The potential of sparse photometric data in asteroid shape modeling

0.3 MB·English

by  J. Hanuš

The potential of sparse photometric data in asteroid shape modeling
J.Hanusˇ∗,J.Dˇurech
AstronomicalInstitute,FacultyofMathematicsandPhysics,CharlesUniversity,VHolesˇovicˇka´ch2,18000Prague,CzechRepublic.
3 Abstract
1
0 WeinvestigatethepotentialofthesparsedataproducedbytheCatalinaSkySurveyastrometricproject(CSSforshort)
2 in asteroidshape and rotationalstate determinationby the lightcurveinversionmethod. We show thatalthoughthe

n photometricqualityoftheCSSdata,comparedtothedensedata,issignificantlyworse,itisinprinciplepossiblethat
a these data are forsome asteroidswith high lightcurveamplitudessufficientfora uniqueshape determination. CSS
J data are available for ∼ 180 asteroids for which shape models were previously derived from different photometric
0
datasets. For13asteroidsfromthissample,wederivetheir uniqueshapemodelsbasedonlyonCSSdata,compare
3
the two independentshape models together and discuss the reliability of models derived from only CSS data. We

] also use CSS data to determine shape models for asteroids with already known rotational period values, derive 12
P
uniquemodelsandcomparepreviouslypublishedperiodswithperiodsdeterminedfromthefull3Dmodelingbythe
E
lightcurveinversionmethod. Finally, we test differentshape resolutionsused in the lightcurveinversionmethodin
.
h ordertofindreliableasteroidmodels.
p
Keywords: Asteroids,Sparsephotometry,Shapemodels,Lightcurves
-
o
r
t
s 1. Introduction (see KaasalainenandTorppa,2001;Kaasalainenetal.,
a
2001, 2002). LI, in a form we use, leads to a convex
[
Orbitalparametersarecurrentlyknownformorethan shapeapproximationofarealasteroid(closetoitscon-

1 400000 asteroids. On the other hand, rotationalstates
vex hull) that has to be a single body (or the possible
v and shapes were determined only for a small fraction
moonhastobesosmallthatitsphotometricsignatureis
5 of them. In the Minor Planet Lightcurve Database1
0 indistinguishablefromthenoise)inarelaxedrotational
(Warneretal., 2009), periods for ∼ 3500 asteroids
2 state(i.e.,rotatesaroundtheaxiswithamaximummo-
7 are stored. Convex shapes with spin axis directions mentum of inertia). Sidereal rotational period and the
. were derived only for ∼ 200 asteroids, these three-
1 directionofthespinaxisaredeterminedsimultaneously
dimensional models of asteroids are available in the
0 with the shape. Typical number of free parameters is
3 Database of Asteroid Models from Inversion Tech- ∼50–80.
1 niques2 (DAMIT, Dˇurechetal., 2010) maintained by
v: the AstronomicalInstitute of the Charles Universityin Weusetwodifferenttypesofdisk-integratedphotom-
i Prague,CzechRepublic. etry: (i) dense-in-time,whichtypicallyconsistsoftens
X
All asteroid models in DAMIT were derived by the
to a few hundreds of individual data points observed
r lightcurve inversion method (LI). This gradient-based
a duringonerevolution(typicallyseveralhours),and(ii)
method is a powerful tool that allows us to derive ba-
sparse-in-time,wherethetypicalseparationofindivid-
sic physicalpropertiesofasteroids(the rotationalstate
ualmeasurementsislargecomparedtotherotationpe-
and the shape) from their disk-integrated photometry
riod. Forsparse data, we usually have a few measure-
mentspernight, such as in the case of astrometricsky
∗Corresponding author. Tel: [phone]. Fax: +420 surveys. Based on the type of the photometry, we use
221912577. theterms“denselightcurves”and“sparselightcurves”.
Emailaddress:[email protected]() Unique asteroid shape models are typically feasible if
1http://cfa-www.harvard.edu/iau/lists/Lightcurve-
wehavedenselightcurvesfromatleastthreeapparition,
Dat.html
2http://astro.troja.mff.cuni.cz/projects/aste- combineddenseandsparse datagiveusmorevariabil-
roids3D ity.
