198 5MNRAS.212. .817S families. Commensurabilities,inparticularwithJupiter, leadtotheabsenceofasteroidsat of thesebodies,reviewedbyChapman,Williams& Hartman(1978),Chapman(1983)andin study. certain solardistancesknownastheKirwoodGaps, and thesehavebeenthesubjectofintense of asteroidshaveshownmanyclustersintheorbital elements,commonlyknownasasteroid Gehrels (1979),haveshownthatdistinctgroupsofcommon genesisexist.Studiesofthedynamics Over thepasttwodecadesourknowledgeofasteroidshasexpandedgreatly.Physicalstudies planets. Wetherill(1976)estimatesthat30000±50 per centlargerthan1kmmaycrossMars’ andJupiter,thereisasignificantpopulation which crossestheorbitsofterrestrial , althoughHelin&Shoemaker(1979)giveafigure of10000±50percent. 1 Introduction -crossers) aretermed Apolloobjects.AdifferenttypeofEarth-crosserisan asteroidofthe Aten class;thesehavesemi-major axesa0.9833 au. (TheEarth’s distance 1.0167au.)Another classofasteroid,theAmorobjects,arearbitrarily selectedbydint orbital eccentricityof0.0167 meansthatithasperiheliondistance0.9833au andaphelion 2 7 Although thevastmajorityofasteroidsinhabit region entirelyboundedbytheorbitsof oflargesemi-major axiswhosepathstakethemclosertotheSunthan 1.0167 au(i.e. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Mon. Not.R.astr.Soc.(1985)212,817-836 Duncan I.SteelandW.J.BaggaleyDepartmentofPhysics, the terrestrialplanets Impacts oftheApollo-Amor-Atenasteroidsupon Accepted 1984September26.Received14;inoriginalformJuly11 Collisions intheSolarSystem-1. population offourAten,34Apolloand38Amorasteroidseachthe University ofCanterbury,Christchurch,NewZealand useful incalculatingtherateofremovalthesebodiesbyterrestrialplanets, terrestrial planetsisdeterminedbyanewtechnique.Theresultingmeancollision Summary. Thecollisionprobabilitybetweeneachofthepresently-known rates, coupledwithestimatesofthetotalundiscoveredpopulationeachclass,is influx totheEarthisfoundbeoneimpactper160000yr,butthisfigurebiased by theinclusionoffourrecently-discoveredlow-inclinationApollos.Excluding and thecrateringrateoneachplanetbybodiesofdiameterinexcess1km.The The impactrateishighestfortheEarth,beingaroundtwicethatofVenus. these fourtheratewouldbeoneper250000yr,inlinewithpreviousestimates. rates forMercuryandMarsusingthepresentsampleareaboutoneper5Myr one per1.5Myrrespectively. 198 5MNRAS.212. .817S 7 818 D.LSteelandW.J.Baggaley which willevolvesothattheybecomeEarth-crossers,andwouldthenbeclassedasApollos for eachplanet,andthemoon.Thisrequiresanestimateoftotalpopulationclass; of havingperiheliondistance1.0167<#<1.3au.ManyAmorasteroidsareknowntohaveorbits Helin &Shoemaker(1979)findthenumberstoabsolutevisualmagnitudeF(l,0)=18be: asteroids andeachoftheterrestrialplanetsisinterest.Thefirstanestimatecrateringrates 700±300 Apollos; (Shoemaker etal.1979;hereafterSWHW). population; theabovefigureassumesthatonlyoforderoneintwentyApolloshasyetbeen criticized byKresák(1978a,1981),whofindsthatonlyabout250Apollosshouldexist.Thereis discovered. ThemethodsformakingtheestimateshavebeendiscussedbyWetherill(1976)and , butthesemightbetakentoasteroidsofdiameter1kmorgreater(Wetherill1976). Conversion fromavisualmagnitudetophysicalsizeisseverelydependentupontheassumed the Apollo-Atenpopulation(SWHW). —5000 ‘Mars-grazers’. 1000-2000 Amors; —100 Atens; interest isindeducingtheirlifetimesandhenceproductionrate.Althoughasteroidbeltcollisions also somedisagreementbetweenlunarandterrestrialcrateringratestheexpectedratefrom 10000±5000 Mars-crossers;and inner SolarSystemisalsoappreciable.Previousestimatesfortheirdynamicallifetimesareofthe are themostsignificantforApollosandAmors,lossrateduetoplanetarycollisionsin order of2x10yr(Wetherill1967,1976,1979;Tedescoetal.1981).Thisismuchshorterthanthe age oftheSolarSystemsothatacontinuousproductionisrequired.Possiblesourcesare has beendisputed(Levin&Simonenko1981;Kresák1981).Theoverallknowledgeofthe extinct cometarynuclei.Thelatterisgenerallyfavoured(SWHW;Wetherill1979)althoughthis belt,wherecloseencountersmightthrowthecollidingobjectsintoeccentricorbits,or population. Thesealsohavethehighestcollisionprobabilitieswithplanets,sothatoverall close totheeclipticplane,abiastowardslow-inclinationobjectsexistsindiscovered asteroid populationhasbeenreviewedbyHughes(1981). years asanexplanationformassextinctionsofterrestrial biota(Napier&Clube1979;Alvarezet in thispaper.ThisproblemhasbeenstudiedbyKnezevic (1982). distribution mayhavealowermeancollisionratethanthatdeducedbypreviousresearchers,and al. 1980).Inparticular,recentspeculationshavesuggested a26-Myrcyclicitytocometarywaves Hut &Muller1984).Thisperiodiscomparabletothe dynamicallifetimesoftheplanet-crossing entering theinnerSolarSystem(Rampino&Stothers 1984;Whitmire&JacksonDavis, present Apollo-Amor-Atenobjectsarenotindicative ofasteady-statepopulation,butare merely theremnantsofapreviouscometarywave. This hasbeendiscussedinsomedetailby presently-discovered populations (4Atens,34Apollos,38Amors)aretaken tohaveorbital collisional lifetimes,and oftheimpactratewitheachterrestrial planets.The Clube &Napier(1982;1984). asteroids. Iftheseasteroidsarereallyextinctcometary nucleithenitisseenthatpossiblythe distributions typicalofeach class.Thetotalpopulationsaretakentobe100Atens, 700Apollos There aretwomainreasonswhythecollisionprobabilitybetweenApollo-Amor-Aten The methodsofsearchingforplanet-crossingasteroidshavealargeeffectuponthededuced The secondreasonthatthecollisionprobabilityofApollo-Amor-Atenobjectsis A fewmorepertinentpointscanbemadehere.Sinceasteroidsearchestendtocarriedout The asteroidandcometinfluxtotheEarthhasalsoreceived muchattentionoverthepastfew It