Disc evolution and the relationship between $L_{\rm acc}$ and $L_\ast$ in T Tauri stars PDF
Preview Disc evolution and the relationship between $L_{\rm acc}$ and $L_\ast$ in T Tauri stars
Mon.Not.R.Astron.Soc.000,000–000(0000) Printed2February2008 (MNLATEXstylefilev2.2) Disc evolution and the relationship between L and L in T Tauri acc ∗ stars I. Tilling, C. J. Clarke, J. E. Pringle and C. A. Tout 8 InstituteofAstronomy,TheObservatories,MadingleyRoad,Cambridge,CB30HA 0 0 2 n 2February2008 a J 4 ABSTRACT 1 WeinvestigatetheevolutionofaccretionluminosityLaccandstellarluminosityL∗inpre- ] mainsequencestars.WemaketheassumptionthatwhenthestarappearsasaClassIIobject, h p the major phase of accretion is long past, and the accretion disc has entered its asymptotic - phase.Weuseanapproximatestellarevolutionschemeforaccretingpre-mainsequencestars o based on Hartmann, Cassen & Kenyon, 1997. We show that the observed range of values r t k = Lacc/L∗ between 0.01 and 1 can be reproduced if the values of the disc mass fraction as Mdisc/M∗ at the start of the T Tauriphase lie in the range 0.01– 0.2,independentof stellar [ mass.We alsoshowthattheobservedupperboundof Lacc ∼ L∗ isagenericfeatureofsuch disc accretion. We conclude that as long as the data uniformly fills the region between this 1 upperboundandobservationaldetectionthresholds,thenthedegeneraciesbetweenage,mass v and accretion history severely limit the use of this data for constraining possible scalings 0 betweendiscpropertiesandstellarmass. 0 1 Keywords: accretion,accretiondiscs–stars:pre-mainsequence–planetarysystems:pro- 2 toplanetarydiscs . 1 0 8 0 : 1 INTRODUCTION 2 v Xi Theclaimedrelationshipbetweenaccretionrate,M˙,andmass,M∗, 1 in pre-mainsequence stars (e.g. Natta, Testi & Randich 2006 and r references therein) originates from the more direct observational a result that in Class II pre-mainsequence stars the accretion lumi- 0 nosity, L ,issimilarlyfoundtocorrelatewithstellarluminosity, acc L∗.Theaccretionluminosityisdeducedfromtheluminosityinthe -1 c emissionlinesofhydrogen(inNattaetal.,2006,thesearePaβand c a Brγ),togetherwithmodelingoftheemissionprocess(Nattaetal. L -2 = L * etd2riori0eatb0sigue4Lrss;a.ta.mcCTcInhaaalninovsddreidtsLmee∗dra,totkionatilneid.sgeb2nduy0eus0cpec4eleosa).stfchaitTnerhyhgeqeotutorhameebntoietocidtbaaiebeljleslipcenrMtget˙oi-nmaadpnaaepdidHenuMasecrer∗esqtzfuttrshoeopenmrbcsueetntehgtrlerle-aaRlqrcauuktpiassvrsnoefetoplily-r-l log --43 LL aa cc cc== LL **// 11 00 0 non-accretingstars(e.g.D’Antona&Mazzitelli1997). L a c c Theresultofthisexerciseindicatedacorrelationbetween M˙ -5 and M∗ oftheform M˙ ∝ M∗α,whereαiscloseto2.Thesimplest theoreticalexpectation(inthecaseofdiscaccretionwhereneither -6 -3 -2 -1 0 1 thefractionofmaterialinthediscnorthedisc’sviscoustimescale log L * scalesystematicallywithstellarmass)isinsteadthatM˙ ∝ M∗.The claimed steeper than linear relationship thus motivated a variety Figure1.ThedistributionofClassicalTTauristarsinOphiuchusintheL∗, oftheoreticalideasaboutspecificscalingsofdiscparameterswith Lacc plane.Detections (filledsquares)andupperlimits(crosses)deduced stellarmass(Alexander&Armitage2006,Dullemondetal2006). fromemissionlinedata.AdaptedfromNattaetal2006. Clarke&Pringle(2006)however suggestedthattheclaimed correlation between M˙ and M∗ could be understood as follows. 