PreprintsubmittedtoPlanetaryandSpaceScience January31,2013
Our long-term strategy is to enlarge the number of ResponseSystem, Hodappetal., 2004), andlateralso
asteroids with known shapes and rotational states, be- fromthe Gaia satellite (Perrymanetal., 2001), andthe
causeit can helpus in understandingthe physicalpro- LSST(LargeSynopticSurveyTelescope, Ivezic´etal.,
cesses that take place in the asteroid’s populations, 2008). UnderstandingtheCSSdatawithrespecttothe
such as Near Earth Asteroids (NEAs), Main Belt As- asteroidshapemodelingwillspeedupfutureprocessing
teroids (MBAs) or even asteroids in individual fami- anduseofthenewsparsedata.
lies (e.g., Koronis family, Slivan, 2002; Slivanetal.,
2003). Currentdistributionof periodsandspin axesis
2. Photometry
the direct result of the evolutionof these objects start-
ing with theirformationuntilthe presenttime (several For a unique5 shape determination of an asteroid,
hundreds of Myr to ∼ 4 Gyr for most studied aster- wetypicallyneeddensephotometricdatafromatleast
oids). The most prominent processes acting on aster- threeapparitionsandagoodcoverageofthesolarphase
oidsarecollisionsandmassshedding,andtheYORPef- angle6 (i.e., different viewing geometries), ∼ 20 such
fect3(Rubincam,2000;Vokrouhlicky´etal.,2003). The dense lightcurves are usually sufficient. The biggest
knowledge of the shape can be used for several pur- limitingfactorofthesedataistheirinsufficientamount.
poses, e.g.: (i) in the construction of a thermal model Denselightcurveswereobservedfor∼ 3500asteroids.
(e.g., Mu¨lleretal., 2011), where values for geometric Only in ∼ 100 cases, asteroid shape models were de-
albedo, size and surface propertiescan be determined, rived.Determinationofnewmodelsbasedonthedense
(ii)asampleofrealshapesinsteadofsyntheticonescan data is now possible only if new lightcurves are ob-
beusedforthestatisticalstudyofthenon-gravitational served. Manyamateursorsemi-professionalsusetheir
forces(Yarkovsky4andYORPeffects),or(iii)incombi- telescopesforasteroidobservations,buttheirmaingoal
nationwithstaroccultationsbyasteroids. Theseevents is the determination of the synodic period, which can
(observedforhundredsof asteroids)giveus additional bederivedalreadyfromoneapparition. Ifaperiodfor
informationabouttheshape(e.g. non-convexities)and a particularasteroidisthensecure,theyusuallydonot
canlead to a size estimationwith a typicaluncertainty observethisobjectanymore. Wehavetoobservesuch
of10%(see Dˇurechetal.,2011). objectbyourselves.Thisapproachisdemandingonob-
Mostofthecurrentlyavailablephotometricdatawere servational time and also on fundings. Models based
already used in the LI. The only significant excep- only on dense data are important in testing the relia-
tionaredata fromthe CatalinaSkySurveyastrometric bility of models based on sparse data (due to possible
project(CSSforshort, Larsonetal.,2003). Inthispa- comparison).
per,weinvestigatethepossibilityofdeterminationofre- Currently available sparse photometry is accessible
liableasteroidshapemodelsfromonlyasmallamount via the Asteroids – Dynamic Site database7 (AstDyS),
of sparse-in-time distributed low-quality photometric wheredata fromhundredsofastrometricobservatories
measurements(∼ 100)by the LI. Such data fromCSS are stored. The photometry is mostly a by-product of
projectareavailableforseveralthousandsofasteroids. astrometric measurements and in most cases, asteroid
We wantto findoutif thesedataareforasteroidswith magnitudesaregiventoonlyonedecimalplace,sothe
high lightcurve amplitudes of a sufficient amount and accuracyis0.1magatbest. Whetherornorthisissuffi-
qualityforauniqueshapedetermination,andifso,how cientforauniqueshapedeterminationforareasonable
reliable these asteroid models are. The investigation numberofasteroidswasstudiedinHanusˇetal.(2011).
ofsparsedatacapabilitiesandthereliabilityofderived The authors have found 7 observatories with quality
modelsis important, because (i) it can lead to a deter- data and used these data in combination with relative
minationofnewasteroidmodelswithoutaneedofob- lightcurvesforasteroid shape modeling. The most ac-
servinganyadditionalphotometriclightcurves,and(ii) curatesparsephotometryisfromtheU.S.NavalObser-
inafewyears,anotherhugeamountofsparsedatafrom vatoryinFlagstaff(USNO-Flagstaffstation,MPCcode
three astrometric surveys will be available – from the
Pan-STARRS(PanoramicSurveyTelescopeandRapid
5Wedefineauniquesolutionasfollows: (i)thebestperiodhasat
least10%lowerχ2thanallotherperiodsinthescannedinterval,(ii)
forthisperiod,thereisonlyonepolesolutionwithatleast10%lower
3Yarkovsky–O’Keefe–Radzievskii–Paddack effect, a torque χ2thantheothers(withapossibleambiguityforλ±180◦),and(iii)
causedbytherecoilforcefromanisotropic thermalemissionwhich thissolutionfulfillsallouradditionaltests(discussedin Hanusˇetal.,
canalterrotationalperiodsandorientationofspinaxes 2011).