istheaimofthispaperto makeanestimate,usinganewmethod,oftheApollo-Amor-Aten © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 198 5MNRAS.212. .817S Wetherill (1967)generalizedthismethodtocoverthecasewhereneitherorbithaszero Shoemaker (1983). Previous techniquesforevaluatingcollisionprobabilitieshavebeenderivedfromthoseofÖpik 2 Collisionalprobabilitycalculation and 1500Amors,followingHelin&Shoemaker(1979),Wetherill(1982), equations inaMonteCarlosimulation,andrecentlyZimbelman(1984)hasusednumerical eccentricity, andSWHWhavealsodevelopedthistheory.Arnold(1965a,b)usedÖpik’s (1951,1963,1966). Öpiktookoneoftheorbitingobjects(i.e.Earth)tohaveacircularorbit. averaging oftheequationstodeducecollisionprobabilitiesbetweenlong-periodcometsandeach cross-section isä(afunctionofvelocity,duetogravitationalfocusing),andthemeanspatial of theplanets. it totheoutermoonsofJupiter,showingthatfourprogradehadself-collisional unit volume,onaverage).InanimportantpaperKessler(1981)outlinedthismethodandapplied method basedupontheconceptof‘spatialdensity’eachobject(thenumberobjectsper method, whichisanalogoustothekinetictheoryofgases,extremelypowerfulandhaswide general notice,hismethodhasbeenexpandedanddetailedinanAppendixtothispaper.The lifetimes muchshorterthantheageofSolarSystem.SinceKessler’stechniquehasescaped general use.Itisbaseduponthefactthatcollisionprobabilitybetweentwoobjectsgivenby: bodies. ThemeanrelativevelocityatthepositionofvolumeelementisV,,collision where thesummationisperformedoverallvolumeelementsAi/whichareaccessibletoboth assumed thatthetwoobjectsdonotavoideachotherduetolong-livedcommensurabilities.Details densities atthispositionareSandj.Theselatterparametersderivablefroma P^SySyVjdjAUf random: precessionandsecularperturbationsleadtoavariationintheargumentofperiapsison generalization ofÖpik’sformulae,beingdependentuponthesemi-majoraxis(a),eccentricity(e) time-scale whichisshortcomparedtodynamicallifetimes(Öpik1951;SWHW).Italso and inclination(/)ofeachthetwoorbits.Theargumentperiapsisforbodyistakentobe of eachstep,andthemethodnumericalcomputation,aregiveninAppendix. ; The collisionprobabilitieswitheachoftheterrestrial planetsfortheAten,ApolloandAmor in theAppendix.Theplanetarymeanorbitalelements weretakenfromtheExplanatory asteroids areshowninTables1,2and3respectively. The methodofcalculationwasasdescribed 3 CollisionprobabiltiesfortheApollo-Amor-Aten asteroids ; 1;2 The problemofEarth-orbitingartificialsatellitesledKessler&Cour-Palais(1978)todevelopa © Royal Astronomical Society • Provided by theNASA Astrophysics Data System i Table 1.TheAtenasteroids. 2062 Aten0.9665 2100 Ra-Shalom0.8321 NUMBER NAME 2340 Hathor0.8439 1954XA 0.7772 a=(A.U.) 0.3454 3.930.685 0.1826 18.940.950 0.4364 15.760.759 0.4498 5.860.775 e =i=(degrees)'P-(years) ELEMENTS Collisions intheSolarSystem-I.819 9 VENUSEARTH COLLISION PROBABILITYPER10ORBITS 0.44 5.734.64 3.23 15.411.4 35.5 41.2 7.61 198 5MNRAS.212. .817S consequence. Anexception tothiscommentwouldbeapreliminaryorbitwhicherroneously gave lifetimes oftheplanet-crossing bodies,theuncertaintiesinorbitsare ofnomajor 820 D.I.SteelandW.J.Baggaley a verysmallinclination,or aperihelion/aphelionclosetooneoftheplanets, sincethenthe for individualasteroids.However, sinceallthatisrequiredanestimateof thedynamical date. Thismeansthattheorbital elements(a,e,i)usedheremightbeconsidered tobeimprecise numbered asteroidsassoontheorbitisknownsufficiently welltopermitrecoveryatanyfuture un-numbered) ApollosandAmorshavebeenincluded. Mostofthesewouldbecomenamedand Supplement (1974)andthephysicalcharacteristics(mass andradius)fromSmith&West(1983). The asteroidalmassandradiusareinsignificantinthis respect. Notes (1)TorohasacommensurablemeanmotionwiththeEarth,precludingcollisioninpresent Since itisdesirabletohaveaslargeasamplepossible, severalrecentlydiscovered(andhence Table 2.TheApolloasteroids. Number 2063 2201 2329 1862 2101 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 1620 2135 1981 2212 1566 2102 1866 1685 1865 1864 1863 (3) Theorbitof1959LMisuncertainsinceittakenfromonly ashortarc,andsoisnot (2) 1866isadoubtfulEarth-crosser(Shoemakeretal.,1979; Shoemaker 1983). Apollo 6 Toro Midas Geographos Orthos Aristaeus Hermes Adonis Antinous 6743P-L Bacchus Cerberus Daedalus Hephaistos Tantalus 1982HR 1983VA 1979XB 1981 VA 1947XC Sisyphus 1983TF2 1983LC 1982TA 1974MA 6344P-L 1978CA 1973NA 1983TB 1979VA 1982DB 1982BB (1937 UB) Icarus 1950DA (IpandMehra,1973).However,closeencounterswithMarswilldisplaceTorofromthis unaffected (WilliamsandWetherill,1973;Shoemakeretal.,1979). resonance withinabout3x10years,sothatthecollisionprobabilitygivenislargely included here. NAME 2.4600 2.6316 2.1734 2.6143 1.5996 1.3672 1.0776 2.2624 1.7759 a=(A.U.) e= 2.4042 1.2100 1.6805 1.4712 1.6393 2.3030 1.0779 1.1248 2.4272 1.3428 1.0801 1.8749 1.7752 2.1637 1.2715 2.6354 1.2900 2.2602 1.8930 1.2446 1.4609 1.6834 1.4893 2.6186 1.4070 0.4359 0.7092 0.3495 0.7133 0.7118 0.3227 0.6925 0.5237 0.5600 0.7439 0.6499 0.6236 0.4669 0.5394 0.3355 0.3871 0.6586 0.5037 0.6148 0.7710 0.7638 0.8351 0.8268 0.8903 0.6381 0.7620 0.5020 0.6273 0.6411 0.3548 0.2984 0.6065 0.2148 0.3602 ELEMENTS i=(degrees) P=(years] 24.87 22.02 39.84 24.39 23.04 22.91 68.00 41 .15 16.25 16.09 22.16 12.12 37.79 22.04 26.12 11.89 64.02 13.32 12.15 20.94 18.42 9.37 9.42 6.35 6.22 1.52 2.52 2.69 7.90 7.84 1.36 2.78 4.65 1.42 4.269 3.403 1.784 3.858 3.204 2.367 4.227 2.179 2.023 1.119 2.605 3.728 1.331 1.599 2.099 2.365 3.183 1.434 4.278 4.237 3.781 1.389 1.556 1.123 3.495 2.567 1.119 2.184 1.766 1.818 1.465 3.398 1.193 1.669 MERCURY .VENUSEARTHMARS COLLISION PROBABILITYPER10’ 1.69 1.03 2.82 1.71 