such a transition isspread over a wide range (e.g. the slope transition atmagnitude 19 affects the cumulative distribution out toat least magnitude 21).We can fit the differential distribution of our aster-oid magnitudes (i.e. the histograms of Figure 9) from R = 20 . α d = 0 . R = 20 to 25.3, exploringslope breakpoints in a wide range from R = 21 . R = 24 . R = 23 .
5, which yieldedthe lowest χ values. Results for March 31 and the two-night analysis, given in Table 6, are very similar.We explored the robustness of the broken power lawby re-binning the completeness-corrected histograms inlarger bins of width 0.2, 0.3, and 0.4 mag and fittingboth constant and broken power laws to each re-binnedhistogram. In all twelve cases (four bin sizes for threedifferent data sets), the best fit was obtained for a powerlaw break point at either R = 23 . R = 23 . R = 21 . R = 23 .
5, we have adoptedthis value in all cases for the fits presented in Figure12 and Table 6. The best-fit slopes are also very con-sistent for different bin sizes. All but one of the con-stant power law fits are rejected with >
95% confidence (much greater in most cases), while all of the brokenpower law fits are accepted by the same criterion. Hence,this analysis strongly favors the broken power law. Italso appears to validate the predictions of Bottke et al.(2005b), de El´ıa & Brunini (2007), and others of an up-turn in the size-frequency distribution of MBAs in theregime of nonzero tensile strength. We will present afuller comparison of observations with theory, includingan estimate of the actual size at which the transition oc-curs, in our companion paper on the absolute magnitudedistribution (Heinze et al., in prep.).We obtain our final measurements of the slopes andtheir uncertainties by taking averages and standard de-viations of the values in Table 6 for each data set. Theseaveraged values are given in the ‘Log of Corrected His-togram’ rows in Table 7. In every case the fitted slopesare consistent across the data sets. Averaging across alldata sets, we find α d = 0 . ± .
005 for the constantpower law and α d = 0 . ± .
003 and 0 . ± .
008 re-spectively for the bright and faint regimes of the brokenpower law.The best-fit slopes in the brighter magnitude regime ofthe broken power law are remarkably shallow. Formally, α d = 0 .
203 implies a differential size slope b d = 2 . b c = 1 . R = 20 or fainter than R = 23 .
5. Hence,it cannot be seen in the cumulative but only in the dif-ferential distribution. In fact, by integrating our best-fit broken power law for the differential distribution, wehave determined that it is entirely consistent with theslope α c ∼ .
27 found for the cumulative distribution byboth us and Gladman et al. (2009) in the same magni-tude regime.The implications of our very shallow slope measure-ment remain surprising. Since the apparent magnitudedistribution is effectively a weighted average of abso-lute magnitude (and hence size) distributions over themain belt, our measurement of α d ∼ .
203 can only beexplained if the absolute magnitude distribution has aslope at least this shallow somewhere in the main belt.This in turn requires that the differential size slope re-ally is as shallow as b d ∼ . ANALYSIS AND FITS FOR THE FAINTESTASTEROIDSSince our fake asteroid simulation indicated a de-tection rate above 10% out to R mag 25.6 (see Fig- aint Asteroids with DECam
20 21 22 23 24 25 − . . . . . . . March 30 Fixed Power Law
Measured 'R' band magnitude
Log ( A s t e r o i d s pe r . m ag b i n ) Corrected Data and FitDifference Plot S l o p e = . Fit c = with degrees of freedom
50% completenesslimit at mag 25.36
20 21 22 23 24 25 − . . . . . . . March 30 Broken Power Law
Measured 'R' band magnitude
Log ( A s t e r o i d s pe r . m ag b i n ) Corrected Data and FitDifference Plot S l o p e = . S l o p e = . break atmag 23.5 Fit c = with degrees of freedom
50% completenesslimit at mag 25.36
Figure 11.
Differential power law fits to the magnitude histogram of asteroids detected on March 30, showing that the datafrom R = 20 . Left:
Constant power law.Curvature in the difference plot is evident to the eye and manifests itself through the large χ value. The probability of getting χ ≥ . × − . Right:
Broken power law with a slope transition at R = 23 .
