1982MNRAS.200..313F Meteor astronomersregardtheGeminidsasareliableshower.Atmaximumactivity,around back toad401;thereare12recordedapparitions ofthePerseidsbetweenad36and1451. December 14,avisualobservercanseehighrate ofabout60perhourandthisactivity shower isnotanoldshower.TheLyridswerefirst seenin687bc,theDeltaAquaridsgo shows littlevariationfromyeartoyear.Itisinteresting tonotehoweverthattheGeminid [The slightnoteofcautionisintroducedheredue to thethreedatesofhighmeteoractivity, The Geminids,however,werefirstmentionedwith anycertaintyin1862(seeKing1926). 1 Introduction to Wartmann(1841).Since therearenorecordedphenomenonwhichcould accountforthe ad 901Nov30,93029and1571Dec8 mentionedbyNewton(1863).Changing assess whethertheseareGeminidsornot]This ‘newness’isacharacteristicthatthe Geminids haveincommon withtheQuadrantidshowerwhichwasfirstseenin 1835according these datestothecommonepochofad1850,Newton calculatedthatthesecorresponded to Dec13.3,11.6and11.5respectively.Asnoradiant informationisgivenitdifficultto that thedate ofmaximumactivity theshowerappearstohave changedsubstantiallyover formation ofthestream at thesedates,wemustsuspectthatorbitalchanges broughtthese streams ontoanintersection pathwiththeEarth.Thisconclusionissupported bythefact © Royal Astronomical Society • Provided by theNASA Astrophysics Data System -1 planetary inducedgravitationalperturbationonthestreamresultsinapre- Summary. TheGeminidswerefirstreliablyrecordedin1862.Observationsof whereas streamswithinclinationslessthan90°havenegativedSl/dtis that streamswithinclinationsgreaterthan90°havepositivedSlo/dtvalues at anSIvalueof261.38±0.11.Atheoreticalanalysistheeffect this streamoverthelast118yearsindicatethatascendingnodeisstationary are discussedandoneoftheseturnsouttobeverysatisfactory. disobeyed bytheGeminids.Variouspossibleexplanationsforthisdiscrepancy dicted decreaseintheascendingnodeby1.57degcentury.Thegeneralrule Received 1981November26 Queen MaryCollege,MileEndRoad,London,El4NS David W.HughesDepartmentofPhysics,UniversitySheffield, The evolutionoftheorbitGerrinidmeteorstream Mon. Not.R.astr.Soc.(1982)199,313-324 Ken FoxandIwanP.WilliamsDepartmentofAppliedMathematics, Sheffield, S37RH 1982MNRAS.200..313F -1 -1 so thesenseofmasssegregationissimilarforbothstreams. McIntosh &Simek1974).ThismasssegregationisanothercharacteristicthattheGeminids given by the 110yearintervalandalsodependsstronglyonmagnituderangeofmeteorsbeing Hughes &Kaiser1972)whereastheGeminidsisskew(Kresak1974),fallawayfrom stream doesdifferbetweenthetwo.TheQuadrantidactivityistypicallyGaussian(Poole, maximum activitybeingmuchswifterthantherisetoit. An analysis(Spalding1981,privatecommunication)indicatesthat\fortheGeminidsis longitude ofthemaximumactivityQuadrantidmeteorsmagnitudeAf,Hughes&Taylor observed (Plavcova1962;Millman&McIntosh1964;1966;Simek1973;Hajduk, have incommonwiththeQuadrantids(seeHughes,Williams&Fox1981).IfXqissolar 2 Orbitalperturbation Argument ofperihelion324.8° The meanorbitoftheGeminidsistakentobe(seeHughes1978); (1977) foundthat Period 1.57yr. Perihelion distance0.140au Ascending node260.3° 314 K.Fox,I.P.WilliamsandD.W.Hughes X =(261.55±0.5)-(0.0780.025)M(2) in aseriesofrunsonanICL2980usingtheGauss—Jacksontechnique.Theorbiteach hasbeenreplacedbytestparticlespositionedaroundtheorbitgivenaboveat