2.84 26.5 18.1 31.9 15.8 20.5 4.96 4.85 3.63 4.20 6.24 5.63 3.10 3.16 6.11 2.35 115.0 1 2 32.2 29.6 31.7 17.2 71.5 20.0 6.74 2.02 7.67 2.16 7.54 2.92 7.49 7.34 3.35 1.98 6.24 2.64 3.53 1.33 4.12 2.30 2.51 9.48 3.45 1.54 2.24 3.45 1.98 5.31 3.66 5.52 5.16 0.20 0.79 0.22 0.62 0.60 0.16 0.39 0.25 0.87 0.79 0.22 0.29 0.25 0.61 2.15 0.77 0.45 0.25 0.38 0.15 0.27 0.18 3.30 3.18 0.42 0.52 0.28 0.16 3.00 3.79 0.35 1.90 1.02 198 5MNRAS.212. .817S favour thediscoveryofsuchobjects. calculated collisionprobabilitywouldbeconsiderably enhanced.Infact,searchtechniques high eccentricity(e>0.7)could dosoinprinciple.Threeofthesebodieswerelisted bySWHW. Palomar-Leiden Survey(van Houtenetal.1970). (Bender 1979),theMinor PlanetCirculars,theEphemeridesofMinorPlanets, andthe The orbitalelementsused, andlistedinTables1-3,weretakenvariouslyfromthe TRIADfile The fourknownAtenasteroids (Table1)donotcrosstheorbitofMars,although anAtenof © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Table 3.TheAmorasteroids. Notes (1)Themethodusedtodeterminethecollisionprobability (finitevolumeelements)leadstoa NUMBER 2608 2061 1943 1917 1915 2202 1221 1980 1580 1627 2059 2368 1036 1916 433 887 719 9 (4) 1983SAalsocrossesJupiter'sorbit,havingacollisionprobability of224.0per10orbits. (2) 1982YAcrossestheorbitofJupiter,withwhichishasacollision probabilityof525.0per (3) AccordingtotheTRIADfile(Bender,1979),719Albertis a lostasteroid. Anza 1982XB Anteros QuetzalcoaU Cuyo 1980PA 1978DA 1980AA Tezcatlipoca Alinda Amor 4788P-L 1981QB 1983RD 9 Albert Betulia 1972RB Ganymed Eros Pele 1982YA 1983RB 9 1981QA 1981CW 1977VA 1982RA 19820V 1983LB 1980YS 1983SA Ivar 1977RA 1980WF 1979QB 1982FT 1982RB 1963UA 1953RA aphelion of1.01670.Theresultantprobabilityis27.6per10 orbits. non-zero resultfor1982XB,whichhasaperihelionof1.01673 comparedtotheEarth's 10 orbits. NAME a=(A.U.) 2.4783 2.2647 1.8379 1.9263 2.1488 2.5286 1.4307 1.8915 2.2391 2.0888 2.2233 1.7096 1.9206 2.6117 2.4949 2.1487 4.2292 2.1515 1.8779 2.1967 2.2898 2.0329 2.2909 2.5839 1.4583 3.1030 1.8646 1.5748 1.8636 2.1041 2.6625 2.6252 2.2728 1.8153 2.2308 2.3300 2.1024 1.7517 0.5866 0.4468 0.4586 0.5372 0.4435 0.5048 0.2560 0.4863 0.5773 0.5181 0.5578 0.4875 0.5070 0.4343 0.3651 0.5587 0.5124 0.6409 0.4571 0.3687 0.2229 0.2838 0.3969 0.4895 0.4786 0.4486 0.3939 0.3212 0.7147 0.5404 0.5374 0.5259 0.4499 0.4132 0.5141 0.4423 0.2777 0.3946 ELEMENTS i=(degrees) P=(years) 23.99 15.64 20.50 Collisions intheSolarSystem-1. 26.85 37.16 52.03 33.22 19.43 11.89 10.83 32.98 30.78 25.40 10.82 10.96 26.45 24.99 20.07 10.99 12.84 4.18 2.16 3.88 8.70 3.74 9.51 8.79 8.44 9.25 5.21 8.41 4.78 5.93 2.97 2.28 5.26 6.41 3.38 3.901 2.492 4.021 2.601 3.408 2.674 3.150 1.711 3.351 3.019 3.941 3.150 3.315 2.662 2.235 4.221 2.899 3.256 3.465 5.466 4.153 2.573 2.546 2.544 8.697 3.156 1.761 1.976 4.344 3.467 2.446 4.253 2.318 3.048 3.426 3.052 3.332 3.557 COLLISION PROBABILITYPER 9 10 ORBITS 1 2 3 4 0.329 1.37 0.258 3.22 0.911 1.37 1.27 0.543 0.274 0.288 0.210 0.546 0.993 0.300 0.463 0.588 0.885 0.480 0.211 0.208 0.826 0.326 0.661 0.282 0.502 0.661 2.03 0.230 0.279 1.24 0.788 0.528 0.486 3.57 0.449 0.344 1.12 1.98 MARS 821 198 5MNRAS.212. .817S 822 D.I.SteelandW.J.Baggaley 4 They calculatedimpactprobabilitieswiththeEarthusingequationsofÖpik(1951),andby their ownderivativeofthatmethod.Threesetsresultscanthereforebecompared: own limitedmethod.Themorepreciseresultsobtainedinthispaper,usingamethodofwider with theEartharerecentdiscoveries.Thisisdiscussedinmoredetaillater. general use,areseentobeinlinewiththepreviousestimatesofSWHW. are Mars-crossers.ItisseenthatthemajorityofApolloshavingveryhighcollisionprobabilities Hulkower &Bender1984;theorbitalelementsgiveninthatpaperwereerror).Notonlydoes Earth-collision especiallylikely.Withreferencetothetechniqueusedhere,asdescribedin DB isverymuchenhancedat1au,andclosetotheecliptic.Thecollisionallifetimeof1982 Section 2aboveandintheAppendix,thisisseentobebecausemeanspatialdensityof1982 for severeorbitaldisruptionbyaplanetorcollisionwithbeltasteroids,isofcourseverymuch shorter. against animpactwiththeEarth(orMars)isthereforeonlyabout15Myr;dynamicallifetime, 1982 DBhaveasmallinclinationbutalsoitsperiheliondistance(#—0.95au)makesan but ingeneralcomparableorbitalelementsrendercollisionprobabilities.Fromthose have beenusedhere.Thereforeadirectcomparisonofresultsisnotpossibleforeveryasteroid, have beenselectedasexamplesofEarth-impactprobabilities: asteroids forwhichtheelementsusedhereandbySWHWarebasicallysame,following used inthispaperutilizesosculatingorbits,noaccount istakenofthegradualevolution Amor orbitsduetosecular perturbations.ThusonlyMarsoutoftheterrestrial planetshasa Marsden 1970;Wetherill& Shoemaker1982).AbouthalfoftheAmorsbecomeEarth-crossers at non-zero collisionprobability. InfactEarth-approachingasteroidscanevolveinto Earth-crossers some stage,themajority ofthesebeingshallowcrossers(SWHW).This would implyan (i.e. AmorsevolveintoApollos) onatime-scaleofonly~10yr(Wetherill& Williams1968; SWHW cautionedagainsttheapproximationusedinÖpik’soriginalderivation,andthustheir Especially anomalousinthisrespectis1982DB,potentiallytheclosestofApollos(Helin, Of the34ApollosinTable2,fivecrossorbitofMercury,15thatVenus,andallbutone For thoseasteroidswhichwerealsoincludedinthelistofSWHW,somecasesupdatedorbits Table 3showsthecollisionprobabilitieswithMarsfor 38Amorasteroids.Sincethemethod © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Aten 6.96.48.0 Ra-Shalom 6.36.76.1 Hathor 1414.7 Daedalus Icarus Tantalus Hermes Midas Cerberus (Öpik) (SWHW)(thispaper) Pi Pi (Öpik) (SWHW)(thispaper) 2.2 2.5 2.5 3.8 1.0 1.6 9 9 (All