4athermalforceactingonarotatingasteroidwhichcanaltersemi- 6theSun–asteroid–Earthangle
majoraxes 7http://hamilton.dm.unipi.it/astdys/
2
689). These data were alreadystudied in Dˇurechetal. Test 2. The main inconvenientwhen computing mod-
(2009)andbasedonthesedata,theauthorsderivedcon- els from sparse data is thatsynodic periodsare known
vex shape models for 24 asteroids. The biggest dis- only for ∼ 3500 asteroids (determined from dense
advantage of the USNO-Flagstaff station data is that lightcurves and stored in the Minor Planet Lightcurve
they are available only for about 1000 brightest aster- Database).Thisaprioriinformationisusedintheshape
oids. Data from Hipparcos satellite (ESA, 1997) and modeling: we search for the model only near the as-
RoquedelosMuchachosObservatory,LaPalma(MPC sumedperiod(e.g.,wetesttheinitialperiodvaluesfrom
code 950) are also accurate (compared to the other an interval (0.95P, 1.05P), where P is the published
observatories), but their amount of measurements is periodvalue). Typically,wetestahugenumberofini-
evensmallerthanfortheUSNO-Flagstaff station. The tialperiodvalues(usuallythousands),becausethemod-
largest amount of sparse photometry is available from eled parameter space is full of local minims and the
the Catalina Sky Survey observatory (CSS for short, gradient-basedLImethodconvergestothenearestlocal
Larsonetal., 2003). These data were already used in minimumcorrespondingtotheinitialperiodvalue.The
the LI by Hanusˇetal. (2011), but only in combina- minimal differencebetween two local minims is given
tion with other photometric data. Their typical photo- mainly by the timespan of the observational data (see
metric accuracy was ∼ 12%. Although this accuracy Eq. 1) andis usually verysmall (. 10−3 h). Havinga
of the CSS photometry seems very low, the data are short period interval saves a lot of computationaltime
valuableforasteroidswithlightcurveamplitudeshigher andgivesusconfidencethatthemodeliscorrectif the
than & 0.3 mag, which is about a half of all aster- derived period value is close to P. Unfortunately, for
oids (based on the lightcurvedata of the Minor Planet most asteroids, we do not know the period (cannot be
Lightcurve Database, Warneretal., 2009). There are directlydeterminedfromsparsedatawithoutthemodel
dense lightcurves for only ∼ 700 asteroids with num- computationasinthecaseofthedensedata),sowehave
bers over ten thousand, on the other hand, we have at tosearchforthemodelonaperiodintervalofallpossi-
least 100 sparse data points from CSS for ∼ 6000 of bleperiodvalues,typically2–100hours.Thisconsumes
theseasteroids. alotofcomputationaltimeandweloosethepossibility
ofcomparingthetwoperiodvalues.
