Debiased orbit and absolute-magnitude distributions for near-Earth objects

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Debiased orbit and absolute-magnitude distributions for

near-Earth objects

Mikael Granvik, Alessandro Morbidelli, Robert Jedicke, Bryce Bolin, William

Bottke, Edward Beshore, David Vokrouhlicky, David Nesvorny, Patrick Michel

To cite this version:

Mikael Granvik, Alessandro Morbidelli, Robert Jedicke, Bryce Bolin, William Bottke, et al.. Debiased

orbit and absolute-magnitude distributions for near-Earth objects. Icarus, Elsevier, 2018, 312,

pp.181-207. �10.1016/j.icarus.2018.04.018�. �hal-02407887�

ContentslistsavailableatScienceDirect

Icarus

journalhomepage:www.elsevier.com/locate/icarus

Debiased

orbit

and

absolute-magnitude

distributions

for

near-Earth

objects

Mikael

Granvik

a,∗

_,

_Alessandro

_Morbidelli

b

_,

_Robert

_Jedicke

c

_,

_Bryce

_Bolin

b,c,d,e,1

_,

William

F.

Bottke

f

_,

_Edward

_Beshore

g

_,

_David

_{Vokrouhlický}

h

_,

_David

_Nesvorný

f

_,

Patrick

Michel

b

a Department of Physics, 0 0 014 University of Helsinki, P.O. Box 64, 0 0 014, Finland b Observatoire de la Cote d’Azur, Boulevard de l’Observatoire, Nice Cedex 4 F 06304, France c Institute for Astronomy, 2680 Woodlawn Dr, HI 96822, USA

d B612 Asteroid Institute, 20 Sunnyside Ave, Suite 427, Mill Valley, CA 94941, USA

e Department of Astronomy, University of Washington, 3910 15th Ave NE, Seattle, WA 98195, USA f Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, USA g University of Arizona, 933 North Cherry Avenue, Tucson, Arizona 85721-0065, USA

h Institute of Astronomy, Charles University, V Holešovi ˇckách 2, Prague 8 CZ 180 0 0, Czech Republic

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 4 October 2017 Revised 5 April 2018 Accepted 15 April 2018 Available online 25 April 2018

Keywords: Near-Earth objects Asteroids Dynamics Comets Resonances Orbital

a

b

s

t

r

a

c

t

The debiasedabsolute-magnitude and orbitdistributions as well as source regions fornear-Earth ob-jects(NEOs)provideafundamentalframeofreferenceforstudiesofindividualNEOsandmorecomplex population-levelquestions.We presentanewfour-dimensional modeloftheNEOpopulationthat de-scribesdebiasedsteady-statedistributionsofsemimajoraxis,eccentricity,inclination,andabsolute mag-nitudeHintherange17<H<25.Themodelingapproachimprovesuponthemethodologyoriginally de-velopedbyBottkeetal.(2000,Science288,2190–2194)inthatitis,forexample,basedonmorerealistic orbitdistributionsandusessource-specificabsolute-magnitudedistributionsthatallowforapower-law slopethatvarieswithH.Wedividethemainasteroidbeltintosixdifferententranceroutesorregions (ER)totheNEOregion:theν6,3:1J,5:2Jand2:1JresonancecomplexesaswellasHungariasand

Pho-caeas. Inaddition weincludetheJupiter-family cometsas the primarycometarysource ofNEOs. We calibratethemodelagainstNEOdetectionsbyCatalinaSkySurveys’stations703andG96during2005– 2012,andutilizethecomplementarynatureofthesetwosystemstoquantifythesystematicuncertainties associatedtotheresultingmodel.Wefindthatthe(fitted)Hdistributionshavesignificantdifferences, al-thoughmostofthemshowaminimumpower-lawslopeatH∼ 20.Asaconsequenceofthedifferences betweentheER-specificHdistributionswefindsignificantvariationsin,forexample,theNEOorbit dis-tribution,averagelifetime,and therelative contributionofdifferentERsas afunctionofH.Themost importantERsaretheν6 and 3:1Jresonancecomplexes withJFCscontributingafewpercentofNEOs

onaverage.AsignificantcontributionfromtheHungariagroupleadstonotablechangescomparedtothe predictionsbyBottkeetal.in,forexample,theorbitdistributionandaveragelifetimeofNEOs.We pre-dictthatthereare962+52₋₅₆(802+48₋₄₂× 103_{)NEOswithH<17.75(H<25)andthesenumbersarein}

agree-mentwiththemostrecentestimatesfoundintheliterature(theuncertaintyestimatesonlyaccountfor therandomcomponent).BasedonourmodelwefindthatrelativesharesbetweendifferentNEOgroups (Amor,Apollo,Aten,Atira,Vatira)are(39.4,54.4,3.5,1.2,0.3)%,respectively,fortheconsideredHrangeand thattheseratioshaveanegligibledependenceonH.Finally,wefindan agreementbetweenour esti-matefortherate ofEarthimpactsbyNEOsandrecentestimates intheliterature,butthereremainsa potentiallysignificantdiscrepancyinthefrequencyofTunguska-sizedandChelyabinsk-sizedimpacts.

© 2018TheAuthors.PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

∗ _{Corresponding author.}

E-mail address: mgranvik@iki.fi(M. Granvik).

1 B612 Asteroid Institute and DIRAC Institute Postdoctoral Fellow

1. Introduction

Understandingtheorbital andsizedistributions aswell asthe source regions for near-Earth objects (NEOs; for a glossary of https://doi.org/10.1016/j.icarus.2018.04.018

Table 1

Glossary of acronyms and terms. Acronym/ Definition term

703 Catalina Sky Survey (telescope) AICc corrected Akaike Information Criteria CSS Catalina Sky Survey (703 and G96) ER escape/entrance route/region G96 Mt. Lemmon Survey (part of CSS) HFD H -frequency distribution IMC intermediate Mars-crosser JFC Jupiter-family comet MAB main asteroid belt MBO main-belt object ML maximum likelihood MMR mean-motion resonance MPC Minor Planet Center

MOID minimum orbital intersection distance

NEO near-Earth object (asteroid or comet with q < 1.3 au and a < 4.2 au) PHO NEO with MOID < 0.05 au and H < 22

RMS root-mean-square SR secular resonance

YORP Yarkovsky-O’Keefe-Radzievskii-Paddack effect Amor NEO with 1.017 au < q < 1.3 au

Apollo NEO with a > 1.0 au and q < 1.017 au Aten NEO with a < 1.0 au and Q > 0.983 au Atira NEO with 0.718 au < Q < 0.983 au Vatira NEO with 0.307 au < Q < 0.718 au

a semimajor axis

e eccentricity

i inclination

 longitude of ascending node

ω argument of perihelion

M0 mean anomaly

H absolute magnitude in V band

D diameter

q perihelion distance

Q aphelion distance

acronymsandterms,seeTable1)isoneofthekeytopicsin con-temporaryplanetaryscience(Binzeletal.,2015;Harrisetal.,2015; Abelletal.,2015).Herewepresentanewmodeldescribingthe de-biasedabsolute-magnitude(H) andorbital (semimajoraxis a, ec-centricitye, inclination i) distributions forNEOs. The model also enablesaprobabilistic assessmentofsourceregionsforindividual NEOs.

We follow the conventional notation and define an NEO as an asteroid or comet (active, dormant or extinct) with perihe-lion distance q<1.3au and semimajor axis a<4.2au. The lat-ter requirement is not part of the official definition, which has no limit on a, but it limits NEOs to the inner solar sys-tem and makes comparisons to the existing literature easier (cf. Bottke et al., 2002a). The population of transneptunian ob-jects maycontain asubstantial number ofobjects withq<1.3au that are thus not considered in this work. NEOs are further di-videdintotheAmors(1.017au<q<1.3au),Apollos(a>1.0auand

q<1.017au), Atens(a<1.0au andapheliondistanceQ>0.983au), Atiras (0.718au<Q<0.983au) that are detached from the Earth, andtheso-calledVatiras(0.307au<Q<0.718au)thataredetached fromVenus(Greenstreetetal.,2012a).NEOs arealsoclassifiedas potentiallyhazardousobjects (PHOs)whentheirminimumorbital intersectiondistance(MOID)withrespecttotheEarthislessthan 0.05auandH_<22.