per10yr) (All per10yr) 2.0 3.1 0.7 3.7 1.6 1.5 0.7 3.5 3.1 2.5 1.3 1.8 198 5MNRAS.212. .817S -9 -9 4 3 9 -9 9 9 -9 found ameancollisionprobabilitywiththeEarthof~1x10peryearforpopulation—500 probability withtheEarth,asillustratedbynoteconcerning1982XBinTable3.SWHW . Earth-crossing Amors.Thiscompareswiththeirvalueof2.6xl0peryearfortheaverage asteroidal perihelioncloseto1au,andconsequentlyanenhancedspatialdensitycollisional ö>1.0167 auandtheywouldthenbeclassifiedasAmors.Thedistinctionisseentoarbitrary true forsomeApollos.Thesemayoscillateinsemi-majoraxissuchthatepochs and thepartitionbetweenthreeclassificationsinpresentsetisassumedtobemuch same asinanyotherepoch.Thetime-variationnumberoftheseobjectsisdiscussedlater. these apheliondistancesof5.09and7.25aurespectively,sothattheyareJupiter-crossers.This The lifetimeagainstcollisionwithJupiterisaboutlOMyrfor1982YAand40Myr1983SA: between theJovianorbitalextremes.Eventhoughonlytwoof38AmorscrossJupiter, results inahugecollisionprobabilitywithJupiter,especiallysofor1982YAsinceithasaphelion Jupiter ofonly~10yr.Itmustbereiteratedthatthisassumesthat:(1)thepreliminaryorbital mean planetarycollisionprobabilityforthewholesetistotallydominatedbygiantplanet. since acloseapproachresultingingrossorbitaldisruptionisatleast~10timesmorelikely commensurabilities orothercauses. perihelion canbetakentorandom;and(3)theasteroidsdonotavoidJupiterdue elements arereasonablyprecise;(2)precessionoccursatsucharatethattheargumentof 4 Discussion (Weidenschilling 1975),thesetwoobjectshavelifetimesagainstcatastrophicdisruptionby Table 4showsthemeancollisionprobability(per10yr)forseparateAten-Apollo-Amor categories againsteachoftheterrestrialplanets.Itisusefultocomparetheseresultsthose of SWHWwhocalculatedtheEarth-collisionrates. impact probabilityhereisPi=22xl0collisionsperyear(spreada„_=26xl0,theunbiased 9.1 x1CTbySWHW,thediscrepancyismainlyduetoinclusionoflow-inclinationobject estimate ofthestandarddeviationprobabilitiesforallfourAtenasteroids)comparedto the largespread,isinclusionhereofseveral recently-discovered Apolloshavinglarge here thevalueis5.8x10“peryear(spread11.3x).Themajorreasonfordifference,and 1954XA. FortheotherthreeAtensmeancollisionrateis9.6xl0peryear. 1 9 In thesamewayassomeAmorasteroidsareEarth-crossersforpartoftime,converseis An additionalpointfromTable3isthatthepreliminaryorbitsof1982YAand1983SAgive As indicatedpreviously,theagreementisexcellentforAtenasteroids.Althoughmean For theApollosSWHWfoundameanEarth-impactprobabilityof2.6x10“peryear,whereas © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 9 ((J-i) forcollisionsbetweenterrestrialplanetsandtheApollo- Table 4.Meancollisionprobabilities(Äper10yr)andspreads All terrestrial Mars Amor-Aten asteroids. Mercury n Earth planets Atens 2.4 +2.551.0±0.4 43 22 ±26345.811.3 26 ±23155.46.0 P, 34 8.7380.29 33 0.41±0.55380.290.31 71 0.35 Apollos Cottisions intheSolarSystem-1.823 Amors P. n P. asteroids 38 7.5 18 8.8 7 1.4 All i 198 5MNRAS.212. .817S -91 o-91 9 9- 9 824 D.I.SteelandW.J.Baggaley collision probabilityPi—2.9xl0yr,ingoodagreementwithSWHW. which hasaninclinationofonlyI.36Thecollisionprobability11.5x10yrcompareswith have beenpreferentiallydiscoveredduetotheirlow-inclination,Earth-approachingtrajectories. collision probabilities(inparticular1979VA,1982DB,1982HRand1983LC).Itmustbe impact rateis(2.4x50+1.0x100)per10yr,or145Myr.Thisfigureofstatisticalvalue TRIAD file(Bender1979). the valuescalculatedbySWHWof2.9xl0~Öpik’smethodand6.3xl0theirown If thesefourasteroidswererejectedfromthesample,remaining30Apolloshaveamean emphasized thattheseasteroidshaveorbitswhicharenotyetwell-determined,andmay Apollos andAmorcontributions,thesebeingbutasmallfractionofallMars-crossers. method. However,theeccentricityusedbySWHWwasdifferenttothathere,orgivenin larger than1kmcannowbefoundforeachplanet;thefigureMarswillonlyinclude from theEarth.Theabovefigurecouldnowbeusedtofindacratering-rate.craterproduced Atens and100Apolloscandolikewise.UsingthemeanimpactprobabilitiesfromTable4net parameters suchasthetargetmaterialandaccelerationduetogravity,discussedbySWHW, by a1-kmimpactingobjectdependsofcourseupontheincidentvelocityandlocalphysical asteroids ofloweccentricityandsmallsemi-majoraxissincethesehaveachancediscovery only. Itisprobablycorrecttowithinafactorof3or4.Noaccountcanbetakenanyhypothetical Jupiter wouldberapidlyremovedandnotmakeasignificantcontributiontoimpactsupon and elsewhere;thisfacetisnotcoveredhere. not changeappreciablyfromthatpresentsoonaftersuchawave,althoughthenumberwould the terrestrialplanets.ThereforeorbitaldistributionofApollo-Amor-Atenobjectswould mean numberoveralongtime-basecouldthereforebeapplied.SWHWfindevidencefromthe not atypicaloftheorbitaldistributionatanyotherinstantintime,andasimplescalingfor soon tailoff.Ithasbeenassumedforthepurposesofthispaperthatpresentsetelementsis enhancement. ThisisinlinewiththesuggestionsofClube&Napier(1982;1984)whoarguefor mean overthepast3.3xl0yrcontradictinganysteady-stateassumptionandindicatingarecent lunar andterrestrialcrateringrecordthatthepresentnumberofEarth-crossersislargerthan cometary influx.TheEarth-impactratefoundhereishigherthanthatofSWHW,whichmeans periodic replenishmentoftheApollo-Amor-Atenpopulationduetolargeincreasesin imply thatthepresentpopulationofEarth-crossing asteroidsisabovethelong-termaverage; that thediscrepancybetweencalculatedrateandcraterrecordisincreased.Thiswould however, inviewoftheinfluencefewlow-inclination asteroids,suchadeductionmustbe extremely tentative. The onlynumberedasteroidtohaveanobviouslyhighimpactprobabilityis(2101)Adonis Two ofthefourAtensandfive34ApolloscancrossMercury,sothatintotalabout50 Using thepreviously-statedassumedtotalpopulationofeachclass,anetinfluxasteroids If acometarywaveweretoboostthenumberofplanet-crossingasteroids,thosewhichcrossed Following asimilartreatmentfortheotherplanets these impactratesarederived: Note thatthefigureforMercury takesnoaccountofpossibleasteroidsinsmall (a,e)orbits. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Venus Planet Meanimpact Mars Earth Mercury 5 6 5 6 rate forall asteroids 1 per3xl0yr 1 per1.5xl0yr 1 per1.6xl0yr Iper5xl0yr 198 5MNRAS.212. .817S 5 89 9 _1 _1- 78 7 7 8lo Mars-crossers hasnotbeenconsideredbutwillbethesubjectofalaterpaper. The Martianimpactrateisthatfor700Apollosand1500Amorsonly;thelargenumberof parameters oftheseobjectsmaydecreasethecollisionratefoundabove.Thismightalsocome course theApollosandAmorsingeneralcrossasteroidbeltwhere—6.6xl0largeasteroids lifetime of—10yr.Similarlythe38Amorshave0.29collisions(withMarsonly)per10Of lifetime. 43 collisionsper10yrsothattheircollisionallifetime(forcollisonswiththeterrestrialplanets)is found tobeabout6Myr,whereasWetherill&Shoemaker(1982)report3Myr.This factors dominatetheircollisionallifetime,andwillalsobethesubjectsoffuturepapers.Wetherill terrestrial planetscanalsobedetermined.ForthefourAtenasteroids,onaveragetheyhave overall impactrateissmall. is anorderofmagnitudelessthantheratededucedhere.Thecontributionactivecometsto general belief.BystudyingrecentcloseapproachestotheEarthofasteroidsandcomets,Kresák Wetherill &Shoemaker(1982);Kresák(1981)foundthatamuchsmallerpopulationexists, searches showthatalargefractionofplanet-crossingasteroidsarelowinclination. Apollos mentionedabove)shouldbeconsideredamaximum,unlesssystematic,unbiased sample. Clearlythecollisionrateof6Myr(reducibleto4Myrbyrejectionfour about ifmoreasteroidsawayfromtheeclipticwerediscovered,relievinganybiasinpresent doubling isclearlyduetotheinfluenceofrecently-discoveredlow-inclinationApollos were asignificantundetectedpopulationofasteroids towardsitsorbit.Thesefiguresshouldbe planets. Largeperturbationsduetoclosebutnon-collisionalencounterswillbemuchmore minor bodiesortheJovian planets;thisisnotsofortheAtens.Thesefigures shouldagainbe indicative oftheentirepopulation. treated withcautionsincetheyarebiasedtowardsa high impactratebytherecentdiscoveryof taken ofthelargepopulationMars-crossers.Therate forMercurywouldalsobehigherifthere frequent, resultinginsevereorbitaldisruption(Weidenschilling1975). (1979 VA,1982DB,HR,and1983LC).Futurerefinementofourknowledgetheorbital Apollos andAmorstheactual collisionallifetimewouldbedominatedbycollisions withother several Apolloasteroidsofverylowinclination,sothat thesmallsampleusedhereisprobablynot Mercury andMars.ThetotalcollisionrateforMars wouldbemuchhighersincenoaccountis per 160000yr.ForVenustherateishalfofthis,and isatleastanorderofmagnitudelessfor are availableforcollision(Helin&Shoemaker1979);somecanalsointerceptJupiter.Thesetwo although hefindsthatonly—20percentofApollosareextinctcometarynuclei,againstthe greater than1kmforeachplanet.Therateisfoundto behighestfortheEarth,withonecollision of thefourterrestrialplanetshavebeenusedtodeduce collisionratesbyasteroidsofdiameter The collisionprobabilitiesbetweenthepresently-known Apollo-Amor-Atenasteroidsandeach (1978b, c)determinedacollisionrateofonlyoneper1or2Myrforbodieslargerthankm.This 6 Conclusions (1976) foundanetlifetimeof—2xl0yrfortheApollosandAmors. -2.5 x10yr.Sincethesedonotcrosstheasteroidbelt,thisisanestimateoftheirtotalcollisional —2.5xl0yr fortheAtens, —10yrfortheApollos,and~3xl0 Amors. Forthe 9 The 34Apolloasteroidshaveonaverage8.7collisionsper10yr,andaplanetary-collision The aboveestimateoftheEarth-impactrateusedanassumedpopulationidenticaltothat The resultantimpactrateupontheEarthofasteroidslargerthan1kmindiameteristherefore The lifetimesofthevariousasteroidclassesagainstdisruptionbycloseencounterswith The lifetimesoftheseasteroids againstcollisionwithanyoftheterrestrial planetsis For theAtens,ApollosandAmors,lifetimesaboveareonlyforimpactswithterrestrial © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Collisions intheSolarSystem-1.825 198 5MNRAS.212. .817S Clube, S.V.M.&Napier,W.M.,1982.Q.Jl.R.astr.Soc.,23,45. We thankP.M.KilmartinandA.C.GilmoreofMountJohnUniversityObservatoryfor Arnold, J.R.,Astrophys.J.,141,1536. supplying uswithanup-to-datelistofApollo-Amor-Atenasteroidsandorbits. Acknowledgments viewed withdiscretionduetothesmall,andprobablybiased,asteroidsample.Futurediscoveries Alvarez, L.W.,Asaro,F.&Mitchell,H.v.,1980.Science,208,1095. References Arnold, J.R.,1965b.Astrophys.J.,141,1548. of planet-crosserswillallowtheseresultstobere-assessed. 826 D.I.SteelandW.J.Baggaley Chapman, C.R.,1983.Rev.Geophys.SpacePhys.,21,196. Bender, D.F.,1979.Asteroids,p.1014,ed.Gehrels,T.,UniversityofArizonaPress. Clube, S.V.M.&Napier,W.M.,1984.Mon.Not.R.astrSoc.,208,575. Chapman, C.R.,Williams,J.G.&Hartman,W.K.,1978.Ann.Rev.Astron.Astrophys.,16,33. Explanatory SupplementtotheAstronomicalEphemeris,1974.HMSO,London. Davis, M.,Hut,P.&Muller,R.A.,1984.Nature,308,715. Ephemerides ofMinorPlanets,1984.InstituteTheoreticalAstronomy,Leningrad,USSR. Helin, E.F.&Shoemaker,M.,1979.Icarus,40,321. Helin, E.F.,Hulkower,N.D.&Bender,1984.Icarus,57,42. Gehrels, T.,(Ed.)1979.Asteroids,UniversityofArizonaPress. Hughes, D.W.,1981.Phil.Trans,R.Soc.London,Ser.A,303,353. Kessler, D.J.,1981.Icarus,48,39. Ip, W.H.&Mehra,R.,1973.Astr.J,78,141. Kessler, D.J.&Cour-Palais,B.G.,1978.Geophys.Res.,83,2637. Kresák, L.,1978a.Bull.astr.Inst.Czech.,29,149. Knezevic, Z.,1982.Bull.astr.Instr.Czech.,33,267. Kresák, L.,1978b.Bull,astrInst.Czech.,29,103. Kresák, L.,1978c.