Asteroid occultations today and tomorrow: toward the GAIA era

0.15 MB·English

by  P. Tanga

Astronomy&Astrophysicsmanuscriptno.occulto˙v5 c ESO2008
(cid:13)
February2,2008
Asteroid occultations today and tomorrow:
toward the GAIA era
8
0
0 PaoloTanga∗1 andMarcoDelbo1
2
1ObservatoiredelaCoˆted’Azur,BP4229,06304NiceCedex4
n
e-mail:[email protected]
a ∗
J

7 Preprintonlineversion:February2,2008
1
ABSTRACT

]
h
Context.Observationofstaroccultationsisapowerfultooltodetermineshapesandsizesofasteroids.Thisiskeyinformationnecessaryfor
p
studyingtheevolutionoftheasteroidbeltandtocalibrateindirectmethodsofsizedetermination,suchasthemodelsusedtoanalyzethermal
-
o infraredobservations.Uptonow,theobservationofasteroidoccultationsisanactivityessentiallysecuredbyamateurastronomersequipped
r withsmall,portableequipments.However,theaccuracyoftheavailableephemerispreventsaccuratepredictionsoftheoccultationeventsfor
t
s objectssmallerthan 100km.
a ∼
Aims.Weinvestigate current limitsinpredictability and observability of asteroid occultations, and westudy their possible evolution in the
[
future,whenhighaccuracyasteroidorbitsandstarpositions(suchasthoseexpectedfromthemissionGaiaoftheEuropeanSpaceAgency)

1 willbeavailable.
v Methods.We use a simple model for asteroid ephemeris uncertainties and numerical algorithms for estimating the limits imposed by the
4 instruments, assuming realistic CCD performances and asteroid size distribution, to estimate the expected occultation rate under different
8 conditions.
6
Results.Weshowthathighaccuracyephemerideswhichwillbeavailableinthefuturewillextendtowardmuchsmallerasteroidsthepossibil-
2
ityofobservingasteroidoccultations,greatlyincreasingthenumberofeventsandobjectsinvolved.Acompletesetofsizemeasurementsdown
.
1 to 10kmmainbeltasteroidscouldbeobtainedinafewyears,providedthatasmallnetworkofground-based1mtelescopesaredevotedto
∼
0 occultationstudies.
8
0
:
v Keywords.asteroids,astrometry,ephemeris,occultations
i
X
r
a 1. Introduction where pV isthegeometricvisiblealbedoandH istheabsolute
magnitudeoftheasteroid.
An accurate knowledgeof asteroid sizes and size distribution
1329
is a fundamental prerequisite for studying the origin and the D(km)= 10−H/5 (1)
collisional evolution of these bodies, including the formation √pV ×
andthedispersionofdynamicalfamilies.Italsohasimportant NotethatHisaquantitythatcanbeeasilyobtainedfrommea-
implications for our understating of the mechanisms that led surements of the apparentbrightness of the object, for exam-
totheformationoftheterrestrialplanetsofourSolarSystem. ple by meansof photometricobservationsin the visible light.
Furthermore, knowledge of the size of an asteroid allows its The distribution of albedos and its correlation with the taxo-
bulkdensitytobeinferredwhenthemassofthebodycanbein- nomictypesandtheorbitalelementsofasteroidsisakeypiece
dependentlydetermined,forinstancefromtheorbitalperiodof ofdata inunderstandingtheirnatureandtheir origin,andcan
anasteroidsatellite.Informationaboutthedensityofasteroids help to constrain the mineralogical composition of their sur-
iscrucialtostudytheirinteriors,andtoconstrainthecollisional faces. Albedos are also crucial for developingand improving
processesleadingtotheformationandevolutionofthediffer- reliable and truly debiased asteroid population models (e.g.
ent populations of these minor bodies (e.g. Brittetal.2002). Tedescoetal.2005,StuartandBinzel2004).
Moreover, if the size of an asteroid is known, the albedo of Asteroidflybysofspaceprobeshaveprovidednotonlythe
the body can be derivedusing Eq. 1 (e.g. Bowelletal.1989), best“direct”sizedetermination,butalsoawealthofinforma-
tion on shapes and surface propertiesfor a very limited num-
berofthesebodies.However,astatisticallysignificantsample
ofasteroidsizescanbeobtainedonlybyremoteobservations.
Sendoffprintrequeststo:P.Tanga Up to now, mainly indirectmethods have been used to deter-
2 PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow.
mineasteroiddiameters:thevastmajorityofasteroidsizesand thispaper,wealsostudytheobservingrateofasteroidocculta-
albedos have been derived from observations obtained in the tionsexpectedforafixedobservingstationusingaone-meter-
thermalinfrared( 12-100µm)bytheInfraredAstronomical classtelescopeequippedwithacooledCCDcameracapableof
∼
Satellite(IRAS;Tedesco1992,Tedescoetal.2002).However, ahighframerate.
itisknownthatIRASdiametersofasteroidsarenotexemptof Afterhavingassessedthepresentstateofoccultationstud-
systematicerrors.TheywerederivedbymeansoftheStandard ies(Sect. 1)we analyzethe theoreticaloccultationrate thatis
ThermalModel(STMofLebofskyandSpencer1989), whose expected for different asteroid sizes (Sect. 2). We then study
resultsaredependentonthevalueofamodelparametercalled the role of instrumentdetection capabilities(Sect. 3) and that
the beaming factor and indicated with η. The value of η was of ephemeris accuracy (Section 4). The implications of high-
calibrated by forcing the STM to give the correct diameters accuracyephemerisarethendiscussed(Section5).
