The Microlensing and Halo Models of the Galaxy

astro-ph/950513029May1995KUNS1347YITP/K-1108May1995TheMicrolensingandHaloModelsoftheGalaxyYukitoshiKan-ya,RyoichiNishi,DepartmentofPhysics,KyotoUniversityKyoto606-01,JapanandTakashiNakamuraYukawaInstituteforTheoreticalPhysics,KyotoUniversityKyoto606-01,JapanAbstractWeinvestigatedthedependenceoftheopticaldepthofthemicrolensingeventsonmodelparametersoftheGalactichalo.WeonlyconsiderGalacticmassmodelsinwhichtherotationcurveinsidetheSuniscompatiblewiththeobservationandLMCisboundtotheGalaxy.Itisfoundthatvariesuptoafactor2.5fromthestandardsphericalandatrotationhalomodel.ThisimpliesthatonlythemostcentrallyconcentratedhalomodelcanbeconsistentwiththeobservationifthehaloconsistsofonlyMACHOs.WealsocalculatethepowerxofIMFofMACHOconsistentwithTyson’sCCDsurveyaswellasBahcalletal.’sobservationbyHST.Itisfoundthatxisgreaterthan5.1IntroductionGravitationalmicrolensingeventsaredetectedrecentlybythreecollaborations:MA-CHO(Alcocketal.1993[3])andEROS(Aubourgetal.1993[8],Aubourgetal.1995[9])forLMCevents,aswellasOGLE(Udalskietal.(1993)[42])andMACHO(Bennettetal.1994[13])forbulgeevents.FromtheseresultswecandiscussaboutthenatureofthemissingmassinourGalaxy(Paczynski1986[30]).Thekeyquantityinthisproblemistheopticaldepthofthemicrolensing.Thisquantityistheinstantaneousprobabilitythattheeventisoccurringwhenweobservebackgroundstarsrandomly.Observationallyisderivedfromthenumberoftheobservedstars,themeaneventdurationandtheeventrate(Paczynski1986[30]).ForLMCevents,1MACHOcollaborationconcludedthat=8:0+146108in68%condencelevel(Alcocketal.(1995)[5]).Ontheotherhand,severalauthors(e.g.Paczynski1986[30]Griest1991[24])estimatedforthehalomodelwithatrotationcurveintheouterregionoftheGalaxyandobtained’5107,whichsuggestsaMACHOfractionf=0:2+0:330:14andallofthemissingmassofourGalaxymaynotbeMACHOs.Forbulgeevents,theobservationalvalueofis’31061(Udalski1994[43])byOGLEcollaboration,whereistheeciencyoftheobservation,and=3:0+1:50:9106byMACHOcollaboration(Bennett1994[13]).Paczynski(1991)[31],Griestetal.(1991)[25]andKiraga&Paczynski(1994)[26]estimatedforthebulgeas0:11106,whichisatleastfactor3smallerthantheobservationaldata,thatis,f3.Thefraction,f,however,dependsonthetheoreticalestimateofi.e.modelsofourGalaxy.Soitisimportanttoestimatethemodeldependenceof.Themainpurposeofthispaperistodiscusstheoreticalinmoredetail.Althoughthemodelwiththeatrotationcurveisfrequentlytakenasthemassdistri-butioninthehalo,atpresentwecanonlysaythattheGalacticrotationcurveisessentiallyatonlyinsidethesolarneighborhood(e.g.gure2inFich&Tremaine1991[18]).Intheouterregionwehavenodeniterotationcurveatpresentalthoughwecanimposesomeconstraintsaswillbediscussedinsection2.Manyotherspiralgalaxieshaveatrotationcurveuptotheoutermostregion.However,itisreportedthatspiralgalaxieswithitsexponentialdiskscalelengthlessthan3.5kpchavedecliningrotationcurves(Casertano&vanGorkom1991[15]).ForourGalaxyitsscalelengthseemstobemarginal,i.e.,3:5kpc.OurGalaxymayhavenon-atrotationcurvebeyondsolarneighborhood.Theshapeofthehaloisanotherpointtobeconsidered.ItissuggestedfromN-bodysimulationsofthegalaxyformationthatthehalomaybenonspherical(e.g.Aarseth&Binney1978,[1],Aguilar&Merritt1990[2],Binney1994[14]andreferencestherein).Sackett&Gould(1993)[36]andFrieman&Scoccimarro(1994)[19]discussedthattheratiooftheopticaldepthtowardSMCtoLMCisagoodprobefortheshapeofthehalo.Inthispaper,wealsoinvestigatethedependenceofontheshape.AsamodelofthehalowetakethemodelofEvans(1994)[17]thatisapower-lawmodelwitharising,atorfallingrotationcurve.Weimposesomeconstraintsinthismodelandcalculatethedependenceoftheopticaldepthonmodelparameters.ThesamemodelhasbeenusedbyAlcocketal.(1994)[4].Theyconcludedthatchangesuptoafactor10.Inthispaperweusemorestringentconstraintsthantheirs,whichwillbeshownlater.MACHOsmaybethelow-massstars.Richer&Fahlman(1992)[33]suggestedthattheIMFofthelow-massstarlessmassivethan0:5MintheGalacticspheroidstarsismuchsteeperthantheSalpeter’sIMF.Bahcalletal.(1994)[12]observedrecentlyhalostarsinahigh-latituderegionbyHSTandconcludedthatifthedarkhaloconsistsoflow-massstarstheymusthavemasslessthanhydrogen-burninglimit.UsingdataoftheCCDsurveyofTyson(1988)[40]andHSTobservationbyBahcalletal.(1994),wewilldiscussontheconstraintstothepowerofIMFassumingthepowerlawIMF.Theplanofthispaperisasfollows.Insection2wewillshowtheGalacticmodelandimposetheconstraintonit.Insection3wewillderivethedependenceofonourmodelparameter.Insection4wediscussthepowerofIMFconsistentwiththeobservation.2Section5isdevotedtodiscussions.2TheGalacticmodelAsasphericalorspheroidalGalacticmodelwetakeanaxisymmetricpower-lawmodel(Evans1994[17]).Inthismodelthegravitationalpotentialisgiveninthecylindricalcoordinate(R;;z)as=v20Rc=(R2c+R2+z2q2)=2;(1)wherev0,Rc,andqarethenormalizationofpotential,thecoreradiusandtheaxisratioofequipotential,respectively.Therotationvelocityintheequatorialplaneisvc=v20RcR2(R2+R2c)(+2)=2!12R2asR!1:(2)Themassdensityofthehaloisgivenash=v20Rc4Gq2R2c(1+2q2)+R2(1q2)+z2(2(1+)q2)(R2c+R2+z2q2)(+4)=2:(