1987Aj 94. . 18 9Y the Astronomical Journal

Home , 1627 Ivar

1987AJ 94. . 18 9Y 7 THE ASTRONOMICALJOURNALVOLUME94,NUMBER1JULY1987 jects incrateringtheterrestrialplanets,asagentsofmajor tric observationsandcomputations requiredtomaintain bodies areonly3X10yr(Wetherill1976),sothepopula- fundamental prerequisitetoeach stepinthestudyofaster- elements andthegenerationofprediction ephemeridesarea teroids (Marsden1979).Infact, theestimationoforbital ephemeris reliabilityforknown near-Earthandmainbeltas- Additionally, aconsiderableeffort isdevotedtotheastrome- initiated (HelinandShoemaker 1979;Gehrelsetal.1986). eral searchprogramsdesignedtodiscoverthemhavebeen the natureofasubstantialsampleindividualobjects,sev- sus ofthenear-Earthasteroidpopulationandtoascertain biological extinctionsinthepast,andaspotentialthreatsto proximity andaccessibilityofmanytheseobjectsmark objects mightbe,orshare,theparentsofmeteorites.The quiescent cometsaswellmainbeltasteroids.Someofthese the injectionofnewasteroidsintoEarth-crossingorbits. ades andexploitationduringthenextcentury(O’Leary them ascandidatesforexplorationduringthenextfewdec- Sources ofnewnear-Earthasteroidsarethoughttoinclude tion wouldbequicklydestroyedifitwerenotrepopulatedby about 1.6+1.0km)isestimatedtobeapproximately1300 entire populationtoabsolutevisualmagnitude18(diameter 189 Astron.J.94 (1),July1987 0004-6256/87/010189-12$00.90 © 1987Am.Astron. Soc.189 interest recently(Shoemaker1983). our civilizationinthefuturehavearousedagooddealof are fewerthanonehundredknownnear-Earthasteroids,the system’s earlyhistoryisnowfirmlyestablished.Whilethere verse populationwhoseimportanceinthestudyofsolar 1977; NEARReport1986).Finally,therolesoftheseob- (Shoemaker 1983).Typicaldynamicallifetimesforthese Given thisvarietyofmotivationstodevelopareliablecen- Near-Earth asteroidsconstituteanextraordinarilydi- © American Astronomical Society • Provided by the NASA Astrophysics Data System asteroid ephemerides,anuncertaintyanalysishasbeenconductedusingfourasteroidswithdifferent histories ofopticalandradarobservations.Weselected1627Ivarasarepresentativenumberedasteroid few radarmeasurementscanshrinkthepositionalerrorellipsoidbyafactorof2foratleastdecade. we assumedthatradarechoesobtainedduringtheasteroid’smostrecentcloseapproachtoEarth histories areontheorderofmonthsandwhoseorbitsrelativelypoorlyknown.Ineachcasestudy, involve threeunnumberedasteroids(1986DA,1986JK,and1982DB),whoseopticalastrometric with along(56yr)opticalastrometrichistoryandverywell-establishedorbit.Ourothercasestudies In anefforttoassesstheextentwhichradarobservationscanimproveaccuracyofnear-Earth approach toEarthandlosingitentirely.Evenforanasteroidwithextensiveoptical-datahistory,a asteroid couldeasilymeanthedifferencebetweensuccessfullyrecoveringobjectduringitsnextclose provided onlyamodestabsoluteimprovementforthecasewhenlonghistoryofopticalastrometric of-sky) andEarth-asteroiddistanceuncertaintiesasfunctionsoftimethrough2001.Theradardata then studiedthesensitivityofasteroid’spredictedephemerisuncertaintiestohistory(thesched- yielded oneormoretime-delay(distance)and/orDoppler-frequency(radial-velocity)estimates.We oids havingonlyshortoptical-datahistories.Afewradarobservationsofanewlydiscoverednear-Earth data exists(1627Ivar),butratherdramaticreductionsinthefutureephemerisuncertaintiesofaster- data setofopticalandradarobservationstheirassociatederrors,wecalculatedtheangular(plane- ule andaccuracy)oftheradarmeasurementsaswelltoopticalastrometrichistory.Foranygiven Jet PropulsionLaboratory,CaliforniaInstituteofTechnology,4800OakGroveDrive,Pasadena,91109 I. INTRODUCTION RADAR ASTROMETRYOFNEAR-EARTHASTEROIDS D. K.Yeomans,S.J.Ostro,andP.W.Chodas Received 26January1987;revised14March1987 ABSTRACT occultation timingandforradarinvestigations.Themost oids. Especiallyaccurateephemeridesareneededforstellar them duringpost-discoveryreturnsbecausetheirephemer- mainbelt objects.Moreover,itisoftendifficulttorecover than aremainbeltasteroids.Becauseoftheirrelativelyfast spacecraft flybyandrendezvousmissions(Marsden1971; stringent requirementswillariseforasteroidstargeted ides arebaseduponobservationstakenoverrelativelyshort to detectonphotographicplatesthanareequallybright apparent motions,near-Earthasteroidscanbemoredifficult cal observationslessfrequentlyandforshorterdurations oids, forseveralreasons.Theyarefavorablyplacedopti- to developfornear-Earthasteroidsthanmainbeltaster- Friedlander etal1979). turbations bytheinnerplanets. time periodsandtheirmotionsaresubjectedtostrongper- ranging toVenuspermittedimprovementsinourknowledge planetary ephemeridessincetheearly1960s,whenradar of theastronomicalunitbyseveralordersmagnitude have helpedmaintaintheaccuracyofplanetaryephemerides ephemerides ofMercury,Venus,Earth,andMarsensured general relativity(Shapiroetal.1968a,b;Canuto and havepermitteddynamicalstudiesrangingfromtestsof successful guidanceofthefirstinterplanetaryspaceprobes tro etal.1985,1986a,b).However, weareunawareofany information aboutasteroidcharacteristics (Ostro1985;Os- and 29mainbeltasteroidshave furnishedawealthofnew cally since1980,andechoesfrom 17near-Earthasteroids ellites (Chandler1979). ( Ashetal.1967;Newhall1983).Sincethen,radardata (Pettengill andShapiro1965)radarrefinementsofthe 1984) todynamicalinvestigations ofJupiter’sGalileansat- Reliable predictionephemeridestendtobemoredifficult Radar observationshavecontributedtotheaccuracyof Radar observationsofasteroids haveincreaseddramati- 1987AJ 94. . 