(as derived from occultation observations) of the asteroids 1
Ceresand2Pallas(Lebofskyetal.1986).Severalstudieshave
2. Theobservationofasteroidoccultation:where
highlighted, though, that a value of η appropriate for large
wearenow
main belt asteroids, might lead to an underestimation of the
true size when applied to small asteroids whose surface ther- 2.1.Observationalcampaigns
mal characteristics are likely to be different from those of
The first occultation ever recorded is that of 3 Juno, in 1958.
larger bodies (e.g. Spenceretal.1989, Delbo´ &Harris2002,
The thirdone(433Eros),observedin1975,wasalso the first
HarrisandLagerros2002).
tobeobservedbymultiplesites,allowingthedeterminationof
Calibrationofasteroidradiometricdiameterswaspossible
different “chords” of the asteroid (Millis and Dunham 1989).
because,inthepasthalfcentury,thesizesofthelargestaster-
Only17positiveoccultationobservationswereobtainedbythe
oidswerealsoderivedbymeasuringthedurationofstaroccul-
endof1980.
tations.Thisapproachhastraditionallybeenflawedbythepoor
Coordinatedcampaignsandlast-minutesmall-fieldastrom-
accuracyofthepredictionoftheasteroidshadowpathonEarth,
etry, aimed both at improvingthe occultationpath and at dif-
thuskeepingthesuccessrateofasteroidoccultationverylow.
fusingtherelevantpredictionsamongobservernetworks,have
Theavailabilityofprecisestarastrometryobtainedbythemis-
provideda moderateincrease in the rate of successfully mea-
sionHipparcosoftheEuropeanSpaceAgency(ESA),hasim-
sured asteroid occultations events. Up to the early 1990s the
provedthesituationinthelast 10years(Dunhametal.2002),
∼ probabilityofobservinganoccultationchordforagivenevent
stronglyincreasingthenumberofsuccessfullymeasuredaster-
was discouraging. However, consistent efforts were rewarded
oidoccultationchords.Electronicequipments(fastvideocam-
with a total of 60 positive events by 1985, growing to 101 in
eras) have also increased the accuracyof occultation timings,
1990, and increasing more rapidly later, thanks to the avail-
previouslyobtainedmainlybyvisualobservation.However,as-
ability of more precise star astrometry. After the publication
teroidoccultationobservationshaveremainedmainlyanactiv-
oftheHipparcosandTycho-2starcatalogues(Høgetal.2000)
ity reserved to observers equipped with small, movable tele-
thenumberofsuccessfullyrecordedoccultationeventsdoubled
scopes and equipment, ready to accept a rather high rate of
in the decade 1991-2000 (see below for an evaluation of the
“miss”.
currentpredictionaccuracy).Attheendof2004theysummed
Inthisstudy,afterhavingreviewedthepresentstateofas-
up to 658, and today they continueto grow at a rate of about
teroidoccultationstudies,wewillfocusonaverylikelyevolu-
100peryear(Dunham&Herald2005).
tionofthisobservationalapproach,thanksto thefutureavail- ∼
Despitetheapparentlylargenumberofobservedevents,the
abilityofasteroidorbitsandstarpositionswhoseuncertainties
extremely limited numberof chordsfor most of the observed
will be reduced by orders of magnitude. In this respect, the
events results in diameters that remain poorly constrained.
astrometricESAmissionGaia(Perrymanetal.2001)willrep-
Also, the poor knowledge of shapes, rotationalpoles, and ro-
resent a major milestone that will completely change the ap-
tational phases at the epoch of the eventsoften makes the in-
proachto asteroid occultationstudies, openingthe way to the
terpretationofobservationsdifficult.Inpractice,theuseofoc-
possibility of obtaining a more complete mapping of asteroid
cultation diameters found in current datasets1 is not straight-
sizes and albedos. In this paper we address this forthcoming
forward. As an example, ShevchenkoandTedesco2006 were
improvement,exploringthepaththattheapproachto asteroid
able to use just 57 occultation diameters for albedo determi-
occultationswilltakeinthefutureandinvestigatingthelimits
nations, among which only 18 have an estimated accuracy
thatfutureobservationwillbeabletoapproach.
<5%.ShevchenkoandTedesco2006comparedtheir”occulta-
Note thatindirectmethodsof asteroidsize andalbedode-
tionalbedos”againstthealbedosderivedfromIRASobserva-
termination,suchasthosebasedonpolarimetryandthermalin-
tionsinthethermalinfrared,andagainstthepresentlyavailable
fraredobservations,willgreatlybenefitfromoccultationstud-
polarimetricalbedos.However,becausetheirdatasetcontains
iesperformedonalargersampleofasteroids:itwillbepossi-
occultationsizes onlyforasteroidslargerthan 100km,they
ble, for example, to calibrate models used to analyze thermal ∼
werenotabletoexploreanysize-dependenceofsystematicer-
infraredandvisualpolarimetricmeasurementsonalargerand
rorsofpolarimetricandthermalinfraredalbedos.
more accurate sample of calibration asteroid diameters, with
objectsizesextendingdowntosometensofkilometers. 1 Interested readers should refere to the PDS asteroid occultation
Since betterperformingtelescopeswillbe neededto push dataset (AOD), available from the NASA Planetary Data Systems
sizedeterminationtowardsmallerandthusfainterasteroids,in SmallBodiesNode:http://www.psi.edu/pds/archive/occ.html.
PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow. 3
Asteroidoccultationobservations-stronglyencouragedby uncertaintyonthepositionattheepochforwhichtheosculat-
professional astronomers - are supported and secured by or- ingorbitalelementsarecalculated.
ganizations grouping mainly volunteering amateurs, such as NotethattheCEUgivesthelengthofthemajoraxisofthe
the International Occultation Timing Association (IOTA) or uncertainty ellipse (usually very slender) representing the set
the EuropeanAsteroid Occultation Network (EAON), strictly of positionsonthe sky wherethe objectcan befound,at a 1-
cooperatingwith – and including– professionalastronomers. sigma confidence level. This axis is usually lying on the line
Today, for stars currently included in the predictions of as- of variations, or at a small angle with it. The main effect of
teroid occultations (usually brighter than V=12 and listed in this orientation will thus be a major uncertainty on the event
UCACorHipparcoscatalogues)themainsourceofprediction epoch,notontheshadowpathinthedirectionperpendicularto
uncertainties resides in the accuracy of asteroid ephemerides theasteroidshadowmotion.Thislatteruncertaintyisthedom-
(asdiscussed in the nextsection).Currentuncertaintiesin the inatingoneasfarasthe observabilityoftheoccultationevent
ephemerides of asteroids are rarely below 0.5 arcsec (ex- is concerned;in fact, a shift in the predictedepochcan easily
∼
cept for the largest bodies), corresponding to a displacement be compensated by an increased duration of the observation.
