Asteroid 2013 ND15: Trojan companion to Venus, PHA to the Earth

1.3 MB·

by  C. de la Fuente Marcos

Mon.Not.R.Astron.Soc.000,000–000(2014) Printed5February2014 (MNLATEXstylefilev2.2)
Asteroid 2013 ND : Trojan companion to Venus, PHA to the Earth
15
⋆
C. de la Fuente Marcos and R. de la Fuente Marcos
UniversidadComplutensedeMadrid,CiudadUniversitaria,E-28040Madrid,Spain
4
1
0
2 Accepted2014January20.Received2014January19;inoriginalform2013October18

b
e
F ABSTRACT
Venushas threeknownco-orbitals:(322756)2001CK32, 2002VE68 and 2012XE133. The
4 first two have absolute magnitudes 18 < H < 21. The third one, significantly smaller at
H = 23.4 mag, is a recent discoverythat signals the probable presence of many other sim-
]
P ilar objects: small transient companions to Venus that are also potentially hazardous aster-
E oids (PHAs). Here, we study the dynamical evolution of the recently discovered asteroid
. 2013 ND15. At H = 24.1 mag, this minor body is yet another small Venus co-orbital and
h PHA,currentlyclosetotheLagrangianpointL4andfollowingthemosteccentricpathfound
p sofarforobjectsinthisgroup.ThistransientTrojanwillleavethe1:1meanmotionresonance
-
o within a few hundredyears althoughit could be a recurrentlibrator. Due to its high eccen-
r tricity (0.6), its dynamics is different from that of the other three known Venus co-orbitals
t even if they all are near-Earth objects (NEOs). A Monte Carlo simulation that uses the or-
s
a bitaldata anddiscoverycircumstancesofthe fourobjectsasproxiesto estimate the current
[ sizeofthispopulation,indicatesthatthenumberofhigh-eccentricity,low-inclinationVenus

co-orbitalNEOsmayhavebeengreatlyunderestimatedbycurrentmodels.Threeoutoffour
2
v knownobjectswerediscoveredwithsolarelongationatperigeegreaterthan135◦evenifvisi-
3 bilityestimatesshowthatlessthan4percentoftheseobjectsareexpectedtoreachperigeeat
1 suchlargeelongations.Ourcalculationssuggestthatthenumberofminorbodieswithsizes
0 above150mcurrentlyengagedinco-orbitalmotionwithVenuscouldbeatleastoneorderof
5 magnitudelargerthanusuallythought;thenumberofsmallerbodiescouldeasilybeinmany
1. thousands.Thesefigureshavestrongimplicationsonthe fractionof existingPHAsthatcan
0 barelybedetectedbycurrentsurveys.Nearly70percentoftheobjectsdiscussedherehave
4 elongationatperigee<90◦and65percentareprospectivePHAs.
1
: Keywords: celestialmechanics–minorplanets,asteroids:individual:2002VE68 –minor
v planets,asteroids:individual:2012XE133 –minorplanets,asteroids:individual:2013ND15
i
X –minorplanets,asteroids:individual:322756–planetsandsatellites:individual:Venus.
r
a
1 INTRODUCTION theyarealsomembersofthenear-Earthobject(NEO)population
thatexperiencescloseapproachestoourplanet.Theminimumor-
Venus hasno known satelliteslarger than about 0.3km inradius
bitintersectiondistances(MOIDs)withtheEarthof322756,2002
(Sheppard&Trujillo2009).Asforunboundcompanions,multiple
VE68 and 2012 XE133 are 0.08, 0.03 and 0.003 au, respectively.
secularresonancesmakethepresenceofprimordial(long-termsta-
Thispropertymakesthemobjectsofsignificantinterest.
ble) Venusco-orbitals veryunlikely (Scholl,Marzari &Tricarico
Potentiallyhazardousasteroids(PHAs)arecurrentlydefined
2005). In contrast, numerical simulations consistently show that
ashavinganEarthMOIDof0.05auorlessandanabsolutemagni-
present-daytemporaryco-orbitalcompanionstoVenusshouldexist
tude,H,of22.0orless(Marsden1997);followingthisdefinition
(Mikkola&Innanen1992;Michel1997;Christou2000;Tabachnik and at the time of this writing, there are 1451 known PHAs.1 If
& Evans 2000; Brasser & Lehto 2002). This theoretical prospect
thesizelimit(>150m,foranassumedalbedoof13percent)is
hasbeenconfirmedwiththeidentificationof(322756)2001CK32, loweredorignored(hereafter, thePHAregime),thereareseveral
ahorseshoe-quasi-satelliteorbiter(Brasseretal.2004),2002VE68, thousandsof knownPHAs.Loweringthesizelimitappearstobe
a quasi-satellite (Mikkola et al. 2004) and 2012 XE133, a former wellsupportedbytheanalysisoftheeffectsoftherecentimpactof
L5 Trojanmovingintoahorseshoe orbit(delaFuenteMarcos& asmallasteroidincentralRussia,theChelyabinskEvent(seee.g.