5. The differenceplot is much flatter, and a χ value this large occurs with 10% probability.
20 21 22 23 24 25 − Logarithmic Difference Plots for Constant Power Laws
Apparent 'R' Magnitude
Log ( A s t e r o i d s pe r B i n ) M i nu s F i t March 30 DataMarch 31 Data c = with d.o.f.0.3 mag bins, c = with d.o.f.0.4 mag bins, c = with d.o.f.0.2 mag bins, c = with d.o.f.0.3 mag bins, c = with d.o.f.0.4 mag bins, c = with d.o.f.
20 21 22 23 24 25 − Logarithmic Difference Plots for Broken Power Laws
Apparent 'R' Magnitude
Log ( A s t e r o i d s pe r B i n ) M i nu s F i t March 30 DataMarch 31 Data c = with d.o.f.0.3 mag bins, c = with d.o.f.0.4 mag bins, c = with d.o.f.0.2 mag bins, c = with d.o.f.0.3 mag bins, c = with d.o.f.0.4 mag bins, c = with d.o.f. Figure 12.
Logarithmic difference plots for constant and broken power laws with asteroids binned in different ways, showingthat the constant power laws consistently fail to yield a satisfactory fit while the broken power laws consistently succeed.
Left:
Constant power laws. The probabilities of getting the χ values as high as those listed from a true χ distribution with thecorrect number of degrees of freedom range from 5 × − to 4 × − : they are all excluded with greater than 99.5% confidence. Right:
Broken power laws with the break at R = 23 .
5. The probabilities of these χ values range from 14% to 82%: the fits areexcellent and none would be rejected even at a 90% confidence level. Heinze et al.
Table 6.
Differential Power Law Fits to the Magnitude Histogram
Constant Power Law Broken Power Law
Date Bin Size Slope a χ d.o.f Prob d Slope1 b Slope2 c χ d.o.f Prob d March 30 0.1 mag 0.2637 96.5 52 2 × − × − × − × − × − × − × − e e × − e × − e × − aα d from Equation 13. Applies from R = 20 . b α d from Equation 13 for R = 20 . c α d from Equation 13 for R = 23 . d Probability that a true χ distribution with the same number of degrees of freedom would produce avalue greater than or equal to the χ of the fit. All but one of the constant power laws are excludedwith at least 95% confidence (usually much more), and all of the broken power laws are accepted bythe same criterion. e Two-night refers to our analysis that considered only asteroids that were each detected on both nights aint Asteroids with DECam R = 23 .
5, and sug-gested the cause was the different handling of fake vs.real asteroids. In their analysis, real asteroids were de-tected by automated software but additionally screenedby a human and rejected if the detections seemed tobe dubious (i.e., had very low significance); while fakeasteroids were not subjected to the same human screen-ing. Hence, they suggested that very faint real asteroidswould probably be discarded while fake asteroids at thesame magnitude might be flagged as detected. This ex-planation does not appear viable for our data, since wesubjected both fake and real asteroids to essentially thesame manual screening, as well as identical significancethresholds for detection (see § that affects real astronomical objects whenbrightnesses are measured near the detection thresh-old of a flux-limited survey. The bias arises from thefact that all measured fluxes are affected by randomnoise, and that objects fainter