of 350 km on the Bessel plane for an object at 1 AU from It is thus likely that, by retaining the CEU as an estimate for
∼
theEarth.ThecorrespondinguncertaintyonEarthhassimilar thepositionuncertaintyweare,ingeneral,ratherconservative.
values, thus requiringa large numberof appropriatelyspaced The exception is probably represented by a small number of
observerstoincreasethesuccessprobabilityofactuallyobserv- largeasteroids,havingasmallerandmorecircularuncertainty
ing the occultation.Furthermore,in the comingyearsa slight ellipse,forwhichtheCEUcorrespondmorestrictlytothepre-
degradation in the accuracy of occultation prediction can be dictionaccuracy(Bowell,privatecommunication).
expected, as a consequence of the uncertainty on the proper Undertheseassumptions,wecanthenstatethatwithaCEU
motionsofstars. largerthantheapparentangularsizeoftheasteroid,occultation
The Euraster network data2 show that in the year 2005, eventsaredifficulttopredict.Wethereforeusedtheratioofthe
15%oftheoccultationsobservedunderaclearskywassuc- CEU to the apparent angular radius θ of the asteroid (given
∼
cessful.Thissimpleevaluation,however,doesnotincludeany by the ephemeris at the CEU epoch) as a statistical indicator
informationon the real position of the observerswith respect for the current predictability of occultation events. For other
to the predicted path on Earth, nor a distinction based on the epochs,ofcourseindividualCEUvalueswillbedifferent,but
presenceofotherdegradingfactors,suchasasmallfluxdrop, theoverallstatisticsremainvalid.
ephemeris uncertainty, short duration, and bad atmospheric For our computations, the apparent sizes of asteroids θ
conditions.Itwouldthusbedifficultandtime-consumingtoes- were calculated from their IRAS diameters included in the
tablishamoreprecise“average”successrate,sinceitdepends SupplementalIRASMinorPlanetSurvey(SIMPS)ofTedesco
ondetailsthatwouldneedacase-by-caseevaluation. etal.(2002).NotethattheSIMPScontainsdiametersforonly
the first 2200 numbered asteroids. For the large majority of
∼
bodiesinoursample,diametersarethusnotknown.Forthem,
2.2.Currentaccuracyofasteroidephemeris
weestimatedsizesfromasteroidH magnitudes(whicharein-
cludedintheastorb.datfile)andanassumedgeometricvisible
The uncertainty in asteroid ephemeris can be evaluated by
albedoviaEq.1.However,theuseofaconstantalbedothrough
using information provided by the Bowell’s Asteroid Orbit
theentireMainBeltwouldbeanoversimplification:thealbedo
Database(ftp://ftp.lowell.edu/pub/elgb/astorb.html).Theaster-
rangeforMBAsisverylarge(0.03-0.5),anditisknown,for
oidephemerisuncertaintyinthissourceiscalculatedbymeans
instance,thataninversecorrelationbetweenalbedoandhelio-
of the procedure described by Muinonen and Bowell (1993),
centric distance exists. We used, therefore, the procedure de-
anditisexpressedintermsofasmallsetofparameters,namely
scribed by Tedesco et al. (2005) to assign realistic albedosto
the CEU (Current Ephemeris Uncertainty), the PEU (Peak
thenumberedasteroidsnotincludedintheSIMPSdatabase.
Ephemeris Uncertainty, and its date of occurrence), and the
According to this model (and using the numbering as in
PEUoverthefollowing10years.Allquantitiesarereferredto
Tedescoetal.2005),themainbeltisdividedintothreezones:
theconstantlyupdatedepochoftheosculatingorbitalelements.
i.e.zone2for2.064 < a < 2.50;zone3for2.50 < a 2.82;
ThemethodadoptedforthecalculationoftheCEUisconsid- ≤
zone 4+5 for 2.82 < a < 3.278, where a is the orbital semi-
ered to give meaningfulestimates of the currentephemerides
major axis of the asteroid. To assign an albedo, we first di-
uncertaintyformulti-oppositionMain BeltAsteroids(MBAs)
videMBAsaccordingtothezonetowhichtheybelong(2,3or
but,ontheotherhand,itisassumedtobeunreliableforEarth
4+5)and to their mean apparentvisible magnitudeat opposi-
crossers or asteroids with poorly determined orbits. Since we
tion,givenbyV(a,0)= H+5log(a(a 1)).Eithertheobserved
focus our study on the observation of occultations events of −
orthebias-correctedalbedodistributionofTedescoetal.2005
MBAs 3 we can assume that the CEU is representativeof the
(Table 3) is assigned to an asteroid, depending on whether
the value of V(a,0), is < 15.75 or 15.75, respectively.
2 http://www.euraster.net/results/index.html ≥
The same table gives the zonal distribution for four different
3 AccordingtoTedescoetal.(2005)wedefineMBAasthoseaster-
classes of albedos:i.e. low (0.020 < p 0.089),intermedi-
oidshavingorbitsbetweenthe4:1and2:1mean-motion resonances V
≤
with Jupiter, i.e., those with orbital semimajor axes between 2.064 ate (0.089 < pV 0.112), moderate (0.112 < pV 0.355),
≤ ≤
and3.278AUandwithmodestinclinations(<250)andeccentricities andhigh(0.355< pV 0.526).Wethenweightthenumberof
≤
(<0.3). asteroidsaccordingtothecorrespondingzonalalbedodistribu-
4 PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow.
Fig.1. Plot of the ratio CEU/θ versus the object diameter (in km, lower X-axis) for the numbered main belt asteroids known
andobservableatthedateofwriting:Feb20,2007.Thesolidcurve,superimposedontheplot,showsthecumulativenumberof
asteroids(upperX-axis)whichhaveaCEU/θratiosmallerorequalthanthecorrespondingvalueoftheY-axisontheleft-hand
sideofthefigure.Thecurveclearlyshowsthatthereareabout80-90asteroidswithaCEU/θvaluesmallerthan1.Weexpect
that Gaia will improvethe accuracy of asteroid ephemerisprediction by a factor 100 (dotted horizontalline), increasing by
∼
almostthreeordersofmagnitudesthenumberofasteroidsforwhichCEU θ.
