Mon.Not.R.Astron.Soc.000,1–??(2011) Printed18January2012 (MNLATEXstylefilev2.2) On the dust abundance gradients in late-type galaxies: ff I. E ects of destruction and growth of dust in the interstellar medium 2 1 Lars Mattsson1⋆, Anja C. Andersen1 & Joakim D. Munkhammar2 0 2 1DarkCosmologyCentre,NielsBohrInstitute,UniversityofCopenhagen,JulianeMariesVej30,DK-2100,CopenhagenØ,Denmark 2Dept.ofEngineeringSciences,SolidStatePhysics,UppsalaUniversity,Box534,S-75121Uppsala,Sweden n a J 6 1 ABSTRACT ] Wepresentbasictheoreticalconstraintsontheeffectsofdestructionbysupernovae(SNe)and O growth of dust grains in the interstellar medium (ISM) on the radial distribution of dust in C late-type galaxies. The radial gradientof the dust-to-metalsratio is shown to be essentially flat (zero) if interstellar dust is not destroyed by SN shock waves and all dust is produced . h in stars. If there is net dust destruction by SN shock waves, the dust-to-metals gradient is p flatterthanor equalto themetallicity gradient(assumingthegradientshavethe samesign). - o Similarly,ifthereisnetdustgrowthintheISM,thenthedust-to-metalsgradientissteeperthan r orequaltothemetallicitygradient.Thelatterresultimpliesthatifdustgradientsaresteeper st thanmetallicitygradients,i.e.,thedust-to-metalsgradientsarenotflat,thenitisunlikelydust a destructionbySNshockwavesisanefficientprocess,whiledustgrowthmustbeasignificant [ mechanism for dust production. Moreover, we conclude that dust-to-metals gradients can be used as a diagnostic for interstellar dust growth in galaxy discs, where a negativeslope 1 v indicatesdustgrowth. 5 Key words: Galaxies: evolution, ISM; ISM: clouds, dust, extinction, evolution, supernova 7 remnants; 3 3 . 1 0 2 1 INTRODUCTION late-typegalaxiesseemstoworknicelywithoutsignificantnetde- 1 structionofdust(Inoue2003;Hirashita1999).Infact,noorlittle The lifetime of dust grains in the interstellar medium (ISM) is a : net dust destruction makes it easier to explain the dust-to-gas ra- v critical parameter for the evolution of the dust component in a tios,sincestellardustproductionislikelynotsufficientforneither Xi galaxy.Shock-wavesoriginatingfromsupernovae(SNe)arguably theMilkyWay,noranyoftheotherlate-typelocalgroupgalaxies containenoughenergytodestroy(oratleastshatter)dustgrainsas r (Hirashita1999;Zhukovska,Gail&Trieloff2008). a these waves propagate through the ISM. The time scale for such dust destruction depends on several physical conditions, where Observational constraints imply dust production in SNe is the supernova rate (SNR) and efficiency of dust destruction in a ratherinefficient(Kotaketal.2006,2009),whichsuggestthehigh SN-shock are the most important (McKee 1989; Draine 1990). dust masses detected in some, relatively old, SN remnants (see, Shockdestructionofdustgrainshasbeenconsideredquiteefficient e.g., Morgan&Edmunds 2003; Morganetal. 2003; Dunneetal. in many studies, e.g., Jones,Tielens&Hollenbach (1996); Jones 2009; Gall,Hjorth&Andersen 2011) could be the result of sub- (2004); SerraD´ıaz-Cano&Jones (2008),suggest agrain lifetime sequent dust growth, and/or heating of pre-existing dust, rather ofafewtimes108 yrformanydustspecies,butnotethatarecent than dust production in the actual SN. However, since SN rem- re-evaluationofdustlifetimesbyJones&Nuth(2011)showedthat nantswithlargedustmassestypicallycontainvastamountsofcold thelifetimesofsilicategrainsmaybecomparabletotheinjection dust, somedegreeof growth appear tobenecessary even ifthere timescaleofsuchgrains. isacomponentofheatedswept-updust.Thispictureisconsistent While this high dust-destruction efficiency seems consistent withtheoreticalresultswhichsuggest90%ofthedustproducedin withtheMilkyWay(solarneighbourhood), ithasbeenshownby SNe is destroyed by the reverse shock before it reaches the ISM several authors that very efficient dust destruction is unlikely in (Bianchi&Schneider2007).HenceAGBstarscouldbethemost high-z objects (Dweketal. 2007; Gall,Andersen&Hjorth 2011; importantsourceofstellardustasasignificantfractionofthemet- Mattsson2011).ItmayofcoursebethatdustdestructionbySNe als expelled by these stars is expected to be in the form of dust is less efficient in high-z galaxies, but also modelling of nearby (Edmunds&Eales 1998; Ferrarotti&Gail 2006), which is sup- portedbyobservationaldetectionsofdust(see,e.g.,therecentre- sultsbyRamstedtetal.2011).Itshouldbenoted,however,thatthis ⋆ E-mail:[email protected] picturemayneedtoberevisedduetotherecentdiscoveryofalarge c 2011RAS (cid:13) 2 Mattsson,Andersen &Munkhammar amount of cold dust associated with SN 1987A (Matsuuraetal. Σ dZd =y dΣs +Z (r,t)[G(r,t) D(r,t)]dΣs, (4) 2011). g dt d dt d − dt In models of dust evolution for the solar neighbourhood by whereΣ isthegasdensity,Σ isthedustdensity,andtheyieldy g d i Dwek (1998) and Zhukovska,Gail&Trieloff (2008), the limited isdefinedas dustproductioninstarsandpossibledustdestructionbySN-shock 1 mup wavesaremorethanwellcompensatedbyanefficientdustgrowth y = p(m)mφ(m)dm, (5) i α i intheISM,whichissupportedbyobservationsindicatingtheex- Zmlo istenceof large, micrometer-sizeddust grainsindense molecular for both metals (i = Z) and stellar dust (i = d). In equation (5) clouds (Paganietal. 