delaFuenteMarcos2013).Besidesbeingtemporarilytrappedina
Brownetal.2013),whichdemonstratesthatevensmallerobjects
1:1meanmotionresonancewithVenus,alltheseobjectsareMer-
(under20m)canstillcausesignificantdamageiftheyhitdensely
curygrazers,VenuscrossersandEarthcrossers.AsEarthcrossers,
⋆ E-mail:nbplanet@fis.ucm.es 1 http://neo.jpl.nasa.gov/neo/groups.html
2 C. dela FuenteMarcos andR. delaFuenteMarcos
populatedareas.Inanycase,asteroids322756and2002VE68have anduncertaintiesinTable1istheJetPropulsionLaboratory(JPL)
absolutemagnitudes18<H <21;therefore,2002VE68isaPHA Small-BodyDatabase.2
instrictsenseaccordingtothecurrentdefinition. Its very small relative semimajor axis, |a − aVenus| ∼
Discardingaprimordialoriginfordynamicalreasons,thegen- 0.0002±0.0002 au (the smallest found so far), makes this object
esisofthisgroupofNEOsisfarfromwellunderstood.Theseob- aclear candidate tobe aVenus co-orbital. It completes one orbit
jects exhibit resonant (or near-resonant) behaviour with Mercury around the Sun in 224.79 d or 0.6154 yr when Venus does it in
(9:23), Venus(1:1) and theEarth(8:13), following rather chaotic 224.70dor0.6152yr.Itscurrentorbitisbasedon21observations
paths with e-folding times of the order of 100 yr. Close encoun- withadata-arcspanof26d.Asexpectedofrecentdiscoveries,the
terswiththethreeinnermost planets make thispopulation intrin- qualityoftheorbitsofboth2012XE133and2013ND15iscurrently
sicallytransient;therefore,oneormoremechanismsshouldbeat lowerthanthatoftheothertwoco-orbitals.Itissimilar,however,
worktomitigatethelosses,repopulatingthegroup.Christou(2000) totheoneavailablewhentheothertwoobjectswereidentifiedas
predictedaquasi-steady-statefluxofminorbodiesmovinginand unbound companions to Venus back in 2004 (de la Fuente Mar-
outoftheco-orbitalstatewithVenus.Morais&Morbidelli(2006) cos&delaFuenteMarcos2013).ThisobjecthasH =24.1mag
explained the existence of this population within the steady-state (assumedG=0.15)oradiameterof40to100mforanassumed
scenarioenvisionedbyBottkeetal.(2000,2002)whereNEOsare albedointherange0.20–0.04.
constantlybeingsuppliedfromthemainasteroidbelt.Predictions As a Venus co-orbital candidate, the key parameter to study
fromtheirmodelappeartobeconsistentwiththeavailableobser- istheoscillationofthedifferencebetweenthemeanlongitudesof
vationalevidencebuttheyimplythatcurrentsurveyshavealready theobjectandVenusorrelativemeanlongitude,λr.Themeanlon-
reachedcompletenessforH <21.However,somecrucialobserva- gitudeof anobject isgiven byλ=M +Ω+ω,where M isthe
tionalfactshavenotyetbeentakenintoconsideration.Forinstance, meananomaly, Ωisthelongitudeoftheascendingnodeandω is
howefficientareongoingsurveysexpectedtobeinthecaseofob- theargumentofperihelion.Anobjectisco-orbitalifλr oscillates
jectsthatspendmostofthetimeatsolarelongations<90◦?Are (librates)aroundaconstantvalue;ifλr cantakeanyvalue(circu-
theknownobjectsjustthetipoftheiceberg,representinganunex- lates)thenwehaveapassingobject.Foradditionaldetailsonco-
pectedabundance ofsmallminorbodieswhoseorbitsmakethem orbitaldynamicalbehaviourseee.g.Namouni(1999)andChristou
companionstoVenusandPHAstotheEarth? (2000).