Several papers havereported estimates forthe debiased orbit distribution and/or the H-frequency distribution (HFD) for NEOs over the past 25 years or so. The basic equation that underlies mostofthestudiesdescribestherelationshipbetweentheknown NEOpopulationn,thediscoveryefficiency



,andthetrue popula-tionNasfunctionsofa,e,i,andH:

n

(

a,e,i,H

)

=



(

a,e,i,H

)

N

(

a,e,i,H

)

. (1)

Rabinowitz (1993)derived the firstdebiased orbitdistribution andHFDforNEOs.Themodelwascalibratedbyusingonly23 as-teroidsdiscovered bytheSpacewatchTelescope between Septem-ber1990 andDecember 1991.The modelisvalidinthediameter range 10m࣠D࣠10km, andthe estimatedrate ofEarth impacts byNEOsisabout100timeslargerthanthecurrentbestestimates atdiametersD_{∼ 10}m(Brownetal.,2013).Rabinowitz(1993) con-cludedthat thesmallNEOs havea differentHFDslope compared toNEOs withD࣡100m andsuggestedthat futurestudies should “assess the effect of a size-dependent orbit distribution on the derived size distribution.”Rabinowitz etal.(2000) used methods similar to those employed by Rabinowitz (1993) to estimate the HFDbasedon45NEOsdetectedbytheJetPropulsionLaboratory’s Near-Earth-AsteroidTracking(NEAT)program.Theyconcludedthat thereshouldbe700± 230NEOswithH<18.

Although the work by Rabinowitz (1993) showed that it is possible to derive a reasonableestimate forthe true population, the relatively small number of known NEOs (∼ 102_–104_{) implies} that the maximum resolution of the resulting four-dimensional model is poor and a scientifically useful resolution is limited to marginalized distributions inone dimension.Therefore additional constraintshadtobefoundtoderivemoreusefulfour-dimensional models of the true population. Bottke et al. (2000) devised a methodologywhichutilizesthefactthatobjectsoriginatingin dif-ferentparts ofthe main asteroidbelt (MAB) orthe cometary re-gionwillhavestatisticallydistinct orbitalhistoriesintheNEO re-gion. Assumingthat there is no correlation betweenHand(a, e, i), Bottke et al. (2000) decomposed the true population N(a, e, i, H) into N(H)



Rs(a, e, i), where Rs(a, e, i) denotes the steady-state orbitdistributionforNEOs enteringtheNEO regionthrough entrance route s. The primary dynamical mechanisms responsi-blefordelivering objectsfromtheMABandcometaryregioninto the NEO regionwere already understoodat that time, and mod-els for Rs(a, e, i) could therefore be obtained through direct or-bital integration of test particles placed in, or in the vicinity of, escaperoutesfromtheMAB. Theparameters left tobe fitted de-scribed therelative importance ofthesteady-state orbit distribu-tionsandtheoverallNEOHFD. Fittinga modelwiththreeescape routes fromthe MAB, that is, the

ν

6 secular resonance (SR), the intermediate Marscrossers(IMC),andthe 3:1Jmean-motion res-onance (MMR) withJupiter, to 138 NEOs detected by the Space-watchsurvey,Bottkeetal.(2000)estimatedthatthereare910+100₋₁₂₀ NEOs withH<18. Theirestimatesforthe contributions fromthe different escape routes had large uncertainties that left the rel-ative contributions from the different escape routes statistically indistinguishable. Bottke et al. (2002a) extended the model by also accountingfor objects from theouter MAB and the Jupiter-family-comet (JFC) population, but their contribution turned out to be only about 15% combined whereas the contributions by the inner MAB escape routes were, again, statistically indistin-guishable. The second column in Table 2 provides the numbers predicted by Bottke et al. (2002a) for all NEOs as well as NEO subgroups.

D’Abramoetal.(2001)presentedanalternativemethodfor es-timating the HFDthat is basedon the re-detectionratio,that is, the fraction of objects that are re-detections of known objects rather than new discoveries. They based their analysis on 784 NEOs detected by theLincoln Near-Earth AsteroidResearch (LIN-EAR) projectduring1999–2000.Based onthe resultingHFD, that wasvalidfor13.5≤ H≤ 20.0,theyestimatedthatthereshould ex-ist 855± 101 NEOs withH<18. Harris and D’Abramo (2015) ex-tended the methodand redidthe analysis with11,132 NEOs dis-covered by multiplesurveys. Theyproduced an HFDthat isvalid for 9<H<30.5 and estimated that there should be 1230± 27 (990_{± 20)} NEOs with H_<18 (H_<17.75).Later an error was dis-covered in the treatment of absolutemagnitudes that the Minor

Table 2

The Bottke et al. (2002a) estimate for the number of NEOs with H < 18 and a < 7.2 au, and the known pop- ulation (ASTORB a _{2018-01-30) with H < 18, H < 17 and}

H < 16 as a function of NEO subgroup.

Group B02 Known Known Known

H < 18 H < 18 H < 17 H < 16 Amor 310 ± 38 504 218 81 Apollo 590 ± 71 532 226 84 Aten 58 ± 9 36 17 5 Atira 20 ± 3 3 2 0 Vatira – 0 0 0 NEO 960 ± 120 1075 463 170 a _{ftp://ftp.lowell.edu/pub/elgb/astorb.dat.gz .}

Planet Center (MPC)reportsto onlya tenthofamagnitude. Cor-rectingfortheroundingerrorreducedthenumberofNEOsby5% (Stokesetal.,2017).

Stuart(2001)used1343detectionsof606differentNEOsbythe LINEAR projecttoestimate, inpractice,one-dimensionaldebiased distributions fora,e,i,andHbyusingatechniquerelyingonthe

n(a,e,i,H)/



(a,e,i,H)ratio,whereN(a,e,i,H)hadtobe marginal-izedoverthreeoftheparameterstoprovideausefulestimate for thefourthparameter.Theyestimatedthatthereare1227+170₋₉₀ NEOs withH<18and,intermsoforbitalelements,themostprominent differencecomparedtoBottkeetal.(2000)wasapredictedexcess ofNEOswithi࣡20°.StuartandBinzel(2004)extendedthemodel byStuart(2001)toincludethetaxonomicandalbedodistributions. Theyestimatedthattherewouldbe1090_{± 180}NEOswith diame-terD>1km,andthat60%ofallNEOsshouldbedark,thatis,they shouldbelongtotheC,D,andXtaxonomiccomplexes.

Mainzer et al.(2011) estimated that there are 981_{± 19} NEOs with D>1km and 20,500± 3,000 with D>100m based on in-frared (IR) observations obtained by the Widefield Infrared Sur-veyExplorer(WISE)mission.Theyalsoobservedthatthefraction ofdarkNEOs withgeometric albedopV<0.1isabout40%,which shouldbeclosetothedebiasedestimategiventhatIRsurveysare essentially unbiasedwithrespect topV. Mainzer etal.(2012) ex-tended theanalysis of WISE observationsto NEO subpopulations andestimatedthatthereare4700± 1450PHOswithD>100m.In addition,they foundthatthealbedos ofAtensaretypicallylarger thanthealbedosforAmors.

Greenstreet etal.(2012a) improvedthe steady-stateorbit dis-tributions that were used by Bottke et al. (2002a) by using six timesmoretestasteroidsandfourtimesshortertimestepsforthe orbital integrations. Thenew orbitmodelfocused on thea<1au region anddiscussed,forthe firsttime, the so-calledVatira pop-ulation with orbits entirely inside the orbit of Venus. The new integrations revealed that near-Earth asteroids(NEAs) can evolve to retrogradeorbits evenintheabsenceofcloseencounters with planets (Greenstreet etal., 2012b). Similar retrogradeorbitsmust haveexistedintheintegrationscarriedoutbyBottkeetal.(2000, 2002a),buthaveapparentlybeenoverlooked.ThefractionofNEAs on retrograde orbits was estimated at about 0.1% (within a fac-toroftwo)oftheentireNEOpopulation.Wenotethat themodel by Greenstreet et al. (2012a) did not attempt a re-calibration of the modelparameters butused thebest-fit parameters found by Bottkeetal.(2002a).

Granvik et al. (2016) used an approach similar to

Bottke et al. (2002a) to derive a debiased four-dimensional model of the HFD and orbit distribution. The key result of that paper was the identification of a previously unknown sink for NEOs, most likely caused by the intense solar radiation experi-encedbyNEOsonorbitswithsmallperiheliondistances.Theyalso showedthatdarkNEOsdisruptmoreeasilythanbrightNEOs,and concluded that this explainswhy the Atenasteroids havehigher

albedos than other NEOs (Mainzer et al., 2012). Based on 7952 serendipitious detections of 3632 distinct NEOs by the Catalina SkySurvey(CSS)they predictedthatthereexists1008± 45NEOs withH_<17.75,whichisinagreementwithotherrecentestimates. Tricarico(2017)analyzedthedataobtainedbythe9most pro-lific asteroid surveys over the past two decades and predicted that thereshould exist1096.6_{± 13.7}(920_{± 10)}NEOs withH_<18 (H<17.75). Their method relied, again, on computingthe n(a, e, i, H)/



(a, e, i, H) ratio.The chosen approach implied that the re-sultingpopulationestimateissystematicallytoolow,becauseonly binsthatcontainoneormoreknownNEOscontributetothe over-allpopulationregardlessofthevalueof



(a,e,i,H).Detailedtests showedthattheproblemcausedbyemptybinsremainsmoderate whenoptimizing the binsizes. Tricarico (2017)also showedthat thecumulativeHFDsbasedonindividualsurveysweresimilarand thislendsfurthercredibilitytotheirresults.