∼
tionandrandomlydistributealbedosuniformlywithineachof multi-epochastrometryofasteroids,willwebeabletoextend
thealbedoclasses. occultationstudiesofminorbodiestomuchsmallersizes,with
thepossibilityofderivingoccultationdiametersforseveraltens
Fig. 1 shows the resulting ratio CEU/θ for all numbered
ofthousandofasteroids.Theall-skysurveyprogramthatwill
MBAsknownandobservableatthedateofwriting(astorb.dat
have the best ever astrometric accuracy is the ESA mission
file of Feb 20, 2007) as a function of their diameter. We as-
Gaia,foreseentobelaunchedattheendof2011.
sumed thatan asteroid is “observable”(in a broadsense) if it
has a solar elongation 50o. The distribution shows that only
≥
very few asteroids (less than 90) – mainly brightMBAs with
diameterslargerthan50km–havearatioCEU/θsmallerthan
However,the capability of systematic asteroid occultation
one. This roughly correspondsto the number of asteroids for
observingcampaignsofprovidingaccuratesizedetermination
whichwecanexpecttodaytomakereliablepredictionsofoc-
canbejudgedonlyifwe canestimatetheobservationaleffort
cultation events, or stated in another way, their occultations
required. Thus, we will calculate in the following section the
haveahighprobabilityofbeingobservedinsidethepredicted
theoreticalasteroidoccultationrateforagivensiteonEarth,as
shadowpath.Animprovementofoccultationpredictionsis at
a function of the asteroid diameter. Later, ephemeris and star
presentstillpossiblethroughlast-minuteastrometry,aimingto
positionuncertaintieswillbetakenintoaccounttoestimatethe
constrainthetrajectoryoftheasteroidwithrespecttothetarget
successrateofasteroidoccultationmeasurementsconsidering
starinthehoursimmediatelyprecedingtheevent.
instrumentswithdifferentperformances,fromthoseofa20cm
It is probable that only by the forthcoming all-sky as- telescope,uptothethecaseofaone-metertelescopeequipped
tronomical surveys, which are capable of providing accurate withastate-of-the-artCCDcamera.
PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow. 5
3. Anumericalmodelforcomputingthe
2 5
occultationrate
log (# of asteroids)
10
Inordertoreproducearealisticoccultationrateforasteroidsof 1
differentsizes,weneedtoestimatehowmanystarseachaster- 4
sIoiaotbnhnisbiytgdetisoeoeenrortanrhoduccvseiemchedrourp,baftlr2ootesosttrθnhedc,ioeraunEaoflccnbaiastdurdottbauldhiartennyh.sttiewettTcchtotaehdhailenimitecssntutatediaalitmeannrh(neteeceeeas.degdiiatnc.wopywatonpeinotinarhttfvrehhbieasneedntlta,hattserhaeykaasne)ysndf.diagalibTryffluveelhlieiatcasniwrisaregoblendniclweseciuntaeielhftmaoqhteothruieefsneitdakveottyragnbhwleeyoeseonwfnadtetpartearhekotpiyoaec--t. log(Occultations/object/year)10---3210 VVVssstttaaarrr===112680 23og(Number of asteroids in the bin)10
V =14 l
sweptbyeachasteroideveryday,weextractedthestellarden- star 1
sityfromtherealisticgalacticmodelusedbyCrosta&Mignard -4
V =12
star
(2006).Inthismodelthenumberofstarspersquaredegree,as
afunctionoftheirmagnitudeandtheirgalacticcoordinates,is -5 0
10 100
expressedin termsof a setofChebyshevpolynomials,whose Diameter (km)
coefficientsweredeterminedbyafittothemeasuredstarcount.
Fig.2. Number of occultations per object during one year, as
Note that, in practice, the numberof stars occulted every day
afunctionoftheasteroiddiameter(thinlines).Thesestatistics
byeachasteroidismuchsmallerthanone,sotodecidewhether
arevalidforasinglesiteandarecomputedfordifferentvalues
anoccultationeventtakeplaceornot,wegeneratedarandom
ofthelimitingmagnitudeasindicatedbythelegend.Thethick
number uniformly distributed between zero and one, and we
curve(scaleontheright-handsideyaxis)representsthenum-
tookas positive eventsthose cases wherethe randomnumber
ber of asteroidsfor each bin in the simulated population.The
was found to be greater or equal to the number of occulted
productofthislastnumberwitheachrate/objectcurveprovides
stars.Ifthetestwaspositive,theepochoftheoccultationevent
thetotalnumberofoccultationsperyearforagivensizerange.
was generated randomly inside the time step interval. We re-
peated this procedure every day over a period of 5 years, in
ordertoconsiderdifferentapparitionsforeachMBA,thusin- temporaryfluxdropproducedbytheoccultation.Furthermore,
cludingdifferentskypatchesvisitedbyeachbody. since the scientific goal that interests us is the impact of oc-
We assumed our observer to be situated at the geocenter. cultation observations on asteroid size determination, we re-
Thischoiceisnotalossofgenerality,sincetheoverallstatistics strictedourstudytoeventsfromwhichthethesizeofanaster-
willbethesameforanygivensiteonthesurfaceoftheEarth. oidcanbedeterminedwithinagivenuncertaintylevel.