Æw//.astr.Inst.Czech.,29,114. Minor PlanetCirculars,Center,Cambridge,Massachusetts. Levin, B.J.&Simonenko,A.N.,1981.Icarus,47,487. Kresák, L.,198LAdv.SpaceRes.,1,85. Napier, W.M.&Clube,S.V.M.,1979.Nature,282,455. Marsden, B.G.,1970.Astr.J.,IS,206. Öpik, E.J.,1963.Adv.Astr.Astrophys.,2,219. Öpik, E.J.,1951.Proc.R.Ir.Acad.,Ser.A,54,165. Öpik, E.J.,1966.Adv.Astr.Astrophys.,4,301. Tedesco, E.F.,Tholen,D.J.,Zellner,B.,Veeder,G.J.&Williams, J.G.,1981.Bull.Am.astr.Soc.,13,712. Shoemaker, E.M.,Williams,J.G.,Helin,F.&Wolfe,R.F., 1979.Asteroids,p.253,ed.Gehrels,T.,University Rampino, M.R.&Stothers,B.,1984.Nature,308,709. Shoemaker, E.M.,1983.Ann.Rev.EarthPlanet.Sei.,11,461. Weidenschilling, S.J.,1975.Astr.80,145. van Houten,C.J.,Houten-Groeneveld,L,Herget,P.& Gehrels, T.,1970.Astr.Astrophys.Suppl.,2,339. Smith, R.E.&West,G,S.,(eds),1983.SpaceandPlanetary EnvironmentCriteriaGuidelinesforUseinSpace Wetherill, G.W.,1979.Icarus, 37,96. Wetherill, G.W.,1967.J.Geophys.Res.,72,2429. Wetherill, G.W.,1976.Geochim.&Cosmochim.Acta,40,1297. Whitmire, D.P.&Jackson,A. A.,1984.Nature,308,713. Wetherill, G.W.&Shoemaker, E.M.,1982.Geol.Soc.Am.Spec.Pap.,190,1. Williams, J.G.&Wetherill, W.,1973.Astr.J.,78,510. Wetherill, G.W.&Williams, J. G.,1968.Geophys.Res.,73,635. Zimbelman, J.R.,1984.Icarus, 57,48. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System of ArizonaPress. Vehicle Development,1982Revision,Yol.1,(NASATM82478) andVol.2(NASATM82501). 198 5MNRAS.212. .817S 2 (1951), Wetherill(1967)andShoemakeretal.(1979).Itisanalogoustothekinetictheoryof upon Kessler(1981).Thisiseasilyapplied,andcanbecomparedtothepreviousmethodsofÖpik gases. Derived hereistheprobabilityofacollisionbetweentwoarbitraryorbitingobjects,expanding A1 Introduction Appendix: Theprobabilityofacollisionbetweenarbitraryorbitingobjects perturbations andprecession. out byÖpik(1951),thisassumptionisvalidoverastronomicaltimebasesduetosecular essential featureofthemethodisthatargumentperiapsistakentoberandom.Aspointed of thetwosetsorbitalelements(a,e,/),andphysicalcharacteristicsthesebodies(mass moons orbitingtheirparentplanet.Themethodwasdevelopedtofindthecollisionallifetimeof mass), butitcanbeusedforanyKeplerianorbits,suchasstarsaboutthecentreoftheirgalaxy,or and radius,mr).Therefore,withMthemassofprimary,collisionprobabilityis: artificial satellitesinEarth-orbitbyKessler&Cour-Palais(1978),andwasapplied (1981) totheoutermoonsofJupiter. P=P{M\ aj,eimj,r¡;a,mr). o=x(ri+r) Take thevolumeofspaceinwhichitispossiblefortwoobjectstocollidebesplitupinto A2 Basisofthemethod small volumeelementsAf/,eachofwhichiscomparedtotheuncertaintyinorbital paths oftheobjects.Thenpositionseachobjectwithinanyvolumeelementisessentially random, asarethemoleculesinagas,andfluxofoneobjectis inside ofthisvolumeelementthereareonaverageSbodieshavingorbit,perunitvolume. where visthevelocityofobject,andSitsmean‘spatialdensity’withinvolumeÀI/;thatis, F=Sv (1) u2 2 the effectivecollisionalcross-sectiontobecome(Öpik1951): general theobjectshaveanappreciablegravitationalfieldsothatfocusingincreases the bracketisunityplusratioofsquareplanetary escapevelocitytothesquareof the termoutsideofbracketisjustgeometrical cross-section oftheplanet,andinside encounter velocity,V. N=Fot (3) In thecaseofaminorbodysuchascometorasteroid strikingaplanet,randmaresmall.Then —=Svo. (4) so thatthecollisionrateis, from(1)and(3), 2 N t The orbitofeachobjectisdescribedbysemi-majoraxisa,eccentricitye,andinclinationi.An The collisionprobabilityfortwosecondarybodiesorbitingagivenprimaryobjectisfunction This techniqueisespeciallyvaluableindealingwithobjectsheliocentricorbits(Mthesolar The numberofimpactsuponanareaAintimetisthengivensimplybyFAt.However, The numberofcollisonsintimetisthen © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 1 + 2G(ra+ra) 12 2 (r^r)V 2 Collisions intheSolarSystem-1.827 (2) 198 5MNRAS.212. .817S element A¿7,cristhemutualcollisoncross-sectiongivenby(2),andVisrelativevelocityof —=8^0 MJ(5) calculated fromequation(2).InSectionA3theexpressionforspatialdensityisdeduced. P=j SiSVodU.(6) elements accessibletobothobjects: two atthisposition. where SiandSaretherespectivespatialdensitiesoftwoobjectsinthisparticularvolume there isonlyasmallchancethatbothobjectswillbeinthevolumeelementatsametime,so (4) becomes 828 D.I.SteelandW.J.Baggaley the spatialdensitydoesnotvaryappreciablywithinelement;then executed. Equation(6)canbesolvednumericallybydefiningvolumeelementsofsuchasizethat where thebarsindicatethatmeanvaluethroughoutelementisused. 2 2 A3 Spatialdensity S=S(R,ß). If theargumentofperiapsisisrandom,thenspatialdensityafunctiondistancetoprimary S{R,ß)=Q(R)B(ß) (8) In addition,Sisseparatelydependentuponthesetwoparameters: (R) andeclipticlatitude(ß)only,sothat density atßtothespatialaveragedoveralllatitudes. Qdependsuponthesizeandshape of theorbit,describedbyaande;orequallywellperiapsis distanceqandapoapsisq': Here QisthespatialdensityatRaveragedoverall latitudes,andBistheratioofspatial B dependsonlyupontheinclination: A3.1 RADIALSPATIALDENSITY VARIATION as inFig.Al. N Consider asphericalshellof thicknessA7?,atadistanceRfromtheprimarysuch that (¿7^Æ¿7'), and thetwofunctionsQBarederivedseparately. B(ß)=B(i) Q(r)=Q(a, e)=Q(q,q'). t The totalcollisionprobabilitybetweenthetwoissumofratesinallvolume However, thiswouldassumethatonebodyisheldstationaryinthevolumeelement;fact Once themeanencountervelocityisknown(SectionA4),collisioncross-sectioncanbe This assumesthatthecollisionprobabilityissufficientlysmallsuchmanyorbitsare © Royal Astronomical Society • Provided by theNASA Astrophysics Data System J volume j (7) Collisions in the -1. 829