Schunová-Lilly et al. (2017) derived the NEO HFD based on NEO detections obtained with the Panoramic Survey Telescope and Rapid Response System 1 (Pan-STARRS 1). Their methodol-ogywas, again, based oncomputing the ration(a, e, i, H)/



(a, e, i, H), where the observational bias was obtained using a realis-tic survey simulation (Denneau et al., 2013). Marginalizing over the orbital parameters to providea usefulestimate forthe HFD, Schunová-Lillyet al.(2017) founda distribution that agrees with Granviketal.(2016)andHarrisandD’Abramo(2015).

Estimates for the number of km-scale and larger NEOs have thusconvergedtoabout900–1000objectsbuttherestillremains significantvariation atthe smallersizes(in particular,H࣡23). In what follows we therefore primarily focus on the sub-km-scale NEOs.

Of the models described above only Bottke et al. (2000, 2002a) and Granvik et al. (2016) are four-dimensional models, that is, they simultaneously and explicitly describe the correla-tions between all four parameters throughout the considered H

range. These are also the only models that provide information onthesourceregionsforNEOsalthoughGranviketal.(2016)did not explicitly report this information. Although the model by Bottke et al. (2002a) has been very popular and also able to reproduce the known NEO population surprisingly well, it has some known shortcomings. The most obvious problem is that the number of currently known H<18 Amors exceeds the

pre-dicted number of H<18 Amors by more than 5

σ

(Table 2).

Bottkeet al. (2002a) are also unable to reproduce the NEO, and inparticular Aten, inclination distribution (Greenstreet and Glad-man, 2013). These shortcomings are most readily explained by the limited number of detections that the model was calibrated with,butmayalsobeexplainedbyanunrealisticinitialinclination distribution for the test asteroids which were used for comput-ingtheorbital steady-statedistributions,orbynotaccountingfor Yarkovskydriftwhenpopulatingtheso-calledintermediatesource regionsintheMAB.Anintermediatesourceregionrefers tothe es-cape route from, e.g.,the MAB andintothe NEO regionwhereas

asourceregion refersto theregionwherean objectoriginates.In

what follows we do not explicitlydifferentiate betweenthe two butrefer to both withthe term entrance/escape route/region (ER) forthesakeofsimplicity.Bottkeetal.(2002a)also usedasingle powerlaw todescribe the NEO HFD, that is,neither variation in theHFDbetweenNEOsfromdifferentERsnordeviationsfromthe power-law form of the HFD were allowed. The ERs, such as the intermediate Mars-crossers(IMCs) in Bottkeetal. (2000,2002a), havebeen perceived tobe artificial becausethey are not the ac-tual sources in the MAB. The IMC source also added to the de-generacyof themodelbecause thesteady-stateorbit distribution forasteroidsescaping theMABoverlaps withthe steady-state or-bitdistributionsforasteroidsescapingtheMAB throughboth the 3:1JMMRandthe

ν

6 SR. Onthe other hand,objectscan anddo

escapeoutofamyriadoftinyresonancesintheinnerMAB, feed-ingasubstantialpopulationofMars-crossingobjects.Objects trav-elingrelativelyrapidlyby theYarkovskyeffectare more suscepti-bleto jumping acrossthese tiny resonances, whilethose moving moreslowlycan becometrapped(Bottkeetal., 2002b).Modeling thisportion of the planet-crossing population correctly is there-forecomputationallychallenging.Inthispaper,weemploycertain compromisesonhow asteroidsevolveintheMAB ratherthan in-voketime-consumingfull-upmodelsofYarkovsky/YORPevolution (e.g.,Vokrouhlický etal., 2015).The penalty isthat we maymiss bodiesthat escape theMAB via tiny resonances.Our methodsto deal with this complicated issue are discussed in Granvik et al. (2016,2017)and below.Westressthattheobservationaldataused for the Bottke et al. (2002a) model, 138 NEOs observed by the Spacewatchsurvey, was well describedwith the model they de-veloped.Theshortcomingsdescribedabovehaveonlybecome ap-parentwiththe _>100timeslargersampleofknownNEOs thatis availabletoday(15,624asof2017-02-09).Themostnotable short-comingoftheBottkeetal.(2002a)modelintermsofapplication isthatitisstrictlyvalidonlyforNEOswithH_<22,roughly equiv-alenttoa diameterof ࣡100m.Inaddition, theresolutionof the steady-stateorbitdistributionlimitstheutilityofmodelsthatare basedonit(see,e.g.,Granviketal.,2012).

The improvements presented in this work compared to Bottke et al. (2002a) are possible through the availability of roughly a factor of 30 more observational data than used by Bottkeetal.(2002a),usingmoreaccurateorbitalintegrationswith moretestasteroidsandashortertime step,usingmoreERs (7vs 5), and by using different andmore flexible absolute-magnitude distributionsfordifferentERs.

The(incomplete)listofquestionswewillanswerare:

• What isthetotalnumberofAmors,Apollos,Atens,Atiras,and Vatirasinagivensize-range?

• Whatistheoriginfortheobservedexcess(ascomparedto pre-diction by Bottke et al. (2002a)) of NEOs with 20°࣠i࣠40°? Is there a particularsource orare these orbitsin a particular phaseoftheirdynamicalevolution,liketheKozaicycle? • WhatistherelativeimportanceofeachoftheERsintheMAB? • WhatisthefractionofcometsintheNEOpopulation? • Is there a measurable difference in the orbit distribution

be-tweensmallandlargeNEOs?

• AretheredifferencesintheHFDsofNEOs fromdifferentERs? Whatarethedifferences?

• What istheimplicationoftheseresultsforourunderstanding oftheasteroid-Earthimpactrisk?

• Howdoesthepredictedimpactratecomparewiththeobserved boliderate?

• WhatistheHFDforNEOsonretrogradeorbits?

• How does the resulting NEO HFD compare withindependent estimatesobtainedthrough,forexample,cratercounting? • IstheNEOpopulationinasteadystate?

2. Theoryandmethods

Let us, fora moment, assume that we could correctly model allthesize-dependent,orbit-dependent,dynamicalpathwaysfrom theMABtothe NEOregion,andweknew theorbitandsize dis-tributions ofobjects in the MAB. Inthat case we could estimate thepopulation statistics for NEOs by carrying out direct integra-tionsoftest asteroidsfromtheMAB throughthe NEOregion un-til they reach a sink. While it has been shown that such a di-rectmodeling is reasonably accurate forkm-scale andlarger ob-jects,itbreaksdownforsmallerobjects(Granviketal.,2017).The mostobviousmissingpieceisthatwe donotknowtheorbitand sizedistributionsofsmallmain-beltobjects(MBOs)— thecurrent

bestestimatessuggestthattheinventoryiscompletefordiameters

D࣡1.5km(Jedickeetal.,2015).

Insteadwe take anotherapproach thatcan copewithour im-perfect knowledge and gives us a physically meaningful set of knobs to fit the observations. We build upon the methodology originallydevelopedbyBottkeetal.(2000)byusingER-dependent HFDsthatallowforasmoothlychangingslopeasafunctionofH. Eq.(1)canthereforeberewrittenas

n

(

a,e,i,H

)

=



(

a,e,i,H

)

× NER  s=1 Ns

(

H; N0,s,

α

min,s,Hmin,s,cs

)

Rs

(

a,e,i

)

, (2) whereNERisthenumberofERsinthemodel,andtheequationfor thedifferential Hdistribution allowsfora smooth,second-degree variationoftheslope:

Ns

(

H; N0,s,

α

min,s,Hmin,s,cs

)

=N0,s10 H H₀[αmin,s+cs(H−Hmin,s)2]dH =N0,s10αmin,s(H−H0)+ cs 3[(H−Hmin,s)3−(H0−Hmin,s)3]_{. (3)} Thesteady-stateorbitaldistributions, Rs(a,e,i),areestimated nu-merically by carrying out orbital integrations of numerous test asteroids in the NEO region and recording the time that the test asteroids spend in various parts of the (a, e, i) space (see Sections2.2and5).Theorbitdistributionsarenormalizedsothat foreachERs



Rs

(

a,e,i

)

dadedi=1. (4) In practice the integration over the corresponding NEO orbital spaceinEq. (4)isreplacedwithasimplesummationoveragrid offinitecells.

With the orbit distributions Rs(a, e, i) fixed, the free param-eters to be fitted describe the HFDs for the different ERs: the number densityN0,s at the reference magnitudeH0 (common to all sources andchosen tobe H0=17), theminimum slope

α

min,s of the absolute magnitude distribution, the curvature cs of the absolute-magnitude-slope relation, and the absolute magnitude

Hmin,s corresponding to theminimum slope. Notethat at H=H0 inourparametrization,the(unnormalized)N0,seffectivelytakethe sameroleasthe(normalized)weightingfactorsbywhichdifferent ERs contribute to the NEO population inthe Bottke etal.(2000, 2002a)models.BecauseHFDsareER-resolvedinourapproach,the relativeweightingatH=H0isnotexplicitlyavailablebuthastobe computedseparately.