2010). There is further evidence from dust- above, p isthe fraction of the initial mass m of astar ejected in i evolution modelling (see, e.g. Mattsson 2011; Pipinoetal. 2011; theformofnewlyproducedmetalsordust,αisthestellarlock-up Valianteetal. 2011) along with some observational constraints fraction (i.e., the fraction of the baryon mass being locked up in (see,e.g.Michalowskietal.2010)whichsuggestsaneedforsig- long lived stars) and φ(m) is the mass-normalised IMF, with m , lo nificantdustgrowthintheearlyUniverseaswell.Dustgrowthap- m beingtheloweranduppermasscuts,respectively.Combining up pearstodominateoverdustdestructionalsointhelocal,present- equations(3)and(4),wehave dayUniverse(Hirashita1999;Inoue2003;Hirashita&Kuo2011; ∂Z y +Z [G(r,t) D(r,t)] Asanoetal.2012).Itisdifficulttoseparateonescenariowheredust d = d d − , (6) ∂Z y growth is totally dominating over dust destruction from another Z wherethereisverylittledustdestructionandlessdustgrowth.But whichthushavenoexplicitdependenceonthegasmassdensityΣ g whether thereisnet growth/destruction itshould have observable orthestellarmassdensityΣ,althoughG,DandZ ofcoursemay s d consequences,however. have implicit dependencies on theamount of gas and starsbeing We propose here a diagnostic tool for determining whether presentinacertaingalacticenvironment. thereisnetdustgrowthornetdustdestructionintheISMofalate- typegalaxyforwhichdust-to-gasaswellasmetallicitygradients canbederived.Aswewillshowinthispaper,thechange(gradient) ofthedust-to-metalsratioalongagalacticdisciscloselyconnected 3 CONSTRAINTSONDUST-TO-METALSGRADIENTS togrowthanddestructionofdustintheISM.Inanassociatedpa- We will now prove some basic properties of dust-to-metals (ζ) per (Mattsson&Andersen 2011, hereafter cited as Paper II), we gradientsrelativetothemetallicity(Z)gradient.For’logarithmic’ investigatetheimplicationsofobserved dust-to-metalsprofilesin dust-to-metalsandmetallicitygradientsweusethefollowingnota- aselectionoflate-typegalaxiesfromtheSINGS(Kennicuttetal. tions, 2003)sample. ∂lnZ 1 ∂Z ∂lnZ 1∂Z ∆ d = d, ∆ = , (7) Zd ≡ ∂r Z ∂r Z ≡ ∂r Z ∂r d ∂ln(Z /Z) ∂lnZ ∂lnZ 1 ∂Z 1∂Z 2 BASICEQUATIONS ∆ d = d = d , (8) ζ ≡ ∂r ∂r − ∂r Z ∂r − Z ∂r d In order to obtain analytical solutions and be able to manipulate whichareusedsincetheybothhavethesameunit([length] 1).The thebasicequationsofthedust-enrichmentprobleminsuchaway − twogradients∆ and∆ canberegardedascoupledthroughafunc- thatwecanderivesomebasicconstraints,weusetheinstantaneous ζ Z tion f which may be seen as a function of a number of physical recycling approximation (IRA, which essentially means all stars parameters,butingeneralwemaysayitisafunctionoftimetand areassumed tohavenegligiblelifetimeswithrespect totheover- radialposition(galactocentricdistance)ralongthegalaxy.Hence, alltimescaleforthebuild-upofmetalsanddust,seePagel1997) weconsider arelationoftheform∆ (r,t) = f(r,t)∆ (r,t).Inthe throughoutthispaper.Nodelayedelementproductionduetostellar ζ Z followingwewillimplicitlyassumeallquantitiesexcepty ,y are lifetimesisconsidered. d Z functionsofrandt.Wewillalsorefertothecaseofazeroderiva- Forconvenience,wealsodefinethedustdestructionraterela- tivewithrespecttorasa’flat’gradient,whichofcoursecouldbe tivetothegrowthrateofthestellarcomponentΣ as s seenasthecaseofnogradient.However,weprefertodescribethe D(r,t)≡Σ˙ISM(r,t) ddΣts −1, (1) grerfaedrisetnotsaagsrabdeiienngte(idtehreirvaptoivsiet)ivwe,hflicahtdoercnreegaasetisvew,iwthhgearela’cntoegceantitvreic’ ! radiusandviceversafor’positve’gradients.Belowwealsousethe awbhleesrer,Σ˙tISaMreisthtehegadluacsttodceesnttrruicctidoinstarnacteedaunedttiomSeN/aegeanredspthecetivvaerliy- signfunctionsgn(x)≡x/|x|todenotethesignof∆ζ and∆Z. (a notation that we will assume is understood in the following). THEOREM.Foraclosed-boxmodel,withoutanypre-enrichment, Similarly,wealsodefine andwheretheIRAandconstant yieldsyZ,yd havebeenadopted, thefollowingalwayshold: G(r,t)≡Σ˙gr(r,t) ddΣts −1, (2) (i) Aflat(noslope)dust-to-metalsgradientcanonlybeobtained ! ifthereisneithernetgrowth,noranynetdestructionofdustinthe whereΣ˙ istherateofgraingrowth(inmassunits)intheISM. ISM(G=D)orifthemetallicitygradientisflat. gr Assuminga’closedbox’,wheredust destructionintheISM (ii) Ifthedust-to-metalsandmetallicitygradientshavethesame isfromSN-shocks,theequationsforthemetallicityZandthedust- sign,therehastobenetgrowth(G>D)ofdustintheISM. to-gasratioZ becomes (iii) Ifthedust-to-metalsandmetallicitygradientshaveopposite d signs,therehastobenetdestruction(G<D)ofdustintheISM. Σ dZ =y dΣs = y dΣg, (3) g dt Z dt − Z dt (cid:3) c 2011RAS,MNRAS000,1–?? (cid:13) Onthedustabundancegradientsinlate-typegalaxiesI 3 Proof. Fromthe basic equations of dust evolution (see section 2) sgn(∆ ) = sgn(∆ ) is possible if (and only if) G > D, which ζ Z onefinds provespart(ii). ∂ζ y Z Z = d + (G D) 1 ζ. (9) (iii) Inthiscase,ifsgn(∆ ) , sgn(∆ ),then f < 0.IfG > D, ∂Z y − y − ζ Z Z " Z # Equation(6)gives Byuseofthechainrule1,weget dZ dZ d > d . (19) dζ = yd 1 +ζ G−D 1 dZ, (10) dr !G>D dr !G=D dr y Z y − Z dr " Z Z !# Analogoustocase(ii)wehave∆ >∆ =0,whichimplies ζ,G>D ζ,G=D whichintermsof∆ζ and∆Z,canbewrittenas f >0.IncaseG<D,Equation(6)gives y 1 Z dZ dZ ∆ = d + (G D) 1 ∆ . (11) d < d . (20) ζ y ζ y − − Z dr dr " Z Z # !G<D !G=D Thefunction f (seedefinitionabove)isthengenerallyexpressed Thus, we have ∆ < ∆ = 0, which implies f < 0 and ζ,G<D ζ,G=D f yd 1 + Z (G D) 1. (12) wthheircfohrperosgvnes(∆pζa)rt,(iiis)g.n(∆Z) is possible if (and only if) G < D, ≡ y ζ y − − Z Z (i) If∆ =0,thenobviously∆ =0asaconsequenceofEqua- (cid:3) Z ζ tion(11).Incasethereisneithernetgrowth,noranynetdestruction ofdustintheISM(G=D),wehave y 1 4 SIMPLEMODELSOFDUSTGROWTHANDDUST f = d 1. (13) DESTRUCTION y ζ − Z Equation(6)gives 4.1 DustgrowthintheISM ∂Z y The most likely dominant type of ’secondary’ dust production is d = d, (14) ∂Z y that by accretion of atoms(or small molecules) ontopre-existing Z interstellardust grains. Dust grainscan inprinciple alsogrow by andagainbythechainrule, coagulation,butthisprocesswillnotaffectthetotaldustmassvery dZ y dZ much since it ismostly smaller dust grains being joined together d = d . (15) dr y dr intolargergrains.Hence,wewillhereonlydiscussdustgrowthby !G=D Z accretion. Integrating equation (15), together with the natural initial condi- We define the rate per unit volume at which the number of tions Z(r,0) = Z (r,0) = 0 (no pre-enrichment), one obtains d atoms N in dust grains grows by accretion of metals onto these ζ =y /y ,or A d Z dustgrainsinasimilarwayas(see,e.g.Dwek1998) y 1 yZd ζ =1. (16) ddNtA = fsπa2nZngrhvgi, (21) Hence, according to Equation (13), we must have ∆ = 0, since ζ wheren andn arethetotalatomicmetalsanddust-grainnumber Z gr f =0,whichprovespart(i). densitiesintheISM,respectively,aisthetypicalgrainradiusand f s isthestickingcoefficient(i.e.,theprobabilitythatanatomwillstick (ii) First,wenotethatifsgn(∆ )=sgn(∆ ),then f >0.Incase ζ Z to the grain). v is the mean thermal speed of the gas particles G>D,Equation(6)gives h gi (includingmetals),whichisdefinedas dZ dZ drd!G>D > drd!G=D. (17) hvgi≡Z0∞vf(v)dv= rπ8mkTA, (22) Then,byEquation(8)andthefactthat f = 0ifG = D,wehave where f(v)istheMaxwelldistribution,kistheBoltzmannconstant, ∆ > ∆ =0,whichimplies f >0.IncaseG < D,Equa- ζ,G>D ζ,G=D T isthekinetictemperatureofthegasandm istheatomicweight A tion(6)gives ofthegasparticles.Intermsofsurfacedensitiesinthemolecular dZ dZ gascloudswherethedustmaygrow,wecanwrite d < d . (18) dr !G<D dr !G=D dΣd = fsπa2Σ˜ZΣdhvgi, (23) Again, using Equation (8) and f = 0 if G = D, we have dt m d h gri c ∆ < ∆ = 0, which implies f < 0. Hence, ζ,G<D ζ,G=D where Σ˜ is the surface density of free (atomic) metals, m is Z h gri themeanmassofthedust grainsintheISMandd isthesizeof c 1 Ifζisafunctionofrandtwithcontinuousfirstpartialderivatives,andif themolecularcloudinwhichthedustisgrowing.Thetimescaleof randtcanberegardedasdifferentiablefunctionsofZ,then graingrowthcanthenbeexpressedas dζ ∂ζ dr ∂ζ dt Z 1 dZ = ∂rdZ + ∂t dZ. τgr =τ0 1− Zd − , (24) Ataspecifictimet=t0wecanthuswrite where (cid:18) (cid:19) dζ dζ dr dZ = drdZ. τ0 = f πhma2gΣridcv ≈ f πah2mZgrΣidc v , (25) s Zh gi s molh gi c 2011RAS,MNRAS000,1–?? (cid:13) 4 Mattsson,Andersen &Munkhammar in which Σ is the surface density of molecular gas, and Z the efficiencyissetbythefraction f ofinterstellardustdestroyedin mol d metallicity. anencounterwithaSNshockwave,whichoccursinthedefinition ForsimplicitywewillassumeΣ Σ ,sincemostofthe of m (McKee1989;Dweketal.2007), mol ≈ H2 h ISMi ggaesn.inWtehealmsooalescsuumlaregΣaHs2ctlroaucdesstihseinsttahre-foformrmatoiofnmroalteec,ui.lea.r,hydro- hmISMi≡Zv0vf fd(vs)(cid:12)ddMvssw(cid:12)dvs, (32) 1dΣ (cid:12) (cid:12) Σ˙⋆ =ηΣH2 = α dts, (26) where Msw is the s(cid:12)(cid:12)(cid:12)wept-u(cid:12)(cid:12)(cid:12)p gas mass (during Sedov-Taylor ex- pansion), v is the shock velocity, and v , v are the initial and s 0 f asindicatedbyseveralobservationalstudies(e.g.,Rownd&Young the final velocity, respectively. Note that in this way δ is similar 1999; Wong&Blitz 2002; Bigieletal. 