2 I. Tilling,C. J.Clarke, J. E.Pringle& C. A. Tout They noted that the distribution of datapoints in the plane of L∗ The pre-mainsequence stars in which we are interested, versus L (Figure1) more or less fillsaregion that isbounded, ClassII(theClassicalTTauristars),allhavethefollowingproper- acc athigh Lacc bytheconditionk = Lacc/L∗ ∼ 1and,atlow Lacc,by ties: observationaldetectionthresholds,whichroughlyfollowarelation (i)Theyhavepassedthroughthemajorphaseofmassaccre- oftheformL ∝L1.6.Theynotedthatwhenthisrelationbetween tionwhichoccursduringtheembeddedstatesClass0andClassI. acc ∗ Lacc and L∗ iscombinedwithspecificassumptionsabouttherela- Thisimpliestypicallythattheyhavebeenevolvingforsometime tionshipbetweenL∗and M∗,andbetweenL∗andstellarradiusR∗ with their own gravitational energy as the major energy source. (basedonplacingthestarsonpre-mainsequencetracksatapartic- Deuteriumburninginterruptsthisbrieflybutonlydelayscontrac- ularage,ornarrowspreadofages)thentheresultingrelationship tionbyafactorof2orso.Thusforthesestarswemayexpectthat between M˙ and M∗ is indeed M˙ ∝ M∗2. They thus claimed that theyhaveanageT ≈tKHroughly. itiscurrentlyimpossibletorejectthepossibilitythattheclaimed, (ii)Themassesoftheiraccretiondiscsarenowsmall,M < disc steeperthanlinear,relationshipisanartefactofdetectionbiases. M∗.Further,accretiondiscmodelsatsuchlatestages–i.e.theso Clarke&Pringle(2006)alsopointedoutthatthedistribution calledasymptoticstagewhenmostofthemassthatwasoriginally of detections in the Lacc – L∗ plane may nevertheless be used to containedinthemhasbeenaccretedontothecentralobject–ex- tell us something about conventional accretion disc evolutionary hibitapowerlawdeclineinM˙ withtimeandthushavethegeneric models.Inthispaperweexploretheextenttowhichthismightbe propertythattheageofthediscisaboutt .Ifmostofthestar’s disc achieved. The first question which needs to be addressed is why lifetimehasbeenspentaccretinginthisasymptoticregime,wemay thereisaspreadofvaluesofLaccatagivenvalueofL∗.Secondly roughlyequatetheageofthediscandtheageofthestarandwrite we need to understand why the upper locus of detections corre- T ≈t . disc spondstoLacc ∼ L∗,sincethereseemsnoobviousreasonwhythis Giventhis,weseeimmediatelythat,forthesestars,weexpect shouldcorrespondtoadetectionthresholdorselectioneffect. tM = M∗/M˙ >Mdisc/M˙ =tdisc≈tKHandthereforethatLacc<L∗. We begin here (Section 2) by setting out a simple argument Inthefollowingsectionwetestthisargument byevolvinga why – as a generic property of accretion from a disc with mass suiteofmodelstar-discsystemsintheL∗−Laccplane. muchlessthanthestellarmass–onewouldexpectthedistribution tobeboundedbytheconditionLacc ∼ L∗.InSection3weusethe ideasofHartmann,Cassen&Kenyon(1997)todevelopanapprox- imateevolutionarycodeforaccretingpre-mainsequence starsand 3 STELLAREVOLUTIONCALCULATIONS demonstrateitsapplicabilitybycomparisonwiththetracksofpre- mainsequencestarscomputedbyTout,Livio&Bonnell(1999).In Ratherthanattemptingtocarryoutfullstellarevolutioncomputa- Section 4 we apply the code to pre-mainsequence stars accreting tionsofaccreting andevolving pre-mainsequence stars(e.g.Tout fromanaccretiondiscwithadecliningaccretionrateandinvesti- etal. 1999), wehere adopt the simplifiedapproach of Hartmann, gatetherangeof discparameters required toobtain theobserved Cassen&Kenyon(1997). rangeofLacc/L∗.WedrawourconclusionsinSection5. WebeginwithastellarcoreofmassMi =0.1M⊙ andradius Ri =3R⊙.WeprescribetheaccretionrateM˙(t)andthusthestellar massasafunctionoftime t 2 THEMAXIMUMRATIOOFACCRETION M∗(t)=Mi+ M˙(t)dt. (5) Z 0 LUMINOSITYTOSTELLARLUMINOSITY Hartmannetal.