In this paper, we show that the recently discovered asteroid To confirm the co-orbital nature of 2013 ND15 with Venus,
2013ND15isanothersmalltransientVenusco-orbitalinthePHA we have carried out N-body simulations using the Hermite inte-
regime(withaMOIDof0.007au,butanH=24.1mag).Together grator(Makino1991;Aarseth2003)inbothdirectionsoftimefor
with 2012 XE133 (H= 23.4 mag), they may signal the presence 10kyr.ThestandardversionofthisdirectN-bodycodeispublicly
of asignificant population of small transient Venus co-orbitalsin available from the IoA web site.3 Relativistic terms and the role
thePHAregime.Toexplorethisinterestingpossibility,theorbital of the Yarkovsky and Yarkovsky–O’Keefe–Radzievskii–Paddack
propertiesoftheseobjectsandtheirdiscoverycircumstancescanbe (YORP)effects(seee.g.Bottkeetal.2006)areneglected.Thenon-
usedtoestimate(i)theirchancesofbeingfoundbyongoingNEO inclusion of these effectshas no impact on theassessment of the
surveysand,indirectly,(ii)theactualsizeofthispopulation.This current dynamical statusof thisobject but mayaffect predictions
paperisorganizedasfollows.InSection2,wepresenttheavailable regardingitsfutureevolutionanddynamicalpast.Inparticular,the
dataon2013ND15 andbrieflyoutlineournumericalmodel.Sec- Yarkovsky effectmayhaveanon-negligible roleonthemedium-
tion3focusesonanalysingthepast,presentandfuturedynamical andlong-termevolutionofobjectsassmallas2013ND15.Proper
evolutionoftheobject.Section4providesacomparativedynami- modellingoftheYarkovskyforcerequiresknowledgeonthephys-
calanalysiswiththeotherthreeVenusco-orbitals.Somerelevant icalpropertiesof theobjectsinvolved(for example, rotationrate,
issuesregardingtheobservabilityoftheseobjectsarediscussedin albedo,bulkdensity,surfaceconductivity,emissivity)whichisnot
Section5.OurconclusionsaresummarizedinSection6. thecasefor2013ND15.Detailedobservationsobtainedduringfu-
tureencounterswiththeEarthorfromtheGaiamissionshouldbe
abletoprovidethatinformationandeventuallyimprovethemod-
ellingpresented here (seee.g. Carbognani et al. 2012; Cellino&
Dell’Oro2012;Delbo´ etal.2012).
Our model Solar system includes the perturbations by the
eight major planets, treatingthe Earth and the Moon as two sep-
2 DATAANDNUMERICALMODEL
arateobjects,thebarycentreofthedwarfplanetPluto–Charonsys-
Asteroid2013ND15 wasdiscoveredbyN.Primak,A.Schultz,T. temandthethreelargestasteroids.Weuseinitialconditions(posi-
GoggiaandK.ChambersobservingforthePan-STARRS1project tionsandvelocitiesinthebarycentreoftheSolarsystemreferredto
fromHaleakalaon2013July13(Wainscoatetal.2013).Theob- theJD2456600.5epoch,t=0coincideswiththisepoch)provided
jectwasfirstobservedusinga1.8mRitchey–Chretientelescopeat by the JPL’s HORIZONS ephemeris system (Giorgini et al. 1996;
anapparentwmagnitudeof20.5;itwasre-observedinsubsequent Standish 1998).4 Relative errors in the total energy at the end of
nightsfromCerroTololoObservatoryandwiththe3.6mCanada– thesimulationsare< 2×10−14.Additionaldetailscanbefound
France–Hawaii Telescope from Mauna Kea. With a value of the indelaFuenteMarcos&delaFuenteMarcos(2012,2013).
semimajoraxisa=0.7235au,veryclosetothatofVenus(0.7233 In addition to the calculations performed using the nominal
au),thisAtenasteroidisanNEOmovinginaveryeccentric,e= orbital elements in Table 1, we have completed 100 control sim-
0.61,andalittleinclined,i = 4◦.8,orbitthatmakesitaMercury,
Venus and Earth crosser. Therefore, its orbit is slightly different
from those of the three previously known Venus co-orbitals (see 2 http://ssd.jpl.nasa.gov/sbdb.cgi
Table1):(322756) 2001CK32,2002VE68 and2012XE133.The 3 http://www.ast.cam.ac.uk/∼sverre/web/pages/nbody.htm
source of the Heliocentric Keplerian osculating orbital elements 4 http://ssd.jpl.nasa.gov/horizons.cgi
A transientTrojancompanionto Venus 3
Table1.HeliocentricKeplerianorbitalelements(andthe1σuncertainty)ofasteroids2013ND15,(322756)2001CK32,2002VE68and2012XE133(Epoch
=JD2456600.5,2013-Nov-4.0;J2000.0eclipticandequinox.Source:JPLSmall-BodyDatabase).