AsshownbyGranviketal.(2016)itisimpossibletofindan ac-ceptablefittoNEOswithsmallperiheliondistances,q=q

(

a,e

)

=

a

(

1_{− e}

)

_, when assuming that the sinks for NEOs are collisions with the Sun orplanets, or an escape from the inner solar sys-tem. The model is able to reproduce the observed NEO distri-bution only when assuming that NEOs are completely destroyed at small, yet nontrivial distances from the Sun. In addition to the challengeswithnumericalmodels ofsuch a complex disrup-tion event in a detailed physical sense, it is also computation-ally challenging to merely fit for an average disruption distance. Granviketal.(2016)performedanincrementalfittoanaccuracyof 0.001au.Theincrementalfitwasfacilitatedbyconstructing multi-pledifferentsteady-stateorbitdistributions,eachwithadifferent assumptionfortheaveragedisruptiondistance,andthen identify-ingtheorbitdistributionwhichleadstothebestagreementwith theobservations.Eachofthe steady-stateorbitdistributions were constructedsothatthetestasteroidsdidnotcontributetothe or-bitdistributionaftertheycrossedtheassumedaveragedisruption distance.Granviketal.(2016)usedtheperiheliondistanceqasthe distance metric. While it is clear that super-catastrophic disrup-tioncanexplainthelackofNEOsonsmall-q orbits,suchasimple

modeldoesnotallowforanaccuratereproductionoftheobserved

qdistribution.Forinstance,Granviketal.(2016)explicitlyshowed thatthedisruptiondistancedependsonasteroiddiameterand ge-ometricalbedo.

Here we take an alternative andnon-physical route to fit the small-perihelion-distance part of the NEO population to improve thequalityofthefit:weuseorbitdistributionsthatdonotaccount fordisruptionsatsmallqandinsteadfitalinearpenaltyfunction,

p(a,e),withanincreasingpenaltyagainstorbitswithsmallerq.Eq. (1)nowreads n

(

a,e,i,H

)

=



(

a,e,i,H

)

× NER  s=1 Ns

(

H; N0,s,

α

min,s,Hmin,s,cs

)

× [1− p

(

a,e

)

]Rs

(

a,e,i

)

 a,e,i[1− p

(

a,e

)

]Rs

(

a,e,i

)

, (5) where p

(

a,e

)

=



_k

₍

_q 0− q

(

a,e

))

forq≤ q0, 0 forq>q0,

andwesolvefortwoadditionalparameters—thelinearslope,k,of thepenaltyfunctionandthemaximumperiheliondistancewhere the penalty is applied, q0. Note that the penalty function does not have a dependence on H although it has been shown that small NEOs disrupt at larger distances compared to large NEOs (Granviketal.,2016).Wechosetouseafunctionalform indepen-dentofHtolimitthenumberoffreeparameters.

IntotalweneedtosolveforNpar=4NER+2parameters.Inthe followingthreesubsectionswe willdescribethemethodsusedto estimatetheorbital-elementsteady-statedistributionsand discov-eryefficienciesaswellastosolvetheefficiencyequation.

2.1. Estimationofobservationalselectioneffects

All asteroid surveysare affectedby observational selection ef-fects in the sense that the detected population needs to be cor-rected inorder to find the true population.The known distribu-tion of asteroid orbitsis not representative oftheir actual distri-bution becauseasteroid discoveryanddetectionis affectedby an object’s size, lightcurve amplitude, rotation period, apparent rate of motion,color, andalbedo, and thedetection system’s limiting magnitude, survey pattern, exposure time, the sky-plane density ofstars,andother secondaryfactors.Combining theobserved or-bitdistributionsfromsurveyswithdifferentdetection characteris-tics furthercomplicatestheproblemunless thepopulationunder consideration is essentially ‘complete’, i.e., all objects inthe sub-populationare known.Jedicke etal.(2016)provideadetailed de-scriptionofthemethodsemployed forestimatingselectioneffects inthiswork.Theirtechniquebuildsuponearliermethods(Jedicke andMetcalfe,1998; Jedickeetal.,2002; Granviketal.,2012) and takesadvantagesoftheincreasedavailabilityofcomputingpower tocalculateafastandaccurateestimateoftheobservationalbias.

The ultimate calculation of the observational bias would pro-vide the efficiencyof detecting an objectasa function ofall the parameterslistedabovebutthiscalculationisfartoocomplicated, computationally expensive, andunjustifiedfor understanding the NEOorbitdistributionandHFDatthecurrenttime.Instead,we in-vokemanyassumptionsaboutunknownorunmeasurable parame-tersandaverageover theunderlyingsystemandasteroid proper-tiestoestimatetheselectioneffects.

The fundamental unit ofan asteroid observation for our pur-poses is a ‘tracklet’ composed of individual detections of the as-teroid inmultiple images ona single night(Kubica etal., 2007). Atthemeantimeofthedetectionsthetracklethasapositionand

rateofmotion(w)ontheskyandanapparentmagnitude(m; per-haps in a particular band). Note that Jedicke et al. (2016) use a differentnotation. Thetrackletdetectionefficiencydependsonall theseparameters andcan be sensitiveto the detectionefficiency inasingleimage duetoskytransparency,opticaleffects,andthe backgroundof,e.g.,stars,galaxies,andnebulae.Weaveragethese effectsover an entirenight andcalculatethe detectionefficiency (



¯

(

m

)

)asafunctionofapparentmagnitudefortheCSSimages us-ingthe system’sautomated detectionofknownMBOs. Tocorrect forthedifferencein apparentratesof motionbetweenNEOs and MBOsweusedtheresultsofZavodnyetal.(2008)whomeasured thedetectionefficiencyofstarsthatwereartificiallytrailedinCSS images at known rates. Thus, we calculated the average nightly NEO detection efficiencyas a function of the observable tracklet parameters:



¯

(

m,w

)

.

Thedeterminationoftheobservationalbiasasafunctionofthe orbital parameters (



(a, e, i, H)) involved convolving an object’s observable parameters (m, w) with its (a, e, i, H) averaged over theorbital angular elements(longitude ofascending node



, ar-gumentofperihelion

ω

,meananomalyM0)thatcanappearinthe fieldsfromwhichatrackletiscomposed.Foreachimagewe step throughtherangeofallowedtopocentricdistances(

)and deter-minethe rangeof angularorbital elementsthat could havebeen detectedforeach(a,e,i,H)combination.Sincethelocationofthe imageisknown((R.A.,Dec.)=(

α

,

δ

))andthetopocentriclocationof theobserverisknown,then,given

and(a,e,i)itispossibleto calculatetherangeofvaluesoftheotherorbitalelementsthatcan appearinthefield.Undertheassumptionthatthedistributionsof theangular orbital elementsare flat it is then possible to calcu-late



(a,e,i,H)inafield andtheninallpossiblefieldsusingthe appropriateprobabilistic combinatorics.WhileJeongAhnand Mal-hotra (2014) have shownthat the argument of perihelion, longi-tude ofascending node andlongitude of periheliondistributions forNEOshavemodestbutstatistically-significantnon-uniformities, we considerthem tobe negligible forourpurposes compared to theothersourcesforsystematics.

2.2.Orbitintegrator

The orbital integrations to obtain the NEO steady-state orbit distributions are carried out with an augmented version of the SWIFT RMVS4 integrator (Levison and Duncan, 1994). The nu-merical methods,in particularthose relatedto Yarkovsky model-ing (not used in the main simulations of this work but only in somecontrol simulationstoattestitsimportance),aredetailedin Granviketal.(2017). Theonly additionalfeatureimplemented in thesoftwarewasthe capabilitytoingest test asteroids(with dif-ferentinitialepochs)ontheflyastheintegrationprogresses,and thiswasdonesolelytoreducethecomputingtimerequired.

2.3.Estimationofmodelparameters

We employ an extended maximum-likelihood (EML) scheme (Cowan,1998)andthesimplexoptimizationalgorithm(Nelderand Mead,1965) when solving Eqs. (2) and(5) forthe parameters P

thatdescribethemodel.

Let

(

n1,n2,...,nN_bin

)

be thenon-zero bins in the binned ver-sion of n(a, e, i, H), and

(

ν

1,

ν

2,...,

ν

N_bin

)

be the corresponding bins containing the expectation values, that is, the model pre-diction for the number of observations in each bin. The joint probability-densityfunction(PDF) forthedistribution of observa-tions

(

n1,n2,...,nN_bin

)

isgivenbythemultinomialdistribution:

pjoint=ntot! Nbin  k=1 1 nk!