However, we divided by a factor of 2 the number of occulta- We investigate the observability of the events by compar-
tionsobtained,totakeintoaccountthehalfcelestialspherevis- ingportableequipment,similartothosecurrentlyusedbymost
ibleinthepresenceofatheoretical,geometricflathorizon.One observers in present days, to a completely different situation,
shouldnote thatthiscorrectioncouldbe different(andproba- i.e.thatofa fixed1-mclasstelescopeequippedwitha cooled
blylarger)inpractice,duetothedifficultyofobservingevents CCD camera. With the larger instrument, much fainter stars
atlowskyelevations.Moreover,tocalculateasteroidpositions (andmuchsmallermagnitudedrops)canbemeasured,extend-
we solved the unperturbed two-body problem: planetary per- ingtheinvestigationofoccultationeventstowardsmalleraster-
turbationsrepresenta levelof refinementnotrequiredfor our oids.Aswewillsee,thiswillinturnallowtheexploitationof
purposes.Asforasteroiddiameters,thesame valuesassigned improvedasteroidephemerisandstellarastrometry.
as describedin Sect. 2.2were adopted.Finally we computed,
foreachofthesimulatedevents,themaximumdurationofthe
4. Themeasurementofanoccultation:limits
occultationτ using the objectangulardiameterand apparent
o
imposedbytheinstrument
motionattheoccultationepoch,theasteroid’sapparentmagni-
tudema,andthemagnitudedropduringtheoccultation. The observation of an occultation event consists of the mea-
The obtained results, of course, have a statistical value. surementofthetemporalfluxvariation,bymeansofrapidpho-
TheyarepresentedinFig.2,whereweshowtheaveragenum- tometry,ofasinglesourcecomposedbytheunresolvedimages
berofoccultationeventsthatoccurforasinglesiteandasin- ofthe star andtheasteroid.A fluxdropoccursduringthe oc-
gleobject,duringoneyear.Itisclearthatincludingfaintstars cultationwhenonly theasteroid hidingthe star is observable.
stronglyenhancestheoccultationrate.Infact,wefindthatany Thedurationoftheoccultationistypicallyofafewsecondsfor
object larger than 20-30 km occults a 20th magnitude star at large(50-100km)asteroids,andthemostfavorableevents,of
leastonceperyear. course,involveabrightstarandafaintasteroid.Inthiscase,the
However,thetheoreticaloccultationratethatwehavecom- fluxdropcanreachseveralmagnitudes.We will usethe sym-
puted so far represents a theoretical value based on geometry bolsm ,m ,andm toindicaterespectivelythemagnitudeof
a s sa
only.Itdoesnottakeinto accountthe“observability”of each theasteroid,themagnitudeofthestarandthemagnitudeofthe
event,thatisthecapabilityofagiveninstrumenttorevealthe unresolved source star + asteroid immediately before and af-
6 PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow.
tertheoccultation.Forthe correspondingmeasuredfluxeswe equipped with a rapid, uncooled video camera (30 frames
adoptthenotationF(m ),F(m ),andF(m ),respectively. per second,exposure1/30sec) with ratherhighreadoutnoise
a s sa
To obtain useful size estimates of asteroids from the ob- (15e /pixel).We consideredaCCD operatingat290oK, ata
−
servationofanoccultationevent,theuseofperformingequip- pixel scale 0.9 arcsec/pixel, with a seeing FWHM around 1
∼
ment,capableofexposuretimesconsiderablyshorterthanthe arcsec.SkybackgroundwasassumedtobeV=12mag/arcsec2.
maximumeventduration,andsensitiveenoughtoallowaccu- WiththehighCCD framerateimposed,thedominatingnoise
rate photometric measurements of the flux drops is required. source is not dark current, but the read-outnoise. The sensor
Still, a single observation of an occultation will not yield, in temperature is thus of secondary importance. By using aper-
general,a directmeasurementofthe object’ssize. This is be- turephotometrywithanoptimizedaperturesize,weobtainthe
causethemeasuredoccultationchorddoesnotnecessarilycor- resultsshowninFig.3.Theintegraloftheoccultationratefor
respond to the largest projected size of the asteroid occulting objectsd >40kmyieldsnomorethanafeweventsobservable
the star. Moreover,in general, the three-dimensionalshape of athighaccuracyeachyearforasinglesite.Thisappearstobe
theoccultingasteroidanditsorientationshouldbeknownfora fullyconsistentwiththepresentobservationstatistics.
meaningfulinterpretationoftheresultsobtainedfromocculta-
tionmeasurements.
As for the definition of the signal to noise ration (S/N)
of the photometric measurements of the occultation event,
Number of occultations/object observable at different accuracies
we adopt the ratio between the amplitude of the flux drop 1
<40%
(F(m ) F(m )) and the photometric uncertainty associated <20%
as a <10%
− <5%
atwShb/ieltNehcFAoflt=ofhrurtreexeFaqrflsa(hgpuσmnaioxvd2Fvaned(siadt)nnuh)ig−rne+iingnciFsσmhgpt(or2Fhm(au(sσogameastiF)n)one(.amgna)n)teeastqirennuitcdsiuptupromunuamcetnseneidrtdnteacftoihs(nraσetrteyFaa.((csacteshe)e)reitbsvhteeielcnoostw,cdct)ehuwteleteraamvttihaoi(niu2nles-): Occultations / object / year (elong. > 50 deg.) 10 e.00 -.000 0.00000.11115
computed – for each event – the minimum exposure time t
m
providingS/N 3. Assuming that the absolute timing is avail- 1e-006
∼ 10 100
able for the CCD frames with negligible uncertainty,we take Diameter (km)
t as the estimate of the occultation timing uncertainty, i.e.
m
σ = t . Since the maximum transit duration is proportional
τ m
totheobjectdiameterD,theratioσ = σ /τ (whereτ rep- Fig.3. Number of observable occultations per object during
D τ o o
resent the maximum occultation duration, on the center line) one year, as a function of the asteroid diameter, for a 20 cm
will represent the relative accuracy expected for the diameter telescope equipped with a cheap fast-readout camera. These
determination. statistics are valid for a single site and are computed for dif-
Oneshouldnotethat,accordingtotheS/Ndefinitionabove, ferentvaluesofthemaximumrelativeuncertaintyonsize,σD,
anexposuretimeyieldingS/N 3willalwaysbefound,butin asindicatedbythelegend.
∼
some cases it will be comparable to – or significantly longer
than – the occultationduration.However,we will consider as
“non–interesting”thoseeventshavingarelativeuncertaintyon The simulation has also been run considering a 1-m tele-
the finaldiameterdeterminationσ > 40%.By adoptingthis scope coupled to a low-noise, photon-countingCCD camera.