18 9Y T in theepochstatebylinearequation computed value,”areassumedtoberelatedvariations<5x residuals £z,definedinthesenseof“observedvalueminus pute animprovedestimatexofthestate.Themeasurement times. Basedonthemeasurementszandanaprioriestimate vector ofmobservationstheasteroidmadeatvarious velocity, measuredinaninertialframe),andletzdenotea the formalismandnotationofleast-squaresestimationthe- weighted sumofsquarestheresidualsisleast-squares measurement errors.Thevalueofxthatminimizesthe element isthepartialderivativeofmeasurementz,with where AisanmX6matrixofpartialderivatives,inwhich of thestatex,least-squaresmethodcanbeusedtocom- oid’s stateatacertainepoch(sixcomponentsofpositionand ory (seeLawsonandHanson1974).Letxdenotetheaster- stages ofasteroidexploration. where Wisaweightingmatrixcomprisedoftheinverses state estimatex,givenby respect tostatevariableXj,andvisanmvectorofunknown potential contributionsof“radarastrometry”tovarious tive, butourresultsdoofferarepresentativepictureofthe sensible andinformativemanner.Theyarefarfromexhaus- signed tosampleahugeparameterspacebriefly,butin observational histories.Ouruncertaintyanalyseswerede- case studiesoffournear-Earthasteroidswithverydifferent ments candecreasethoseuncertainties.Wehaveperformed matrix forerrorsintheepochstateestimate,whichisamea- that theoff-diagonalelementsofWarezero.Thecovariance the measurementvariancesalongmaindiagonal.We cal astrometricmeasurements,andhowradarmeasure- functions oftimeforseveralyearsafteragivenseriesopti- We refertoAWAastheinformationmatrix,anddenoteit sure oftheuncertaintyinestimate,isgivenby assume thatthemeasurementerrorsareuncorrelated,so tainties inpredictedplane-of-skyasteroidpositionsvaryas certainties. Ofspecialinterestisthequestionofhowuncer- the effectsofradarobservationsonasteroidephemerisun- prove asteroidephemerides. thorough assessmentofhowradarmeasurementscanim- from theinformationmatrix,withoutcomputingesti- calculating thestateestimatexisprimaryobjective. of interest.Thiscontrastswithfullstateestimation,inwhich as Á. the measurementnoisestatisticsWandobservationpar- mate. Furthermore,theerrorcovariancedependsonlyon Note thattheerrorcovariancecanbecomputeddirectly 190 YEOMANSETAL.:RADARASTROMETRYOFASTEROIDS surements themselvesinorder to,perform theanalysis. tial derivativesA,sothatitisnotnecessarytohavethemea- with time,itisnecessarytosimulate thetimeevolutionof as theextendedKalmanfilter(Jazwinski 1970)insteadofa this problemistouseasequential estimationtechniquesuch error-covariance matrix.Thecustomary methodforsolving 0 0 “batch” estimationtechniquesuch asthatdescribedabove. 0 0 0 0 0 T1 T-1 8z =AÔx+v,(1) Our uncertainty-analysistechniqueisbestdescribedusing x =+(AWA)-WÔz,(2) P =(AWA).(3) Here wereporttheresultsofnumericalinvestigationsinto In anuncertaintyanalysis,onlytheerrorcovariancePis In ordertostudyhowtheephemeris uncertaintiesvary 0 0 © American Astronomical Society • Provided by the NASA Astrophysics Data System II. UNCERTAINTY-ANALYSISTECHNIQUE x!,...,x arethecomponentsofstateattime with respecttotheepochstatexiscomputedasfollows: iance. Therequiredpartialderivativeajoftheobservation immediately afterthisobservationhasbeenmade.First,the observation. Thefirstfactorintheaboveproductdependson where e.,arethecomponentsofepochstate,and observation isusedtocomputeanewepocherrorcovar- pose thatwewishtocomputetheephemerisuncertainties quential estimationmethods(Chodas1986). has beenobservedtobenumericallymorestablethanse- ed foreachtimeofinterest.Thispseudosequentialtechnique that ajisthekthrowofA. errors attimet.Thismatrixiscomputedalongwiththe which indicateshowerrorsinthestateatepochmapinto cally. Thesecondfactoristhestatetransitionmatrix0(^), the typeofobservationbeingmade,andiscomputedanalyti- then mappingtheresulttotimet.Thecomputationisrepeat- on onlythesubsetofobservationsmadeuptothattime,and tained byfirstcomputingtheepocherrorcovariancebased recompute theinformationmatrixaftereachobservation, first krowsofA.Itisnotnecessary,however,tocompletely of thefirstkobservationscannowbecomputedusing state componentsx.vianumericalintegration.Note error-covariance matrixatanygiventimeofinteresttisob- chose touseavariationofbatchestimationinwhichthe Rather thantakingtheconventionalapproach,however,we where aisthestandarddeviationofmeasurementnoise since thefollowingrecursiverelationcanbeused: from theinertialcoordinatesystemtoeitherR-T-Nor The mappedcovariancematrixC{t)isthentransformed where O(i)isthestatetransitionmatrix,definedearlier. to varioustimesofinteresttviatherelation nothing isknownaboutthestatebeforefirstobservation. error-covariance matrixP,byinvertingAasinEq.