Figure Al. The variation of spatial density with distance from the primary.

The orbiting body has a velocity VB with radial component yr=yBsiny. (9)

o Vx is taken to be positive towards the primary so that from apoasis to periapsis (0 ^y^90°), and from periapsis to apoapsis (270o^y^0°). The shell has an internal distance R, external distance R', mean distance

R=(R+R')/2 and a volume

MJ=AjzR2/SR (10) as long as (kR

At=2AR/VT. (11)

The velocity of the body is

(12)

G being the universal constant of gravitation. Equations (9) and (12) will allow At to be calculated from (11) as long as sin y is known. The angular momentum per unit mass is: At g: 2 (7VBqcosO°=4VBq= (A) <7 At q':

q'VBq' cos0°=<7'VBq' = (B)

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 198 5MNRAS.212. .817S H 1 2 312 f2, ß+Aß. a _2 requirement istofindthe probabilityoffindingthebodywithinabandlatitude fromßto The orbitingbodyisnow definedtobewithinashellofthicknessARat distance R.The which, withsubstitutionofa=(c¡-\-q')/2,gives A3.2 LATITUDINALSPATIAL DENSITYVARIATION Q=At/TAU (R^(l') isdealtwithinSectionA3.5. over alllatitudes.For{Rq'),Q=0.The methodfordealingwithQas(R->q)or Q{R)=yAjtRa[(R-q)(q’-R)]m. ß(R) =l/4^Äfl3/2 where theperiodTis The spatialdensityisjust and then(10),(14),(15)(16)render (1?) cos y=. T =2jr(a/GM)/ ^r= and then qq=2qa—qq a—q' so that =(?+9')/2 However, cosy= =. or 2tfg/tf-2g'-g'/a=|— —jÆcosy RVcosy= 830 At R: BR 2 This istheprobabilityoffindingbodywithinashell ofunitvolumeatdistanceR,averaged Since siny=l—cos,equations(9),(11)and(12)give: © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Equating (A),(B)and(C)gives 2 ^ R{2a-R) R(2a-R) 2 2qa-q Iq'a-q' D. I.SteelandW.J.Baggaley R(2a-R) 1/2 [l-qq’/R{2a-R)} Rcosy. 1/2 (14) (13) (C) 198 5MNRAS.212. .817S 2 21 2 point O,givenby(R,ß),andisthentakenaroundacircularorbitofinclinationi(thesameasthe real orbit)sothatittraversespathWOX.Allpossiblevaluesfortheargumentofperiapsis 2 real orbitarescannedinthisway. imaginary pathiscircular,thentheperiodofthisorbit which thepathmakeswithlineofconstantlatitude.Sinceobjectcrossesthislatitudetwice in everyorbit,thetotaltimespentthisbandperorbitis ß+Aß as r=2jz/ù). Equations (18)and(19)thereforegivethefractionofobject’stimespentbetweenß At'=2Aß/co sina.(19) =At'/T'AU' (21) In asimilarwayto(15),sinceweneedconsiderunitvolumegivenbyequation(10), cosine ofthelatitude.Thus and ALTisrequiredinordertocalculateB. 4jzRAR =\I2jtRsinacos/1 AR At' Aß and putting(20)(22)into(21)renders: A U'=2jzRcosßARAß (22) or B=2/7is\nacosß (23) 4nRAR T Jtsina 2 Referring toFig.A2,XYistheeclipticplane.Thebodyimaginedbeheldstationaryat If theargumentofperihelionmoveswithanangularvelocityco,whichisconstantsince The timetakentocrossthelatitudebandofwidthAßisjust(Aß/cosina)whereaangle The volumeofahemisphericalshellis(2jzRAR),and thevolumeelementdropsoffas © Royal Astronomical Society • Provided by theNASA Astrophysics Data System ß Figure A2.Thevariationofspatialdensitywitheclipticlatitude. Collisions intheSolarSystem-I.831 (18) (20) 198 5MNRAS.212. .817S f,2l/ f 21/ 21/ 2 ß arebettermeasuresthanRandß,write is which isvalidfor Equations (17)and(24)cannowbesubstitutedinto(8)togivethenetspatialdensity.SinceR A3.3 NETSPATIALDENSITYVARIATION we obtain S(R, R',ß,ß)=l/2jraR[(sini-sinß)(R-q)(q'-R)](25) The spatialdensityiszerooutsideoftheselimits,butcaremustbetakenaslimitsare 832 D.I.SteelandW.J.Baggaley and approached. (q^R, R'^q') which isvalidforB=0athigherlatitudes.Since (0°^\ß,ß'\^i). giving cosi=cosacosß The singularitiesoccurringwhen(R->q),(R->q),or(ß-*i)areintegrableasfollows. A3.4 INTEGRATIONMETHODS sin acos[sin¿-sin/?] S= average spatialdensitywithinavolumeis B(ß)=2/jt (sini—sinß).(24) which mustbeseparatedintoradialandlatitudinal terms.Forverysmallvolumeelements S= equation (22)is dU= 2jtRcosßdRdß and thususing(8)equation(26)becomes S(R,R’,ß,ß') =Q(R,R')B(ß,ß'). Since Randßareindependent, theintegralsareseparable: The treatmentofthisequationas(ß-^i)isdescribedinSectionA3.6. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System i II SdU dU 2 Q(R)B(ß)R cosßdRdß 2 R cosßdRdß (26) (28) (27) 198 5MNRAS.212. .817S 1 ß R,1/2 2 any pointcanjustbesettotheaverageradius: Let thethicknessofsphericalshell(AR)besufficientlysmallsuchthatradialdistanceat A3.5 spatialdensityas(R-*q)or latitudes. is theratioofmeanspatialdensitybeweenßandtooverall B(ß,ß')=j B(ß)cosßdßjÇcosßdß(10) and Jr Here is themeanspatialdensitybetweenRandR'averagedoveralllatitudes, Thus (29)becomes JR if (R'^q')thenthesquare bracketbecomes Í 0^' )=,TTTñf[(R-q)andinsideapoapsis After evaluationoftheintegralthisbecomes Q{R, R')=S{R)dR and bysubstituting(17)thisis iR’q')or(R’ßalways,thefactor outside ofthebracketensuresthat © Royal Astronomical Society • Provided by theNASA Astrophysics Data System D. I.SteelandW.J.Baggaley (32) Collisions in the Solar System -1. 835 S(R, R', ß, ßf) /IR'-laV /sinß' arcsin arcsin I —— \ q'-q ) \ sin/ (33) 2jt3aR A iR (sin ^ - sin ß) (2R-2a\ /sin A —arcsin 1 -arcsinl \ q’-q ) \ sin / / This formula is necessary when R or R’ are close to periapsis or apoapsis; and when the absolute values of the latitudes ß or ß’ are close to the inclination of the orbit. Otherwise equation (25) is used to find the spatial density.

A4 Relative velocity

The of the relative velocity of the two bodies in a particular volume element is required. The individual velocities can be calculated by recourse to equation (12), and then the relative velocity from

2 y =y|1+V|2-2VB1yB2Cos0 (34) as long as the encounter angle is known. Fig. A3 represents a coordinate system centred upon the point of intersection, with the XY plane perpendicular to the spherical shell in Fig. Al. The angle can be calculated by spherical trigonometry as: cos (j)=[sin y i sin y2] + [cos y\ cos y2 cos (oq - a2)]. (35)

The values of cosyx and cosy2 are given by equation (13), taking the positive square root in o o each case since (270 =^y^90 ). Hence sin yx and sin y2 can be calculated, but these can take either sign: from apoapsis to periapsis (0°^y^90o) so that sin y is positive. From periapsis to apoapsis the converse is true. In considering Fig. A2 it was seen that

cosi=cosacosß

so that «i and a2 can be calculated from the inclination and the latitude: cos a=cos//cos (36)

where

ß=(ß+ß')/2.

Figure A3. Relative velocity of the two colliding objects.

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 198 5MNRAS.212. .817S 2 iterations inradius(AÆ=0.01Æatworst)and50latitude(Aß=3°.6is to programforaself-adjustingnumberofiterations(i.e.investigatethevolumespace given by(22)withthemeanparametersbeingused,i.e. cross-section iscalculatedfrom(2)usingthemeanrelativevelocity.Finallyvolumeelement mean velocityisfoundfrom(34)using(37)andtheradiusvectorR(12).Thecollision defining ARandAß).Thesummationof(7)isthen over thisnumberofvolumeelements,7. accessible tobothbodiesandthensubdividethisintoasuitablenumberofvolumeelementsby sufficiently preciseforallorbits,bearinginmindthedesirederrorlimit.Itiscomparativelysimple and 05=0°,±/). radial dependence(c/.SectionA3.5).HislatitudebandswereAß=3°.Itwasfoundherethat100 physical usefulness,aprecisionofbetterthan1percentisentirelysatisfactory. precision andthecomputationtimeinvolved.However,sinceanexactcollisionprobabilitylacks are givenby(25)or(33),bearinginmindthelimitationofvalidityafterequation(25).The 836 D.I.SteelandW.J.Baggaley All termsarenowavailablefortheevaluationofequation(7).ThetwospatialdensitiesandS A5 Collisionprobability low-velocity collision.) high velocity,suchaswhentheminorbodyisinaretrogradeorbit,lessthanthatof reciprocal ofthesquarerelativevelocity.Becausethisprobabilityacollisionat when aminorbodyencountersoneofthegiantplanets:enhancementgravitational AU=2jtRcosßARAß. (38) cross-section overthegeometricalcross-section,givenbyequation(2),dependsupon was madeclearbyKessler(1981),thisisnotthesameassayingthatallcollisionvelocitiesare equally likely:thehighervelocitiesrenderalikelihoodofcollision.(Anexceptiontothisis velocity isrequired:thisjusttheaverageoffourpossiblecollisionvelocities.However,as purposes ofcalculatingthemeancollisionprobabilityviaequation(7),encounter Hence therearefourpossibleintersectionvelocitiesresultingfromequation(34).Forthe cos0=±[sinysiny2] +[cosycosy2cos(a±a)].(37) and fourvaluesforintersectionangle