2008; Leroyetal. to the ǫ¯-parameter (average grain-destruction efficiency) used by 2008; Bigieletal. 2011; Feldmann,Gnedin&Kravtsov (McKee 1989), which should not be confused with the ǫ (dust- 2011; Schrubaetal. 2011). Such a relation is also growthparameter)introducedintheprevioussection.Itshouldalso supported by theory and recent numerical experi- bestressedthat f isnotaconstant,butafunctionoftheshockve- d ments (see, e.g. Krumholz,McKee&Tumlinson 2009; locityv. s Krumholz,Leroy&McKee 2011). Moreover, the mean ther- A Larson (1998) IMF and m 1000M (Dweketal. mal speed hvgi is roughly constant in the considered ISM 2007)suggestsδ ∼ 10,whichishlikeISlMyicl∼osetoanu⊙pperlimitfor environment and the typical grain radius does not vary much. δ.Justasinthecaseofǫabove,itisnotabsolutelyclear,however, Hence,thetimescaleτ0 isessentiallyjustasimplefunctionofthe thatδcanbetreatedasaparameterthatdoesnot varyduringthe metallicity, the gas abundance and the growth rate of the stellar courseofevolutionoftheISMinagalaxy,butitseemsinagiven component, environmentafairapproximation. ǫZdΣ τ−01 = Σ dts, (27) g 5 ANALYTICSOLUTIONS the constant ǫ will, in the following, be treated as an essentially free(butnotunconstrained)parameterofthemodel.Theexpected Forsimplicitywehaveassumedaclosedbox(seesection3),i.e., valueisontheorderofafewhundred,whichisrequiredtoobtain noin-oroutflowsto/fromthedisc.Thisisnotinagreementwith τ 107yr,suggestedabove.Wewillhereadopt thewidelyacceptedideasaboutgalaxy-discformation,wherethe gr ∼ baryons (in the form of essentially pristine gas) are assumed to dZd = 1 Zd Zd = Zd, (28) be accreted over an extended period of time. But as shown by dt !gr (cid:18) − Z (cid:19) τ0 τgr Edmunds (1990), the only major effect of unenriched infall is to make the effectiveyield smaller, i.e.,todilute thegas so that the as the rate of change of the dust-to-gas ratio Z due to accretion d metallicity builds up more slowly. As we in this study uses the of metalsonto pre-existingdust grainsintheISM. Notethatthis present-day metallicityasinput, theoverall effectsof assuming a formulationof ’secondary’ dust production differsfromthat used closedboxarerathersmall,andingeneralonlyaccretionofmetal- byEdmunds(2001)andMattsson(2011)inthatitalsodependson enriched gas can affect the dust-to-metals ratio significantly (see thedustabundanceintheISMandthedepletionofmetalsinatomic AppendixB). state. 5.1 Generalsolution 4.2 Dustdestruction Adopting the closed-box scenario, the dust destruction and dust The dominant mechanism for dust destruction isby sputtering in growthmodelsasdescribedabove,resultsinanequationfordust thehigh-velocityinterstellarshocksdrivenbySNe,whichcanbe evolution, directlyrelatedtotheenergyoftheSNe(Nozawa&Kozasa2006). dZ Z dΣ FollowingMcKee(1989);Dweketal.(2007)thedustdestruction Σ d = y +Z ǫ 1 d Z δ s, (33) time-scaleis g dt d d − Z − dt (cid:26) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27) Σ whichcombinedwiththemetallicityZgives τ = g , (29) d hmISMiRSN ddZZd = y1 yd+Zd ǫ 1− ZZd Z−δ , (34) whereΣ isthegasmassdensity, m istheeffectivegasmass Z g h ISMi (cid:26) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27) cleared of dust by each SN event, and RSN isthe SN rate, which whereyZ isthemetalyield.Providedyd < yZ,thegeneralclosed- maybeapproximatedas boxsolution(ofequation34)forthedust-to-gasratioZd interms ofthemetallicityZis(seeAppendixCforasketchyderivation), 100M R (t) Σ˙ (r,t) ⊙φ(m)dm. (30) SN ≈ s Z8M⊙ Z = yd Z δ ϕ11 yyZd,(ǫZǫy−Zδ)2,ǫδy2Z −ϕ11 yyZd,ǫδy2Z,(ǫZǫy−Zδ)2 (35) Tishneoitnltiekgerlayltionveaqruyatmiounch(3o0v)eirstahecodnisstcaneitthweirt,hhreenscpeectthetotitmimees,caanlde d yZ (cid:18) − ǫ(cid:19)2ϕ10(cid:20) yyZd,ǫδy2Z,(ǫZǫy−Zδ)2(cid:21) +ϕ01(cid:20) yyZd,(ǫZǫy−Zδ)2,ǫδy2Z(cid:21)  τdmaybeexpressedas for  h i h i τ−d1 ≈ Σδg ddΣts, (31) ϕij(k,x,y)≡M i+ 2k,i+ 12;2x U j+ 2k,j+ 12;2y . (36) " # " # whereδwillbereferredtoasthedustdestructionparameter.This ThefunctionsMandUaretheconfluenthypergeometricKummer- parameterisdimensionless,andassuchitcanbeseenasameasure Tricomi functions of the first and second kind, respectively oftheefficiencyofdustdestruction.Moreprecisely,however,the (Kummer1837;Tricomi1947,seealsoAppendixD). c 2011RAS,MNRAS000,1–?? (cid:13) Onthedustabundancegradientsinlate-typegalaxiesI 5 In the equations above, y and y are the stellar dust and 6 GRAPHICANALYSIS d Z metal yields, respectively, δ is the ’dust destruction parameter’ UsingthenumericalimplementationofMandUdescribedinAp- (see section 4.2) and ǫ is the ’grain-growth parameter’ (see sec- pendix D we will here demonstrate the general behaviour of the tion4.1). equation (35) issingular at Z = δ/ǫ, which means this dust-to-metalsratioζ =Z /Zusingcontourplots.Unlessanything general solution must be used with care. It is relatively straight d elseisstated, weassumey = 0.02isagood typicalmetal yield forwardtoimplementtheKummer-Tricomifunctionsnumerically Z (whichisconsistentwiththeresultsofPaperII,butnotethat,e.g., (see Appendix D), but there is a regular singularity at the origin Garnettetal.2002,findsalowervalue)andy = 1y forsimplicity. in Kummer’s equation (to which M and U are linearly indepen- d 2 Z dentsolutions)whichcancausepotentialproblemsinthevicinity ofZ=δ/ǫ. 6.1 GeneraleffectsofgrowthdestructionofdustintheISM Incaseofnodustdestruction(δ=0)thedust-to-metalsratiobuilds uptoamaximum(whereζ 1)asǫandthemetallicityZincreases ∼ 5.2 Specialcases (seeFigure1,leftpanel).Atlowmetallicities(halfofsolarorless, inthepresentcase)theeffectofincreasingǫisrelativelysmallonce The general solution presented above is obviously not simple to we get beyond a certain ǫ, while at higher metallicities ζ grows useinpractice,nottheleastbecauseofthesingularityatZ =δ/ǫ. rapidly until the metals reservoir is exhausted and ζ approaches However, inthespecial casey 0(negligable net contribution d → unity(asalsofoundinthemodelsby,e.g.,Hirashita&Kuo2011; ofdustfromstars)thesingularitycanberemovedandthesolution Asanoetal.2012).Incaseofnodustgrowth(ǫ = 0)thedust-to- expressedas(seeAppendixC) metalsratioisonasteep”downhillslope”(approachingζ =0)for Zd =exp ξ2 exp ξ2 +η Z erfi(ξ) erfi(ξ ) −1 (37) essentiallyallmetallicitiesandδ-valuesontheconsideredinterval Z 0 0 d,0 − 0 (seefigure2,leftpanel).Notethatζisverysmallathighmetallicity d,0 (cid:16) (cid:17)n (cid:16) (cid:17) (cid:2) (cid:3)o ifthereissignificantdustdestruction. withZ beingtheinitialdust-to-metalsratio, d,0 1(ǫZ δ)2 1(ǫZ δ)2 πǫ ξ − , ξ 0− , η , (38) 6.2 Dust-to-metalsgradients ≡ s2 ǫyZ 0≡ s2 ǫyZ 0 ≡ r2yZ Asshownbythetheoreminsection3theeffectofdustdestruction and Z the initial metallicity. In the solution above, erfi(z) is the anddustgrowthonthedust-to-metalsgradientinagalaxydiscis 0 imaginary error function, related to the ordinary error function tomakeitsteeperorflatter.Theeffectofgrowthanddestructionof erf(z)aserfi(z)= ierf(iz),whereerf(z)isdefinedas dustintheISMcanbeillustratedinamoreintuitivefashionifwe − considerthespecificeffectsonagivenmetallicityprofile.Wehere 2 x erf(x)= e t2dt. (39) assumethatmetalsinadiscfollowsanexponentialdistribution, − √π Z0 R Physically,onemayinterpretthissolutionasdescribingthesubse- Z(R)=Z0exp −R , (43) quentevolution(wherethedustcontributionfromstarsmaybecon- 0! sidered negligible) after an initial phase of metal enrichment and wherewesetthecentralmetallicitytoZ0 =0.055andthee-folding dustformationleadinguptothepointwhereZ = Z0 andZd=Zd,0. scalelengthR0 issettobetheunitforthegalactocentricdistance. Itmaynotbeentirelyrealistic,butitdemonstratestheinterstellar Therightpaneloffigure1showshowdustgrowthcreatesadust- ”battle” between growth and destruction of dust grains in a very to-metalsgradientthatfallsofwithgalactocentricdistanceandbe- niceway(seeSection6.3andFigure3). comes increasingly steeper asǫ increases(for ǫ = 0thegradient Even when thereis asignificant net contribution from stars, is flat). Similarly, the right panel of figure 2 shows how dust de- wecanstillfindsimplersolutionsforspecialcases.Incasethereis structioncreatesaninwardsgradient,startingfromaflatgradient nodustdestructionbySNe(δ=0)thesolutionreducesto forδ=0. Inthecontextofdust-to-metalsgradientsassignsofeithernet y M 1+ 1yd,3;1ǫZ2 dustgrowthornetdustdestruction,oneshouldaswellnotethatif Zd= yZd M(cid:16) 12yy2Zdy,Z12;221ǫy2ZZ2yZ (cid:17)Z, (40) flthaet,miteitsalmliocirteyogrraledsisenimtapnodsstihbeledutostd-tiost-ignagsugisrhadbieetnwtseaernepeusrseensttieallllayr (cid:16) (cid:17) dustproduction(albeitwithahighstellardustyield)andscenario and if there is dust destruction, but no grain growth in the ISM includingdustgrowthand/ordustdestructionintheISM. (ǫ=0),then y Z Z = d 1 exp δ . (41) d δ " − − yZ!