(3). for observationz.Itisnowsimpletocomputetheepoch the plane-of-skycoordinatesystem.TheR-T-N Note thattheaprioriinformationmatrixAiszero,since system isdefinedbyorthonormalbasisvectorsalongthe N positionuncertaintiesquotedinthenextsectionare The plane-of-skysystemisdefinedwithonebasisvector orbital plane,andthedirectionnormaltoplane. radial vectorfromtheSun,transversedirectionin ers intheplaneofskyasviewedfromEarth.TheR-T- aligned alongtheEarth-asteroiddirection,andtwooth- 6 formed error-covariancematrix C.Notethatthetopleft square rootsofthefirstthreediagonal elementsofthetrans- elements defineaboxwhichjust enclosestheerrorellipsoid. teroid’s position.Thesquareroots ofthefirstthreediagonal constant-probability ellipsoidfor theuncertaintyinas- 0 In thenextsection,“plane-of-sky uncertainty”denotesthe lv6 k u16 3x3 submatrixoftheerror-covariance matrixrepresentsa k fc k 0 RTN 2 T The informationmatrixAresultingfromtheprocessing Consider thekthobservationzmadeattimet,andsup- A*; —A_j, CU) =(Í),t^<(6) Finally, theepocherror-covariancematrixP*ismapped k k Æ kk+x dz* de x dz de k 6 dz dx k x6 ^de de ~dx dxj~ l6m l dx _dxs de 6 x6 , (4) 190 (5) 191 YEOMANS ETAL. : RADAR ASTROMETRY OF ASTEROIDS 191 apparent angular size of the semimajor axis of the error el- in. RESULTS lipse created by projecting the error ellipsoid onto the plane of the sky. Table I outlines the rationale for our four case studies, Although we have described our uncertainty analysis in which together comprise a variety of possible optical and terms of the least-squares method, the technique we actually radar data histories for near-Earth asteroids, and which are used for this study is not least squares, but rather an equiva- described in detail in this section. lent method formulated in terms of the square root of the Our first object of study, asteroid 1627 Ivar, has a long information matrix (Bierman 1977). This alternate method history of astrometric observations dating back to 1929 and is numerically superior to least squares, since it avoids the was observed with the Arecibo radar during its approach to explicit computation of the information matrix, thereby re- within 0.2 AU of the Earth in July 1985. With its extensive ducing the accumulation of error due to numerical roundoff. optical astrometric history and 1985 radar observations dur- The three types of observations used in this study are opti- ing a close Earth approach, 1627 Ivar represents one ex- cal angle (i.e., a right ascension and a declination), radar treme of our sample. As an example of an object with a short time delay, and radar Doppler frequency. We assume a time- optical astrometric history and radar observations made at delay datum to be a measurement of the distance or “range” relatively small geocentric distances, we chose asteroid 1986 between the centers of mass of the Earth and the asteroid, DA. We selected asteroid 1986 JK as an object with radar and a Doppler datum to be a measurement of the time deriv- observations made during an extremely close Earth ative of that distance. (This simplification is appropriate for approach but lacking extensive astrometric observations. our present purposes, but glosses over certain fundamental Finally, we examined the currently lost asteroid 1982 DB in theoretical and experimental considerations; see, for exam- an effort to see how hypothetical radar observations, in early ple, Ash et al. 1967 ). The potential radar Doppler and range 1982, might have improved the chances of recovering this accuracies that can be achieved for spherical bodies having object. rough surfaces have been studied by Jurgens and Bender Our strategy in each study was to investigate the posi- ( 1977). For our current study, realistic standard deviations tional uncertainties for each object during future returns for the Doppler and range measurements were chosen with when the ephemerides are generated (i) using only optical the object’s dimensions and geocentric distance in mind. data obtained through the most recent apparition, (ii) using Large upper bounds upon these values were selected to let us radar data as well as the existing optical data, and (iii) by explore the sensitivity of the uncertainties to widely varying augmenting the “nominal” data sets in (i) and/or (ii) with radar accuracies. hypothetical (simulated) future optical and/or radar data. The standard deviation chosen for the optical measure- For each of our cases, Table II gives the initial conditions at ment noise was either 1" or 1.5'' in each coordinate; these epoch (x0) used to generate a reference ephemeris. Table II values correspond to the root-mean-square residuals ob- also gives the actual and simulated observation intervals, as tained in the recent orbit determinations using only optical well as the assumed error sources and their standard errors data. The times of simulated optical data were chosen such (i.e., the “noise”) in the optical and radar measurements. that the Earth-asteroid distances were small enough and the The intervals spanned by our assumed radar noise levels solar elongation angles were large enough to allow optical overlap values realized in recent radar observations of near- observations to be taken. Earth asteroids with the Arecibo and Goldstone instru- We assume throughout this study that measurement noise ments, but have been widened to reveal the sensitivity of is the dominant error source in the asteroid-orbit improve- ephemeris uncertainties to the accuracy of Doppler and ment process. All other errors, such as those in the observa- range measurements. The observations used for a particular tory locations and planetary positions and masses, have been asteroid in our study do not necessarily represent the most neglected. As a result, the absolute uncertainties given in the complete data set available. next section are somewhat optimistic, although the relative In a typical case study, we augmented optical data with results are valid. radar by first including a single Doppler datum on a certain