Aphelion distance2.56au Eccentricity 0.896 Inclination 23.6° relevance beingJupiter,VenusandtheEarth.Amean wastakenoftheorbitalelements each particleatyearlyintervalsstartingthepresent timeandgoingbothbackforward particle changesundertheinfluenceofplanetaryperturbations,onlythreeplanets taken tobe0.00hron1980Feb.12.Alltheequationsofmotionsweresolvednumerically equal intervalsofeccentricanomaly.Thestartingtimetheperturbationanalysiswas in Fig.1gives equations ofthesefitsaregiveninthefigures. semimajor axis,andperiheliondistanceeccentricity asafunctionoftimeareshownin X =283.24-0.109M.(1) This isdirectlyrelatedtothepositionofascending nodeoftheorbit.Thefittodata orbital parameterstomeasureandthemostaccurate isthetimeofmaximumstreamactivity. Figs 1,2and3respectively.Alinearregressionanalysis hasbeenappliedtothedataand G G Murray 1979). nodal retrogressionof1.568 degcentury.TheQuadrantidstreamalsosuffered anodal 150 years.Thevariationsoftheascendingnodeand argumentofperihelion,inclinationand retrogression butinthis case thevaluewas0.489degcentury(seeHughes, Williams& Q T ismeasuredinyears,r =0being1980Feb.11.0.Thisequationindicates thatthereisa The wayinwhichthespatialdensityofmeteoroidsvariesasEarthpassesthrough © Royal Astronomical Society • Provided by theNASA Astrophysics Data System We willsingleoutoneoftheserelationshipsfor more detaileddiscussion.Theeasiest For thepurposeofevaluatingorbitalevolutionmeanorbit,plethora 260.29-0.01568 TV (3) 1982MNRAS.200..313F as afunctionoftime.T=0is1980Feb11.0. Figure 1.ThevariationoftheascendingnodeandargumentperihelionGeminidmeteorstream function oftime. 7’=0is1980Feb11.0. Figure 3.Thevariationofthe periheliondistanceandeccentricityoftheGeminidmeteor streamasa Figure 2.Thevariationofthesemi-majoraxisandinclinationGeminidmeteorstreamasafunction of time.T=0is1980Feb11.0. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 1850 19001950 1850 19001950 Evolution oftheGeminidmeteorstream 1-349 1-350 1-351 322 325 323 324 CO UJ 315 1982MNRAS.200..313F -1 -1 -1 316 K.Fox,I.P.WilliamsandD.W.Hughes itwouldhavebeenaroundDecember13in1918and12 gives thetimesofGeminidmaximumasafunctionobservationyear.Itmustberemem- & Obrubov(1979)foundabout1.6degcentury.Allthesevaluesarelargeandeffec- Jim Jones(privatecommunication1981)foundavalueof1.62degcentury,Babadzhanov year inthefourleapyear’cycle.Alsoprecessionofvernalequinoxleadsto bered thattheactualtimeofstreammaximuminanyyeardependsonposition tually sayingthatifmaximumoccursnowonDecember14thensimplyduetogravitational theoretical estimationsofthisquantity.Plavec(1950)foundavalue1.625degcentury. in Table1recordthesolarlongitudeX©,formeanequinoxdateand maximum ofthevisualGeminidmeteorstreamasafunctionyearobservation. X© (1950.0)=261.38±0.11,(4) X© (1950.0)correctedtoacommonepoch.Fig.4showsplotofforthe date ofstreammaximummovingearlierintheyearasonegoesbacktime.Othercolumns A linearregressionanalysistotheweighteddataproducesrelationship tion. Notethattheascendingnodeandsolarlongitudehavesamenumericalvalues. and showsthatthereisnoclearvariationasafunctionoft,tbeingtheyearobserva- 1856. Thissuppositioncanbeeasilycheckedbylookingbackatthehistoricalrecord.Table1 visual Geminidsasafunctionoftheyearobservation.The linearrelationshipisgivenbyequation(4). Figure 4.Thesolarlongitude\©(1950.0)correctedto a commonepochforthemaximumactivityof Year maximum. JulianDayhasbeenconvertedto\©(1950.0)by usingEmerson(1978),seeFig.4. Table 1.ObservationsofthemaximavisualGeminid shower.