# 6.3 Dustgrowthsvs.destruction Ifthereisneithergrowth,nordestructionofdustintheISM(ǫ = Growthanddestructionofdustmustlikelyoccurtogether.Aswe δ=0),wehavethetrivialcase describeinAppendixCthegeneralsolutionwithbothǫandδnon- zero, has a singularity at Z = δ/ǫ, which makes the analysis of y Zd= ydZ, (42) how growth and destruction compete somewhat complicated and Z notleastlimited.Hence,wewillhereconsiderthespecial(andnot corresponding topurestellardustproductionandobviouslyaflat entirelyrealistic)casewherestellardustproductionisconsidered dust-to-metals gradient. All thespecial cases above evade thein- negligible(seeequation37)startingfromapointintimewhenthe convenientsingularityatZ=δ/ǫ. metallicityZ = 1.0 10 5 andthedust-to-metalsratioζ = 0.5.In − · c 2011RAS,MNRAS000,1–?? (cid:13) 6 Mattsson,Andersen &Munkhammar figure 3 we show ζ as a function of ǫ and δ for a fixed present- outwardsslope,providedthedust-to-gasratioaswellasthemetal- daymetallicityZ =0.02.Increasingtheefficiencyofdustdestruc- licityhavenegativegradients,i.e.decreaseswithgalactocentricdis- tioncounteractsthedustgrowth,whichisshownbythe”downhill tance.Hence, weexpect dust-to-metalsgradientstohavepositive slope”towardshighδandlowǫ values.Clearly,ahighefficiency (inwards)gradientsifdustdestructionismoreimportantthandust of dust destruction is not likely if there is to be a significant net growth,andifdustgrowthisthemoreimportantprocessweexpect productionofdustwithoutinvokingaridiculouslyshorttimescale themtobenegativeingeneral.Thedust-to-metalsgradientthusap- forthedustgrowthintheISM.Moreprecisely,itisrequiredthat pearstobeausefuldiagnosticfortheexistenceofinterstellardust ǫZ δ,whichincaseZ=0.02andδ=10,wouldimplyǫ 500. growth. ≫ ≫ Withsuchalargeǫthetypicalgrowth-timescaleisdownto 105 Oursimplemodelofdustgrowthhasjustoneadjustablepa- ∼ yrorless.Aswementionedinsection4.1,ǫshouldnotexceedval- rameter.Thisparameter(ǫ)canhavearatherwiderangeofnumeri- uesofafewhundredifthedust-growthtimescaleτ istobecon- calvaluesdependingonwhatoneassumesaboutthephysicalprop- gr sistentwiththesuggestednumbersforthelocalISMoftheMilky ertiesofthedustgrainsaswellasthegasinISM.Inprincipleǫis Way.Itisquitepossiblethatτ canbesignificantlyshorterin,.e.g., proportionaltothegasmassdensityΣ ifallotherquantitiesremain gr g adenserenvironment,butadeeperanalysisofthisgoesbeyondthe constant,butthestarformationefficiencyηislikelyproportionalto scopeofthispaper. Σ raisedtosomepower(Krumholz&McKee2005)andsincethe g cloudsized isalsolikelyrelatedtoΣ ,ǫmaynotbemuchdepen- c g dentonΣ afterall.Moreprecisely,thestar-formationefficiency(or g 6.4 ”Critical”metallicityfordustgrowth timescale)isexpectedtocorrelatewiththefree-falltimescale,i.e., JustasZhukovska,Gail&Trieloff(2008),Hirashita&Kuo(2011) Σ˙⋆ ∝ ΣH2/τff,whereτff ∝ Σg−1/2 (Krumholz,McKee&Tumlinson 2009), assuming Σ ρ (Elmegreen2002), and thescaleof the alincdiyA”sZacnritowehtearl.e(t2h0e1d2u)s,tw-meafissndcotnhtartibthuetiroenefxriosmtag”racirnitigcraolwmthetianl-- acnlodudvsizeisdrcouisghglivygecn∝obnystgaJneatn(sisloetnhgetrhmλalJc∝onΣdg−i1t/i2o.nAs)s, tǫhe∝eτffffedcctiΣveg creases rapidly. But this rapid increase over orders of magnitude h gi dependenceonΣ isexpectedtobeweak,ifnotnegligible.Thus, occursonlyifthestellardustyieldy issignificantlylowerthanthe g d itisfairtoassumethatǫ is(effectively)onlyveryweaklydepen- metalyield(seefigure4,leftupperpanel).Moreover,Z depends crit dentonΣ withinagalaxy,althoughfromonegalaxytoanotherǫ somewhatonthedust-growthtimescale(orǫ),whichcanbeseen g mayvarysignificantly,however(seePaperII).Belowweanalyse infigure4(rightupperpanel).Hence,Z shouldnotbeviewedas crit therangeofpossibleǫ valuesconsideringjustmean/characteristic auniversalconstant.Infact,areasonabledefinitionofZ would crit valuesofΣ ,ηand v . bethemetallicityatwhichstellardustproductionandthenetdust g h gi Intermsoftheincludedphysicalparameters(seesection4.1), growthintheISMcontributeequallytobuild-upoftheinterstellar wefind dustcomponent.Insuchacase,adoptingthemodelusedabove, fπa2 v Zcrit=Zd+ ǫyZdd − δǫ. (44) ǫ≈ αhηishdchihgmigrihΣgi. (45) Ifdustgrowthdominatesoverstellardustproductionanddustde- The lock-up fraction α is 0.6 - 0.8 for a normal IMF (we use stthhtreiusncvtZaioclrunite≈i,nthZtehde,dwuISshtMicehviossluungteiggoelnisgt(iaZbsclrefit,u∼ni.cet.yi,odi/nfyyoZdf./Amǫte≪mtalel1tiaclailtniycd)itsδieh/soǫub≪ledlob1we, whaneerdeasdαinocp=et hthd0ec.i7t,y=pseic5ea0lMpsaiczt.tesTsohofenaa2mv0e1or1laeg,ceufilggaurraricenlo1mu)d,ashdsηcihimsisg1r∼i0o-f11c0Go0uyprrs−ce1, essentially identical to the case of pure stellar dust production - depends on the typical grain size a, where the latter ranges be- withoutanygrowthordestructionofdustgrainsintheISM.Inthe