2013ND15 322756 2002VE68 2012XE133
Semimajoraxis,a(au) = 0.7235±0.0002 0.725457880±0.000000004 0.7236543902±0.0000000005 0.72297±0.00014
Eccentricity,e = 0.6115±0.0006 0.3824353±0.0000002 0.41033045±0.00000005 0.4332±0.0003
Inclination,i(◦) = 4.794±0.009 8.13201±0.00002 9.005960±0.000013 6.711±0.008
Longitudeoftheascendingnode,Ω(◦) = 95.84±0.02 109.5014±0.0002 231.579688±0.000005 281.095±0.006
Argumentofperihelion,ω(◦) = 19.69±0.02 234.0956±0.0002 355.463807±0.000014 337.085±0.007
Meananomaly,M(◦) = 357.57±0.06 1.35441±0.00009 123.30681±0.00003 351.6±0.2
Perihelion,q(au) = 0.2811±0.0005 0.44801718±0.00000011 0.42671696±0.00000004 0.4098±0.0003
Aphelion,Q(au) = 1.1660±0.0002 1.002898585±0.000000006 1.0205591818±0.0000000007 1.0361±0.0002
Absolutemagnitude,H(mag) = 24.1 18.9 20.5 23.4
greater than the usual value of +60◦. The libration centre is dis-
placedfromthetypicalequilaterallocationforeccentricorbits;the
short-termevolutionof theresonant angle,λr,isdisplayedinthe
B-panelsofFig.2.
All the integrated control orbits for 2013 ND15 exhibit Tro-
janlibrationwithrespecttoVenusatt=0.Asanexample,Fig.2
displaystheshort-termdynamicalevolutionofanorbitarbitrarily
closetothe nominal one (central panels) and thoseof tworepre-
sentative worst orbits which are most different from the nominal
one.Theorbitlabelledas‘dynamicallycold’(left-handpanels)has
beenobtainedbysubtractingthricetheuncertaintyfromtheorbital
parameters(thesixelements)inTable1.Itisindeedthecoldest,dy-
namicallyspeaking,orbit(lowestvaluesofa,eandi)compatible
withthecurrentvaluesofitsorbitalparameters.Incontrast,theor-
bitlabelledas‘dynamicallyhot’(right-handpanels)wascomputed
byaddingthreetimesthevalueoftheuncertaintytotheorbitalel-
ementsinTable1.Thismakesthistrajectorythehottestpossiblein
dynamicalterms(largestvaluesofa,eandi).Allthecontrolorbits
Figure1.Themotionof2013ND15 overthetimerange(-200,50)yris exhibitconsistentbehaviourwithinafewhundredyearsoft = 0.
displayedprojectedontotheeclipticplaneinacoordinatesystemrotating Asteroid2013ND15isindeedaVenusTrojanaccordingtothecur-
withVenus(nominalorbitinTable1).TheorbitandpositionofVenusare rent observational uncertainties or at aconfidence level >99 per
alsoindicated. cent.However, itsTrojandynamical statusisonlytemporaryand
thetadpoleorbitthatitcurrentlyfollows(seeFig.1),shortlived.
ulationsusingsetsof orbitalelementsobtained fromthenominal
oneswithintheaccepteduncertainties.Thecomputedsetofcontrol
orbitsfollowsanormaldistributioninthesix-dimensional orbital Ourcalculations(seeFigs3and4,left-handpanels)showthat
parameter space. These synthetic orbital elements arecompatible 2013ND15hasalreadyremainedinthe1:1commensurabilitywith
withthenominal oneswithinthe3σ deviations(seeTable1)and Venusforafewthousandyears.Duringthistime-span,theobject
reflecttheobservationaluncertaintiesinastrometry.Theshort-term may have been a Trojan (L4 and L5), a quasi-satellite (λr = 0◦)
dynamicalevolutionoftheentiresetofcontrolorbitsisconsistent and a horseshoe librator (λr = 180◦). Its future orbital evolution
withtheonederivedfromthenominalorbit. strongly depends on the outcome of flybys with our planet dur-
ingthenext250–500yr.AverycloseencounterwiththeEarthat
nearly0.003 au maytake placeabout 250 yr fromnow. Asteroid
3 CURRENTDYNAMICALSTATUSANDEVOLUTION 2013ND15 maybecomeapassingobjectaftersuchanencounter
butrecurrentco-orbitalepisodesareobservedformostcontrolor-
The motion of 2013 ND15 over the time range (-200, 50) yr as bits.Thisisconsistentwithitspastorbitalevolution.Afuturebetter
seeninacoordinatesystemrotatingwithVenusprojectedontothe orbitmayhelptoreducetheuncertaintiesinthepredicteddynami-
eclipticplaneisplottedinFig.1(nominal orbitinTable1).This calevolutionbutthefactisthatmuchbetterorbitsstilltranslateinto
minorbodyisaVenusco-orbitalcurrentlyfollowingatadpoleor- ratherchaoticorbitalevolutionbothinthepastandthefuturefor
bit around Venus’ Lagrangian point L4; it is, therefore, a Trojan (322756)2001CK32and2002VE68(seee.g.delaFuenteMarcos
(seee.g.Murray&Dermott1999).Duetoitssignificanteccentric- &delaFuenteMarcos2012,2013).Forobjectsfollowingintrinsi-
ityandinaccordancetotheoreticalpredictions(Namouni,Christou callyirregularpaths,animprovedorbitdoesnotmakemedium-or
&Murray1999; Namouni&Murray2000),thelibrationangleis long-termpredictionssignificantlybetterormorereliable.