\©(1950.0)isthesolarlongitudeat 1971 1896 1862 1980 1935 1899 1893 The nodalretrogressionratefortheGeminids,givenabove,agreeswellwithother Royal Astronomical Society •Provided bythe NASA Astrophysics Data System Dec 14.7 Date Dec 12 Dec 11.95 Dec 11 Dec 13.48 Dec 12.25 Julian 2441300.2 2413905.45 2412809.75 2401485.5 2444586.98 2415000.5 \© (1950.0) 261.21 261.197 ±0.11 261.73 261.5 261.72 261.28 259.96 ± 0.25 ± 0.11 ± 1.00 ± 1.0 ±0.25 ± 0.24 Sky andTel(1972) King (1926) Hawkins &Almond(1952) King (1926) Ref. Spalding (1981) Spalding (1981) Spalding (1981) 1982MNRAS.200..313F 1 Figure 5.Aschematicrepresentation oftheeffectplanetarygravitationalperturbation onmeteor streams withinclinations (a)lessthan90°and (b)greaterthan90°. i >90°andnegativefor<90°. inclination ofthestreamtoeclipticanditsproximitymassiveplanets.Aschanges be representedbyanangularmomentumvector.ThisisshowninFig.5fortwocases,one approximation themeteoroidstreamcanberegardedasasimplegyroscopeandsuch be representedbyanannulusofmassdistributedaroundtheorbitplanet.Theeffect the resultscanbeaveragedovermanyorbits.Toafirstorderapproximationplanet The effectofplanetaryperturbationonameteorstreamdependstogreatextentthe 3 Acomparisonwithothermeteorstreams are slowincomparisontoboththeorbitalperiodofstreamandperturbingplanet time. Fori=90°and0°thereisnogravitycouple,d^l/dtexpectedtobepositivefor The senseofthegravitycouplesareshowninFig.5.Fori<90°effectistodecrease where theinclinationislessthan90°andotherinchnationgreater90°. of thismassistogenerateagravitationalfieldtowardstheecliptic.Againfirstorder has shownnoappreciablechangeoverthelast118years. about 1.6degcentury"whereasobservationsofthevisualmeteorsshowthatX©(1950.0) A comparisonbetweenequations(3)and(4)immediatelyrevealsaworryingdiscrepancy. V andftasafunctionoftime.Fori>90°theeffectistoincrease The theoreticalperturbationcalculationsindicatethat«ftdecreasesasafunctionoftime,by © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Evolution oftheGeminidmeteorstream317 1982MNRAS.200..313F between alargenumber ofasteroidorbitsandtheGeminidsorbitwere calculated. One , 132 Aethra,wasfoundtohave aminimumdistanceof approachwhichiseven less attempt tofindifanyknown asteroidpassedclosetotheGeminids. asteroid wouldhaveaneffectontheorbitalevolution. approach tooneoftheasteroidsisapossibility.Needless tosay,acloseapproachlarge the calculations.However,asstreampasses right throughtheasteroidbelt,aclose tion, hasasignificanteffectonthestream.Itcan easilybeshownthatofthenineplanets, retrogressing? only theEarth,JupiterandVenuscanhavemuch effectandthesehaveallbeenincludedin contrive topresentazerodSl/dttheobserver onEarthwheneffectivelythestreamis From theprecedingdataitcanbeseenthatwe have aproblem.HowdotheGeminids 4 Aproblemandpossiblesolutions tion ofthevisualGeminids. together withthemeanpresentdayorbitalparametersofstreamsaregiveninTable2. Only onedSl/dtvaluegoesagainstthegeneralprinciplegivenaboveandthatisobserva- with aviewofobtainingroughideaastothemeandSl/dtvalues.Theresultsthisanalysis performed