3. Reliability of asteroid models derived only from
theCatalinaSkySurveyphotometricdata
Inthistestweusedtheaprioriinformationaboutthe
TheCatalinaSkySurveyastrometricprojecthaspro- periods: forsomeasteroids,weknewtheirperiods,we
duced the largest amount of sparse photometric mea- hadonlyCSSdata(thereexiststheirdenselightcurves,
surements. For most asteroids, no other data of suffi- but we do not have them in an electronic form), and,
cientqualityareavailable. based on that data, we were also able to derive their
models near the published period. We extended the
Test 1. One way how to test the reliability of mod- modelsearchontheperiodintervalof2–100hoursand
els based only on CSS data is to compare them with triedtofindthemodelsolutionsagain(aswouldbedone
models derived from different photometric data set. if the period was unknown). This was performed for
So far, about a hundred of asteroid models were de- 12 asteroids. In 7 cases, the previoussolution derived
rived from dense data (e.g., Kaasalainenetal., 2002; on a shorter period interval was reproduced also on a
Torppaetal.,2003;Slivanetal.,2003;Marciniaketal., larger period interval. For 4 asteroids, we did not get
2007; Dˇurechetal., 2007), another ∼ 100 models a unique period solution on the extended period inter-
were derived from combined dense and sparse data val, and a formally correct (i.e., it fulfilled our condi-
(Dˇurechetal.,2009;Hanusˇetal.,2011). Thesemodels tionsforauniquesolution),butadifferentsolutionwas
are believed to be reliable and thus correct. For most found for one asteroid. On Fig. 1, we show a peri-
of these asteroids, CSS data are also available, so we odogram of asteroid (5647) 1990 TZ, where for each
tried to derive their models from only CSS data itself. initialperiodvalueaχ2-valuecorrespondingtothebest
Wehadatleast50individualdatapointsfor185aster- shape model and pole direction (a local minimum in
oids. In 13 cases, we were able to derive their unique themulti-dimensionalparameterspace)isplotted. This
solutions. Onlyin7casesthesolutionwasinanagree- model computation was based on 87 individual mea-
mentwiththemodelbasedonadifferentdataset,other surementsfrom CSS, derivedperiod P = 6.13867h is
sixsolutionswereformallycorrect,butderivedperiods in agreement with a period P = 6.141 h reported by
weredifferentfromtheexpectedones. BembrickandBolt(2003).
3
1
(5547) 1990 TZ
0.8
0.6
2
c
0.4
0.2
Best solution, P = 6.13867 h
0
10 20 30 40 50 60 70 80 90 100
Period P [h]
Figure1: Periodogram ofasteroid (5647)1990TZ.Eachpoint corresponds toaparticular local minimuminthemodeledparameter spaceof
periods,poledirectionsandshapes.
Test 3. Since we derived several unique models that 4. Discussion
wereclearly incorrect,we performedthistest to check
more carefully the stability of the solution and to de- In Test 1, where we compared the derived models
tect false solutions. The shape of an asteroid is repre- withcorrespondingmodelsbasedondifferentdatasets
sentedbycoefficientsofitsexpansionintothespherical (stored in DAMIT), in 7 cases out of 13, the solution
harmonicfunctions.Thesecoefficientsareoptimizedin wasinawellagreementwithamodelbasedonadiffer-
theLI.Thenumberofcoefficientsisgivenbytheorder ent and higher data set. From this point of view, CSS
n of the shape expansion we use, we call n the shape data,whichrepresentsthelargestdatasetofsparsepho-
resolution. In the LI, we typically use n = 6, which tometry with photometric accuracy theoretically suf-
correspondsto49 coefficients((n+1)2). Forasteroids ficient for a unique shape determination, seem to be
with . 100sparse data pointsfromCSS, so manyco- promising for the asteroid shape modeling. Unfortu-
efficientscould be to much and the LI could then lead nately, several clearly wrong models that fulfilled our
to a random unstable solution that is even unique and conditionsonauniquesolutionwerederived(theyhad
physically correct. Another input parameter of the LI differentsiderealrotationalperiodsthanwasexpected).
istheperiodstepP betweentwosubsequentperiods SimilarproblemoccurredinTest2,wherewecompared
step
fromthe scanned interval. The minimaldifferencebe- thederivedmodelswiththeirpreviouslyknownperiods.
tweentwolocalminimsinthemodeledparameterspace Onemodelwasclearlyincorrect.
ofperiodsisgivenby There are several possible explanationswhy we de-
rived incorrect models from the CSS data: (i) data for
P2
∆P= , (1) asteroids with low amplitudes are more noisy, (ii) low
T amountof data with respectto thenumberof modeled
where P is the periodvalue and T the timespan of the parameters,(iii)systematicerrorsofthesurvey.Incases
observationaldata. Inourcomputations,weuseforthe (i)and(ii),thedataarepoor,containnotenoughinfor-
periodstepavalueP =0.8∆P. mationabouttheperiodandshape,andcouldproducea
step
Wetestedthestabilityofthesolutionondifferentval- random solution with an unrealistic rms value (the fit
ues of the resolution n and on two values of the pe- is “too perfect”). Oszkiewiczetal. (2011) show that
riod step P (0.8∆P and 0.5∆P). Table 1 summa- photometric data sets from astrometric stations (such
step
rizes the results of this test for all asteroids in Test 1 as CSS) exhibitmagnitudevariationswith apparentV-
for which we derived their unique solutions for n = 6 bandmagnitude.Forthebrightestasteroids,theimages
and P = 0.8∆P(correspondsto column6). Correct aresaturated, andforthefaintest, thebackgroundsub-
step
uniquesolutionismarkedby“X”, wronguniquesolu- tractionisimperfect. Thisconsequenceof(iii)isprob-
tionby“x”andnouniquesolutionby“–”. ablypartiallyvisibleinourdata–mostoftheincorrect
4
Table1:TestofthestabilityofthemodelsolutionontheshaperesolutionnandthestepintheperiodPstep.Firstcolumngivestheasteroidnumber,
symbol“X”meansthatforagivennandPstep,bothuniqueandcorrectsolutionwasderived,symbol“x”meansthatauniquebutwrongsolution
wasderived,andfinally,inthecaseifnouniquesolutionwasdeterminedweusethesymbol“–”.