tween0.001µmforthesmallestseedparticlesand 1µmforlarge ∼ rightupperpaneloffigure4,Z = y /y ismarkedbyavertical full-grown dust grains. Hence, it is more convenient to introduce crit d Z dashedline.Atlowermetallicitiesallmodelcurvesareindeedthe the characteristic grain density ρgr = hmgri/hVgri, where Vgr is same. the volume of a dust grain. The grain density ρgr is typically 3.3 ThiscriticalmetallicityZcrithasaninterestingimplicationfor gcm−3(Draine&Li2007)forsilicatesand1.85gcm−3foramor- dust-to-metals/gas gradients, as it predicts the existence of bends phouscarbon(Rouleau&Martin1991),butothervaluescanalso also in logarithmic slopes and the existence of a critical galacto- befoundintheliterature.Takingρgr =2.5gcm−3asrepresentative centricdistanceinbetweenaninnerandanouter”plateau”where figureforcosmicdustingeneral,wearriveat cζoinssceoqnusetanncte(osefethFeigin4t,elroswteellrapradnueslst-)g.rTohwisthnorant-elin(seeaerefeqautautrieonis2th8e) ǫ 4.2 f a −1 hΣgi . (46) ≈ × s µm M pc 2 havinganon-linear(Zd2)term.Althoughequation(24)andequation ! ⊙ − ! (28)togetherrepresentonespecificmodel,allmodelsofinterstel- Assuming that all metals that come in contact with a dust grain lar dust growth will be non-linear as long as they depend on the will stick to that dust grain (f = 1), a small characteristic grain s amountofdustand”free”metals(notlocked-upindust)available. size a = 0.01µm and a relatively high average gas density of Σ =50M pc 2,willresultinanǫofroughly2 104corresponding togatypica⊙lgra−in-growthtimescaleofτ 10·6 yrifthegascon- gr ∼ sumption rate issimilar to that of thesolar neighbourhood. Such 7 DISCUSSIONANDCONCLUSIONS high values of ǫ may be expected in young star-forming systems Wehaveshownthatdustdestructionbyshockwavesfromexplod- (e.g., late-type dwarf galaxies) where one has reasons to believe ingSNeandinterstellardustgrowthactsinoppositewaysonthe thatgasdensitiesarequitehighandthegrain-sizedistributionisbi- dust-to-metalsgradientoveragalaxydisc(seethetheoremproved asedtowardssmallgrains.Thelatterisduetoinsufficienttimefor insection3).Thisishardlysurprising,butstartingfromanexactly extensivegraingrowth,andgrainshatteringowingtoanelevated flatgradient(ornogradient,moreprecisely)dustdestructionwill SNrateandstrongUVradiationasconsequencesofrecentstarfor- overtimecreateaninwardsslope,whiledustgrowthwillcreatean mation.If f =0.1(whichismoreconsistentwithsilicategrowth), s c 2011RAS,MNRAS000,1–?? (cid:13) Onthedustabundancegradientsinlate-typegalaxiesI 7 Figure1.Left:Dust-to-metalsratioζ=Zd/ZasafunctionofthemetallicityZandthedust-growthparameterǫforthecasewherethereisnodustdestruction duetoSNe(δ = 0).Right:Sameastheleftpanel,butasafunctionofthegalactocentric distanceinagalaxydiscassuminganexponentialdistributionof metals. Figure2.Left:Dust-to-metalsratioζ=Zd/ZasafunctionofthemetallicityZandthedust-destructionparameterδforthecasewherethereisnodustgrowth intheISM(ǫ = 0).Right:Sameastheleftpanel,butasafunctionofthegalactocentric distanceinagalaxydiscassuminganexponentialdistributionof metals. a = 1µmandΣ = 5M pc 2,thenǫ 2andthusτ 109 yr. ThemodelofdustdestructionduetoSNshockwaveshasef- Withf =1,a=g0.1µma⊙ndΣ− =10M ≈pc 2,thecorresgpro∼ndingτ fectivelyonlyoneparameteraswell.Thisdustdestructionparam- s g − gr is 108 yrassumingagas-consumpti⊙onrateandametalcontent eterδcanbeexpressedas ∼ similartothatofthelocalGalaxy.Thisnumberisconsistentwith, m 100M estimatesmadeinsomeotherstudies(Jones,Tielens&Hollenbach δ= h ISMi ⊙φ(m)dm, (47) α 1996;Jones2004;Zhukovska,Gail&Trieloff2008;Jones&Nuth Z8M ⊙ 2011), but slightlylonger thanthetimescalessuggested recently where φ is the IMF and α is the lock-up fraction, as previously by Hirashita&Kuo (2011). Note that grain size is the parameter defined.Withα = 0.7andanormalIMF(see,e.g.,Larson1998), thatislikelymostimportantforthevalueofǫ asitcanvaryquite wefind significantly.Thegasmassdensitycanvaryoverseveralordersof m magnitudeaswell,butasdescribedaboveitmaybecancelledout δ 0.018 h ISMi , (48) ≈ × M byotherparameters. ⊙ ! whichsuggest δisoforder ten,if m 1000M .Theactual h ISMi ∼ ⊙ c 2011RAS,MNRAS000,1–?? (cid:13) 8 Mattsson,Andersen &Munkhammar Figure3.Contourplotofthedust-to-metalsratioasfunctionoftheparametersǫ(growth)andδ(destruction)forthespecial(andnotentirelyrealistic)case wherestellardustproductionisconsiderednegligible(seeequation37)startingfromapointintimewhenthemetallicityZ=1.0 10 5andthedust-to-metals · − ratioζ=0.5.Thepresent-daymetallicityisassumedtoberoughlysolar(Z=0.02). efficiencyofdustdestruction,andthustheeffectiveinterstellargas be arbitrarily large. In particular, the dust-to-metals gradient can massclearedofdust,isnotverywellknown.Therefore,itisreason- never become steeper thanthe metallicitygradient only owing to abletotreatδasanessentiallyfreeparameter.Inordertohavenet secondary dustproduction instars(forfurtherdetailsandamore growthofdustintheISM,thevalueofδneedstobeδ<ǫ(Z Z ). quantitativeanalysis,seeAppendixA). − d Thismeansδ 10islikelyattheupperendofpossiblevaluesfor Thelifetimeofstarsmayalsoplayarole,butsincethevery ∼ • such ascenario, assuming Z −Zd ≈ 0.01, which suggest δ = 10 samestarsthatareproducingthemetalsarealsoresponsibleforthe wouldrequireǫ > 1000orτgr . 