fortheEtaAquarid,Orionid,Perseid,LeonidandQuadrantidmeteorstreams Geminids Perseids Quadrantids Leonids Orionids Eta Aquarids Meteor stream Table 2.dSl/dtvaluesandorbitalparametersforaseriesofmeteorstreams. (time intervalusedfordSl/dt) 318 K.Fox,/.P.WilliamsandD.W.Hughes Using themethodoutlined inMurray,Hughes&Williams(1980)theminimum distances A searchwasmadeofthe orbitaldataoftheknownasteroids(seeBender 1979)inan © Royal Astronomical Society • Provided by theNASA Astrophysics Data System One possibilityisthataplanetaryobject,whichhas beenignoredinthenumericalsimula- An analysisofthedatestreammaximaasafunctionobservationyearhasbeen (1862-1980) visual (1828-2028) theory (1950-1974) radar (1835-1970) visual (1866-1967) (36-1980) (585-1980) (405-1910) theory, thispaper theory, Babadzhanov theory, Plavec theory, Jones etal 1 -1.6 -1.57 -1.63 -1.62 - 0.489 -0.81 ±0.4 -0.35 ±0.06 + 1.68 + 0.038±0.027 + 0.611 + 0.35 dtl/dt (deg century") 0.0 * Q 163 0.97 114 0.94 163 0.54 (deg) AU 158 0.70 23.9 0.14 71.4 0.98 46 0.96151 24 0.92174 AU Q e 14.9 0.9387 2.7 0.90324 5.2 0.68168 7.5 0.83109 deg (jj 1982MNRAS.200..313F 2O unknown type.RadiometricobservationsofAethra(Tedescar&Bowell,privatecommunica- Semi majoraxis2.61365au Argument ofperihelion254.391° Ascending node258.476° Inclination 25.072° Eccentricity 0.38356 were carriedoutwithdifferentassumedmassesandtheresultsthenanalysed.Itwas tion 1981)indicatethatithastheratherunusualalbedoof0.23whichwouldleadtoa The massofAethraisunknown.IthasanabsolutemagnitudeB(l,0)10.21and 0.05 and0.01au.Fig.6showstheorbitofGeminidstreamasteroidAethra. elements atepochJuHandate2443800.5. than the0.002aubetweenEarthandstream.Aethrahasfollowingorbital stream intotheobservednearzerod^l/dtitsmasswouldhavetobeonlyslightlylessthan discovered thatinorderforAethratoconverttheprevioustheoreticalretrogressionof diameter ofabout40kmandamassintheregion10g.Anumbercomputerruns that ofJupiter.Thisisclearlyimpossibleandsothissuggestedsolutioncanberuledout. are markedas1,2 and3. and theFirstPointofAries as itsnormal.Thethreecloseapproaches,0.01AU,0.05AU and0.0003AU ecliptic planeandb)projected intotheplanewhichcontainsSunandhasline betweentheSun Figure 6.Theorbitoftheasteroid 132AethraandtheGeminidmeteorstream,a) projected intothe It wasfoundtohavethreecloseapproachesperorbittheGeminidsofdistances0.0003, © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Evolution oftheGeminidmeteorstream319 1982MNRAS.200..313F effects willbenon-linear. produced byplanetaryperturbationsitisworthyofinvestigationbecausethecombined A co= equivalent todisturbingtheKeplerianmotionwithaforcecomponents»S'andTgivenby 9.17 secondspercentury.Thoughthisissmallcomparedtothe1.5degreescentury radians perrevolution,(seeWeinberg1972).FortheGeminidsthisresultsinachangeof planetary ,thisisgivenby known thatgeneralrelativityproducesasmallchangeintheargumentofperihelion 320 K.Fox,I.P.WilliamsandD.W.Hughes in thesamesenseasmotion.Theaboveperturbingforcehasbeenaddedtousual equations ofmotionthestreamwithr,rand6givenatanyinstantby Here Sisalongtheradiusvector;Tinplaneoforbitandperpendicularto r=|r| r = nodal retrogressionrate,theinclusionoftheserelativisticeffectsresultedinaverysmall These equationswereintegratednumerically,usingButcher’smethod,whichisasixthorder, value justastheobservationsshow.Thisscenario