Icarus210(2010)158–181 ContentslistsavailableatScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus Treatment of star catalog biases in asteroid astrometric observations Steven R. Chesleya, James Baerb,*, David G. Monetc aJetPropulsionLaboratory/Caltech,4800OakGroveDrive,Pasadena,CA91109,USA bJamesCookUniversity,SchoolofEngineeringandPhysicalSciences,Townsville,QLD4811,Australia cUnitedStatesNavalObservatory,Flagstaff,AZ86001,USA a r t i c l e i n f o a b s t r a c t Articlehistory: Inthispaper,wediscussthedetectionofsystematicbiasesinstarpositionsoftheUSNOA1.0,A2.0,and Received26January2010 B1.0catalogs,asdeducedfromtheresidualsofnumberedasteroidobservations.Wepresentatechnique Revised29May2010 fortheremovalofthesebiases,andvalidatethistechniquebyillustratingtheresultingimprovementsin Accepted2June2010 numberedasteroidresiduals,andbyestablishingthatdebiasedorbitspredictomittedobservationsmore Availableonline11June2010 accuratelythandoorbitsderivedfromnon-debiasedobservations.Wealsoillustratethebenefitsofdebi- asingtohigh-precisionastrometricapplicationssuchasasteroidmassdeterminationandcollisionanal- Keywords: ysis,includingarefinedpredictionoftheimpactprobabilityof99942Apophis.Specifically,wefindtheIP Asteroids ofApophistobeloweredbynearlyanorderofmagnitudeto4.5(cid:2)10(cid:3)6forthe2036closeapproach. Comets (cid:2)2010ElsevierInc.Allrightsreserved. 1.Introduction referenceframefromtheGuideStarCatalogtotheICRF;itcontains 526,280,881sources(Monet,1998).TheB1.0catalog,introducedin While future asteroid surveys, such as PanSTARRS, promise to 2003, contains 1,042,618,261 objects down to V magnitude 21, routinelyprovideobservationswithuncertaintiesontheorderof withanestimatedaccuracyof0.20arcsec(Monetetal.,2003). 0.1arcsec (Jedicke et al., 2004), many high-precision astrometric Notethatothercatalogs,suchasUCAC2and2MASS,havesmal- applications,suchasmassdeterminationandYarkovskyanalysis, lerestimatedpositionerrors;butthesecatalogseachhavelimita- studyinteractionsfixedinthepast,orrelyuponlongobservational tions relative to B1.0. The UCAC2 catalog, introduced in 2004, baselines to model slowly evolving phenomena. So while new contains 48,330,571 stars with positions accurate to within observationsmaybehelpful,wemustalsousecontemporaneous 0.07arcsec at the limiting magnitude of 16; but it only covers andhistoricalobservations.Ourlong-termgoal,therefore,istocre- the sky from declination (cid:3)90(cid:3) to +40(cid:3) (Zacharias et al., 2004). ate a statistical error model of asteroid observations, providing (Note: The recently introduced UCAC3 catalog (Zacharias et al., realistic, observatory-specific estimates of error correlations and 2004) covers the entire sky, thus resolving this limitation.) The uncertainties that will allow us to make the best possible use of all-sky 2MASS catalog, introduced in 2003, contains 470,992,970 theexistingbodyofobservations. objects, with positions accurate to within 0.07arcsec (Skrutskie Theastrometricreductionofsuchobservationsreliesheavilyon etal.,2006);butitislimitedtoVmagnitude17,meaningthatref- accuratestarcatalogs;backgroundstarsinanimageareidentified, erencestarsmaybeoverexposedinimagesofsmallasteroids. andusedasreferencesagainstwhichthepositionsoftheheadand Whilecreatingthefirstiterationofastatisticalerrormodelof tail of an asteroid trail may be determined. Ideally, star catalogs asteroidobservations,wedetectedsignificantbiasesintheresidu- shouldbe‘‘denseanddeep”,containingagreatmanystarsofvary- alsofthenumberedasteroids,whichweretracedtobiasesinthe ing brightness distributed throughout the entire sky, allowing starcatalogsfromwhichthoseobservationswerereduced. high-precisionastrometryregardlessofasteroidsizeorlocation. Afterdescribingourdiscoveryandinvestigationofthesebiases Inthatcontext,threeall-skyUSNavalObservatorystarcatalogs (Section2),wewilldescribetheavailableasteroidastrometricdata areamongthemostuseful.TheUSNOA1.0catalog,introducedin (Section3).Next,wepresentatechniquetodebiastheseobserva- 1996,contains488,006,860sourcesdowntoVmagnitude20,with tions(Section4),anddemonstratethatitsubstantiallyeliminates anestimatedaccuracyof0.25arcsec(Monetetal.,1998).TheA2.0 systematicerrorsinorbitfitsofthenumberedasteroids(Section5). catalog, introduced in 1998, is an update of A1.0 that moves the Wevalidatethisnewdebiasingtechniquebydemonstratingthatit producesorbitsthatpredictbetter,whichisthecrucialtestofthe quality of an orbit estimate (Section 6). Finally, we illustrate the application of catalog debiasing to issues such as estimating the * Correspondingauthor. mass of perturbing asteroids, and assessing the probability of an E-mailaddresses:[email protected](S.R.Chesley),jimbaer1@earthlink. net(J.Baer). impactforpotentiallyhazardousasteroids(Section7). 0019-1035/$-seefrontmatter(cid:2)2010ElsevierInc.Allrightsreserved. doi:10.1016/j.icarus.2010.06.003 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED 2010 2. REPORT TYPE 00-00-2010 to 00-00-2010 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Treatment of star catalog biases in asteroid astrometric observations 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Jet Propulsion Laboratory/Caltech,4800 Oak Grove REPORT NUMBER Drive,Pasadena,CA,91109 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT In this paper, we discuss the detection of systematic biases in star positions of the USNO A1.0, A2.0, and B1.0 catalogs, as deduced from the residuals of numbered asteroid observations. We present a technique for the removal of these biases, and validate this technique by illustrating the resulting improvements in numbered asteroid residuals, and by establishing that debiased orbits predict omitted observations more accurately than do orbits derived from non-debiased observations. We also illustrate the benefits of debiasing to high-precision astrometric applications such as asteroid mass determination and collision analysis, including a refined prediction of the impact probability of 99942 Apophis. Specifically, we find the IP of Apophis to be lowered by nearly an order of magnitude to 4.5 106 for the 2036 close approach. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE Same as 24 unclassified unclassified unclassified Report (SAR) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 S.R.Chesleyetal./Icarus210(2010)158–181 159 Withthecatalogbiasesresolved,afuturepaperwilldescribethe 1000 subsequent development of a statistical error model for asteroid 800 observations,demonstrateitsvalidity,anddetailitsapplications. 600 400 2.Evidenceforbias 200 0 Star catalog biases were encountered during the development −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 of a statistical error model for astrometric asteroid observations, Mean of Normalized RA Residuals as described in Carpino et al. (2003). Such a model requires that the observational residuals be unbiased, normally distributed, 400 anduncorrelated. 300 Intestingthefirstiterationofourerrormodel,however,itbe- cameclearthattheseassumptionswerebeingviolated. 200 AsillustratedinFig.1,forinstance,probabilitydistributionsof 100 the nominal declination residuals for the numbered asteroids re- vealed a clear bias of approximately +0.16arcsec. Significantly, 0 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 no such bias is evident in the right ascension residuals; this led Mean of Normalized DEC Residuals usinitiallytobelievethatonlythedeclinationobservationswere affected. Fig.2. Meanofnormalizedpostfitresidualsfor1649numberedasteroidsunder Additionally,wecalculatedakurtosisofapproximately4.6for automatedorbitmaintenanceasofmid-2008,includingallnumberedNEAs,plusa boththeRAandtheDECresidualprobabilitydistributionsdepicted handfulofothertargetsofinterest,suchasspacemissiontargets.Theresiduals werenormalizedbydividingeachresidualbyitscorrespondingweight;andsince inFig.1;thiscomparestoakurtosisof3.0forthestandardnormal the vast majority of observations are weighted at 1arcsec, the abscissa is distribution, indicating that the RA and DEC distributions are approximatelyinarcsec. slightlypeaked,thusdeviatingsomewhatfromtheexpectednor- mal distribution. Adjusting the chi-square observation rejection 0.057arcsec, the declination histogram is decidedly nonGaussian threshold in the orbit determination algorithm failed to reduce andasymmetric,withameanof0.074arcsecandastandarddevi- eitherkurtosistoexpectedlevels.