 ν

k ntot

nk , (6)

where

ν

k/ntotgivestheprobabilitytobeinbink.InEMLthe mea-surementisdefinedtoconsistoffirstdetermining

ntot=

Nbin



k=1

nk (7)

observationsfromaPoissondistributionwithmean

ν

tot andthen distributingthoseobservationsinthehistogram

(

n1,n2,...,nN_bin

)

. That is, the total number of detections is regarded as an addi-tionalconstraint. Theextended likelihoodfunctionLisdefinedas thejoint PDF forthe totalnumber ofobservations ntot andtheir distribution in the histogram

(

n1,n2,...,nN_bin

)

. The joint PDF is thereforeobtainedby multiplyingEq.(6)witha Poisson distribu-tionwithmean

ν

tot=

Nbin



k=1

ν

k (8)

andaccountingforthefact thattheprobabilityforbeinginbink

isnow

ν

_k/

ν

tot:

p_joint=

ν

ntot

tot exp

(

−

ν

tot

)

ntot! ntot! Nbin  k=1 1 nk!

 ν

k

ν

tot

nk =

ν

ntot

tot exp

(

−

ν

tot

)

Nbin  k=1 1 nk!

 ν

k

ν

tot

nk =exp

(

−

ν

tot

)

Nbin  k=1 1 nk!

ν

nk k (9)

Neglectingvariablesthatdonotdependontheparametersthat aresolved for,the logarithmofEq.(9),that is,the log-likelihood function,canbewrittenas

logL=−

ν

tot+

Nbin



k=1

nklog

ν

k, (10)

wherethefirst termon therighthand side emergesas a conse-quenceofaccountingforthetotalnumberofdetections.

Theoptimumsolution,inthesenseofmaximumlog-likelihood, logLmax, is obtained using the simplex algorithm (Nelder and Mead,1965) whichstartswithNpar+1random Npar-dimensional solutionvectorsP_l(l₌1_,Npar+1)whereNparisthenumberof pa-rameterstobesolvedfor.Thesimplexcrawlstowardstheoptimum solutionintheNpar-dimensionalphasespacebyimproving,ateach iterationstep,theparameter valuesoftheworst solutionlogL_min

towardstheparametervaluesofthebestsolutionlogLmax accord-ingtothepredefinedsequenceofsimplexsteps.Theoptimization endswhenlogLmax− logLmin<



andall

|

Pi,m− Pj,m

|

<



1,where

i,j refer to differentsolution vectors, m is the indexfor a given parameter, and



1∼ 2× 10−15. To ensure that an optimum solu-tionhasbeenfoundwerepeatthesimplexoptimizationusingthe currentbestsolutionandNrandomsolutionvectorsuntillogLmax changes by less than



2 in subsequent runs, where



2∼ 10−10. Wefoundsuitable valuesfor



1 and



2 empirically. Largervalues wouldpreventtheoptimumsolutionstobefoundandsmaller val-ueswould not notably change theresults. Finally,we employ 10 separate simplex chains to verify that different initial conditions leadtothesameoptimumsolution.

Asan additionalconstraintweforcethefittedparametersPto be non-negative. The reasoning behind this choice is that nega-tiveparametervaluesareeitherunphysical(N0,s,

α

min,s,cs,k, q0) orunconstrained(Hmin,s).AminimumslopeoccurringforHmin,s< 0 is meaningless because all NEOs have H࣡9.4 and we fit for 17<H<25. Hence Hmin,s0 is an acceptable approximation in the hypothetical case that the simplex algorithm would prefer

Hmin,s<0.

Inwhatfollowsweuselow-resolutionorbitdistributions(

δ

a= 0.1au,

δ

e=0.04,

δ

i=4◦)tofitforthemodelparameters,because itwassubstantiallyfasterthanusingthedefaultresolutionofthe steady-stateorbitdistributions(

δ

a₌0_.05au_,

δ

e₌0_.02_,

δ

i₌2◦).In both casesweusea resolutionof

δ

H=0.25magforthe absolute magnitude. We combinethe best-fitparameters obtained inlow resolutionwiththeorbitdistributionsindefaultresolutionto pro-vide ourfinal model.We think thisisa reasonableapproach be-causetheorbitdistributionsarefairlysmoothregardlessof resolu-tionandthedifferencebetweenfittinginlowordefaultresolution leadstonegligibledifferencesintheresultingmodels.

AlthoughGranviketal.(2017) identifiedabouttwo dozen dif-ferentERs,concernsaboutdegeneracyissuespreventedusfrom in-cludingalltheERsseparatelyinthefinalmodel.Insteadwemade educated decisions in combining the steady-state orbit distribu-tions intolarger complexesby,e.g., minimizing Akaike’s Informa-tionCriteria(AIC;Akaike,1974)withacorrectionformultinomial dataandsamplesize(Eq.7.91inBurnhamandAnderson,2002):

AICc=2Npar− 2logLmax+

2Npar

(

Npar+4

)

4Nbin− Npar− 4. (11)

3. DistributionofNEOsasobservedbyCSS

The Mt. Lemmon (IAU code G96) and Catalina (703) stations ofthe CatalinaSky Survey (CSS; Christensenet al., 2012) discov-ered or accidentally rediscovered 4035 and 2858 NEOs, respec-tively,duringthe8-yearperiod2005–2012.Themotivationfor us-ingthedatafromthesetelescopesduringthistimeperiodisthat one of these two telescopes was the top PHO discovery system from2005 through 2011 and the two systems havea long track record of consistent,well-monitored operations. The combination ofthesetwofactorsprovidedusreliable,high-statisticsdiscoveries ofNEOssuitedtothedebiasingprocedureemployedinthiswork.

Detailsof CSSoperationsandperformance can be found else-where(e.g.Christensenetal., 2012;Jedicke etal.,2016)but gen-erally, theG96 site withits 1.5-mtelescope can be considered a narrow-field ‘deep’ survey whereas the 0.7-m 703 Schmidt tele-scope is a wide-field but ‘shallow’ survey. The different capabil-ities provide an excellent complementarity for this work to val-idate our methods as described below. To ensure good quality data we used NEO detections only on nights that met our cri-teria (Jedicke et al., 2016) for tracklet detection efficiency (



0), limiting magnitude(Vlim), anda parameter related to the stabil-ity of the limiting magnitude on a night (V_width). About 80% of all 703 fields and nearly 88% of the G96 fields passed our re-quirements.Theaveragetrackletdetectionefficiencyforthefields thatpassedtherequirementswere75%and88%for703andG96, respectively, while the limiting magnitudes were V=19.44 and

V =21.15(Jedickeetal.,2016).

AllNEOs thatwere identifiedintrackletsinfieldsacquiredon nights that met our criteria were included in this analysis. It is important to note that the selection of fields and nights was in nowaybased onNEO discoveries.The listofNEOs includesnew discoveriesandpreviouslyknownobjectsthatwereindependently re-detectedbythesurveys.TheeclipticcoordinatesoftheCSS de-tectionsatthetimeofdetectionshowthattheG96survey concen-tratesprimarily ontheeclipticwhereas thewide-field703survey imagesovera muchbroaderregion ofthesky(Fig.1,top 2 pan-els). Itisalsoclearfromthesedistributionsthat bothsurveysare located in thenorthern hemisphere as noNEOs were discovered witheclipticlatitudes<−50◦_.

The detected NEOs’a, e, i,andH distributionsare alsoshown inFig.1anddisplaysimilardistributionsforbothstations.The en-hancementneartheq=1aulineinthe(a,e)plotsispartlycaused by observational biases. The smallNEOs that can be detected by

-80 -60 -40 -20 0 20 40 60 80 -150 -100 -50 0 50 100 150 G96

Topocentric ecliptic latitude [deg]

Topocentric ecliptic longitude wrt opposition [deg]

-80 -60 -40 -20 0 20 40 60 80 -150 -100 -50 0 50 100 150 703

Topocentric ecliptic latitude [deg]

Topocentric ecliptic longitude wrt opposition [deg]

10 15 20 25 30 H magnitude 15 30 45 60 75 Inclination [deg] 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 G96 Eccentricity

Semimajor axis [au]

10 15 20 25 30 H magnitude 15 30 45 60 75 Inclination [deg] 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 703 Eccentricity

Semimajor axis [au]

5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 G96 H

Perihelion distance [au]

5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 703 H

Perihelion distance [au]

Fig. 1. Ecliptic coordinates at discovery for NEOs detected by (top left) G96 and (top right) 703. Observed ( a, i, e, H ) distributions for NEOs detected by (middle left) G96 and (middle right) 703. The gray line in the ( a, e ) panels corresponds to q = 1 au . Observed ( q, H ) distributions for NEOs detected by (bottom left) G96 and (bottom right) 703.

ground-basedsurveysmustbeclosetoEarthtobebrighterthana system’s limitingmagnitude, andobjectson orbitswithperihelia neartheEarth’sorbitspendmoretimeneartheEarth,thereby en-hancingthenumberofdetectedobjectswithq∼ 1au.Thiseffectis obviousinthebottompanelsinFig.1inwhichitisclearthat de-tectionsofsmallNEOs(H_࣡25)arecompletelylackingforq_>1.1au (q>1.05aufor703);inotherwords,verysmallobjectscanbe de-tectedonlywhentheyapproachclosetotheEarth.Thus,theNEO modeldescribedhereinisnotconstrainedbyobservationaldatain thatregionof(q,H)space.Insteadtheconstraintsderivefromour understandingofNEOorbitaldynamics.