4 C. dela FuenteMarcos andR. delaFuenteMarcos
Figure2.Comparativeshort-termdynamicalevolutionofvariousparametersforanorbitarbitrarilyclosetothenominaloneof2013ND15 asinTable1
(centralpanels)andtworepresentativeexamplesoforbitsthataremostdifferentfromthenominalone(seethetextfordetails).ThedistancefromtheEarth
(A-panels);thevalueoftheHillsphereradiusoftheEarth,0.0098au,isdisplayed.Theresonantangle,λr (B-panels).Theorbitalelementsa−aVenus
(C-panels),e(D-panels),i(E-panels)andω(F-panels).Thedistancestothedescending(thickline)andascendingnodes(dottedline)appearintheG-panels.
Earth’sandMercury’saphelionandperiheliondistancesarealsoshown.
4 ASTEROID2013ND15INCONTEXT the‘-’signisforthedescendingnode.Thepositionsofthenodes
areplottedintheG-panels.Theevolutionovertimeofthelocation
Fig.4displaysthecomparativeevolutionoftheosculatingorbital
ofthenodesof(322756)2001CK32,2002VE68and2012XE133is
elementsandother parametersof interestofalltheknown Venus
verysimilar;theyremainconfinedbetweenMercury’sandEarth’s
co-orbitals (nominal orbits in Table 1). It is clear that the higher
aphelia, 0.4667 au and 1.0167 au, respectively. In sharp contrast,
eccentricity(mainly)andslightlylowerinclinationoftheorbitof
thenodesof2013ND15 exhibitawideroscillationrange,0.3–1.2
2013 ND15 are responsible for the obvious differences observed au.Thelowerinclinationtranslatesintomorefrequentflybyswith
in the figure. One common feature is also evident, all these ob-
theEarth.Theseencountersarenotasdeepasinthecaseofsome
jectsswitchbetweenresonantstatesrelativelyfrequently.Transfers
oftheotherthreeobjectsbuttheyoccurmoreoften.Asteroid2013
betweentadpole,horseshoeandquasi-satelliteorbitsaretriggered
ND15 seemstobeanextremeversionof2012XE133 anditsorbit
bycloseencounterswiththeinnerplanetsandthosearetheresult
isevenmoreunstable.
ofthelibrationofthenodes(Wiegert,Innanen&Mikkola1998).
Chaos-inducedtransferfromoneLagrangepointtoanotherappears
tobeadynamical propertycommontotemporarilycapturedTro-
jansoftheterrestrialplanets(Schwarz&Dvorak2012).
5 BUT,HOWMANYAREOUTTHERE?
Foranobjectfollowinganinclinedpath,closeencounterswith
major planets are only possible in the vicinity of the nodes. The Themodel inMorais &Morbidelli (2006) onlystrictlyapplies if
distance between the Sun and the nodes is given by r = a(1− H <22anditpredictstheexistenceoftwoVenusco-orbitalswith
e2)/(1±ecosω),wherethe‘+’signisfortheascendingnodeand absolute magnitude brighter than 21 mag. There are two known
A transientTrojancompanionto Venus 5
approximation (e.g. Murray & Dermott 1999) to find the Earth-
objectMOIDorperigee.Thenweestimatethesolarelongationat
perigee(assumedtobethemostlikelydiscovery time)andstudy
theresultingstatistics.TheorbitoftheEarthusedinthissimulation
wascomputedatEpochJD2456600.5 bythe HORIZONSsystem.
Therandomvaluesoftheorbitalparametersoftheobjectsfollow
thetrendsobserved inTable1,namely0.720< a(au)<0.727,
0.3< e <0.7and0< i(◦) <10,withbothΩandω∈(0,360)◦.
Theresultsof107 testorbitsareplottedinFig.5(thevalueofthe
geocentricsolarelongationiscolourcoded).Onlyperigeesofless
than0.1au(∼91percent,65percentareinthePHAregime)are
considered.Thefractionoftheseorbitsreachingperigeewithelon-
gation<90◦(dawntodusksky)is72.5percent;only3.6percent
oftheobjectsreachperigeewithelongations>135◦.