107 yr.Highvaluesofδmaybe stellarproductionofdust,thiseffectcannotbedominant.Infact,it foundinstarburstenvironments,wherehighSNratesandpossibly shouldbenegligible. alsotop-heavyIMFsareexpected.However,ingeneralδislikely small,sincehighratesofdustdestructionaresomewhatinconsis- tentwiththefactthatdustisubiquitousthroughouttheUniverse. Thus,weconcludethatdust-to-metalsgradientscanbeused Although simplifying assumptions have been made in this asadiagnosticforinterstellardustgrowthingalaxydiscs,where study inorder to obtainareasonably simpleparametric model in anegativeslopeindicatesdustgrowth. terms of ǫ and δ, a clear outwards slope is unlikely to be the re- sult of any other mechanism than dust growth in the ISM. Other Dust growth has a non-linear nature as the time scale for mechanisms,whichhoweverappearlesseffective: it must depend on both the metallicity and the amount of avail- able seed grains. As a consequence there is a ”critical” metallic- Accretion of dust free material onto the galactic disc may ity (which depends on the dust-growth and dust-destruction time • affect the dust-to-metals ratio if the infalling gas contains some scalesaswellasthedust-to-gasratio)atwhichthedustproduction fractionof atomicmetals(seeAppendix Bforfurtherdetailsand byinterstellargraingrowthexceedsstellardustproductionandthe workedoutexamples).Themetallicityoftheaccretedgasislikely dust-to-gasratiodivergesfromthesteadyincreaseobtainedincase muchlessthanthatoftheISM,sotheeffectcannotbeverylarge thedust massisowingtostarsonly. Thisallowsforbends inthe anditwouldalsomimictheeffectof dust destructionrather than logarithmicslopesof thedust-to-metalsprofileeven ifthemetal- dustgrowth. licityfollowsan exponential fall-off with galactocentric distance. Secondary dust production in stars, i.e., a stellar dust yield DustdestructionintheISMduetoSNemayalsoaffecttheshape • whichincreasesasthemetallicityofstarsincreases,mayinprinci- of the dust-to-metals profile, creating a central depression as the plecreateadust-to-metalsgradientalongagalaxydisc.However, dust-to-gasratio,themetallicityandtheintegratednumberofSNe therelativeincreaseofthestellardustyieldalongthedisccannot typicallyincreases inthecentral partsof agalaxy disc compared c 2011RAS,MNRAS000,1–?? (cid:13) Onthedustabundancegradientsinlate-typegalaxiesI 9 Figure4.Effectsofthecriticalmetallicityfordustgrowthdominationanditsdependenceonthestellardustyieldanddustgrowthparameterǫ.Theupper panelsshowtheevolutionofthedustmassasafunctionofmetallicity forvariousvaluesofthestellardustyieldyd withafixedǫ = 200(leftpanel)and variousvaluesofǫwithafixedstellardustyieldyd =5.0·10−3yZ (rightpanel).Thelowerpanelsshowthecorrespondingplotsofthedust-to-metalsratioζ asafunctionofgalactocentricdistanceassumingane-foldingdecayofthemetallicityalongthedisc(seeequation43). totheouterdisc.However,sincedustgrowthincreasesaswell,the interstellardustgrowthbeingthedominantdustproductionmech- expectedneteffectisanincreaseddust-to-metalsratioinanycase. anisminlate-typegalaxies. Finally, we note that combining recent observational results (Munoz-Mateosetal.2009;Moustakasetal. 2010) one findsthat dust-to-metalsgradientsinlate-typegalaxydiscsappearrelatively steep(andnegative),i.e.,showaclearfall-offwithgalactocentric ACKNOWLEDGMENTS distance,whichsuggestinterstellardustgrowthismoreimportant than stellar dust production. In Paper II of this series, where we The authors thank the reviewer, Anthony Jones, for constructive comparetheoretical modelsandobservational resultsinmorede- andhelpfulcommentsandcriticismthatgreatlyhelpedtoimprove tail,wereturntothisfactandlookformorequantitativeevidenceof thepresentation.L.M.acknowledgessupportfromtheSwedishRe- c 2011RAS,MNRAS000,1–?? (cid:13) 10 Mattsson,Andersen& Munkhammar searchCouncil(Vetenskapsrådet).TheDarkCosmologyCentreis Leroy A. K., Walter F., Brinks E., Bigiel F., de Blok W. J. G., fundedbytheDanishNationalResearchFoundation. 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