(Note:Thedefinitionofkurtosis ationof0.095arcsec. usedinthispaperresultsinanormaldistributionhavingakurtosis UponanalysisoftheRosettaspacecraft’sencounterwithAster- of3.0.) oid2867Steins,Morley(T.Morley,privatecommunication)inde- Finally, as will be discussed in Section 5.3, the residuals from pendently deduced a bias of +0.212arcsec in the declination closely-spaced observations of the same asteroid made by the observationsofthatasteroid.However,tofullyaccountfortheob- same observatory appeared highly-correlated. While the correla- servedencountergeometry,arightascensionbiasof+0.092arcsec tionsassociatedwitheachspecificobservatorydifferedsomewhat was also necessary. This was the first indication that a bias was inmagnitude,theyremainedsignificantevenforobservationssep- present in both coordinates. As Morley noted, a systematic bias aratedbyseveraldays. in the right ascension observations of asteroids would not be Basedonthesedata,wehypothesizedthatthereweresystem- immediately obvious, since a least-squares orbit determination aticdeclinationbiasesinthestarcatalogsusedtoreducetheaster- algorithmwouldsimplyrotatethecalculatedorbitintheequato- oidobservations.Andasweinvestigatedfurther,weencountered rialframeaboutthecelestialpoletominimizetheRMSerror,thus evidenceoutsideofourownwork. eliminatingtheevidenceofanetrightascensionbias.Sincenoneof Fig. 2, dating to mid-2008, depicts histograms of the postfit theotherorbitalparameterscouldbemanipulatedsoastoelimi- means of right ascension and declination residuals for the 1649 natethedeclinationbias,itappearedtobetheonlysystematicer- numbered asteroids under automated orbit maintenance at that rorintheobservations. time.Whiletherightascensionhistogramshowsasymmetricdis- As will be described more fully in Section 7.2.1, Mauna Kea tributionwithameanof0.004arcsecandastandarddeviationof Observatory (MPC observatory code 568) independently noted a persistentpositivedeclinationbiasinobservationsof99942Apo- phis (D. Tholen, private communication). In the discussions that Raw RA and Dec Residuals followed, Tholen referred us to da Silva Neto et al. (2005), who 1 RA hadobservedameandeclinationbiasofapproximately0.11arcsec 0.9 Dec intheUSNOB1.0positionsoftheopticalcounterpartsof64ICRF sources. Using the more accurate UCAC2 catalog as a reference, 0.8 comparisons of the positions of stars appearing in both B1.0 and 0.7 UCAC2wereusedtoderivelocalcorrectionstotheB1.0positions; y 0.6 implementing these local corrections, the B1.0 declination biases c n for the ICRF sources were reduced to approximately 0.03arcsec. ue 0.5 Intime,thisideaprovedmostuseful. q e fr 0.4 3.Theasteroidastrometricdata 0.3 0.2 The Minor Planet Center (MPC), hosted by the Smithsonian Astrophysical Observatory at Harvard Univ., Cambridge, Mass., 0.1 operates under the auspices of Division III of the International 0 AstronomicalUnionastheclearinghouse forasteroidastrometric −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 residual (arcsec) measurements,bothopticalandradar(Williams,2009).Asapart ofthisfunction,theMPCarchivesanddistributestheavailabledata Fig.1. Nominalresidualsofthenumberedasteroids,illustratingDECbias. onintervalsofapproximatelyonemonth. 