Ourresultsconcerningthevisibilityoftheseobjectsarecon-
sistent with those in Morais & Morbidelli (2006); however, our
interpretation is rather different. Assuming that the brightest ob-
Figure3.Theresonantangle,λr,forthenominalorbitof2013ND15 in jects are the easiest to spot and that, at elongations > 135◦, the
Table1(redline)andtwoofthecontrolorbits.Thebluelinecorrespondsto discoveryefficiencyofNEOsurveysapproaches100percent(but
aparticularcontrolorbitthatwaschosenclosetothe3σlimitsoitsorbital seebelow),thediscoveryofonesinglebrightobject,2002VE68,
elementsaremostdifferentfromthenominalones.Thethirdorbit(green atelongation150◦.9in12yearsofobservations stronglysuggests
line)hasosculatingelementswithin1σofthoseofthenominalorbit.
that at least 28 other similarly bright objects are still waiting for
discovery. Besides, southern declinations under -30◦and northern
declinationsabove+80◦areoutsidethecoverageofmostsurveys.
Table2.Equatorial coordinates (J2000.0),apparentmagnitudes (withfil- Nearly30percentofthestudiedorbitshaveMOIDunder0.05au
ter),solarelongation, θ,andphase,φ,atdiscoverytimeofknownVenus anddeclinationunder-30◦.Thesefactssuggestthatthenumberof
co-orbitals.Source:MPCDatabase.
asteroids with sizes above 150 m currently engaged in co-orbital
motionwithVenusis,atleast,oneorderofmagnitudelargerthan
Object α(h:m:s) δ(◦:′:′′) m(mag) θ(◦) φ(◦)
usuallythought. Thenumber ofsmallerbodiescouldeasilybein
2001CK32 16:54:52.51 +33:02:23.9 17.2 83.8 90.0 manythousands andtheymaybesecondary fragments, theresult
2002VE68 01:05:00.91 +26:34:22.8 14.1(R) 150.9 28.2 oftidalorrotationalstresses(seee.g.Richardson,Bottke&Love
2012XE133 07:32:51.54 +59:02:11.9 18.5(V) 137.0 40.5 1998).
2013ND15 21:21:58.739 -26:52:18.69 20.5(w) 154.3 22.3 Fig.6shows thefrequency distributioninequatorial coordi-
nates at perigee for the set of orbits studied here. From this and
Table2,itisobviousthatnoneoftheknownobjectswerediscov-
objects within that limit: (322756) 2001 CK32 and 2002 VE68. ered within the regions of the sky where the density of perigees
Morais&Morbidelli(2006)alreadyconsideredsurprisingthatthe of these objects isexpected tobe thehighest (the eclipticpoles).
currentlyknownsampleoftheseobjectswasapparentlycomplete Fig.7providesthedistributioninequatorialcoordinatesoftheac-
up to H < 21. This result is even more extraordinary when we tualperigees(colourcoded).Oncemoreitisclearthattheknown
realizethatongoing NEOsearchprogrammes arenotspecifically objects have been discovered far from the optimal locations. The
targetingthisclassofobjects.Alookbackintothediscoverycir- smallestMOIDsarefoundintheregionwithinα∈(14,22)hand
cumstancesofindividualobjects,ascompiledinTable2,mayhelp δ∈(20,−70)◦.Only2013ND15hasbeendiscoveredclosetothat
ustounderstandbetterthisintriguingissue.Inparticular,thevalue areaofthesky.Thisstatisticalanalysiscanbeusedtoimplement
ofthesolarelongationoranglebetweentheSunandtheobjectas better strategies tofind theseobjects. In any case, optimized sur-
seenfromtheEarthatthetimeofdiscoverycanprovidevaluable veysrequiretheuseofspace-basedtelescopes.TheupcomingGaia
cluesonhowabundanttheseobjectsreallyare.Thesourceofthe mission6 representsaverygoodopportunitytostudythispopula-
datainTable2istheMinorPlanetCenter(MPC)Database.5 tionandconfirm/rejectourcurrentviews.Gaiacanobserveclose
Threeoutoffourobjectswerefoundatelongations>135◦. totheSun(downtoanangulardistanceof45◦fromtheSun)and
Observationally speaking, this is not unusual as NEO surveys shouldbeabletoobserveover77percentoftheseobjects,ifthey
mainlyfocusontheregionsoftheskyatelongationsinthatrange. do exist. In sharp contrast, nearly 70 per cent of the objects dis-
However,andintheory,mostoftheobjectsdiscussedhereareun- cussedherehaveelongationatperigee<90◦and,therefore,cannot
likelytoroutinelyreachperigeeatsuchlargeelongations asthey be observed from the ground. The search for Trojan asteroids in
spendmostofthetimeintheunobservable(daytime)sky.Morais theinnerSolarsystemingeneral,andusingGaiainparticular,has
&Morbidelli(2006)estimatedthat78percentofVenusco-orbitals beendiscussedbyTodd,Coward&Zadnik(2012).