D
constraint,“unobservable”events(i.e.thoserequiringt τ ) These cameras, based on an electron multiplier in the read-
m o
arediscarded.Wethenclassifiedtheremainingeventsas≤func- out register, operate at very low temperatures ( 180 oK) and
∼
tionofσ . presently represent the state–of–the–art technology for astro-
D
Ofcourse,discardedeventscouldstillbeofscientificvalue, nomical use. Considering a larger pixel (22 µm) and a faster
forexampleforthesearchofasteroidsatellites,howeverinthis focal ratio (f/6), the image scale (0.76 arcsec/pixel) is not far
studywedonotaddressindetailthisotherwiseinterestingsub- fromthatofthe20cmtelescopeequippedasabove.Read-out
ject. noise is consideredto be (1 e−/pixel);all other conditionsare
The predictions of occultation events that are currently notvariedrelativelytothepreviousnumericalexperiment.The
distributed to observers include target stars not fainter than increase in camera performance and photon flux provides an
V=12-13, thus reducing the occultation rate at about 0.1 improvementinthenumberofeventsatagivensizerange,by
events/objet/yeareven for the largest bodies. As a further se- afactorof1to2ordersofmagnitude(Fig.4).
lection,requirementsontheminimumoccultationdurationand Inthefollowingsection,weusethestatisticsobtainedwith
magnitude drop mean that to only the 1000 asteroids larger thesetwoextremeinstrumentsetupsasareference,andwein-
∼
than 40 km are taken into account. We ran a first simula- cludein ouranalysisthe contributionof uncertaintiesonboth
∼
tion considering the performance of a 20 cm, f/10 telescope asteroidsandstarpositions.
PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow. 7
In a secondstep, we computethe ratio p = 2θ/σ , (we
path
recallherethat2θ is the apparentasteroiddiameter)averaged
Number of occultations/object observable at different accuracies
100 this over all the events occurring in a given diameter range.
<40%
<20% This can be considered to be a “prediction efficiency factor”.
<10%
Occultations / object / year (elong. > 50 deg.) 0 0. 00. 001. 11110 <5% apscwaaIsfttecaaairmrttwtonruhthaoudape,irgalhonalearhiyslnsdgyicdstebhoiudsbeompmtseeininuvmpse=cgcilaaaocetathtren1xiebeoasslsdtiinoGemditfpseathopotrdruefroleiisoisbbdutesbhpuaisnasaeebtcenaisrieorstlodrav1iinttontie-,yaasns.iattlienhrgnIrvituneambdienetuonhrascttcetheeeocacredrdunlotet,iaσlnhtsttaiyhfiaaenps,ntaredttitvytenoheharinnneooσhtcginearpCdeuvsaatEteelehhntenpoUeGvrthoe.ebaoebpclTeumcasr(hsecen6tseisuras8irvekflai%tocnasenaotltnti)lauditoonohaolenwddnes-f,
viation. Thus p cannot be considered more than a reasonable
0.0001 guess of the success probability for each event. On the other
10 100
Diameter (km) hand, one could also note that 1/p, for p < 1, is an estimate
ofthenumberofeventsthatshouldbeobservedtowitnessone
positive event, on average. For those events with p 1 the oc-
∼
Fig.4. Number of observable occultations per object during cultation prediction is accurate and, in those cases where we
oneyear,asafunctionoftheasteroiddiameter,fora1mtele- have p >> 1 the occultationpath is knownso accuratelythat
scopeequippedwithacooled,low-noise,fast-readoutcamera. itcouldbeevenpossibletodeterminethepositionofthechord
Thehistogramiscomputed,asinFig.3,fordifferentvaluesof relativetotheobjectbarycenter.
σ . In the case of the after-Gaia statistics, different assump-
D
tions are neededfor both CEU and σ . Since the dependence
∗
of the ephemeris accuracy on the asteroid size should pre-
5. Roleofastrometricaccuracy serve the general features shown in Fig. 1, we simply as-
sume an improvement factor of 100 in the ephemeris uncer-
Uptonowwehavediscussedthestatisticsofoccultationevents
tainty of each object, i.e. a post-Gaia value of the CEU re-
andtheirobservabilityasifwewereabletoknowasteroidand
ducedbya factorof100with respecttothepresentone.This
starpositionswithinfiniteaccuracy.Inreality,theyareaffected
isareasonableassumptioninagreementwithdifferentanalyt-
byastrometricuncertainties.Asaconsequence,inprinciple,an
ical estimates and simulations of the improvements in aster-
observerhastomonitoralargernumberofcandidateeventsin
oidorbitalelementsthatGaiawillallow(Virtanenetal.2004,
orderto reach the countof the positive occultationspredicted
Muinonenetal.2004, Hestroffer&Berthier2005). We also
by the theoretical rate. The number of events that need to be
assume that the end-of-missionaccuracy on star positions (in
monitored can be estimated by assuming as “impact section”
mas)canberepresentedby:
for each asteroid the apparent size given by its diameter plus
thewidthoftheephemerisuncertaintyattheoccultationepoch. σ (mas)=4 10 2e0.44(V 20)+1.35, V >12 (3)
− −
Addinganuncertaintyonthestarpositionwillfurtherincrease ∗ ×
the“impactsection”. whereVisthestarmagnitude.Avalueσ =17µasisobtained
Totakethiseffectintoaccount,foreacheventwecomputed atV=154.Inthiscaseσ saturatestothev∗aluereachedatV=12
∗
anestimateofthetotaluncertaintyontheoccultationpathas: forbrighterstars.