Table I. Rationale for selecting asteroid case studies.

Optical Radar Case Asteroid astrometric astrometric study history history

1627 Ivar Excellent Range and Doppler observations in 1985 at 0.2 AÜ from Earth 1986 DA Good Range and Doppler observations 2 months after discovery and at 0.2 AU from Earth. 1986 JK Fair Doppler observations 3 weeks after discovery and at 0.03 AU from Earth. 1982 DB Poor None attempted. Post- discovery detection not possible with available instrumentation.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1987AJ 94. . 18 9Y 192 YEOMANSETAL.:RADARASTROMETRYOFASTEROIDS from TableIIthat,comparedtotheopticaldatasets, is zero. the timeofEarthcloseapproach whentheradialvelocity vations (fromanephemeris-improvement viewpoint)isat the Earth-asteroidradialvelocity (absoluteDoppler)islar- the optimumtimeformakingDopplerobservationsiswhen ties towhenradarobservationsweretakenandfoundthat We alsoexaminedthesensitivityofephemerisuncertain- radar datasetsareverysmallandspanshorttimeintervals. radar rangingalsoyieldaDopplermeasurement.)Note of thosedates.(Ingeneral,thewaveformsusedinplanetary days later,andthenincludingarangedatumononeorboth date, thenincludingasecondDopplerdatumondatefew it isbettertotake themallonoptimaldaysrather thanto gest. Conversely,theworsttime formakingDopplerobser- In addition,foragivennumber ofDopplerobservations, © American Astronomical Society • Provided by the NASA Astrophysics Data System Observations : Asteroid Observations : Simulated Optical Employed Optical Orbit Employed: Reference for Observations : Simulated Radar Error Sources: The orbitalelementsgiveninthetableareappropriateforepochofosculation. inclination (/).Theangularelementsarereferredtotheeclipticplaneandequinoxof1950. AU, eccentricity(e),argumentofperihelion(¿y),longitudetheascendingnode(fl),and elements listedaretheperihelionpassagetime(T)inephemeristime,distance(q) Range datanoise Optical data Doppler datanoise Í2 (deg.,1950) 03 (deg.,1950) Number Dates Number Date 7/5/854/10/86 Dates e q (AU) T (E.T.) Date 7/9/854/20/86 Epoch (E.T.) i (deg.,1950) Geocentric dist.0.21AU Geocentric dist.0.20AU0.21 noise (arcsec) (km) (cm/s) 9/25/29 - Yeomans(1986a) 9/24/98 - 9/30/93 - 6/19.0/86 0.3966274 132.67498 7/14.1782/85 1/29/90 - 167.33513 1.1240960 1/23/95 - Table II.Error-analysisassumptions. MPC 10752 8.44331 3/3/99 6/12/95 7/18/85 1/8/94 12/15/90 1627 Ivar 10 242 1.0 40 Notes toTableII 4/2/86 - 8/5/96 - 2/5/86 - 0.586325 4/7.6197/86 6/19.0/86 7/3/91 - 0.5, 10.0 126.7066 11/4/94 - 1.166000 11/18/99 - Urata(1986) 64.5487 1986 DA IAU Cire. 4.3009 8/12/91 4/20/86 3/16/86 9/14/96 12/28/99 12/14/94 1, 30 33 4186 18 1.0 spread themoutoverlessfavorabledays(i.e.,whenthe and alsoallows us tocalibratecertainaspectsof ourmethod- the Earthensuresthatitwillbe wellobservedinthefuture. its 25September1929discovery atJohannesburgbyE. are addedtotheDopplermeasurements,wecouldfindno radial velocityisrelativelylow).Whenrangeobservations Hertzsprung, andthefrequency ofitscloseapproachesto ment timesasfreeparametersinthefollowingcasestudies. lations manageable,wehavenottreatedtheradarmeasure- dar observations.Inanefforttokeepthescopeofourcalcu- general conclusionsconcerningtheoptimaltimesforra- This objecthasbeenobservedduring 12apparitionssince Table IIIsummarizesourresults forasteroid1627Ivar, 0.896892 5/4/86 - 6/19.0/86 Yeomans(1986b) Yeomans 9/7/86 - 232.4143 0.684269 7/1.8000/86 10/10/95 - 62.2412 0.028 AU 0.031 AU 2, 20 5/30/86 5/28/86 1986 JK IAU Circ. 5/27/86 2.1494 9/17/86 29 1.0 10/30/95 4220 5 a) CaseStudy#1:1627Ivar 0.3, 30.0 9/25/92 - 8/27/90 - 0.953219 0.13 AU 2/28/82 - 314.2073 0.361080 0.15 AU 2, 20 3/14/82 3/7/82 157.9308 1/15.5838/82 1/11.0/82 Unpublished 1.5 5/14/82 19 10/26/90 10 1.4136 1982 DB 1/3/93 192 >H 00 193 YEOMANS ETAL. : RADAR ASTROMETRY OF ASTEROIDS 193 O'! Table HI. Ephemeris uncertainties for 1627 Ivar for the dates 24 June 1985, 9 May 1990, and 3 May 1995. When radar observations are included in the data set, both Doppler and range observations were assumed to have been made on 5 and 9 July 1985. r- oo Plane of sky Optical uncertainty (arc-sec) Range uncertainty (km) astrometric June June Nay June May Nay data (no.) 1985 1985 1990 1985 1990 1995 Actual Pred. Pred. Pred. Pred. Pred. 1962-1984 83 4.4 3.3 1.9 500 439 600 optical

1929-1985 242 0.6 0.39 0.20 52 30 40 optical 1929-1985 242 (0.39) 0.13 (52) 12 optical + all radar

1929-1995 0.39 0.19 52 30 36 optical 1929-1995 271 (0.39) 0.13 (52) 11 15 optical + all radar Note to Table III The entries in parentheses do not reflect the radar data because the reference time (June, 1985) is prior to the assumed date of the radar observations (July, 1985).