alsofitsotherknownfactsaboutthe meteor streamwiththeeclipticplane.Consider the schematicdiagraminFig.7.Dueto increase. Thusgeneralrelativitycanberuledout. seven stageRunge—Kuttatechnique.Unfortunately,ratherthanhelpinslowingdownthe value shown in Fig.8.Toincludethem allonthesamediagram wouldsimplythicken the gravitational perturbationthepointP,whichcan be takenasbeingthecentreofstream Earth’s path,willhaveremainedatP.Thusthesolar longitudewillhavemaintainedthesame the cross-sectionwillhavedecreasedbetween1862 andtodayjustastheorypredicts. as seenbytheEarthtoday,willhavebeenatP*in 1862.Theascendingnodeoftheorbit required bytheexplanation givenabove. This isshowninFig.8.It canbeseenthatin1862theGeminidsorbitwasinside theEarth’s stream. Usingtheadoptedorbitalelementsof mean streamitispossibletocalculatethe However, thesolarlongitude,whichisobserved centreofthemeteoractivityalong that passesthroughPis261°whileofP'263° thustheascendingnodeofanypointin distance ofclosestapproachbetweenthestreamand theEarth’sorbitasafunctionoftime. andithassubsequently movedoutwardsandcrossedoverourplanet’s orbit,justas . |rxv| Hagihara (1972)hasshownthattherelativistictreatmentoforbitssmallbodiesis © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Another possibilityisthatgeneralrelativitymightappreciablyaffecttheorbit.Itwell 2 2 The finalpossibilityinvolvestheshapeofcross-sectionintersection The variationinthedistance foreachindividualparticleisextremelysimilar tothemean GM f2GM3ri2GMd rc(l -2GM/rc) 2 r •v 0 0 2 r r Lrcc{\-IGMo/rc)] a(l-e)c 6nGM 0 2GMrd 0 (6) (5) 1982MNRAS.200..313F from beingoutsidetoinsidetheEarth’sorbit. the Earthisshownasafunctionoftime.T=01980Feb 11.0.Thesignoftheclosestapproachdistance Figure 8.TheclosestapproachdistancebetweentheperturbedGeminidmeteorstreamandorbitof changes frompositivetonegativewhenthepointon Geminid orbitwhichisclosesttoEarthchanges Figure 7.Schematicrepresentationofthecross-sectionstreamthrougheclipticandinter- action withtheEarth’sorbit. line shown.Thisdiffersconsiderablyfromthe Quadrantid caseshowninMurrayetal. comparison totheQuadrantidstreamwhichhasJupiter nearaphelion.Theaccuracyofthe (1980) Fig.3,duetothefactthattherearenolarge planetsclosetotheGeminidorbitin orbit at7=0.(TheEarth movesthisdistanceinthreehours.)Fig.8does showthatthe gravitational perturbation. was firstseenthisdistance wasabout—0.0115auandthereisnodoubt thattherecent closest approachdistance isvaryingrapidlyasafunctionoftime.In1862 when thestream discovery oftheGeminids isduetothefactthattheyarebeingsweptpast theEarthby assumed orbitisindicatedbythefactthatstream isonly0.002auawayfromtheEarth’s 11 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Evolution oftheGeminidmeteorstream321 1982MNRAS.200..313F 2 -3 2 r = n dm °cr~dt However, someinsightintothelikelyshapeofcross-sectioncanbeobtainedbyproducing where isthechangeinaphelion--cometangle.Thus, a streamasdataisonlyavailableaboutthemeteoroidswhichcollidewithEarth. nucleus ofacomet. where ristheheliocentricdistanceofcomet. Sun duringthattime,i.e. from thecometperunittime,dt,isproportionaltoradiativeenergyreceived a numericalsimulationofthebreak-upcometmovingonmeanorbitstream. dm ccdd. where Ristheradiusofcometinkm,l/nfractionsolarradiationutilizedfor 322 