ItisalsoworthnotingthecleardepletionofobjectswithH_{∼ 22}

inthetwobottomplotsofFig.1(alsovisibleintheH− a panels in the middleof thefigure). The depletion band inthis absolute magnitude range is clearly not an observational artifact because thereisnoreasontothinkthatobjectsinthissizerangearemore challengingtodetect thanslightlybiggerandslightlysmaller ob-jects. Theexplanationisthat theHFDcannot bereproduced with

asimplepower-lawfunctionbuthasaplateauaroundH∼ 22.This plateaureduces their numberstatistics simultaneously and com-binedwiththeir smallsizesreducestheir likelihoodofdetection. Goingtoslightly smallerobjects will increase thenumber statis-ticsandtheyarethereforedetectedingreaternumbersthantheir largercounterparts.

4. ObservationalselectioneffectsofCSS

The observed four-dimensional (a, e, i, H) distributions in Fig. 1 are the convolution of the actual distribution of NEOs withthe observational selection effects



(a, e, i, H) asdescribed in Section 2.1. The calculation of the four-dimensional



(a, e, i, H) is non-trivial but was performed for the Spacewatch survey (Bottke et al., 2002a), for the Catalina Sky Survey G96 and 703 sitesemployedherein (Jedickeetal.,2016),andmostrecentlyfor a combinationof manyNEO surveys(Tricarico,2016; 2017). It is impossibletodirectly comparethecalculated detectionefficiency

Fig. 2. A 2-d slice through the 4-d detection efficiency, ( a, e, i, H ), with i = 2 ◦_and

H = 22 . 875 for the G96 survey ( H = 22 . 875 corresponds to objects of about 100 m diameter). To enhance the regions with small efficiency the fig shows log ( a, e, i,

H ) as a function of semi-major axis and eccentricity.

inthesepublicationsbecause they referto differentasteroid sur-veysfordifferentperiodsoftime.ThefactthatTricarico’s(2017) fi-nalcumulativeHdistributionisinexcellentagreementwiththeH

distributionfoundinthisworksuggeststhatbothbiascalculations mustbeaccuratetowithintheavailablestatistics.

The



(a,e,i,H)sliceinFig.2illustratessomeofthefeaturesof theselectioneffectsthataremanifestedintheobservationsshown in Fig. 1. The ‘flat’ region with no values in the lower-right re-gionrepresents bins that donot containNEO orbits. The flat re-gion in the lower-left corresponds to orbits that can not be de-tectedbyCSSbecausetheyareusuallytooclosetotheSun.There isa‘ridge’ofrelativelyhighdetectionefficiencyalongtheq=1au linethatcorresponds tothe enhanced detectionofobjectsinthe

evs. a panels inFig. 1.That is,higher detectionefficiencyalong theridgemeansthat moreobjectsare detected.Perhaps counter-intuitively,the peak efficiency for this (i, H) combination occurs forobjectswith(a_{∼ 1.15}au,e_{∼ 0.05)}whileobjectsonorbitswith 1.0au≤ a≤ 1.1au are lessefficiently detected. This is because the synodic period betweenEarth andan asteroid with a=1.1au is 11 years but only about 7.5 years for objects with a=1.15au,

closetothe8-year survey time periodconsidered here.Thus, as-teroids with a=1.1au are not detectable as frequently asthose witha=1.15au.Furthermore,thosewithsmallersemi-majoraxis havefasterapparentratesofmotionwhentheyaredetectable. In-terestingly,thedetectionefficiencyisrelativelyhighforAten-class objectsonorbitswithhigheccentricitybecausetheyareat aphe-lionandmovingrelativelyslowlywhentheyaredetectableinthe nightskyfromEarth.

The bias against detecting NEOs rapidly becomes severe for smaller objects (Supplementary Animation 1 and Supplementary Fig.1),andonlythosethat havecloseapproachestotheEarthon low-inclinationorbits areeven remotelydetectable. Formore de-tailsonthebiascalculationandadiscussionofselectioneffectsin generalandfortheCSSwereferthereadertoJedickeetal.(2002, 2016).

5. OrbitdistributionsofNEOs

5.1. IdentificationofERsintheMAB

In order to find an exhaustive set of ERs in the MAB, Granvik et al. (2017) used the largest MBOs with H magnitudes below the assumed completeness limit and integrated them for 100MyroruntiltheyenteredtheNEOregion.

Granvik etal. (2017) startedfrom theorbital elementsandH

magnitudes of the 587,129 known asteroidsas listed on July 21, 2012intheMPC’s

MPCORB.DAT

file.ForMBOsinteriortothe3:1 MMR with Jupiter (centered at a∼ 2.5au) they selected all non-NEOs(q>1.3au)thathaveH≤ 15.9.Exteriortothe3:1MMRthey selectedall non-NEOsthathaveH≤ 14.4anda<4.1au.Toensure thatthesampleiscompletetheyiteratively adjustedthesecriteria toresultinasetofobjects thathadbeendiscoveredprior toJan 1, 2012—that is,no objects fulfillingthe above criteriahad been discoveredinthe ∼ 7-monthperiodleadingtotheextractiondate. Toguaranteeareasonableaccuracyfortheorbitalelementsand

HmagnitudesGranviketal.(2017)alsorequiredthattheselected objectshavebeenobservedforatleast30days,whichtranslatesto arelative uncertaintyofabout1%forsemimajor axis,eccentricity andinclinationforMBOs(Muinonenetal.,2006).Itiswellknown thattheHmagnitudesmayhaveerrorsofsometenthsofa magni-tude.However,whatisimportantforthepresentstudyisthatany systematiceffectsaffecttheentiresampleinthesameway,sothat theHcutisdoneinasimilarfashionthroughouttheMAB.Inthe endGranviketal.(2017)were leftwithasampleof92,449MBO orbits whereHungaria andPhocaea test asteroidswere cloned 7 and3 times,respectively (Fig.3, top andmiddle). They then as-signedadiameterof100mandarandomspinobliquity of ± 90° to each test asteroid. The test asteroids were integrated with a 1-daytimestep for100 Myrunder the influence of a Yarkovsky-driven semimajor-axis drift and accounting forgravitational per-turbations by all planets (Mercury through Neptune). During the course of the integrations 70,708 test asteroidsentered the NEO region(q<1.3au)andtheirorbitalelementswererecordedwitha timeresolutionof10kyr.

Theorbitalelements(a,e,i,



,

ω

,M0)attheMBO-NEO bound-ary(q=1.3au)define thelocationsoftheescaperoutesfromthe MBandformtheinitialconditionsfortheNEOresidence-time in-tegrations(Fig.3,bottom).Weclonedthetestasteroidsassociated withthe

ν

6,o SR,andthe7:2Jand8:3JMMRs5timestoincrease thesampleintheseotherwiseundersampledERs. Thetotal num-beroftestasteroidswasthusincreasedto80,918.

Ourapproachtolimitourselvestoonly100-m-diametertest as-teroids could be problematicbecauseYarkovsky drift inthe MAB may affect the resultant NEO steady-state orbit distribution. The effectwouldarisebecausedifferentdriftratesimplythatasteroids driftingintoresonances willspend a differentamountoftime in or close to the resonances. In cases where the bodies are drift-ing slowly, they could become trapped in tiny resonances and pushed out of the MAB prior to when our modelresults predict (e.g.,Nesvorný andMorbidelli,1998;Bottkeetal.,2002b).Inother cases, SRs such as the

ν

6 withthe adjacent

ν

16 can change the inclinationof theasteroid(FroeschleandScholl,1986;Scholland Froeschle,1986)andtheamountofchangedependsonthetimeit takesfortheasteroid to evolveto theNEO region.Unfortunately, we are not yet at the point where full-up models including ac-curate representations of the Yarkovsky and YORPeffects can be included for tens of thousands ofasteroids across the MAB. Our work inthispaperrepresentsa compromise betweengettingthe dynamicsascorrectaspossibleandensuringcomputational expe-diency.Ourmainconcernhereisthatusingadriftratethatvaries withsizecouldleadtosteady-stateorbitdistributionsthatare cor-relatedwithasteroiddiameter.

Fig. 3. (Top) Initial ( a, i ) distribution of test asteroids. (Middle) Initial ( a, i ) distribu- tion for test asteroids that entered the NEO region ( q = 1 . 3 au ) during the 100-Myr integration. (Bottom) ( a, i ) distribution of test asteroids at the time they entered the NEO region ( q = 1 . 3 au ). The color coding in the middle and bottom plots cor- respond to the nominal set of ERs defined in Section 6.1 . The ERs were defined based on initial orbital elements (Hungarias and Phocaeas) or on orbital elements at the epoch when the test asteroids enter the NEO region (the ν6 , 3:1J, 5:2J, and

2:1J complexes). (For an accurate interpretation of the color coding in the middle and bottom plots we refer the reader to the electronic version of this article.)