ology. Each row in Table III corresponds to a particular set similar trend is evident in the values of the plane-of-sky un- of data used in our simulations, and the last six columns give certainties predicted by our covariance analysis (a decrease plane-of-sky and Earth-asteroid distance uncertainties for from 3.3" to 0.39"), but the corresponding actual rms re- the dates 24 June 1985, 9 May 1990, and 3 May 1995. The siduals are several tens of percent larger than our predicted geocentric distances were 0.2,0.5, and 0.8 AU, respectively, uncertainties. This calibration experiment and others like it on these three dates (see Fig. 1), which were chosen to be indicate that our covariance calculations yield plane-of-sky representative of times when the asteroid could be observed uncertainty predictions whose relative values are very reli- easily with optical telescopes. able but whose absolute values are likely to be optimistic by For the June 1985 date, Table III compares the actual rms several tens of percent. residual for the optical astrometric observations made dur- Ivar’s extensive astrometric history allowed some experi- ing that month to the rms uncertainty that our analysis pre- mentation to determine the sensitivity of future ephemeris dicted for the same period. As expected, the size of the rms uncertainties to optical astrometric data intervals. We came residual decreases (from 4.4" to 0.6") as the astrometric to the expected conclusion that extending the data interval data arc was expanded from 1962-1984 to 1929-1985. A substantially improves the uncertainties of future ephemer-

© American Astronomical Society • Provided by the NASA Astrophysics Data System >H 00 194 YEOMANS ETAL. : RADAR ASTROMETRY OF ASTEROIDS 194 O'! ides. For example, the plane-of-sky angular ephemeris un- rendezvous spacecraft arriving more than a decade after the certainties in June 1985 and May 1990 were 3.3" and 1.9", radar observations were made. respectively, when 1962-1984 optical, astrometric data were oor- used. The inclusion of data over the longer 1929-1985 inter- b) Case Study #2:1986 DA val reduced the 3.3" uncertainty to 0.39" in June 1985 and 0.20" in May 1990. (For a given linear position uncertainty M. Kizawa ( 1986) discovered this asteroid in Shizuoka, at the asteroid, the angular uncertainty increases with de- Japan, on 16 February 1986. Pre-discovery observations are creasing geocentric distance. Hence the angular uncertainty available from February 5, and we have included actual ob- is larger in 1985 than it is in 1990 because the asteroid comes servations through March 16 and simulated observations a factor of 2 closer in 1985. ) through April 20 in our analysis. Radar observations of Ivar, conducted at Arecibo during Figure 2 shows the Earth-asteroid distance as a function 5-10 July 1985 by Ostro et al. (1985), have yielded echoes of time, and Fig. 3 shows the 1986-2001 progression of ephe- resolved in Doppler frequency and, for one date, in time meris uncertainties in the asteroid-centered R-T-N coordi- delay. These data will provide velocity and distance esti- nate system (see discussion in Sec. II) after the last observa- mates with uncertainties of about 10 cm/s and 1 km. For our tions were made in 1986. The radial (Sun-asteroid) position analysis, we assumed that two pairs of radar observations uncertainty reaches a minimum when the radial velocity of (Doppler and range) were made on 5.0 July and 9.0 July the asteroid passes through zero (at aphelion and perihe- 1985. When these simulated radar data are included in our lion), while the transverse asteroid position uncertainty covariance analysis, the plane-of-sky uncertainty in the May reaches a maximum at perihelion when the transverse veloc- 1990 ephemeris is reduced from 0.20" to 0.13". Hence, aug- ity peaks. In the absence of astrometric observations beyond menting 242 optical-angle measurements with just two radar 1986, the radial and transverse position uncertainties grow range/Doppler measurements achieves a 65% reduction in with time, but certainly not in a linear fashion. The position the plane-of-sky ephemeris uncertainty five years after the uncertainty normal to the asteroid orbit plane is always very radar observations were taken. Although the addition of ra- small relative to those in the two other coordinates. Figure 3 dar observations allows only a small absolute improvement shows the general trend for the time history of ephemeris in Ivar’s long-term ephemeris, this improvement might uncertainties of Earth-approaching asteroids in the absence prove useful in refining future predictions of stellar-occulta- of further observations. The absolute size of the uncertain- tion track locations. ties will change with the asteroid under study, but the trends The last three columns of Table III give our calculated of the uncertainties, expressed in this coordinate system, will uncertainties in the Earth-Ivar distance for three appari- remain qualitatively the same. tions and five different astrometric data sets. Radar astro- Radar-range and Doppler observations of 1986 DA were metry reduces range uncertainties to approximately 40% of made during 18-21 April 1986 at Arecibo (Ostro et al. the values corresponding to optical-only data sets. For both 1986a), but we have used either April 10 or April 10 and 20 the plane-of-sky and range ephemerides the improvement as the radar observation epochs in our simulations. Table IV due to radar astrometry lasts for at least a decade past the gives the ephemeris uncertainties of asteroid 1986 DA on 3 epoch of the radar observations. July 1991—a date near the middle of a several-week period Ivar is one of several objects under consideration for a when the asteroid should be optically observable. spacecraft-rendezvous mission during the 1990s (NEAR Table IV shows that the uncertainties are insensitive to Report 1986). One of the most practical implications of our Doppler accuracy when only one Doppler observation is calculations is that, even for an extensively observed aster- made, whereas there is a rather dramatic sensitivity to oid, a few radar measurements can shrink the positional er- Doppler accuracy when two Doppler observations are ror ellipsoid by a factor of 2 for many years—perhaps result- made. Adding a single range measurement to two Doppler ing in more accurate targeting of a future flyby or observations reduces the plane-of-sky uncertainties by a fac-

D < O z < COh- Û 9 o cc Fig. 2. 1986 DA: Asteroid’s geocentric dis- LU tance (AU) versus calendar date. eofr- < X fr- ÛC LU<