arehiddenintheunobservablesky(theirregionIII)andjustthree Atthispoint,itmaybearguedthatourfindingsaresimplya
per cent arefound intheobservable sky(theirregionI).Inorder corollaryof thestrong, but known, observational biasagainst de-
tocomputeourownestimates,weperformedaMonteCarlosimu- tectingAtenspointedoutbye.g.Mainzeretal.(2012).Ifweper-
lation(Metropolis&Ulam1949)inwhichweusedtheequations formanotherMonteCarlocalculationforobjectsmovinginAten-
oftheorbitsofboththeEarthandtheobject under thetwo-body typeorbits (a < 1au, Q >0.983 au, withQ = a (1 + e)
5 http://www.minorplanetcenter.net/db search 6 http://sci.esa.int/gaia/
6 C. dela FuenteMarcos andR. delaFuenteMarcos
Figure4.ComparativedynamicalevolutionofvariousparametersforthefourknownVenusco-orbitals.ThedistancefromtheEarth(panelA);thevalueof
theHillsphereradiusoftheEarth,0.0098au,isdisplayed.Theresonantangle,λr(panelB)forthenominalorbitinTable1.Theorbitalelementsa−aVenus
(panelC),e(panelD),i(panelE)andω(panelF).Thedistancestothedescending(thickline)andascendingnodes(dottedline)appearinpanelG.Earth’s
andMercury’saphelionandperiheliondistancesarealsoshown.
beingtheapheliondistance,andassuminge <1.0andi <60◦, perigeeatsolarelongation<90◦;incontrastandforhypothetical
withboth Ω and ω in the range 0–360◦) and plot the results, we Venusco-orbitalsofthetypediscussedhere,thefractionis65.9per
obtainFig.8.Itistruethatthereisindeed astrongobservational cent.Thedifferenceislargeenoughtowarrantspecificconsidera-
biasagainstdetectingAtens.Itsactualimpactishoweverunknown tion,separatedfromthebulkoftheAtens.Only29.3percentofthe
becausewedonotknowtherealdistributionoftheorbitalelements Aten-typeorbitsstudied herearemoving inthePHAregime; for
ofobjectsmovinginAten-typeorbits.Ourdiagrams(hereandin Venusco-orbitalsofthetypediscussedherethisfractionamounts
thepreviousfigures)assumethatthedistributionoftheorbitalele- to65.1percent.Itisobviousthattheoverallprobabilityofcollision
mentsisuniformbutitcouldbethecasethatlow-eccentricity,low- with our planet is much higher for the type of Venus co-orbitals
inclinationorbitsarelesslikelytooccurthaneccentricandinclined discussed in this work than for the typical Aten population. The
onesduetofrequentplanetarycloseencounters.Wesimplydonot meanimpactprobabilitiesassociatedwithAtensandotherdynam-
know; acompletesampleofobjectsisneeded toprovideabetter icalclasseshavebeenstudiedpreviously(seee.g.Steel&Bagga-
answer.However, iftheorbitaldistributionisuniformor closeto ley1985;Chyba1993;Steel1995;Dvorak&Pilat-Lohinger1999;
uniform as assumed here, the predictions from our simple mod- Dvorak & Freistetter 2001) and it is well established that Atens
ellingshowthattheactualnumberofAtensshouldbesignificantly have the shortest lifetimes against collision, closely followed by
higher than the current tally. Consistently, the number of objects Apollos(afactor2–5longer).Iftheobjectsstudiedherehavelife-
movinginorbitssimilartothoseofknownVenusco-orbitalsmust timesagainst collision withour planet even shorter than those of
behighertoobuttheirdetectabilityismoreseriouslyaffectedthan typicalAtens,thispopulationmustbeabletoefficientlyreplenish
that of Atens (compare frequencies in Figs 5 and 8 for an equal thelossesviasomeunknownmechanism(s).
numberoftestorbits,107).