160 S.R.Chesleyetal./Icarus210(2010)158–181 Table1 Table2 Opticalobservationdatatypes. CatalogflagsfoundforCCDobservationsonMPCdata file. Type MPCflag Count Catalog MPCflag Count CCD C,c 58,083,989 FormerB1950.0 A 662,222 USNOA1.0 a 173,556 Photographic 303,325 USNOSA1.0 b 31,044 Transitcircle T 26,968 USNOA2.0 c 26,071,594 Micrometer M 12,081 USNOSA2.0 d 1,327,783 Approximate X 10,763 UCAC1 e 243,111 Hipparcos H 5494 Tycho-1 f 0 Occultation E 991 Tycho-2 g 299,648 Satellite S 277 GSC1.0 h 0 Rovingobserver V 177 GSC1.1 i 10,055 Mis-taggedobs. b 1 GSC1.2 j 11,665 GSC2.2 k 180 Total – 59,106,288 ACT l 13,866 GSCACT m 364,575 TRC n 0 WehavebasedthepresentstudyonthecompleteMPCastro- USNOB1.0 o 6,799,942 PPM p 0 metric data set dated 2008 December 12. In this data set there UCAC2-beta q 0 are59,106,288opticalobservationsofasteroids,anumberthatin- UCAC2 r 12,315,889 cludes48,807,317opticalobservationsof202,855numberedaster- USNOB2.0 s 445 oidsandanadditional10,298,971opticalobservationsofasteroids UCAC3-beta t 0 UCAC3 u 0 temporarilydesignatedbutnotyetnumbered.Therearealsoradar NOMAD v 2763 delayandDopplerobservations,whichweneglectforthepresent CMC w 337,053 discussion.TheopticaldatasetisdominatedbyCCDobservations, Hip2 x 0 whichcomprise98.3%oftheobservations. GSC(generic) z 1537 The MPC astrometric data are stored in 80-column ASCII re- Unspecified – 10,079,283 cords.Anearlycompletedescriptionofthedataformatisavailable Total – 58,083,989 ontheMPCwebsite,1whichwebrieflysummarizehere.Thereisone recordperobservation,exceptforSatelliteandRovingobservations, whichrequireasecondobservationrecordtodescribethepositionof the observer. The primary information on the optical astrometry Table3 Observer-suppliedcataloginformation. data record is the observation time and sky position, given as (J2000) right ascension and declination. The measurement tech- Obs.code Timeframe Catalog nique,orobservationtype,isrecordedbyacodeincolumn15,the 704 Before2000-January-01 USNOA1.0 countsforwhicharegiveninTable1.Theobservatorycodeincol- 704 After2000-January-01 USNOA2.0 umns 78–80 is a three-character identifier that uniquely identifies 691 1991-August-31–1999-September-28 GSC1 691 1999-September-29–2000-December-21 USNOA1.0 the observing location, and sometimes the observing program. (In 691 2000-December-22–2006-December-26 USNOA2.0 caseswheremultipleobservingprogramssharethesameobserva- 703 Before2005-January-01 USNOA2.0 tory code, e.g., Mauna Kea or Mt. Palomar, there may be an addi- tional field that records the observing program.) The observatory codes and the names and locations of the associated observatories infofor59%oftheuntaggedastrometry.Table3liststherulesthat aretabulatedbytheMPC.2Inthispaper,forbrevity,wewilloften wehaveappliedtootherwiseuntaggedastrometry. refer to the observatories only by their observatory code. There is Many of the individual catalog tags given in Table 2 can be alsoafieldfortheobservation’sMinorPlanetCircularorSupplement grouped togetherfor our purposes. For instance,the USNO SA2.0 reference, and a field that allows the observer to provide a terse catalogismerelyasubsetofUSNOA2.0,andUCAC2isanextension annotationregardingtheobservationqualityorreductiontechnique. toUCAC1.Wehavemergedthevariouscolumn-72symbolsasde- Photometryinformationmayalsobepartoftherecord. tailedbyTable4.Wenotethatover90%oftheastrometryiscom- Forthepresenteffort,themostimportantpartoftheobserva- prisedfromfourcatalogs,USNOA1.0,USNOA2.0,USNOB1.0,and tion record is the undocumented annotation, in column 72, of UCAC. An additional 7.2% still have catalog unknown, although thestarcatalogthatwasusedinthereductionoftheastrometry mostofthesearefrommajorNEOsurveyprograms,andsothese (Gareth Williams, private communication). Table 2 lists the field alsoareprobablydominatedbythesamefourcatalogs.Aswedis- valuesandtheirmeanings,alongwiththenumberofCCDobserva- cussbelow,onecaninferwhichcatalogwasusedinsuchcasesbya tionsforeachfieldvalue.Aninspectionofthedataindicatesthat statisticalanalysisoftheresiduals.Fig.3depictsthehistoryofcat- theMPCdidnotstartrecordingthecatalogintheobservationre- aloguseforthedatagiveninTable4.Somecatalogsappearearlier