than the 50% complete-ness limit are likely to be detected only if their fluxes areboosted by a large positive realization of the randomnoise. Therefore, the faintest objects will be detectedonly under circumstances that will also cause their mea-sured fluxes to be offset brightward of the true values.It follows that our fake asteroid simulation correctlyindicates our detection completeness as a function oftrue magnitude, but does not account for the fact thatthe faintest real objects, if detected at all, will be mea-sured to be significantly brighter than they really are.Hence, for the faintest asteroids, the completeness cor-rection is not an ‘apples to apples’ comparison. Un-avoidably, it compares input, fake-asteroid magnitudesthat are free of flux overestimation bias with measuredmagnitudes of real asteroids that do experience the bias.This is why the corrected counts of real asteroids fallfar below the power law fit at magnitudes slightly fainterthan 25.3. The completeness test accurately indicatesthat more than 10% of real asteroids are detected evenat magnitude 25.6, but does not account for the fact thatthese objects are measured as being brighter than they https://old.ipac.caltech.edu/2mass/releases/allsky/doc/sec5 3a.htmlprovides a further interesting discussion of flux overestimation bias. really are and hence do not populate the 25.6 magnitudebin in the histogram of real asteroids. It is interesting tospeculate that the similar effect seen by Gladman et al.(2009), which they attribute to the imperfect realism oftheir fake asteroid test, may in fact represent this sameunavoidable bias.10.1. Modeling and Correcting Flux OverestimationBias
We wish to correct the bias in our measured mag-nitudes in order to probe the statistics of the faintestasteroids. We do this by adopting a simple but physi-cally motivated three-parameter analytical model of thedetection process, and using statistical simulations tosolve for the model parameters that best describe theresults of the fake asteroid simulation.The dominant source of noise in our flux measure-ments of faint objects is the Poisson shot noise of the skybackground. Even for a very dark sky with a brightnessof 22 magnitudes per square arcsecond, the sky con-tributes 50 times more photons within a photometricaperture of radius 1 arcsecond than does a 25th magni-tude star or asteroid. Thanks to this huge number of skyphotons, the Poisson noise of the sky background can bewell approximated by a Gaussian random variable.In our model, we therefore allow the true flux of an as-teroid to be modified by random sky noise with a Gaus-sian distribution: the standard deviation σ sky of thisdistribution is the first of the model’s three parameters.Presumably, σ sky will be related to the square root of thetypical number of sky photons detected over the angularsize of the PSF. The asteroid is detected if its modifiedbrightness exceeds σ sky by a factor N thresh , which is thesecond parameter of our simulation and is expected tobe in the range 7–8, since it should correspond approx-imately to the detection thresholds of 7 . σ and 7 . σ derived in §