P.O.S. ANGULAR UNCERTAINTY (arc-min) POSITION UNCERTAINTY (km) 195 YEOMANSETAL.:RADARASTROMETRYOFASTEROIDS © American Astronomical Society • Provided by the NASA Astrophysics Data System (millions) Table IV.Plane-of-skyangular-positionuncertaintiesfor1986DAon3July 1991.Radar observations weresimulatedeitheron10Aprilor201986,and no astrometric observations beyond1986wereassumed. 2 Dopplerobs.& 2 Dopplerobs.& 2 Dopplerobs.& No radarObservations 2 Dopplerobs.& 2 Dopplerobs. 1 Dopplerobs. (range noise= (range noise= (range noise= (range noise= Dataset Doppler noise= 0.5 km) 0.5 km) 2 rangeobs. 2 rangeobs. 10 km) 1 rangeobs. 10 km) 1 rangeobs. 10.1 10.6 11.1 1 cm/s30 0.42 0.91 0.43 0.95 0.43 3.40 0.43 3.40 Plane-of-sky 1.47 9.01 uncertainty (arc-min.) (units of10km) uncertainty 0.247 0.532 0.249 0.556 0.249 2.01 0.852 5.26 0.249 2.01 6.50 5.90 6.22 1 cm/s30 Range values wereassumedtobe0.5kmand30cm/ measurements. TherangeandDoppler-noise sensitivity ofuncertaintiestonumberrange tainties versuscalendardate.Curvesshow s. Fig. 4.1986DA:Plane-of-skypositionuncer- centered coordinatesystemisdefinedbythe verse unitvectordefinedasT=NXR. tainties versuscalendardate.Theasteroid- Fig. 3.1986DA:Asteroid’spositionuncer- Sun-asteroid unitvector(R),the (N) normaltotheorbitplane,andatrans- 195 >H 00 196 YEOMANS ETAL. : RADAR ASTROMETRY OF ASTEROIDS 196 O'!

oor-

Fig. 5.1986 DA: Plane-of-sky position uncertainties versus calendar date. Curves show dramatic improvement when 1991-1999 astrometric data are included in ephemeris predictions. The range and Doppler-noise values were assumed to be 0.5 km and 30 cm/s.

DATE

tor of 3. A single range observation is just as good as two strength of this object’s actual optical and radar data sets, when Doppler accuracy is high, but a second range measure- then the inclusion of 1991 optical observations in this ob- ment becomes very valuable when the Doppler accuracy is ject’s orbit will cause the plane-of-sky position uncertainties coarse. The sensitivity of the uncertainties to achievable val- to drop precipitously (see Fig. 5 ). Furthermore, inclusion of ues of range accuracies is negligible. Note that the uncertain- the simulated optical observations through 1999 (see Table ties in the distance predictions are nearly proportional to the II) keeps the uncertainties at the arcsecond level throughout plane-of-sky uncertainties. That is, the effect of radar astro- the 1990s. However, it is interesting to note that during that metry is to reduce the future error ellipsoid’s size without decade the positive effect of the 1986 radar data remains significantly affecting its shape. evident, halving the uncertainties attainable with the optical Figure 4 demonstrates the time dependence of the plane- data alone. For example, during the 1995 return, we find the of-sky position uncertainties for the three cases where there maximum plane-of-sky uncertainty to be about 4" if we ex- are two Doppler observations (noise = 30 cm/s) and zero, clude the radar data and about 2" if we include it. The poten- one, or two range observations (noise = 0.5 km). While the tial value of such an improvement to searches for stellar oc- nonlinear growth of position uncertainties is evident, the cultation events could be appreciable. percentage improvement caused by the radar observations is Recently, Gehrels (1986a) provided optical astrometric fairly constant with time. data for 1986 DA in July 1986 so that the recovery of this As Table IV and Fig. 4 demonstrate, inclusion of 1986 object in 1991 should be assured. However, these data were radar astrometry greatly reduces the ephemeris uncertain- not included in our analysis. ties in the period following the radar observations. This reduction is far more dramatic (in absolute as well as relative c) Case Study #3:1986 JK terms) than that for Ivar because 1986 DA’s optical-data history is much shorter than Ivar’s. If we assume that 1986 This asteroid was discovered by C.S. and E.M. Shoemaker DA is recovered in 1991, as seems likely in view of the on 4 May 1986 (Shoemaker and Shoemaker 1986), and we

Fig. 6. 1986 JK: Asteroid’s geocentric distance (AU) versus calendar date.

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Fig. 7. 1986 JK: Plane-of-sky position uncertainties versus calendar date. Curves show position uncertainties when September 1986 optical, astrometric data are included, and when they are not.

have used 29 observations over the May 4-27 period. Simu- 23'. That is, even without any optical observations beyond lated observations were also assumed to have been made in the discovery month, the radar Doppler measurements ap- September 1986 and when the asteroid’s computed motion pear capable of refining our knowledge of this object’s orbit suggests another Earth approach in October 1995. Figure 6 so much that they ensure optical recovery in 1995. Clearly, shows the Earth-asteroid distance as a function of time over 1986 JK demonstrates the powerful capability of radar ob- the interval 1986-2001, and Fig. 7 shows the plane-of-sky servations to improve the ephemerides of newly discovered, position uncertainties as a function of time with and without near-Earth asteroids. optical astrometry in September 1986. Note the considerable Recently, recovery of this object in 1995 became far more benefit of the longer astrometric data interval provided by likely when optical astrometric positions of 1986 JK became the September 1986 observations. available for July, August, and September 1986 (Gibson During the extremely close approach of this asteroid to 1986; Gehrels 1986b). Table V presents results that apply if the Earth (to within 0.029 AU) in late May 1986, successful we assume that the astrometric data interval for 1986 JK radar Doppler observations were made at the Goldstone an- extends from May 4-27 with two additional simulated ob- tenna in California (Ostro et al. 1986b); the actual dates of servations being made in September 1986. the radar measurements were used in our calculations. Similar to the analysis for 1986 DA, when two Doppler If we assume that the asteroid was observed only optically observations were taken, there is a great sensitivity of the during May 4-27, and that no radar or subsequent astromet- 1986 JK plane-of-sky ephemeris uncertainties to Doppler ric observations are made, the plane-of-sky position uncer- accuracy. For this object, there is also some sensitivity to tainty on 10 October 1995 is 97° (Fig. 7). However, the addi- Doppler accuracy when only one Doppler observation is tion of two Doppler observations ( with noise values of 2 cm/ processed—a sensitivity that was not obvious with 1986 DA. s) on 28 May and 30 May 1986 reduces this uncertainty to The uncertainties are sensitive to the addition of one range

Table V. Plane-of-sky angular-position uncertainties for 1986 JK on 10 October 1995. Radar observations were simulated either on 30 May and/or 28 May, 1986, and no astrometric observations beyond 1986 were assumed. When one radar observation was simulated, it was taken on 28 May (Doppler) or 30 May 1986 (range).