cord until mid-2001,and even after that many observers did not than their release date because older images werein some cases reportthecataloginformationtotheMPC.Thusforabout17%of remeasured withnew software and star catalogs. USNOA2.0 has the astrometry there is no catalog information available on the been used heavily, with more than 2 million observations every MPC data record. Fortunately, muchof the missingcatalog infor- year since 2000, which happened to be the last year that USNO mation is from only a few observatory codes, and for data from A1.0 saw extensive usage. But starting in 2005, the UCAC and prior to mid-2001. Thus, although the MPC did not record the USNOB1.0catalogshavetogetheraccountedforthemajorityofre- information,someoftheobservingprogramsstillhavetheirobser- portedastrometry. vationlogsfromthatperiodandsowewereabletosupplycatalog 4.Twoapproachestodebiasing 1 http://www.cfa.harvard.edu/iau/info/ObsFormat.html. Therearetwoobviouswaytoquantifytheeffectofstarcatalog 2 http://www.cfa.harvard.edu/iau/lists/ObsCodesF.html. bias in asteroid orbit determination. One can proceed indirectly, S.R.Chesleyetal./Icarus210(2010)158–181 161 Table4 UCAC2declinationobservations.Giventhestatedaccuracyofthe Mergedcatalogcounts. UCAC2 star positions, this was clearly unrealistic. We concluded Catalog MPCflags Count % thatthebiasesintheothercatalogswerelikelyresultinginbest- fitorbitsthatmadetheUCAC2residualsappearmuchlargerthan USNOA2.0 c,d 30,786,427 53.0 UCAC e,q,r,t,u 12,559,000 21.6 their intrinsic errors. Therefore, we decided to repeat the second USNOB1.0 o,s 6,800,387 11.7 iteration, but with no corrections applied to the UCAC2 derived Unknown – 4,171,601 7.2 positions. USNOA1.0 a,b 2,205,452 3.8 Theresultsfromtherevisedseconditerationshowedsignificant GSC1 h,i,j,z 543,037 0.9 improvement;formostoftheobservationsets,thebiaseshadbeen GSCACT m 364,575 0.6 CMC w 337,053 0.6 reducedtobelow0.06arcsec.AndyetthebiasesintheUCAC2de- Tycho f,g 299,648 0.5 rived observations remained above 0.2arcsec. A third iteration ACT l 13,866 0.0 yieldedlittlefurtherreduction. NOMAD v 2,763 0.0 By this point, additional analysis had demonstrated that the GSC2 k 180 0.0 asteroid residuals did indeed contain a right ascension bias; and Total – 58,083,989 – wesuspectedthatourfailuretoaccountforthiswasresponsible for the slow convergence in eliminating the observed bias in the UCAC2 residuals. We therefore implemented a two-dimensional throughasystematicstudyofthepostfitresidualsofastrometric bias model for all future iterations, with a grid spacing of 1h in asteroidobservations,ordirectly,bycomparingstarcatalogposi- right ascension and 10(cid:3) in declination. As before, the mean ob- tions with those in a more accurate and precise reference. Since servedbiasineachgridsquarewasusedtodebiastheobservations the problem first arose for us from the perspective of asteroid in the next iteration. But unfortunately, two further iterations residuals,webeganwiththefirstapproach. yielded little further improvement; the biases in the UCAC2 de- rivedobservationsremainednear0.2arcsec. 4.1.Debiasingfromindirectreferencetopostfitastrometricresiduals Despite these difficulties, the bias corrections that we derived throughthisindirectapproachdosharemanyofthegrossproper- As shown in Fig. 1, the observations of numbered asteroids tiesseeninthecorresponding(andmuchmoredetailed)skymaps demonstratedaclearobservationalbiasindeclination;butnosuch derivedfromthedirectapproachseeninthenextsection.Inretro- biasappearedinrightascension.Wethereforebeganbydebiasing spect,webelievethat,hadwestartedwithatwo-dimensionalbias onlythedeclinationcoordinates.Aftercalculatingtheorbitsofthe modelinthefirstplace,perhapswithsmallerbinsonthesky,this numbered asteroids and compiling the ‘‘observed–predicted” iterativetechniquemighthaveyieldedadequateresults;butwedo residuals, we divided the sky into 18 zones of declination, each