Totestthisscenarioweselectedthetestasteroidsinset’B’in Granvik et al. (2017) that entered the NEO region and produced steady-statedistributions forD=0.1kmandD=3kmNEOs that escapethroughthe3:1JMMRandthe

ν

6SR.Therewasthusa fac-tor of30difference insemimajor-axis driftrate. Tosavetime we decidedtocontinuetheintegrationsforonlyupto10Myrinstead of integrating all test asteroids until they reach their respective sinks.SincetheaveragelifetimeofallNEOsis ࣠10Myrthischoice neverthelessallowed mosttest asteroidsto reachtheir sinks. We then discretized and normalized the distributions (Eq. (4)), and

computedthe difference betweenthe distributions for smalland largetestasteroids.

Inthecaseofthe3:1JMMRwefoundnostatisticallysignificant differencesin NEO steady-state orbitdistributions betweenlarge andsmalltestasteroidswhencomparedtothenoise (Supplemen-tary Fig. 2). For

ν

6 we found that while the “signal” is stronger comparedto3:1JsoisthenoisefortheD₌3kmcase.Aprioriwe wouldexpect a stronger effect forthe

ν

6 resonance because the semi-majoraxisdrift induced bythe Yarkovskyeffectcan change thepositionoftheasteroidrelativetoanSR,butnotrelativetoan MMR,whichreacts adiabatically.However, basedon our numeri-calsimulationswe concludedthat theYarkovsky driftintheMAB resultsinchangesintheNEOsteady-stateorbitdistributionsthat arenegligibleforthepurposesofourwork.Thereforewedecided tobasetheorbital integrations,thatare requiredforconstructing thesteady-stateNEOorbitdistributions,onthetestasteroidswith

D₌0_.1kmthatescapetheMAB(Granviketal.,2017).

5.2.OrbitalevolutionofNEOsoriginatingintheMAB

Nextwe continuedtheforwardintegrationoftheorbitsofthe 80,918100-meter-diametertestasteroidsthatenteredtheNEO re-gion with a slightly different configuration as compared to the MBO integrations described in the previous subsection. To build smooth orbit distributions we recorded the elements of all test asteroids with a time resolution of 250 yr. The average change inthe orbital elementsover 250 yr based on our integrations is

a=0.004au,

e=0.006, and

i=0.728◦. The averagechange setsa limit onthe resolutionofthe discretized orbitdistribution in order to avoid artifacts, although the statistical nature of the steady-stateorbitdistributionsoftensdiscontinuitiesintheorbital tracksofindividualtestasteroids.

A non-zero Yarkovsky drift in semimajor axis has been mea-suredfortensofknownNEOs (Chesleyetal.,2003;Nugentetal., 2012; Farnocchia et al., 2013; Vokrouhlický et al., 2015), but the common assumption is that over the long term the Yarkovsky effect on NEO orbits is dwarfed by the strong orbital perturba-tionscausedbytheirfrequentandcloseencounterswithterrestrial planets.Forakm-scaleasteroidthetypicalmeasureddriftin semi-major axiscaused by theYarkovsky effect is ∼ 2× 10−4_auMyr−1 or∼ 5× 10−8_au

₍

_250yr

₎

−1 _{whereas the average change of} semi-majoraxisforNEOs from(size-independent)gravitational pertur-bationsis∼ 4× 10−3_au

₍

_250yr

₎

−1_{.Therateofchangeofthe} semi-major axis causedby Yarkovsky is thus severalorders of magni-tude smallerthan that caused by gravitationalperturbations. We concludedthattheeffectoftheYarkovskydriftontheNEO steady-stateorbitdistribution isnegligiblecomparedto thegravitational perturbations causedby planetaryencounters. Hence we omitted Yarkovsky modeling when integrating test asteroids in the NEO region.

Integrations using a 1-day timestep did not correctly resolve closesolarencounters. Thisresultsinthesteady-stateorbital dis-tribution atlow a and large eto be very “unstable”, asone can seeby comparing the distributions in thetop left corners of the (a, e) plots in Supplementary Fig. 1, obtained by selecting alter-nativelyparticleswithevenoroddidentificationnumbers.Sowe reduced the nominal integration timestep to 12 h and restarted theintegrations.These NEOintegrations requiredonthe orderof 2 million CPU hours and solved the problem (see top left cor-nersofleft-hand-side (a,e) plots inFigs. 4and5). We note that Greenstreetetal.(2012a)usedatimestepofonly4hourstoensure thatencounterswithVenusandEarthare correctlyresolved even in the fastestencounters. Considering that the required comput-ingtimewouldhavetripledifwehadusedafour-hourintegration stepandconsidering that all the evidencewe havesuggeststhat the effectis negligible, we saw no obvious reasonto reduce the

Fig. 4. Steady-state orbit distributions (left) and the corresponding uncertainty distributions (right) for NEOs originating in asteroidal ERs: Hungarias (top panel), ν6 complex

(middle panel), and Phocaeas (bottom panel). (For an accurate interpretation of the color coding, the reader is referred to the electronic version of this article.)

timestepbyanadditionalfactorof3.Westress,however,that the integration step is automatically substantially reduced when the integratordetectsaplanetaryencounter.

The orbitalintegrationscontinueduntilevery testasteroidhad collidedwiththeSunora planet(Mercurythrough Neptune),

es-capedthesolarsystemorreachedaheliocentricdistanceinexcess of100au.Forthelastpossibilitywe assumethatthelikelihoodof thetestasteroidre-enteringtheNEOregion(a<4.2au)is negligi-bleasitwouldhaveto crosstheouterplanetregion without be-ing ejectedfromthe solarsystemor collidingwitha planet.The

Fig. 5. Steady-state orbit distributions (left) and the corresponding uncertainty distributions (right) for NEOs originating in asteroidal ERs: 3:1J (top panel), 5:2J complex (middle panel), and 2:1J complex (bottom panel). (For an accurate interpretation of the color coding, the reader is referred to the electronic version of this article.)

longest lifetimes among the integrated test asteroids are several Gyr.

Outofthe80,918test asteroidsabouttoentertheNEOregion wefollowed79,804(98.6%)totheirrespectivesink.Theremaining 1.4%ofthetest asteroidsdidnotreach asinkforreasonssuch as

endingup inastableorbitwithq>1.3au.As theYarkovsky drift wasturnedoff theseorbitswere found toremainvirtually stable overmanyGyrandthusthetestasteroidswereunabletodriftinto resonancesthat wouldhave broughtthemback into theNEO re-gion.Inaddition,theoutputdatafilesofsometestasteroidswere

corrupted, and in order not to skew the results we omitted the problematictestasteroidswhenconstructingtheNEOsteady-state orbitdistributions.

5.2.1. LifetimesandsinksofNEOs

As expected, the mostimportant sinksare (i) a collision with theSunand(ii)anescapefromtheinnersolarsystemafteraclose encounterwith,primarily,Jupiter(Table3).

TheestimationofNEOlifetimes,thatis,thetimeasteroidsand comets spend in the NEO region before reaching a sink when startingfrom the instant when they enter the NEO region (that is, q≤ 1.3au for the first time), is complicated by the fact that NEOsare alsodestroyed by thermaleffects(Granviketal., 2016). Thetypicalheliocentricdistanceforathermaldisruptiondepends onthesize ofthe asteroid.Forlarge asteroidswithD࣡1km the typical perihelion distance at which the disruption happens is

q∼ 0.058au. We see a 10–50% difference in NEO lifetimes when comparingtheresultscomputedwithandwithoutthermal disrup-tion(Table3). Forsmallerasteroidsthe differenceisevengreater becausethedisruptiondistanceislarger.

We define the meanlifetime of NEOs to be the average time ittakes for test asteroidsfrom a given ER to reach a sink when startingfromthe time that they enter the NEO region. Our esti-mateforthe meanlifetimeofNEOs originating inthe3:1J MMR is comparableto that provided by Bottke et al. (2002a), but for

ν

6 andthe resonances in the outer MAB our meanlifetimes are about 50% and 200% longer, respectively (Table 3). These differ-encescan,potentially,be explained by differentinitial conditions andthe longer timestep used forthe integrations carried out by Bottkeetal.(2002a).Alongertimestepwouldhavemadethe or-bitsmoreunstablethantheyreallyareandcloseencounterswith terrestrialplanetswouldnothavebeenresolved.Anaccurate treat-mentofcloseencounterswouldpullouttestasteroidsfrom reso-nanceswhereasaninabilitytodothiswouldleadtotestasteroids rapidlyendingupintheSunandtherebyalsotoashorteraverage lifetime.

5.2.2. Steady-stateorbitdistributionsandtheiruncertainties

Wecombinedtheevolutionarytracksoftestasteroidsthat en-ter theNEO region through12 ERs andfrom 2additionalsource regionsinto6steady-stateorbitdistributionsby summingupthe time that thetest asteroidsspend in variousparts ofthe binned (a,e,i)space(leftcolumninFigs.4and5).