(1997)writetheenergyequationas Fromatheoreticalpointofviewtherearethreerelevanttimescales inthecaseofstargainingmassbyaccretionfromadisc.Thefirst 3GM2 1 M˙ R˙ is L∗=−7 R∗3M∗ + R+LD. (6) tM =M∗/M˙, (1) Inderivingthisequationtheyassumethatthestarisalwaysinhy- drostaticbalanceand,beingfullyconvective,canbetreatedasan whichisthecurrenttimescaleonwhichthestellarmassisincreas- n = 3/2 polytrope. Account is taken of the conversion of grav- ing.Thesecondis itationalenergyintothermalenergyvia pdV workaswellenergy t =M /M˙, (2) inputfromdeuteriumburningLDandenergylossviaradiativecool- disc disc ingatthephotosphereL∗.Intheformwrittenabove,itisassumed whichisthecurrenttimescaleonwhichthediscmassisdecreasing. thatthematerialaccretingontothestararriveswithzerothermal ThethirdistheKelvin-Helmholtztimescale energy. Thisisprobably agood approximation tothecase where material enters the star via a radiative magnetospheric shock or GM2 t = ∗, (3) from a disc boundary layer. The rate of energy input to the star KH R∗L∗ deliveredbydeuteriumburningisparameterised(followingtheex- whichisthetimescaleonwhichthestarcanradiateitsthermalen- pressiongivenbyStahler1988forann=3/2polytrope)by ergy.Itisalsothetimescaleonwhichastarcancomeintothermal setqauriliisbrmiuamin.lyOndutheetopriet-smcaoinntrsaecqtuioenncuen,dwehregnrathveitylu,mtinoissitaylsooftthhee LD/L⊙=1.92×1017XfXD MM⊙∗!13.8 RR⊙!−14.8, (7) KH evolutionarytimescale. where X isthemassfractionofhydrogenand f isthefractionof Weseeimmediately(sinceLacc∼GM∗M˙/R)that deuterium remaining in the star relative to the accreted material. L t Weassume that the accreted material has the same abundance as acc = KH. (4) L∗ tM the star before deuterium begins burning and we take this initial The ratioof L to L in TTauriStars 3 acc ∗ 0.5 parameterisationofthephotosphericconditiongivenby(9)).The resultsareillustratedinFig.2,foraccretionrates M˙ = 10−6,10−7 and10−8M⊙yr−1.Oneachtrack(wherepossible)weplotthepoint at which the stellar mass is M∗/M⊙ = 0.2,0.5 and 1.0 together 0 withthecorresponding points onthetracks. Bothsetsof compu- tationshadthesameinitialvaluesof M∗ = 0.1M⊙ andR = 3R⊙. ThestarsfirstdescendaHayashitrack,whilebeingdrivenacross tohigher effectivetemperaturebytheaccreting matter.Whenthe -0.5 * coretemperaturereachesabout106Ktheyevolveupthedeuterium- L burning main sequence. Once deuterium fuel is exhausted they g o resume their descent towards the zero-age main sequence, leav- l -1 ing their Hayashi tracks when the stellar luminosity drops to the pointthattheKelvintimescalebecomesgreaterthantheaccretion timescale.Atthispoint,accretionisagainimportantandthetracks arethendriventohighereffectivetemperatures.Uptothesepoints -1.5 oursimplifiedmodelappearstobeinreasonablygoodagreement withToutetal(1999)’smodels,evenatearlytimesandlowmasses, whentheequationsareformallyincorrect.Allthepointsareaccu- -2 ratetowithinafactoroftwoinluminosityandtowithin10percent 3.7 3.65 3.6 3.55 3.5 3.45 3.4 log T ineffectivetemperature.Weconcludethatthissimplifiedmodelis eff adequateforthetaskinhand. Figure2.Accretingpre-mainsequence trackscomputedusingthesimpli- fiedevolutionaryscheme,with fmin=0.001.Trackscorrespondtoconstant accretionratesof(fromtoptobottom)10−6,10−7 and10−8M⊙yr−1.Ini- 4 APPLICATIONTOTTAURISTARS tiallythemassis0.1M⊙andtheradiusis3R⊙.Thetracksareterminated whentheapproximationsceasetobevalid(becausearadiativecorehasde- In view of our ignorance about formation history of pre- velopedorhydrogenburningbeginsinthecore)oratamaximumageof mainsequencestarsintermsoftheiraccretionhistory,weneedto 20Myr.Pointsareplottedalongthetracksatmassesof0.1,0.2,0.5and make some assumptions about theform of M˙(t). Indoing so, we 1.0M⊙uptothemaximummassallowedbyeachtrack.Thecorresponding areguidedbysomesimplephysicalideas.Itneedstobebornein pointsfromtheToutetal.