notbelievethatitcanbeaseffectiveasthedirectapproachthatwe 10(cid:3) wide; within each zone, we calculated the mean declination havedeveloped,whichwedescribenow. biasesforUSNOA1.0,A2.0,B1.0,UCAC2,andobservationsofvari- ous eras whose reduction catalogs were unknown. We then cre- 4.2.Debiasingfromdirectreferencetostarcatalogs ated best-fit polynomials for each of these groups to model the declinationbiasesineachzone,andusedthemtodebiastheobser- Anassessmentofthesystematicerrorsinstarcatalogastrome- vationsforthenextiteration. tryisthekeyingredientinthedirectapproachtoquantificationof Initially, the approach seemed promising; the biases resulting systematicerrorsinasteroidastrometryfromthissource.Thuswe fromtheseconditerationwerereducedtoonethirdoftheirprevi- neededtoproduceaspatiallyresolvedastrometriccomparisonof ousvalues.However,inordertoachievethesereductions,wehad variouscatalogsthathavebeenusedasreferencecatalogsforthe toapplysignificantcorrections(ontheorderof0.35arcsec)tothe astrometry of solar system objects. Because of its wide dynamic range and its rigorous ties to the ICRS (Arias et al., 1995) using x 106 Observatory Code: ALL theTycho-2catalog(Høgetal.,2000),the2MASSstarcatalog(Skr- 8 utskieetal.,2006) was chosenas thecatalogto whichallothers were compared. According to its documentation, the 2MASS sys- USNO A2.0 tematic position errors are 70–80milliarcsec with respect to the 7 UCAC USNO B1.0 ICRS. The 2MASS catalog does not contain proper motions, but Unknown the mean epoch of the 2MASS observations is very close to USNO A1.0 6 Other 2000.0.Hence,thecomparisonofallofthecatalogsignoredproper motions. 5 Toachievethedesiredspatialresolution,theJPLHEALPix3tes- sellation (Górski et al., 2005) was used. Adopting SIDE=64 with theHEALPixpackageproduces49,152tilesonthesky,calledHEAL- 4 Pix,eachwithanareajustabitsmallerthanonesquaredegree.This seemstobeareasonablecompromisebetweenhighresolutionand 3 havingenoughstarsineachtile.Giventhistessellationscheme,the tilenumberforeachentryofeachstarcatalogwascomputed,and 2 thecatalogsweresortedonthisindextoexpeditethecalculation. The numerical processing was the same for all catalogs under consideration. For each HEALPix, the stars in the catalog and in 1 2MASSwereextracted,andaspatialcorrelationusingaradiusof 2.0arcsecwasusedtoidentifystarsincommon.Thesizeofthera- 0 diuswaschosentobelargeenoughtoaccommodatetheexpected 19901992199419961998200020022004200620082010 Fig.3. Year-by-yearcountsofstarcatalogusageforCCDastrometry. 3 http://healpix.jpl.nasa.gov. 162 S.R.Chesleyetal./Icarus210(2010)158–181 systematicdifferences(upto1.0arcsec),butsmallenoughtomin- Table5 imize the number of spurious correlations. Once the pairs were Inter-catalogsystematicerrors,withrespectto2MASS. identified,themeanandstandarddeviationintherightascension Catalog Mean±std.dev. and declinationdirections were computed. These provide a mea- RA(mas) DEC(mas) sureofthesystematicandrandomlocalpositionerrors.Notethat Tycho-2 (cid:3)1±28 (cid:3)12±24 thecalculationwasdoneinthetangentplane.ForeachHEALPix,a UCAC2 2±23 (cid:3)7±20 tangentpointatthecenterofthetilewasadopted,andthetangent USNOB1.0 (cid:3)16±123 126±123 plane positions for the catalog and 2MASS stars were computed. USNOA2.0 63±180 142±189 Hence,theresultingastrometricshiftsand dispersionshouldmi- USNOA1.0 (cid:3)41±419 (cid:3)34±352 micthoseneededtocorrecttheanalysisofobservationaldata. We performed the analysis for five of the catalogs listed in Ofcourse,thediscussionsofardoesnotrevealanythingabout Table 4: USNO A1.0, USNO A2.0, USNO B1.0, UCAC2, and Tycho. thestructureonthesky.Ifhalfoftheskyhasapositiveoffsetand The output of the comparison of these catalogs with 2MASS was halfoftheskyhasanegativeoffsetthenthemeanoffsetcouldstill asinglefilecontaining49,152lineseachwiththenominalposition besmall.Toaddressthisquestion,Fig.5depictsskymapsoftheRA oftheHEALPixonthe skyand valuesfor themeanand standard