To understandthe statisticaluncertainty ofthe orbit distribu-tions,wedividedthetestasteroidsforeachorbitdistributioninto even-numberedandodd-numberedgroupsandestimatedthe un-certaintyoftheoverallorbitdistributionby computingthe differ-encebetweentheorbitdistributions composedofeven-numbered andodd-numberedtest asteroids. The differencedistribution was thennormalizedbyusingthecombinedorbitdistributionsoasto resultina distributionwiththesameunitsasthecombined dis-tribution(rightcolumninFigs.4and5).

5.2.3. Steady-stateorbitdistributionofJupiter-familycomets

There are several works in the published literature which computed the steady-state orbital distribution of Jupiter-family comets by integratingparticles coming fromthe trans-Neptunian region up to their ultimate dynamical removal. The pioneer-ing work was that of Levison and Duncan (1997), followed by Levisonetal.(2006),Di Sistoetal.(2009),andBrasserand Mor-bidelli (2013). The resulting JFC orbital distributions have been kindlyprovidedtousbytherespectiveauthors.Wehavecompared them and selected the one from the Levison et al. (2006) work becauseit isthe onlyone constructed usingsimulations that ac-countedforthegravitationalperturbationsbytheterrestrial plan-ets.Thus,unliketheotherdistributions,thisoneincludes“comets”

Fig. 6. JFC steady-state orbit distribution. (For an accurate interpretation of the color coding, the reader is referred to the electronic version of this article.)

on orbits decoupled from the orbit of Jupiter (i.e., not under-going close encounters with the giant planet at their aphelion) such as comet Encke. We think that thisfeature is important to modelNEOs oftrans-Neptunianorigin.Weremindthereaderthat Bottke etal. (2002a) used the JFC distribution from Levisonand Duncan(1997),giventhattheresultsofLevisonetal.(2006)were not yetavailable. Thus,this isanotherimprovementof thiswork overBottkeetal.(2002a).TheJFCorbitaldistributionweadopted isshowninFig.6.

Thereisanimportantdifferencebetweenwhatwehavedonein thisworkandwhatwasdone intheearliermodelsoftheorbital distribution ofactive JFCsbecause they includeda JFC fading pa-rameter.Inessence,acometisconsideredtobecomeactivewhen its periheliondistance decreases belowsome threshold(typically 2.5au)forthefirsttime.Thateventstartsthe“activityclock”. Par-ticles are assumed to contribute to the distribution of JFCs only uptoatimeT_active oftheactivityclock.Limitingthephysical life-timeisessentialtoreproducetheobservedinclinationdistribution ofactiveJFCs,asfirstshowninLevisonandDuncan(1997).What happens after Tactive is not clear. The JFCs might disintegrate or theymaybecomedormant.Onlyinthesecondcase,ofcourse,can thecometcontributetotheNEOpopulationwithanasteroidal ap-pearance.Webelievethesecondcaseismuchmorelikelybecause JFCsarerarelyobservedtodisrupt,unlikelongperiodcomets. Be-sides,severalstudiesarguedfortheexistenceofdormantJFCs(e.g., Fernández etal., 2005; FernándezandMorbidelli, 2006). Thus,in orderto build thedistribution showninFig.6 wehave usedthe original numerical simulations of Levison et al. (2006) but sup-pressedanylimitationonaparticle’sage.

6. DebiasedNEOorbitandabsolute-magnitudedistributions

6.1. Selectingthepreferredcombinationofsteady-stateorbit

distributions

We first needed to find the optimum combination of ERs. To strikea quantitative balancebetweenthegoodness offitandthe numberofparametersweusedtheAICcmetricdefinedbyEq.(11). Wetested9differentERmodelsoutofwhichallbutonearebased on different combinations of the steady-state orbit distributions described inSection 5. The one additional model is the integra-tionof Bottke-likeinitial conditionsby Greenstreet etal.(2012a). The combination of different ERs was done by summing up the residence-time distributions of thedifferent ERs, that is, prior to normalizing theorbit distributions. Hence the initial orbit

distri--22650 -22600 -22550 -22500 -22450 -22400 -22350 -22300 -22250 -22200 G12 5_ERs 6_E R_s 7_ERs 8_ERs 9_E R_s 10 ERs 23ERs 44600 44700 44800 44900 45000 45100 45200 45300 45400 Log-likelihood AIC ER model Log-likelihood AIC

Log-likelihood with common HFD AIC with common HFD

Fig. 7. The log-likelihood of the best-fit solution (left axis and solid line) and the corresponding AICc metric (right axis and dashed line) as a function of the number of ERs included in the model. G12 stands for the five-component orbit model by

Greenstreet et al. (2012a) . For reference we also show the results when assuming a common HFD for the model with 7 ERs.

butionandthedirectintegrationsprovidedtherelativeshares be-tweenthedifferentorbitdistributionsthatwerecombined.

Asexpected,themaximumlikelihood(ML)constantlyimproves asthesteady-stateorbitdistributionisdividedintoalarger num-ber of subcomponents (Fig. 7). The ML explicitly showsthat our steady-state orbit distributions lead to better fits compared to the Bottke-like orbit distributions by Greenstreet et al. (2012a), evenwhenusingthesamenumberofERs,that is,fourasteroidal andone cometaryER.The somewhat unexpectedoutcome ofthe analysis is that we do not find a minimum for the AICc metric, whichwouldhavesignaledan optimumnumberofmodel param-eters (Fig. 7). Instead the AICc metric improves all the way to the most complex model testedwhich contains 23 different ERs and hence 94free parameters! The largest dropin AICc per ad-ditional source takes place when we go from a five-component modeltoasix-componentmodel.Thedifferencebetweenthetwo beingthattheformerdividestheouterMABintotwocomponents andlackHungariasandPhocaeaswhereas thelatterhasa single-component outer-MAB ER and includes Hungarias and Phocaeas. Continuing to theseven-component modelwe againsplit up the outer-MABERintotwocomponents(the5:2Jand2:1Jcomplexes). Thedifferencebetweenthefive-componentmodelandthe seven-componentmodelisthustheinclusionofHungariasandPhocaeas in the latter. The dramatic improvement in AICc shows that the HungariasandPhocaeasarerelevantcomponentsofanNEO orbit model.

Althougha modelwith94free parameters isformally accept-able,we hadsome concernthat itwould leadto degeneratesets of modelparameters. It mightalso (partly)hide real phenomena that are currentlynot accountedforandthus shouldshow up as adisagreementbetweenobservationsandourmodel’spredictions. Thereforewetook aheuristicapproachandcomparedthe steady-state orbit distributions to identify those that are more or less overlapping and can thus be combined. After a qualitative eval-uation ofthe orbit distributions we concluded that it is sensible to combinethe asteroidal ERs into sixgroups (Figs. 4 and5). Of thesesixgroupstheHungariaandPhocaeaorbitdistributions are uniquely defined based on the initial orbitsof the test asteroids whereasthefourremaininggroupsarecomposedofcomplexesof escape routes (

ν

6, 3:1J, 5:2J, and 2:1J) that produce overlapping steady-stateorbitdistributions.

In principle one could argue, based on Fig. 7, that it would makesensetouse9ERsbecausethenthelargestdropintheAICc metric wouldhavebeenaccounted for.Thedifference between7 and9 ERs is that the4:1J hasbeenseparated fromthe

ν

6

com-Fig. 8. Steady-state orbit distributions for ν6 and 7:2J (top) and 4:1J (bottom). (For

an accurate interpretation of the color coding, the reader is referred to the elec- tronic version of this article.)

plex(Fig.8) andthe

ν

6 componentexterior tothe 3:1Jhasbeen separatedfromthe3:1J(Fig.9).However,thedifferencesbetween the

ν

6complexandthe4:1Jorbitdistributionsaresmallwiththe mostnotabledifference beingthat the

ν

6 distributionextends to largera.Similarly,the

ν

6componentexteriortothe3:1Jhassome clearstructurecomparedtothe3:1Jcomponentbutthisstructure isalsoclearlyvisiblein thecombined3:1J orbitdistribution(top panelinFig.5).Thereisthusasubstantialoverlapintheorbit dis-tributionsandwe thereforedecidedto includeone cometaryand sixasteroidalERsinthemodel.

6.2.Thebest-fitmodelwith7ERs

Havingsettledonusing7steady-stateorbitdistributionsforthe nominal model we then turned to analyzing the selected model in greater detail. As described in the Introduction, it is impossi-bleto reach an acceptable agreement betweenthe observed and predictedorbitdistributionunlessthedisruptionofNEOs atsmall

q isaccountedfor(Granviketal., 2016).Here we solvedthe dis-crepancybyfittingforthetwoparametersthatdescribeapenalty functionagainstNEOs withsmallq (seeSection2), inadditionto theparametersdescribingtheHdistributions.The best-fit param-eters for the penalty function are k=1.40± 0.07au−1 and q0= 0.69± 0.02au.Althoughadirectcomparisonofthebest-fitpenalty function and the physical model by Granvik et al.(2016) is im-possiblewe find thatthe penalty function p=0.86± 0.05 atq= 0.076au and, of course, even higher forq<0.076au. Considering