6. However, we do not expect the likeli-hood of detection to drop discontinuously from 100% to0% when the modified flux drops below N thresh σ sky . Itshould instead exhibit a gradual decrease in detectionprobablility — albeit less gradual than the histogram-based completeness curves shown in Figure 8, since theyexhibit additional broadening due to the sky noise.We invested considerable thought in choosing a one-parameter mathematical form for this gradual sensitiv-ity decrease, and we finally adopted a chi-square distri-bution. We choose the chi-square distribution becauseour detection process ( § S is the sample standard deviation for n dof Heinze et al. − . − . . . . Fake Asteroid Photometry for March 31
Input 'R' band magnitude M ea s u r ed − i npu t b r i gh t ne ss ( m ag ) ●●● ●● ●● ●● ●● ●● ● ● ●● ●● ●●●● ●● ●● ●● ●● ●●● ●●●●● ●●● ●●● ●●● ●● ●● ●● ●● ● ● ●●●● ●●● ●●●●● ●● ●●● ● ●●● ● ● ● ●●● ●● ●● ● ●●● ●● ●●●● ●● ●●● ●●● ● ●● ● ●● ●●● ●●● ● ●●●● ●●● ● ●●●●● ●● ●●● ●● ●●● ● ● ● ●●● ●● ● ● ●● ● ●● ● ●● ●●● ●●● ● ● ●● ●● ●●● ●● ● ●● ● ● ●●● ●●● ●● ●●●● ●● ● ●●● ●● ● ● ●● ●●● ●●● ●● ● ●●● ●● ● ●● ●●● ● ●● ● ●● ●● ●● ●●●● ●● ●● ●● ● ●●●● ●●● ● ●●● ● ● ●●● ● ●●● ●● ●●●●●● ● ●●●●●●● ● ●●● ●●● ●●● ● ●●● ●●● ● ● ●●● ●● ●● ●●● ● ● ●● ●●● ●●● ●●●● ●● ●● ●● ●● ●●● ●●● ● ●● ●● ●●●● ●● ●● ●● ● ●● ●●●● ●● ●●● ● ●●●● ●●● ●●● ●●●●● ●● ●● ●● ●● ●● ●● ●●● ●● ● ●●● ●● ●●● ● ●●● ●● ● ●● ●● ●● ● ●●● ● ●● ●●●● ●●● ● ●● ● ●● ●● ● ●●●● ●● ●● ●● ●●●● ● ● ●●● ●● ●●●●● ●● ●● ●● ●● ● ●● ●● ●● ●●● ●● ●● ●●●●●●● ● ● ●● ●●●● ● ● ●● ●●● ●● ● ●● ●●● ● ●●● ● ●● ● ●● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ●● ● ●●● ● ●● ● ● ●●● ●●● ●● ●● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ●●● ● ●● ●●● ●● ●● ●● ●● ● ●● ●● ●● ●● ●●● ●●●●● ● ●●● 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Two−Night Fake Asteroid Photometry
Input 'R' band magnitude M ea s u r ed − i npu t b r i gh t ne ss ( m ag ) ●●●● ●●● ●●● ●●● ●●●●●●● ●●●● ● ●● ●● ● ●● ●● ●● ●● ●●● ● ●●●● ●●●● ● ●●●●● ● ● ●● ● ●●●● ● ●●● ● ● ●●●●● ●●● ●● ●●● ●●●● ● ●●● ● ●● ● ● ●●● ● ●●●● ●● ●●● ● ●● ● ●● ●●● ●●● ● ●●●● ● ●●●●● ●●● ● ●● ●● ●●● ●● ● ●●●● ●●●●●● ●● ●●●●● ●● ● ●●● ●●●●● ●● ● ●●●● ●●● ● ●● ●●●● ●●● ●●●● ● ● ●● ● ●● ●●● ●● ●●● ●● ●● ●● ●●● ● ● ●● ●●● ● ●● ●● ●●● ●● ● ●●●●● ●● ●●● ●● ●● ●●● ●● ● ●● ●●● ●●● ● ●●●● ●● ●● ●●● ●● ●● ● ● ●●●●● ●●● ●● ● ● ●●●● ● ●●● ●● ● ●● ●●●● ●● ●● ●● ●●● ● ●●● ●●● ●●● ●● ●●● ●●● ●● ● ●●● ●● ●● ● ●●●●● ●●●● ●●● ●● ● ● ●● ● ●● ●● ●● ● ●● ●● ● ●●● ● ●● ●●● ● ●● ●● ●● ●● ● ●● ●●●● ● ● ●●●●● ●●● ●●●● ● ● ●●● ●●● ●● ●●● ●●● ●●● ●● ●● ●● ● ●● ● ●●●● ●●● ●●● ● ●●● ● ●● ●● ●●● ● ●● ●● ●●● ● ● ●●●● ●●● ●● ● ● ●● ● ●● ●● ●●● ● ●●●● ● ●●● ●● ●●● ● ●● ● ●● ●●● ●●●● ●● ● ● ●● ●● ●●● ● ●● ●●● ●● ●● ●●● ● ●● ● ●● ● ●●● ● ●● ●●● ●●● ● ●● ● ● ●● ● ● ●●●● ● ●●●● ● ● ●● ● ● ●● ●●●● ●● ●● ●● ● ●● ●● ●● ●● ●●● ● ●● ● ●● ●●● ● ●● ● 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Figure 13.
Flux overestimation bias for measured magnitudes in our fake asteroid simulation. This is not a flaw of thesimulation: instead, it reveals a measurement bias that affects the real asteroids too. A very faint asteroid will be detected onlyif it falls on a positive fluctuation of the sky noise, and hence its measured magnitude will be too bright. Gray lines indicatemagnitude differences of 0.0 and ± .