andDECdifferenceswithrespectto2MASSforthefivecatalogsun- deviation for the astrometric offset (excepting for UCAC2 which derconsideration;whileFig.6depictsthestandarddeviationsin doesnotcovertheentiresky).Whiletheprimaryuseofthecom- thoseRAdifferences.Notefirstthatthecolorscaleisdifferentfor parisonfileisintendedtobeaschemebywhichindividualobser- different catalogs in order to show as much of the structure in vationscanbecorrectedtotheastrometricreferenceframedefined thecatalogbiasaspossible.Intheseplots,regionswithrelatively by the 2MASS catalog, various visualization tools were used to small biases are colored green, while red regions show positive understandthegrosspropertiesofthevariouscatalogs. bias and blue regions have negative bias. The USNO A1.0 catalog Fig.4includesplotsoftheprobabilitydensityofthemeandif- doesshowsignificantstructure,despiteitsratherinnocuouspre- ferences between the various catalogs and 2MASS among the sentation in Fig. 4, and yet the hemispheric differences are rela- 49,152HEALPixcells.Table5liststhemeanandstandarddeviation tively modest. On the other hand, the USNO A2.0 and B1.0 ofthesamedistributions.Fromthese,weobservethatthepreci- catalogs, which together account for approximately 70% of all sionandaccuracyoftheTychoandUCACcatalogsarecomparable, asteroidastrometry, are moreproblematic.We notein particular and vastly superior to the other catalogs. Indeed, the differences thatthedeclinationmapsforthesetwocatalogsareplainly‘‘hot” with respect to 2MASS appear small enough that it is difficult to overall,inagreementwithFig.4.Also,theRAmapsshowevidence know whether they should be attributed to the subject catalogs, forhemisphericbias,withalargecoldregion(negativeoffset)cen- to 2MASS or whether the differences are a blend of errors from tered around 45(cid:3) RA extending from pole to pole, and an even bothsidesofthecomparison. stronger hot region opposite, which is, however, roughly limited But, Tycho and UCAC have relatively few stars and are too tothenorthernhemisphereforA2.0andthesouthernhemisphere sparse for most asteroid astrometry work, and so much of the forB1.0.Incontrast,theUCACandTychoskymapsshowverylim- asteroid astrometry has been and continues to be measured itedstructure,moreconsistentwithrandomfluctuations. againstthemuchmoredenselypopulatedUSNOfamilyofcatalogs. Theprocedurefollowedinthispaperistousethemeancatalog HereweseethattheUSNOA1.0hasrelativelylargedispersions,of differencesoneachHEALPixcelltodebiasastrometryfromtheass- order400mas,butoverallitexhibitsarelativelymodestbias.On sociated catalog in that cell. Specifically, for a given astrometric theotherhand,theUSNOA2.0andB1.0catalogsrevealsignificant observation with known catalog, we first compute which of the biases overall,especiallyindeclination,withmeanoffsetsof142 49,152HEALPixskytilescontainstheastrometryandlookupthe mas and 126 mas, respectively. These values are comparable to associated RA and DEC offsets, which are subtracted from the thestandarddeviationsofthesamples.Inrightascension,B1.0ap- rawastrometrytoproducedebiasedastrometry.Inlightofthefore- pearsrelativelyunbiasedoverall,whileA2.0doesshowamodest goingdiscussiononthelowsystematicerrorsseenintheUCACand netRAbias. Tycho catalogs, we consider that astrometry reduced with these catalogsshouldnotshowsignificantcatalog-basedsystematicbias. Therefore, we apply catalog bias corrections only for astrometry USNO B1.0 Tycho2 4 20 derived from the USNO A1.0, A2.0 and B1.0 catalogs. The catalog comparisonfileandancillarydocumentationareavailabletoother 2 10 researchersviaFTP.4 0 0 −0.5 0 0.5 −0.1 0 0.1 5.Atestonallnumberedasteroids USNO A2.0 UCAC2 20 2 To evaluate the effectiveness of the debiasing technique out- 1 10 lined in the previous section, we have calculated the orbits of asteroids, both withoutand with debiasing corrections,to obtain 0 0 −0.5 0 0.5 −0.1 0 0.1 whatwecall‘‘raw”and‘‘debiased”fits,respectively.Welimitthe objects considered to those