(1999)tracksarejoinedtothesebydottedlines. mind,however, thatalthough our simpleassumptions giveriseto resultswhichprovideareasonabledescriptionofthedata,thisdoes massfractiontobeX = 3.5×10−5 asdidToutetal.(1999).We notimplythattheassumptionsarenecessarilycorrectnorthatthey D fixthehydrogenmassfractionatX=0.7.1 aretheonlyoneswhichmightwork. Thevariationof f isgivenby(equivalenttoStahler1988). Weassumethatmostofthemasswhichhasaccretedontoa TTauristarhasbeenprocessedthroughanaccretiondisc.Thisas- df M˙ L = 1− f − D , (8) sumption,coupledwiththeobservationalfactthatprotostellardiscs dt M∗ QDXDM˙ ! areatmost10–20percentofthestellarmassandusuallymuch whereQ =2.63×1018ergg−1istheenergyavailablefromfusion less,tellsusthattheevolutionoftheaccretiondischasprogressed D of deuterium. Because of timestep problems we set f = 0 once intoitslate, asymptotic phase. Lynden-Bell &Pringle(1974; see the value of f falls below some small value f . Except where also Hartmann et al. 1988) present similaritysolutions for accre- min explicitlymentioned,wetake f =10−6. tiondiscevolutionusingthesimpleassumptionthatthekinematic min Forafullyconvectivestar,thestellarentropyiscontrolledby viscosityvariesasapower lawof radius. Theyfindthat forsuch the outer (photospheric) boundary condition which takes account discs,thediscaccretionratevariesatlatetimesasapowerlawin ofthestrongtemperaturesensitivityofthesurfaceopacityforstars timewith M˙ ∝ t−η, for some η > 1. Thisis true more generally on the Hayashi track. Hartmann et al. (1997) adopt a fit to the fordiscsinwhichtheviscosityalsovariesaspowersoflocaldisc (non-accreting) stellar evolutionary tracks of D’Antona & Mazz- properties, such as surface density (Pringle 1991). Here we take itelli(1994)intheform η = 1.5, corresponding to kinematicviscosity being proportional toradius(Hartmannetal.1988). 0.9 2.34 TL∗h/eLy⊙st=ress0.tM5hMa∗t⊙thisfi2tRRis⊙only.approximateandisadequateonly(9in) cto=nsitWd0eearfatfesorsuutrmheseeottsnhsaoetftTmofToafdouerrlmsi saitntaiortsneroamnnlsdyotbfaekctehoemt0de=isvci5smib×ales1s0a5tfryaarc.ttiWimonee f = M /M attimet ,where M isthefinalstellarmass.We themassrange0.36 M∗/M⊙ 61. cdoinscsider dfisc =f 0.2,0.1,00.037 and 0f.014. In order to stay within Inordertoprovidesomeestimateoftheaccuracyofthissim- disc thecredibilitylimitsofourevolutionaryschemeweconsiderfinal plified evolutionary scheme, we compared the results against the fullstellarevolutioncalculationsofToutetal.(1999),forthecase massesonlyintherange0.46 Mf/M⊙61.0. We also need to define the stellar accretion rate from time ofconstantaccretionrate(havingfirstsatisfiedourselvesthatstars intheappropriatemassrangeinToutetalareindeedwellfitbythe t = 0. At t = 0 we start with M∗ = Mi = 0.1M⊙. In principle, theevolutionofthestarfort > t candependcriticallyontheac- 0 cretionratepriortothispoint.Thisissimplybecausetheevolution 1 Our notation differs from that of Stahler (1988) and Hartmann et al. timescaleforapre-mainsequencestarisalwayscomparabletoits (1997)becauseweprefernottousetheirnon-standarddefinitionof[D/H]. age,sothatithasneverhadtimetoforgetitsinitialconditions.In WerecovertheirβD≡QDXDinequation(8)withXD=3.5×10−5. ordertominimisethiseffect,weassumethattheaccretionratedoes 