Theresultofthecomputationof pisgiveninFig.5forthe
σ = σ2 +σ2, whereσ is assumed to be equalto
path q CEU ∗ CEU present and post-Gaia asteroid ephemerides and star position
theCEUfortheasteroiddiscussedabove,whileσ represents
uncertainties.Itisclearthatthepresentoccultationpredictabil-
∗
the uncertainty on the star position. The prediction lists usu-
ityappearsreasonable(p>0.1)forthelargestMainBeltaster-
allydiffusedtotheamateurcommunityincludethestarsofthe
oidsonly(D>100km),butinthe“post-Gaia”eraanequivalent
catalogues FK6 (I+III), Hipparcos, Thyco 2, and UCAC2 for
accuracycanbereacheddownto 15km.
starsofmagnitude10<V<12.5(Dunham&Herald2005).For ∼
stars with V<10 the UCAC2 positions, if available, are used
to replace those given in the Thyco 2. A reasonable estimate 6. Discussionandperspectives
of star position uncertainties is of about 1 mas for the FK6,
Letusconsidera10-20kmasteroidasarepresentativeexam-
10masfortheHipparcos,70masforTycho2,and20masfor
ple. By multiplyingthenumberofobjectsin themodelatthe
theUCAC2,forsourcesintherangeofmagnitudesV 10-14,
∼ correspondingbin(104)withthecurrentnumberofoccultation
whereasthe uncertaintyis of about40 mas outside this range
per object per year for a small instrument (Fig. 3), one finds
(Zacharias et al. 2004). For each simulated event we thus at-
thatabout0.1occultations/yearoccurata5-10%levelofsize
tributetotheoccultedstaranuncertaintytypicalofThyco-2or
UCAC2 stars,with equalprobability(thisroughlyreflectsthe 4 This accuracy is obtained from simulations of the on-board de-
current distribution of catalogue exploitation for predictions, tection(Arenou&Lim2003)andfromtheexpectedmissionperfor-
seeDunham&Herald2005). mances.
8 PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow.
1
1
Prediction efficiency factor (p) 0 0.0.11 Fraction of events (15-25 km asteroids) 0.1
today
post-Gaia
0.001 0.01
10 100 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0
Diameter (km) V star
Fig.5. Predictionefficiencyfactorfor“presentday”and“post
Gaia”accuraciesonasteroidephemerisandstarpositions. 1
ab(Fcecigou.bra5sec)ryivsefosdor(oalonwsianv(g∼elr0ea.g0sei0t)e2t.)oFthouabrtttahsieenvrmeornoaerlesh,uupcnrcdeerdseisdc.tIieotvniesnetthffisuscshieosniumcldy- m asteroids) 0.1
opd(Vcliycc<tuiOa1mlbt2naip)ltit,oithosyoesnbiiobdsstleihearre,amvrwtahebhiettlaerhenrstgdwhoi,iniosftohdtehrqi(taus∼hsisept0iem.zr2seoa)eimrndaatsenn,tdgwoeeiqb.otuhrbiitgpDahmint≃easn1tcta0c,r0uorokcacmctceuuratlhstaetet0irpoo.r0nied1s- Fraction of events (15-25 k 0.01
∼
times per object per year (at high accuracy).Monitoring sev-
eral objects by several sites will be sufficient to obtain, over
a period of years, a sample of observed positive events. This 0.001 0.2 0.4 0.6 1.0 1.7 2.8 4.6 7.7
is compatiblewith a lower limit at 100km for the currently Max occultation duration (s)
∼
availableoccultation-deriveddiameters.
Oneshouldnotethatobservingoccultationswithlargerdi-
Fig.6. Distributionsof the occulted star V magnitude(upper
ameterscanimprovetheaccuracyofphotometry,andincrease
panel)andofthedurationoftheevents(lowerpanel)foralloc-
thenumberofcandidateeventsbyincludingsmallerfluxdrop.
cultationsinvolvingasteroidsinthediameterrange15-25km.
However,withpresentdaysasteroidephemeridesuncertainty,
the predictability of the occultation events remains poor,thus
preventingefficientobservationsatsmallsizes. theasteroidsizedistributionforpredictableoccultations.Also,
Much more interesting is the case of the post-Gaia sce- we computedthat 70%of the eventshave a flux droplarger
∼
nario,in which asteroid orbitsare improvedby a factor 100 than1magnitude.However,inordertoexploitthefullpotential
∼
and stellar astrometry makes a revolutionarystep forward for of asteroid occultationsin the post-GAIA era, more powerful
star brightness down to V 20. In fact, in this case the pre- instruments must be devoted to this activity. Figure 4, shows
∼
diction efficiencyremains p=1 (or larger)down to a diameter that – by multiplying the predicted rate by the corresponding
of20km.Thisimpliesnotonlyaperfectpredictabilityofthe number of objects – 20 to 50 occultations/year are theoreti-
eventsthemselves,butalsoanuncertaintylowerthanthewidth callyobservablewiththehighestsuccessprobabilitiesforeach
oftheoccultationshadowonEarth.Inotherwords,observers 10-20kmobject,byusinga1m telescope,withdiameterun-
ofoccultationsofasteroidwith D >20kmwillhavesomein- certainties<10%.Allknownobjectsinthissizerangewilllie
dicationsof the position of the chordrelative to the projected within the CEU/θ 1 limit if a two-orders of magnitude or-
∼
shapeoftheobject.Thisrepresentsamajorchangeintheway bitimprovementisreachedafterGaia(Fig.1).Thisresulthas
asteroidoccultationscanbeapproached,andcouldmakethem impressive implications, since it means that, in principle, all
anevenmorepowerfulinstrumentforasteroidstudies. objects larger than that limit could be measured by monitor-
InFig.6weprovidethedistributionoftheoccultationdu- ing occultations. Furthermore, the relatively large number of
rations and star magnitudes for events involving asteroids in events could allow a single site to collect good statistics for
the 15-20 km range. Comfortably long occultations and rela- eachobject,thusallowinganaccuratediameterdetermination.
tivelybrightstarsarestillpresenteventowardthesmallendof In practice, the large number of asteroids involved (104) im-
PaoloTangaandMarcoDelbo:Asteroidoccultationstodayandtomorrow. 9
pliesthatafewhundredeventsshouldbemonitoredeachday. Crosta,M.T.,Mignard,F.2006,Class.QuantumGrav.,23,4853
Ofcourse,thisisahugeeffortforasingleinstrumentorteam, Delbo´,M.,&Harris,A.W.2002,MeteoriticsandPlanetaryScience,
butit remainsfeasibleif spreadovera few yearsorif several 37,1929
siteswithsimilarinstrumentsarecooperating. Dunham, D.W.and Goffin,E.,Manek, J.,Federspiel, M.,Stone, R.,
Owen,W.2002,Mem.S.A.It.73(3),662
Thenumberofeventsatsmallerasteroiddiameterswillstill
Dunham,D.W.andD.Herald2005.AsteroidOccultationList,NASA
be interesting, although completeness will be out of reach. A
Planetary Data System, EAR-A-3-RDR-OCCULTATIONS-
selection of objects with the lowest ephemerides uncertainty
V2.0:OCC-OCCLIST-200405
will represent a good sample for extending size distributions
Harris,A.W.,Lagerros,J.S.V.,2002,inAsteroidsIII,ed.W.Bottke,
towardkm-sizedasteroids.