Our analysisindicatesthattheexistenceofarelativelylarge
Only 26.4 per cent of the simulated Aten-type orbits reach population of Venus co-orbitals with sizes under 150 m is very
A transientTrojancompanionto Venus 7
Figure5.Distributioninequatorialcoordinatesatperigeeoforbitssimilar Figure8.SimilartoFig.5butforobjectsmovinginAten-typeorbits.See
to those ofknown Venus co-orbitals as a function of the (colour coded) thetextfordetails.
geocentricsolarelongation(toppanel).Afrequencyanalysisofthesame
data(bottompanel).Onlyperigees<0.1auareconsidered.
likely.Thissmallsizesuggeststhattheycouldbefragmentsoffrag-
ments.Asteroidaldecaycouldbeinducedbycollisionalprocesses
(e.g.Ryan2000)butalsobethecombinedresultofthermalfatigue
(e.g.Cˇapek&Vokrouhlicky´2010)andtidal(e.g.To´th,Veresˇ&Ko-
rnosˇ2011)orrotationalstresses(e.g.Walsh,Richardson&Michel
2008).Theselastthreeprocessescanefficientlyproducesecondary
fragments.TherecentstudyofMainzeretal.(2014)maypointin
that direction. Using NEOWISE (the asteroid-hunting portion of
the Wide-field Infrared Survey Explorer mission) data they have
foundapopulationoftinyNEOscharacterizedbyincreasingalbe-
doswithdecreasingsize.Thismaybetheresultofaselectionbias
against finding small, dark NEOs but it may also be the specific
signatureofadistinctivepopulationamong smallobjectsthatare
theresultofsecondaryfragmentation.
Figure6.Frequencydistributioninequatorialcoordinates(rightascension, 6 CONCLUSIONS
toppanel,anddeclination,bottompanel)atperigeeoforbitssimilartothose
Inthispaper,wehaveidentifiedyetanotherhigh-eccentricityVenus
ofknownVenusco-orbitals(asinFig.5).Thebestareastosearchforthese
objectsarelocatedtowardstheeclipticpoles. co-orbital and PHA, 2013 ND15. This object is the first known
VenusTrojan.OurcalculationsindicatethatthissmallNEOmoves
inatransitional,highlychaotictrajectory;ithasbeenaco-orbital
companiontoVenusforjustafewthousand yearsanditmaybe-
comeapassingobjectwithinthenextfewhundredyears.Although
its future orbital evolution is not easily predictable beyond that
time-scale,theobjectmayexperiencerecurrentco-orbitalepisodes
with Venus. However, all the calculations consistently show that
2013 ND15 iscurrently in the Trojan dynamical state. Due to its
veryeccentricorbit,itsdynamical evolutionismarkedlydifferent
from that of the three previously known Venus co-orbitals, being
evenlessstable.Ontheotherhand,thescarcityofdetectedVenus
co-orbitalsappearstobetheresultofobservationalbias,notphysi-
callackofobjects.Ourcalculationssuggestthatthenumberofmi-
norbodieswithsizesabove150mcurrentlyengagedinco-orbital
Figure 7. Distribution in equatorial coordinates of the (colour coded)
motionwithVenuscouldbeatleastoneorderofmagnitudelarger
perigeesofobjectsmovinginorbitssimilartothoseofknownVenusco-
thanusuallythought;thenumberofsmallerbodiescouldeasilybe
orbitals(asinFig.5).Thegreenpointsrepresentthediscoverycoordinates
inmanythousands.Theseobjectsarebetteridentifiedusingspace-
ofthefourobjectsinTable2.
based telescopes because nearly 70 per cent of them have solar
elongationatperigee<90◦.
8 C. dela FuenteMarcos andR. delaFuenteMarcos
ACKNOWLEDGEMENTS ToddM.,CowardD.M.,ZadnikM.G.,2012,Planet.SpaceSci.,73,39
To´thJ.,VeresˇP.,KornosˇL.,2011,MNRAS,415,1527
The authors would like to thank S. J. Aarseth for providing one
WainscoatR.J.,MicheliM.,ForshayP.,PrimakN.,SchultzA.,GoggiaT.,
of thecodes used in thisresearch and the anonymous refereefor ChambersK.,ListerT.,2013,MPECCirc.,MPEC2013-O04
his/her constructive and useful report. This work was partially WalshK.J.,RichardsonD.C.,MichelP.,2008,Nature,454,188
supported by the Spanish ‘Comunidad de Madrid’ under grant WiegertP.A.,InnanenK.A.,MikkolaS.,1998,AJ,115,2604
CAMS2009/ESP-1496. Theauthorsalsothank M. J.Ferna´ndez-
Figueroa,M.RegoFerna´ndezandtheDepartmentofAstrophysics
of theUniversidad Complutense deMadrid(UCM)for providing
computingfacilities.Mostofthecalculationsandpartofthedata
analysis were completed on the ‘Servidor Central de Ca´lculo’ of
theUCM.Inpreparationofthispaper,theauthorsmadeuseofthe
NASAAstrophysics DataSystem, theASTRO-PHe-printserver,
theMPCdataserverandtheNEODysservice.
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