with well-determined orbits, namely USNO A1.0 the numberedasteroids, to ensure that randomerrors do not—in 1.5 RA themean—contaminatethe results in asignificantway. Also, be- 1 0.5 DEC causetheobjectivehereisnotnecessarilytoobtainthebestpossi- bleorbits,butrathertoevaluatetheefficacyofdebiasing,wehave 0 −1 0 1 usedonlydatathatcomeswithinthescopeofthedebiasingtech- nique,namelyastrometryknowntobereducedwithrespecttoone Fig.4. Probabilitydensitiesofinter-catalogsystematicerrors,ascomparedtothe ofthefivecatalogsdiscussedintheprevioussection.Thisamounts 2MASScatalog.Foreachplot,theabscissaisthedifferenceinarcsecbetweenthe givencatalogand2MASSandtheordinateistheassociatedprobabilitydensityin arcsec(cid:3)1.Notethattheplotsarenotallonthesamescale. 4 ftp://ssd.jpl.nasa.gov/pub/ssd/debias/debias.tgz. S.R.Chesleyetal./Icarus210(2010)158–181 163 Fig.5. HEALPixmapsofRAandDECbiaswithrespectto2MASSforfivestarcatalogs,asdescribedintext. 164 S.R.Chesleyetal./Icarus210(2010)158–181 to48,621,261observationsof201,804asteroidsovertheinterval 1949-11-19to2008-12-12.(Note:The1949-eraobservationsare likelyprecoveryattemptsreducedusingoneofthefivecatalogs.) Outlierrejectionwasusedintheorbitcalculation,accordingto thetechniquedescribedbyCarpinoetal.(2003),withseparatefits fortherawanddebiasedcases. Thedistributionofthechangeinorbitalelementsbetweenthe rawanddebiasedfitsisdepictedinFig.7.Thereweseesignificant changesintheinclinationsofthedebiasedorbits,relativetotheir ‘‘raw” orbit counterparts. The inclinations generally declined, whichisconsistentwiththefactthatthemajorityoftheastrome- tryfallsinthenorthandhasapositivedeclinationbias,whichwill tend to erroneously increase the estimated inclination. There are smallerchangesinthesemimajoraxisandeccentricityestimates, butstillmanycaseswith>1sigmadeviations. 5.1.Correlationofpostfitresidualswithskyposition Fig.8depictsthepostfitresidualsfortherawfits,wherethesky mapreflectsthemeanpostfitresidualineachHEALPixcell.Asis thecasewiththeasteroidsthemselves,themajorityofnumbered asteroiddetectionsarefoundalongtheecliptic.Onlycellswithat least 10 residual measurements are plotted. For the USNO A1.0, A2.0 and B1.0 catalogs, the raw astrometric residuals show a remarkably similar structureand magnitudetothat ofthecorre- spondingcatalog(Fig.5).Thissuggeststhat,wheretheobservation densityissufficient,theindirectapproachtodebiasing(byrefer- encetopostfitresiduals)mayhaveutilityincaseswherethecata- logbiasispoorlyconstrained. Thereisalsoasignificantbiasinthepostfitresidualsofastrom- etry reduced with the Tycho and UCAC catalogs, despite the low systematicerrorsseenwiththesecatalogs.Thisisnotparticularly surprising,giventhatthevastmajorityofobservationsarisefrom thesignificantlybiasedA2.0andB1.0catalogs;thesecatalogshave skewedtheorbitestimatesfortherawfits,leadingtohigherresid- ualsintheunbiasedTychoandUCACastrometry.Inparticular,po- sitivedeclinationbiaseswilltendtoskewtheorbitalinclinationso thatthedeclinationresidualsfromunbiasedobservationswillalso bepositive. Thepostfitresidualsforthedebiasedorbitalfitsaredepictedin Fig.9, wherethesignatureseeninFig. 8isnowsubstantially re- duced, indicating that the biases have been largely eliminated (note,however,theevidentstripingintheUCACplots,whichwe 1 Δa/σ a Δe/σ 0.8 e Δi/σ i 0.6 0.4 0.2 0 −3 −2 −1 0 1 2 3 Fig.6. HEALPixmapsofrightascensionstandarddeviationwithrespectto2MASS. Fig.7. Distributionoforbitalelementvariationbetweenrawanddebiasedfitstoall Declinationdispersionsarenotsignificantlydifferent. numberedasteroids. S.R.Chesleyetal./Icarus210(2010)158–181 165 Fig.8. Meanresidualsforfitsofrawastrometrytonumberedasteroids.Theplotsdepict49,152equalareacellsusingtheHEALPixalgorithm(Górskietal.,2005).Cellswith 10orfewerobservationsarenotplotted. 166 S.R.Chesleyetal./Icarus210(2010)158–181 Fig.9. Meanresidualsforfitsofdebiasedastrometrytonumberedasteroids,asinFig.8.ThedebiasinghassubstantiallyremovedthecatalogbiassignalseeninFig.8.