K.Fox,LP.WilliamsandD.W.Hughes so thattheCartesiancomponentsofejectionvelocity aregivenby Taking n=1andfixings,pRtobe0.1cm,0.8gcm10kmrespectivelyenables particle onthecometaryorbit.Theradialdistancerofthispointiscalculatedfrom number vintherange(0,In)isselectedwhichdeterminespointofejectionadust the ejectionspeed,tobecalculatedasafunctionofradialdistance. sublimation andsparetheradiusdensityofsphericaldustparticlesincgsunits. velocity, whileitsposition isgivenbythepointofejection.Fromthisposition andvelocity rdQ exdt The ejectedparticle’soverall velocityiscalculatedbytheadditionof ’sorbital ~,’ V = tion, byletting ?!, ?2>itherange(0,1)arethengenerated.These areusedtoproducethedirectionofejec- so thattheejectionspeedVcanbecalculatedbyusingequation(7).Tworandomnumbers c the orbitof the ejectedparticlecanbe obtained.Astheparticle issmall,radiationpressure c cos d—12?2, 0=277?! E E V =cosO. Ey =Ksin0sin0 V =sin6cos0 zE E xE Whipple (1951)givesthefollowingexpressionforejectionspeedofdustfrom © Royal Astronomical Society • Provided by theNASA Astrophysics Data System Earth basedobservationsgiveverylittleinformationabouttheshapeofacross-section Consider thefollowingsimplifiedmodel.Supposethatamountofdust,dm,released Due toKepler’ssecondlaw 2 The break-upofthecometcannowbemodelledinfollowingway.Firstarandom 1 +ecosv a(l —e) 9/4 (-±- \nspr /2-1 0.013 ^jRlx656cms c v 1/2 1982MNRAS.200..313F 5_11 been approachingtheEarth’s orbitduringthelast150years,andpassedabout 15yearsago. + representstheintersectionpoint ofanejectedparticleandtheecliptic. Figure 9.Thecross-section obtained fromacomputersimulationofthebreak-up thecometwhich formed theGeminidstream. • istheintersectionpointofmeanstreamandecliptic andthishas the orbitalelementsinsteadofGM©. the descendingnode,isinvicinityofEarth’sorbit.Thexandycoordinates in cgsunits.Hencethegravitationalparameterß=isusedcalculationof descending nodearegivenby force oftheSun’sgravitybyafactor1—Bwhere will haveaneffectonthisorbit.Immediatelyafterejectionpressurereducethe y =—rsin£2. have alreadybeendetermined. B =5.8x10"sp" lie inaconcentratedbaratanangleofabout25°totherestcross-sectionandthis will varyasafunctionoftime,thestreamissweptpastEarth’sorbit. This istherightshapeandorientationtogiveobservedresults.Alsomostofparticles could alsoexplaintheskewshapeofGeminidstream.Notethatdegreeskewness = Where x =—reos£2 is orthogonalandintheplaneofecliptic.The(x,y)coordinatescanthusbecalculated for eachparticleejectedfromthemodelcometsinceorbitalelementsa,e,coand£2 and wherethexdirectionisfromSuntowardsfirstpointofAriesy r © Royal Astronomical Society • Provided by theNASA Astrophysics Data System The Geminid‘comet’intersectstheeclipticintwopointsbutonlyoneofthesepoints, The ejectionofonethousandrandomparticlesresultinthecross-sectionshownFig.9. 2 1 —ecosco a{\ -e) Evolution oftheGeminidmeteorstream323 324 K. Fox, I. P, Williams and D. W. Hughes Thus by considering the effects of the shape of the cross-section of the Geminid meteor stream on the observations of the solar longitude its anomalous behaviour appears to be explained.

Acknowledgment

One of us (KF) acknowledges the award of an SERC research studentship during the tenure of which this work was carried out.

References

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