05 mag. The green curves show the models of this effect that we describe in § samples drawn from a Gaussian distribution with truestandard deviation σ , the quantity ( n dof − S /σ isdistributed according to a chi-square distribution with n dof − n dof becomes thethird and final parameter in our analytical model of thedetection process.Given an input ‘true’ flux for a faint asteroid, the de-tection process is modeled as follows. First, the trueflux is modified by the addition of a Gaussian randomvariable with mean zero and standard deviation σ sky .Then, the detection threshold N thresh σ sky is multipliedby a scale factor equal to p X chi / ( n dof − X chi is a random variable drawn from a chi-square distribu-tion with n dof − σ sky , N thresh , and n dof ) of our analytical detection model, we first attempta statistical fit to the results from the fake asteroid sim-ulation. We perform a statistical simulation aimed atmatching both the histogram-based completeness curvesplotted in Figure 8, and the mean offsets between mea-sured and input magnitudes which we have plotted inFigure 13. Judging the completeness to be near 100%at magnitudes brighter than 24.9, and so low at magni- tudes fainter than 25.6 that the fake asteroid results areunreliable, we attempt to fit the results only in the mag-nitude range 24.9–25.6. This range includes eight mag-nitude bins, in each of which we fit two data points (com-pleteness and mean magnitude), so the three-parameterfit is heavily overconstrained. For the two-night datawith its greater sensitivity, we fit 9 bins from magnitude24.9–25.7.We perform the fitting using a statistical simulation inwhich the actual input magnitudes of the ∼ aint Asteroids with DECam § σ sky converted to an equivalent magni-tude. The analytical model fits to the detection ratesand mean magnitudes are plotted as green curves in Fig-ures 13 and 8, respectively. The blue curves in Figure8 also show our chi-square model of the detection rateas function of modified magnitude. The broader dropoffof the actual detection rate relative to this chi-squaremodel is due to the sky noise (parametrized in our modelby σ sky ), which is the actual cause of the flux overesti-mation bias. The best-fit values obtained for σ sky and N thresh are near the range we would expect, and leadto limiting magnitudes very close to those we derivedfrom the histogram-based completeness curve. The bestfit values of n dof are much smaller than the number ofsamples used to probe the sky noise ( ∼ § n dof , the smallbest-fit values of n dof indicate that additional effectsbesides the variation in the sample standard deviationcontribute to the gradual dropoff of detection efficiency,which is hardly surprising. It is probably best to thinkof n dof simply as parameterizing the gradual sensitivitydecrease, and not as the number of degrees of freedomof in a literal chi-square distribution that is realized atany stage of our actual detection process. This does notdetract from the value of our analytical model as a use-ful (though necessarily simplified) representation of ourdetection process.10.2. Apparent Magnitude Distribution from theAnalytical Detection Model
We use the analytical detection models derived fromour fake asteroid simulations to model the detection ofreal asteroids assuming that the true distribution is a power law (or broken power law) of the form given inEquation 13. We attempt to match the histogram of realasteroids in 57 bins of 0.1 magnitude width covering theinterval from magnitude 20.0 to 25.6. Since positive skynoise fluctuations can move faint asteroids into brighterbins on the magnitude histogram, we simulate an in-put power law distribution that extends all the way tomagnitude 26.0. We perform fits using both a constantpower law and a broken power law with a break pointat R = 23 .
5. For the constant power law, we solve forthe single slope value α d by simply probing a finely sam-pled range of values and selecting the one that producedthe smallest χ value for the fit. For the broken powerlaw, we solve for the values of α d in the two magnituderegimes by a 2D grid search. As in the analytical de-tection model applied to the fake asteroids, hundreds oftimes more random asteroid clones are generated thanare actually detected in the real data, in order to en-sure the statistical noise from the simulation makes nosignificant contribution to the final uncertainty.By construction, the simulation explicitly models bothincompleteness and photometric bias to predict the rawmagnitude histogram that should be observed given aninput power law. Hence, while the much simpler anal-ysis of § § n in each bin, with uncertaintygiven by σ n = √ n .Figure 14 shows the best broken power law fits ob-tained using our analytical model, demonstrating thatour model has achieved its main objective: the faintestdata points are no longer statistical outliers, indicatingthat we have successfully modeled the phenomena thatlead to very low numbers of asteroids being measuredin these bins. The model therefore represents a sig-nificant improvement over the simple, histogram-basedcompleteness correction presented in § α d = 0 . ± . . ± . . ± .