4 I. Tilling,C. J.Clarke, J. E.Pringle& C. A. Tout thevalues oft andof M˙ corresponding totheseparameters. We s i 1 notethatinallourmodelst ≪ t ;thus,bythetimethatthestar s 0 isdeemedtobeaClassicalTTauristar,theaccretionratehasbeen decliningasapowerlawoverthemajorityofthesystem’slifetime. FromFig.3weseethat our data pointsmoreor lessfillthe 0 range10−2 < Lacc/L∗ < 1intherange ofvaluesof L∗ whichare accessibletoourcomputations.Itisreasonabletoexpectthatwith a more detailed evolutionary scheme, valid at lower masses, this -1 trendwouldcontinuetolowervaluesofL∗.Thusthesesimpleas- sumptions, coupled with this simplified evolutionary scheme are capable ofproducing thetypical observed spread of pointsinthe Lacc−L∗diagram(seeFig. 1). -2 The stars in Fig. 3 are evolving at essentially constant mass because, by assumption, Mdisc ≪ M∗ and have already con- sumedalloftheirinitialdeuterium.Atconstantmass,Equation9 -3 gives L∗ ∝ R2.34. For a star evolving on the Kelvin-Helmholtz timescale this implies R∗ ∝ t−1/3.34 and thus that L∗ ∝ t−0.7. Since Lacc = GM∗M˙/R∗ andwehave assumed M˙ ∝ t−η wehave L ∝t−(η−1/3.34).Fromthiswededucethatastimevaries,wehave acc -4 approximately -1.5 -1 -0.5 0 0.5 L ∝L(η−0.3)/0.7. (12) acc ∗ Forη=1.5thisisL ∝L1.7.ThustheslopeofthetracksinFig.3 Figure3.Accretion luminosity Lacc against stellar luminosity L∗ forthe acc ∗ isaresultoftheratiooftheratesatwhichthestarandtheaccretion variousrunswithparametersgiveninTable1.Thefilledsymbolsleadto finalmassMf =1.0M⊙andtheopensymbolstoMf =0.4M⊙.Eachlineof discevolve.Owingtothephotosphericboundaryconditionthisra- symbolsrepresentsoneevolutionarysequence,startingattoprightattime tiojustdependsonη. t0=5×105yrandfinishingbottomleftattimet=107yr,withonesymbol It might be thought suggestive that the relation Lacc ∝ L∗!.7 plottedevery5×105yr.Foreachmass,thediscmassfractionsattimet0 (which holds in the case η = 1.5, at least over the range of L∗ are fdisc = 0.2(uppermosttrack)= 0.1,= 0.037and= 0.014(lowermost for which (9) is valid), is very close to the mean relationship in track). The dashed diagonal lines represent Lacc/L⊙ = 1.0,0.1 and 0.01 the observational data. It is then tempting to see this as observa- respectively. tionalsupportforaviscositylawwiththisvalueofthepowerlaw index.However,westressthatwhenweexperimentedwithadif- notincreasewithtime.Thisisprobablyreasonableforstarswhich ferentviscositylaw,whichchangedtheslopeoftheindividualstel- forminisolation(e.g.Shu,Adam&Lizano1987),butmaynotbe lartrajectoriesaccordingtoequation(12),thedistributionofstars validformoredynamicalmodesofstarformation,whenlateinter- inthe L∗ −Lacc plane was indistinguishable, filling, asit did, the actionandmergingmaybenormal(e.g.Bate,Bonnell&Bromm availableparameterspacebetweenLacc∼L∗andobservationalde- 2002). To be specific we assume that the accretion rate is of the tectionthresholds. form We note that at fixed L∗ there is a trend for lower values of L tocorrespondtolowervaluesof M ,thediscmassfraction M˙ =M˙ =const., 06t6t (10) acc disc i s atthetimewhenthestarisdeemedtoappearasaclassicalTTauri andthat star. Thus in this picture the spread of values of Lacc/L∗ at fixed t −η L∗comesaboutbecauseofthespreadofdiscmassesatlatetimes. M˙ =M˙ , t 6t, (11) Suchaspreadcouldbeduetoanumberofreasonssuchasarange i t ! s s ofinitialdiscsizes(i.e.initialangularmomenta)oravarietyofdisc wheret andM˙ arechosentoensurethattheintegralofM˙ between truncationscausedbydynamicalinteractions. s i