P =0.8∆P P =0.5∆P
step step
Ast n=2 n=3 n=4 n=5 n=6 n=2 n=3 n=4 n=5 n=6
685 X X – X X X X – – X
1022 X X – – X X X – – X
1419 X X X X X X X X X X
1568 X X X X X X X X X X
2156 X X X X X X X X X X
3678 – – X X X – – X – X
4483 X X X X X X X X X X
52 – – x – x – – x – –
68 – – – – x – – – – x
216 – – – x x – – – – –
455 – – x x x – – x x x
984 – – – – x – – – x –
1382 – – – – x – – – – –
modelsare brighter(have higherabsolute magnitudes) andP = 0.5∆P. Themodelisthenreliableifweget
step
thanthecorrectones.Wearenotabletodistinguishbe- similaruniquesolutions.
tween the effects of the amplitude and the noise from We rerun the computation for all 8 successfully de-
each other only from the sparse data itself without a rivedmodelsfromTest2withn=3andP =0.5∆P.
step
model computation, so there is no way how to detect Previouslycorrectsolutionswereconfirmed,theincor-
the case (i). The amount of data sufficient for a cor- rectonewasruledout.
rectuniqueshapedeterminationisdependentmainlyon Our method assumes a single body in a relaxed ro-
thetimeepochs(geometryofobservations)andalsoon tation state (rotates around the principal axis with the
the signal-to-noise ratio, which we do not know. So, maximummomentumofinertia).Objectsliketumblers
the sufficient amountof sparse measurements strongly or binary asteroids are not identified by the used form
variesfromoneasteroidtotheother. of the LI, their model determination usually fails be-
cause thesingle periodmodeldoesnotcoverthemore
complex period features. The only exception are fully
As seen from Table 1, models derived correctly for
resolution n = 6 and period step P = 0.8∆P are synchronous close binary systems with similar-sized
step
components,suchas91Antiope(Merlineetal.,2000).
mostlydeterminedforotherresolutions nandbothpe-
Theseobjectsarereproducedashighlyelongatedsingle
riodstepsP . Nowrongsolutionappeared.Inseveral
step
bodies,andbecausetheyhavetypicallyhighlightcurve
cases, a uniquesolutionwas notfound,butthe correct
amplitudes,theycouldbeeasilypresentinthepossible
period was still detected (we had multiple pole solu-
sampleofmodelsbasedonCSSdata.
tions). On the other hand, for incorrect models from
Test1,wegotuniquesolutionsfordifferentresolutions
nandperiodstepsPstep onlyrarelyandneverforn = 2 5. Conclusions
or n = 3. All these uniquesolutionswere always also
incorrect. There is a weak dependence on the period SparsedatafromtheCatalinaSkySurveyarelowef-
step, P = 0.5∆P seems to be more convenient be- ficient in unique model determination and seem more
step
cause when used, less incorrect models were derived. accurate for less brighter asteroids. Incorrectly deter-
Tominimizethedeterminationofincorrectmodelsthat minedmodelscouldbeindicatedbyunrealisticrmsval-
satisfyourrequirementonuniquesolutions,weshould ues,butnotgenerally.
use the values n = 6 and P = 0.5∆P for resolution Correctperiodsolutionsforasingleasteroidbutdif-
step
andperiodstep,respectively.Ifauniquesolutionisthen ferentresolutionnwerealwaysstable,thedispersionin
determined,thecomputationshouldbererunwithn=3 theperiodwastypicallylowerthan∆P(givenbyEq.1),
5
whichisforaperiodvalueof10hoursandthetimespan tronomicalInstituteoftheCharlesUniversityinPrague
of the observations 10 years ∼ 10−3 h (typical values (http://sirrah.troja.mff.cuni.cz/tiger).
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