020 for March 30, 31, and the two-nightdata respectively. The new values are consistent withbut slightly steeper than the average value of α d =6 Heinze et al.
Table 7.
Slope α d for Best-Fit Power LawsConstant Power Law Broken Power LawData Set Method R >
20 mag R = 20 to 23.5 mag R > . a . ± .
004 0 . ± .
004 0 . ± . b . ± .
013 0 . ± .
023 0 . ± . . ± .
010 0 . ± .
018 0 . ± . a . ± .
002 0 . ± .
002 0 . ± . b . ± .
018 0 . ± .
028 0 . ± . . ± .
013 0 . ± .
024 0 . ± . a . ± .
005 0 . ± .
003 0 . ± . b . ± .
020 0 . ± .
035 0 . ± . . ± .
020 0 . ± .
026 0 . ± . . ± .
015 0 . ± .
026 0 . ± . a Uncertainties obtained for this method are probably unrealistically small. b This model, which corrects for flux overestimation bias, is described in § Table 8.
Analytical Detection ModelsDate σ sky N thresh n dof phot offset 50% completeness limitMarch 30 27.620 mag 8.040 29 0.0073 mag 25.357 magMarch 31 27.470 mag 6.825 25 0.0309 mag 25.385 magBoth Nights 27.880 mag 9.200 49 0.0376 mag 25.471 mag . ± .
005 obtained for constant power law fits tothe corrected histograms in § χ values of thenew constant power law fits are 84.9, 64.2, and 58.4 forMarch 30, 31, and the two-night data respectively, with55 degrees of freedom for the single-night data and 56for the two-night fits, which reach 0.1 mag fainter. Theprobabilities of getting χ values at least this high withthese numbers of degrees of freedom are 6 × − , 19%,and 39%, respectively. Hence, in contrast to the simpler,logarithmic fits of § R = 20 . α d = 0 . ± . . ± . . ± .
035 for March 30, 31,and the two-night data respectively, while in the faintregime ( R = 23 . α d =0 . ± . . ± . . ± .
033 for therespective data sets. Interestingly, the slopes in bothregimes have become less extreme than for the simplerfits in § χ = 71 .
9, 50.7, and 41.6 for the re-spective data sets, where the single-night fits have 53 de-grees of freedom and the two-night fit has 54. The prob-abilities of getting these values are 4.3%, 56%, and 89%respectively. Hence, while the other two fits are excel-lent, the one for March 30 remains somewhat marginaleven for a broken power law (though greatly improvedrelative to the constant power law).Although the evidence for the broken power law doesnot seem as strong under the new type of fit, it clearlyremains a better description of the data than a con-stant power law. The evidence for a break in the powerlaw appears still more compelling when we consider theupturn in the cumulative distribution from Figure 10,which is hard to explain any other way and which hap-pens consistently on both nights at a magnitude consid-erably brighter than the sensitivity limit. Finally, thefact that the new fits produce less extreme slopes in thebright regime makes the broken power law seem morebelievable. As mentioned in § α d ∼ .
203 found by our earlier analysis seems some-what improbable a priori. aint Asteroids with DECam
20 21 22 23 24 25
Analytical Detection Model for Two−Night Asteroids
Measured 'R' magnitude R ea l a s t e r o i d s pe r . m ag b i n Raw CountsAnalytical modelBest fit broken power law
20 21 22 23 24 25 − − Success of Analytical Model shown by Logarithmic Difference Plots
Measured 'R' band magnitude
Log ( A s t e r o i d s pe r b i n ) , D a t a m i nu s F i t March 30March 31Two−night
Histogram−based correctionHistogram−based correctionHistogram−based correctionAnalytical modelAnalytical modelAnalytical model
Figure 14.
Success of our analytical detection model ( § § Left:
The same magnitude histogram of two-nightasteroids as shown in Figure 9, this time fit with the analytical detection model. The single-night results (not shown) lookvery similar.