zeroandinfinityis Mf −Mi andthattheintegralof M˙ fromt0 to ThereisalsoatrendforlowervaluesofstellarmassM∗tocor- infinityis fdiscM∗(seeTable1).Wenotethat,sincet0≫ts,thedisc respondtolowervaluesofL∗.Itisevident,however,thatthetracks isalwaysintheasymptoticregimeatthestagethatitisclassifiedas in this diagram intermingle to some extent. At the times plotted aTTauristar.Westressthattheformoftheevolutionaryprescrip- Mdisc ≪ M∗,soeachtrackcorrespondsessentiallytoafixedstel- tionthatwehaveadoptedforM˙(t)priortot0isnottheonlyonethat larmass.ThusitisclearthatatagivenvalueofL∗therecanbea producesagivendiscandstellarmassatt andthat-dependingon spreadofstellarmassesdependingontheassumedstellarage. 0 thetemporalhistoryandlocationofmatterinfallontothedisc-a varietyofpriorhistories,withthesameintegratedmass,arepossi- ble.Ourintentionhereishowevernottoexploreallpossibilitiesbut 5 CONCLUSIONS toadoptaplausibleandeasilyadjustableprescriptionwithwhich onecaninvestigatethesystem’sevolutionintheLacc−L∗plane. Wehaveshown,usingsimplifiedstellarevolutioncalculationsfor InFig.3weshowtheresultsofthesecomputationsshowing starssubjecttoatimedependentaccretionhistory,thataplausible the accretion luminosity (Lacc) as a function of stellar luminosity morphologyoftheL∗−LaccplaneforTTauristarshasthefollowing (L∗).WeplotresultsfortwofinalmassesMf/M⊙=1.0and0.4and features, forthefourinitialdiscmassfractions fdisc =0.014,0.037,0.1and (i)anupperlocuscorrespondingtoL∗≈Laccand 0.2.Weplotthepositionsstartingattimet = 5×105yrandthen (ii)diagonaltracksforfallingaccretionratesasstarsdescend atintervalsof5×105yruntiltimet = 107yr.InTable1wegive Hayashitracks. The ratioof L to L in TTauriStars 5 acc ∗ Table1.TheparametersfortherunsshowninFig.3.Mfisthefinalstellar mass. fdisc isthefractionofMf whichisstillinthediscattimet0 = 5× 105yr.TheaccretionratetakesonaconstantvalueofM˙iuntiltimetsand thereafterdeclinesas∝t−ηwithη=1.5 Mf/M⊙ fdisc ts/yr M˙i/M⊙yr−1 1.0 0.2 5.5×104 5.4×10−6 1.0 0.1 1.4×104 2.2×10−5 1.0 0.037 1.9×103 1.5×10−4 1.0 0.014 2.8×102 1.1×10−3 0.4 0.2 8.0×104 1.2×10−6 0.4 0.1 2.0×104 5.0×10−6 0.4 0.037 2.8×103 3.6×10−5 0.4 0.014 4.0×102 2.5×10−4 Wehavearguedthat(i)isagenericfeatureofstarswhichare Finally,wenotethatwehavechosentorelateourstudytoob- accretingfromadiscreservoirwithmasslessthanthecentralstar servational data in the Lacc − L∗ plane because the ratio of these coupledwiththeassumptionthatthemassdepletiontimescaleof quantitiesisstraightforwardlyderivablefromemissionlineequiv- thedisciscomparabletotheageofthestar.2.Thefirstisnecessary alent widths. Thus this comparison is more direct than the alter- forstarstobeclassifiedasClassicalTTauristarswhilethesecond native, i.e. the comparison with observational data that has been resultsfromthegradualdecayoftheaccretionrate. transformedintotheplaneofM˙ versusM∗viatheuseofisochrone Weshowthattheslopeofofthediagonaltracks(ii)isrelated fittingandtheapplicationoftheoreticalmass-radiusrelations.Itis tothe index of thepower law decline of accretion ratewith time neverthelessworthnotingthattheupperlocusof Lacc ∼ L∗ corre- at latetimes. This itself depends on the scaling of viscosity with spondstoalinewith M˙ ∝ M∗.Althoughitisoftentakenasself- radiusinthedisc.Forreference, ifν ∝ R,then L ∝ L1.7 along evidentthatthisistheexpectedratio(inthecasethatthemaximum acc ∗ suchtracks,whereasifν∝R1.5thenL ∝L2.4. initial ratio of disc mass to stellar mass is independent of stellar acc ∗ Wehoweverarguethat,aslongasobservationaldatapointsap- mass),thisactuallyonlycorrespondstoM˙ ∝M∗inthecasethatthe peartofillthewholeregionoftheL∗−Lacc planethatisbounded disc’sinitialviscoustimeisindependentofstellarmass.Giventhe bytheupperlimit(i)andluminositydependentsensitivitythresh- possiblevarietyofviscosityvaluesandradiiofprotostellardiscsit olds,itwillremainimpossibletodeducewhatviscositylaworwhat isnotatallevidentthattheseshouldconspiretogethertogivethe dependenceofdiscparametersonstellarmasswillberequiredin sameviscoustimescale–until,thatis,thatonerealisesthatalldiscs ordertopopulatethisregion.Thisisbecauseoftheseveredegen- intheasymptoticregimehaveviscoustimescalewhichisequalto eracies that exist between stellar mass, age and accretion history thediscage.Theconclusionthattheupperlocusshouldfollowthe (ourresultsshowthatinordertopopulatethedesiredregion,itis relationM˙ ∝ M∗isthusaverygeneralone. necessary that there is, at the least, a spread in both stellar mass Insummary,weconclude,asinthestudyofClarke&Pringle and initial disc properties). It is not, however, necessary to posit (2006), that the observed population of the Lacc − L∗ plane, and anyparticularscalingofdiscpropertieswithstellarmassinorder theconsequentrelationshipbetween M˙ and M∗,isstronglydriven to populate the plane, in contrast to the hypotheses of Alexander byluminositydependentsensitivitythresholdsandanupperlocus &Armitage2006 andDullemondetal2006. Likewise,although, at Lacc ≈ L∗.Themainnew ingredient injected byour numerical asnotedabove,thetrajectoriesfollowedbyindividualstarsinthis calculationsandplausibilityargumentsisthatwenowunderstand plane are a sensitive diagnostic of the disc viscosity law, this in- thatthisobservedupperlocuscanbeunderstoodasaconsequence formationiswashedoutinthecaseofanensembleofstarswhich ofplausibleaccretionhistories,giventhecurrentpropertiesofthe simplyfillthisplane. relevantpre-mainsequencestars.Inaddition,ourcomputationsin- Wethereforeconcludethatsuchdatacouldonlyyielduseful dicatetherangeofinitialdiscpropertiesthatarerequiredinorderto insightsifregionsoftheplaneturnedouttobeunpopulated(orvery populatetheregionofparameterspaceoccupiedbyobservational sparselypopulated).ThereisashadowofasuggestioninFigure1 datapoints:weareabletoreproducetheobservedrangeofvaluesof thatatlowL∗thedatafallsfurtherbelowthelocusLacc=L∗thanat Lacc/L∗atagivenvalueofL∗byassumingoncetheTTauriphase higherluminositiesalthough,giventhesamplesize,thisdifference hasbeenreached,thediscmassfractionslieroughlyintherange is not statistically significant (Clarke & Pringle 2006). Any such 0.016 Mdisc/M∗60.2,independentofstellarmass. edge in the distribution could in principle re-open the possibility ofconstrainingdiscmodels. Wehowever emphasise thatthislow luminosity regime canin any case not be explored by thesimple ACKNOWLEDGMENTS evolutionarymodelsthatweemployhere,andthat–ifitbecomes necessarytoinvestigatethisregime–itwillbeessentialtousefull CATthanksChurchillCollegeforhisFellowship.Weacknowledge stellarevolutioncalculations. usefuldiscussionswithLeeHartmannandKeesDullemond. 2 Wemayinfactarguethattheexistenceofsuchanupperlimitinthedata REFERENCES isagoodsignatureoftemporallydecliningaccretionthroughadisc,sinceif ClassicalTTauristarsinsteadaccretedexternalgasataconstantrate(e.g. Alexander,R.D.,Armitage,P.J.,2006.ApJL639,83 Padoanetal2005),nosuchfeaturewouldbeapparentinthedata BateM.R.,BonnellI.A.,BrommV.,2002,MNRAS,336,705 6 I. Tilling,C. J.Clarke, J. E.Pringle& C. A. 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