Plane-of-sky Range uncertainties uncertainties (arc-minutes) (units of 105 km) Doppler noise 2 cm/s 20 cm/s 2 cm/s 20 cm/s Dataset No radar Observations 76.6 76.6 12.1 12.1 1 Doppler obs. 35.4 65.6 5.58 10.4 2 Doppler obs. 7.39 33.8 1.17 5.33 2 Doppler obs. & 1 range obs. 2.29 5.52 0.361 0.870 (range noise = 2 km) 2 Doppler obs. & 1 range obs. 2.24 5.47 0.354 0.862 (range noise = 0.2 km) 2 Doppler obs. & 2 range obs. 2.28 5.32 0.360 0.838 (range noise » 2 km) 2 Doppler obs. & 2 range obs. 2.24 5.22 0.354 0.823 (range noise « 0.2 km)

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Fig. 8. 1986 JK: Plane-of-sky position uncertainties versus calendar date. Curves show sensitivity of position uncertainties to Doppler accuracy.

point but not to the accuracy of that range point, and there Table VI shows the effects of simulated radar astrometry, appears to be little improvement with the addition of a sec- taken on 7 and 14 March 1982 upon the asteroid’s ephemeris ond range observation. uncertainties for 27 August 1990, during the next favorable Figure 8 presents the time history of the position uncer- opportunity for optical recovery. Acquisition of a single, tainties when two Doppler observations are made at two high-precision Doppler datum reduces the plane-of-sky un- different noise values. (Note that the curve representing no certainty by more than an order of magnitude, once again radar observations is the lower curve in Fig. 7 drawn to a demonstrating the potential of radar astrometry in securing different scale). The bottom curve in Fig. 8 shows the ease the orbits of newly discovered near-Earth asteroids. with which 1986 JK can be recovered in 1995 if the existing From Table VI and Fig. 9, we note that the plane-of-sky Doppler observations can be included in the orbit-determin- position uncertainties are sensitive to Doppler accuràcy ation process. when one or two Doppler measurements are made. As in the other case studies, while the results are sensitive to the addi- d) Case Study #4:1982 DB tion of one range point to two Doppler observations, there is little sensitivity to range accuracy and only a modest sensi- This asteroid was discovered on 28 February 1982 (Helin tivity to the addition of a second range observation. 1982), when it was coming out of solar conjunction, 0.1 AU Figure 10 displays the geocentric distance of 1982 DB as a from the Earth and receding in the southern sky. Had it been function of time and shows that another close Earth ap- discovered a few weeks earlier, radar measurements would proach is due in late 2001. In fact, a separate numerical inte- have been possible and the resulting astrometry could have gration confirms that the computed motion of the asteroid ensured recovery after its discovery apparition. Unfortu- will bring it to within 0.028 AU of the Earth on 31 December nately, 1982 DB exited the Arecibo and Goldstone observa- 2001—the first New Year’s Eve in the next millennium. Us- tion windows prior to discovery. Attempts to recover this ing just the 1982 optical data (i.e., the only data actually object in 1984 and 1986 were unsuccessful and it is now available now), the plane-of-sky and range uncertainties considered lost. during the close Earth approach are 55° and 9.4 X105 km, or

Table VI. Plane-of-sky angular-position uncertainties for 1982 DB on 27 August 1990. No astrometric observations beyond 1982 are assumed. Radar observations were simulated on 14 March 1982 and/or 7 March 1982.

Plane-of-sky Range uncertainties uncertainties (arc-min.) (units of 105 km) Doppler accuracy 2 cm/s 20 cm/s 2 cm/s 20 cm/s Dataset No radar Observations 40.2 40.2 2.06 2.06 1 Doppler obs. 3.5 6.7 0.173 0.340 2 Doppler obs. 2.3 4.9 0.114 0.248 2 Doppler obs. & 1 range obs. 0.69 2.7 0.0404 0.143 (range tiolse = 30 km) 2 Doppler obs. & 1 range obs. 0.69 2.7 0.0402 0.143 (range noise = 0.3 km) 2 Doppler obs. & 2 range obs. 0.68 1.34 0.0400 0.0728 (range noise = 30 km) 2 Doppler obs. & 2 range obs. 0.63 0.64 0.0374 0.0378 (range noise = 0.3 km)

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Fig. 9. 1982 DB: Plane-of-sky position uncertainties versus calendar date. Curves show sensitivity of position uncertainties to Doppler accuracy when only one radar observation is made.

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0.0063 AU. Hence, if this asteroid is not recovered in 1990, Once asteroids are recovered optically on their second its location on the sky during the 2001 close approach will be (and subsequent) apparitions, their orbits are vastly im- essentially unknown and the uncertainty in the closest-ap- proved and generally secure, with or without radar observa- proach distance will be about 25% of the distance prediction tions. However, even for asteroids with very secure orbits, itself. Because of the reasons outlined in Sec. II, even this the several-fold improvement in the accuracy of future ephe- large “formal” uncertainty is likely to be much smaller than merides from radar observations would be potentially valu- the “true” uncertainty. Of course, optical recovery in 1990- able for occultation predictions, dynamical studies, or if the 1992 could reduce these uncertainties by three orders of asteroid should become a target for a future flyby or rendez- magnitude. vous space mission. The percentage improvement in future ephemerides due to the addition of radar data is nearly con- IV. CONCLUSIONS stant for at least a few decades. From an ephemeris-improvement standpoint, the optimal For first apparition, close Earth-approaching asteroids, time to make Doppler astrometric observations with a given radar observations provide an extremely powerful technique level of accuracy is when the Earth-asteroid radial velocity for improving the accuracy of their orbits and future ephe- is largest—unfortunately not at the closest-approach point, merides. For newly discovered asteroids with short observa- where echoes are strongest. When only one Doppler obser- tion intervals, optical astrometric data alone are often not vation is made, the sensitivity of the ephemeris uncertainties sufficient to prevent their being lost. Successful radar obser- to the Doppler accuracy is not as impressive as when two vations could secure such an object’s orbit and ensure its Doppler observations are made. Future ephemeris uncer- recovery during a subsequent Earth approach. tainties are very sensitive to the addition of one range obser-