Right:
Logarithmic difference plots of the best-fit broken power laws using the old histogram-based completenesscorrection (e.g. Figure 11) compared to those using the new analyical detection model. With the new model, the faintest pointsare no longer statistical outliers: instead, the model enables us to probe the abundance of these extremely faint asteroids.
In contrast to the steeper bright-end slopes, the faint-end slopes are considerably shallower under the newanalysis. Concerned that this might indicate a bug inthe analytical model (or a real dearth of extremely faintasteroids), we probed both possibilities. Constrainingthe fit to R ≤ . √ n . Both methodsare approximations because they implictly assume sym-metrical Gaussian uncertainties, when the number n ofasteroids per bin would actually be expected to have aPoisson distribution. The approximations are reason-able and difficult to avoid, but their differing imperfec-tions likely explain the different slope and χ values ofthe fits in this section as compared to those in § R = 25 . § ±
15 asteroids persquare degree for
R < . ±
18 as-teroids per square degree in this magnitude range, andhence the total sky density of asteroids brighter than R = 25 . ±
23 per square degree.
CONCLUSIONWe have used the technique of digital tracking to lever-age the remarkable wide-field capability of DECam onthe 4m Blanco telescope and perform the deepest eversurvey of main belt asteroids. In a single DECam field(about 3 square degrees), we have detected 3234 distinctasteroids, of which 3123 are confirmed on two consecu-tive nights.8
Heinze et al.
20 21 22 23 24 25 − − Logarithmic Difference Plots of Constant vs. Broken Power Laws
Measured 'R' band magnitude
Log ( A s t e r o i d s pe r b i n ) , D a t a m i nu s F i t March 30March 31Two−night
Constant power lawConstant power lawConstant Power LawBroken power lawBroken power lawBroken power law
Fit c = with d.o.f.Fit c = with d.o.f.Fit c = with d.o.f.Fit c = with d.o.f.Fit c = with d.o.f.Fit c = with d.o.f. Figure 15.
Logarithmic difference plots using our analyticaldetection model ( § § χ values as large as those observedare 6 × − , 19%, and 39% for constant power law fits to therespective data sets, and 4.3%, 56%, and 89% for the brokenpower laws. Formally, one out of three fits is unacceptable atthe 95% confidence level in each case, so the broken powerlaw does not exhibit the same level of dominance here asin our earlier analysis (e.g. Figure 12). Nevertheless, thebroken power law is preferred in every case. We have analyzed our detection rate as a function ofmagnitude using a carefully constructed fake asteroidsimulation, which allows us to correct for incompletenessin our asteroid counts out to R magnitude 25.3. Fainterthan this, our completeness correction falters due to fluxoverestimation bias. This bias arises because the faintestobjects are detected only if their flux is augmented bya positive realization of random noise (mostly Poissonnoise from the sky backround in our case). Hence, whendetected at all, the faintest asteroids are measured assystematically brighter than they really are. Building onour fake asteroid simulation, we construct an analyticalmodel of our detection process that fits and corrects theflux overestimation bias, enabling our statistical analysisof asteroids down to R magnitude 25.6.We find a sky density of 697 ±
15 asteroids per squaredegree brighter than R = 25 . ±
23 brighter than R = 25 . R = 19than it is at magnitudes fainter than R = 20. From R = 20 to the faint limit of our survey at R ∼ . α d ∼ . ± . R = 23 .
5. This implies that ex-tremely faint asteroids are more abundant than extrap-olating power laws fit to brighter objects would lead usto expect — contrary to some reports claiming a reduc-tion in asteroid abundance at magnitudes fainter than23 .
5. The average best fit slopes we find for this brokenpower law are α d = 0 . ± .
026 for the bright regime( R = 20 to 23.5 mag), and α d = 0 . ± .
025 for thefaint regime ( R = 23 . § ACKNOWLEDGMENTSWe acknowledge essential support and assistancefrom John Tonry and Larry Denneau of the Asteroid aint Asteroids with DECam
Facility:
4m BlancoREFERENCES