A. Cellino, P. Paolicchi, R.P. Binzel, (Univ. of Arizona Press,
Gaia will not only provide accurate astrometry, but also Tucson)205
photometry and spectroscopy. Experiments of lightcurve in- Hestroffer, D., Berthier, J. 2005 in ESA SP-576: The Three-
versionfromsparedata(Cellinoetal.2006)showthatshapes DimensionalUniversewithGaia,297
and rotation vectors could be retrieved for potentially all the Høg,E.,Fabricius,C.,Makarov,V.V.,Urban,S.,Corbin,T.,Wycoff,
main belt asteroids from Gaia observations. Earth-based sur- G.,Bastian,U.,Schwekendiek,P.,Wicenec,A.2000,A&A,355,
veys such as Pan-STARRS could provide a significant con- L27.
Lebofsky, L. A. and Spencer, J. R. 1989, in Asteroids II, ed. R. P.
tribution, although photometric accuracy will be much lower.
Binzel(Univ.ofArizonaPress,Tucson)128
However,the availability of shapes and physicalephemerides
Lebofsky,L.A.,Sykes,M.V.,Tedesco,E.F.,Veeder,G.J.,Matson,
willallowextractionofamaximumofinformationfromoccul-
D.L.,Brown,R.H.,Gradie,J.C.,Feierberg,M.A.,Rudy,R.J.
tationstudies,sincetheorientationoftheasteroidasprojected
1986,Icarus68,239-251.
ontheskyattheoccultationepochwillbeknown.
Mignard,F.2005.inESASP-576:TheThree-DimensionalUniverse
Severalasteroidsatelliteswillalsobediscoveredandtheir withGaia5.
relativeobitsdetermined,thusenrichingevenmorethepoten- Millis, R.L., Dunham, D.W. 1989. in Asteroids II, ed. R. P. Binzel,
tialoffurtheroccultationevents. (Univ.ofArizona,Tucson)148
Inconclusion,allthestatisticalargumentspresentedinthis Muinonen,K.,Bowell,E.1993,Icarus104,255
study indicate that, when Gaia observationswill be available, Muinonen, K.,Virtanen,J.,Granvik, M., Laakso,T. 2004, ESASP-
asmallsetofdedicated1mtelescopeswillbeabletoprovide 576:TheThree-DimensionalUniversewithGaia,223
Perryman,M. A.C.,deBoer,K.S.,Gilmore,G.,Høg,E.,Lattanzi,
– within a few years – a complete census of asteroid diame-
M.G.,Lindegren,L.,Luri,X.,Mignard,F.,Pace,O.,deZeeuw,
ters down to 10 km in size. The equipment foreseen to be
∼ P.T.2001,A&A,369,339
used for future asteroid occultation studies such as fast, low
Shevchenko,V.G.,Tedesco,E.F.2006,Icarus184,211-220.
noise CCD cameras,representthe state-of-the-artpresent-day
Souza,S.P., Babcock, B.A., Pasachoff, J.M., Gulbis, A.A.S., Elliot,
technology, but it is reasonable to predict that in 10 years
J.L.,Person,M.,Gangestad,J.W.2006,PASP,118,1550
∼
from now (the after-Gaia era) even more optimized and per- Spencer,J.R.,Lebofsky,L.A.,Sykes,M.V.1989,Icarus78,337-354.
formingdeviceswillbeavailable.Today,lownoiseCCDcam- Stuart,J.S.,Binzel,R.P.2004,Icarus170,295-311.
erasarebeginningtobeusedinprofessionaloccultationequip- Tedesco,E.F.,Cellino,A.,Zappala´,V.2005,Astron.J.,129,2869.
ment(Souzaetal.2006) and we can expectthat they will en- Tanga, P. 2005, in ESA SP-576: The Three-Dimensional Universe
counterawidediffusioninthefuture.Consideringtheconstant withGaia,243.
progressmadeinparallelbyinstrumentsaccessibletothenon- Tedesco, E. F., Noah, P. V., Noah, M., Price, S. D. 2002, Astron. J.
123,1056-1085.
professionalcommunity,itisalsohighlylikelythattheroleof
Tedesco E. F., ed. (1992). The IRAS Minor Planet Survey. Tech.
dedicatedamateur astronomerswill continueto be a precious
Rpt. PL-TR-92-2049. Phillips Laboratory, Hanscom Air Force
sourceofscientificinformationforasteroidstudies.
Base, Massachusetts (online version available at pdssbn.astro.
umd.edu/sbnhtml/asteroids/physical param.html)
7. Acknowledgments Virtanen, J., Muinonen, K., Mignard, F., 2004, ESA-SP576: The
Three-DimensionalUniversewithGaia,325
The work of Marco Delbo was supported by the European Zacharias,N.,Urban,S.E.,Zacharias,M.I.,Wycoff,G.L.,Hall,D.
SpaceAgency(ESAExternalFellowshipProgram).Aspecial M.,Monet,D.G.,Rafferty,T.J.2004,Astron.J.127(5),3043-
thankgoesto F. Mignardforhavingprovidedthe star density 3059
model,andforthealwaysstimulatingdiscussions.
References
F.Arenou,J.C.Lim2003,GaiaTechnicalNote,ESA,OBD-FAJCL-
001
Bowell,E.,Hapke,B.,Domingue,D.,Lumme,K.,Peltoniemi,J.,and
Harris, A. W. 1989, in Asteroids II, ed. R. P. Binzel, (Univ. of
Arizona,Tucson)524
Britt, D. T., Yeomans, D., Housen, K., Consolmagno, G. 2002,
AsteroidsIII485-500.
Cellino,A.,Delbo`,M.,Zappala`,V.,Dell’Oro,A.,&Tanga,P.2006,
Adv.SpaceRes.,38,2000