© American Astronomical Society • Provided by the NASA Astrophysics Data System 200 YEOMANS ETAL. : RADAR ASTROMETRY OF ASTEROIDS 200 vation, but there is little sensitivity to range accuracy and 1983) would render the ratio of actual error to calculated only a modest sensitivity to the addition of a second range error much larger for comets than for asteroids. observation. Our results, which underscore the potential of radar astro- We would like to thank R. F. Jurgens for initiating this metry for improving the future ephemerides of Earth-ap- project and providing helpful discussions. The research de- proaching asteroids, also apply to Earth-approaching, short- scribed in this paper was carried out by the Jet Propulsion period comets. However, various additional systematic Laboratory, California Institute of Technology, under con- sources of error in making the optical measurements and in tract with the National Aeronautics and Space Administra- modeling nongravitational phenomena (Yeomans et al. tion.

REFERENCES Ash, M. E., Shapiro, 1.1., and Smith, W. B. ( 1967). Astron. J. 72, 338. Near-Earth Asteroid Rendezvous (NEAR) Science Working Group Re- Bierman, G. J. ( 1977). Factorization Methods for Discrete Sequential Esti- port (1986). JPL Reorder number 86-7 (Jet Propulsion Laboratory, mation (Academic, New York). Pasadena, CA). Canuto, V. M., Goldman, I., and Shapiro, 1.1. ( 1984). Astrophys. J. 276,1. Newhall, XX, Standish, E. M., and Williams, J. G. (1983). Astron. As- Chandler, J. F. (1979). Determination of the Dynamical Properties of the trophys. 125, 150. Jovian System by Numerical Analysis, Ph.D. thesis (Massachusetts Insti- O’Leary, B. ( 1977). Science 197, 363. tute of Technology, Cambridge, MA). Ostro, S. J. (1985). Publ. Astron. Soc. Pac. 97, 877. Chodas, P. W. (1986). Combined Orbit and Attitude Determination for a Ostro, S. J., Campbell, D. B., and Shapiro, 1.1. ( 1985). Science 229, 442. Low-Altitude Satellite, Ph.D. thesis (University of Toronto, Toronto). Ostro, S. J., Campbell, D. B., and Shapiro, 1.1. ( 1986a). Bull. Am. Astron. Friedlander, A. L., Wells, W. C, Davis, D. R., Housen, K., and Wilkening, Soc. 18, 796. L. L. ( 1979). Asteroid Mission Study, Report No. SAI 1-120-839-M11. Ostro, S. J., Yeomans, D. K., Jurgens, R. F., and Goldstein, R. M. ( 1986b). (Science Applications, Inc., Schaumberg, Illinois). Meteoritics 21, 475. Gehrels, T. (1986a). Minor Planet Circ. No. 11014. Pettengill, G. H., and Shapiro, 1.1. ( 1965). Annu. Rev. Astron. Astrophys. Gehrels, T. ( 1986b). Minor Planet Circ. No. 11327. 3, 377. Gehrels, T., Marsden, B. G., McMillan, R. S., and Scotti, J. V. (1986). Shapiro, I. I., Pettengill, G. H., Ash, M. E., Stone, M. L., Smith, W. B., Astron. J. 91, 1242. Ingalls, R. P.,andBrockelman,R. A. ( 1968a). Phys. Rev. Lett. 20,1265. Gibson, J. ( 1986). Minor Planet Circ. Nos. 11127 and 11221. Shapiro, 1.1., Ash, M. E., and Smith, W. B. (1968b). Phys. Rev. Lett. 20, Helin, E. F. (1982). IAU Circ. No. 3675. 1517. Helin, E. F., and Shoemaker, E. M. ( 1979). Icarus 40, 321. Shoemaker, C. S., and Shoemaker, E. M. ( 1986). IAU Circ. No. 4215. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory (Aca- Shoemaker, E. M. ( 1983). Annu. Rev. Earth Planet. Sei. 11,461. demic, New York). Urata, T. ( 1986). IAU Circ. No. 4186. Jurgens, R. F., and Bender, D. F. ( 1977). Icarus 31, 483. Wetherill, G. W. ( 1976). Geochim. Cosmochim. Acta 40, 1297. Kizawa, M. ( 1986). IAU Circ. No. 4181. Yeomans, D. K. ( 1986a). Minor Planet Circ. No. 10752. Lawson, C. L., and Hanson, R. J. ( 1974 ). Solving Least Squares Problems Yeomans, D. K. ( 1986b). IAU Circ. No. 4220. (Prentice-Hall, Englewood Cliffs, NJ). Yeomans, D. K., Jacobson, R. A., Williams, B. G., and Chodas, P. W. Marsden, B. G. ( 1971 ). In Physical Studies of Minor Planets, edited by T. ( 1983). In Cometary Exploration, edited by T. I. Gombosi (Central Re- Gehrels, NASA SP-267 (U.S. GPO, Washington, DC), p. 639. search Institute for Physics, Hungarian Academy of Sciences, Budapest, Marsden, B. G. (1979). In Asteroids, edited by T. Gehrels (University of Hungary), Vol. 3, p. 95. Arizona, Tucson, AZ), p. 77.