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1991Apj. . .377. .318B the Astrophysical Journal, 377:318-329

1991ApJ. . .377. .318B (1987) thatstarsof~1-2.5Mmaylosesignificantamounts ments. Thisisincontrastto standardsolarmodels,which (R) atthesolarage.Theirmasslossexposedlayerswhere the could reproducethepresentsolarluminosity(L)andradius might alsodrivemassloss,perhapsaidedbytheeffects of note thatthepulsatingôScutistarslieinthisrange.Earlier, main sequence,atspectraltypesfromearlyAtomid-F; they during post-main-sequenceevolution(Willson&Bowen they arguedthatpulsationmaydrivemostmass-lossepisodes Cepheid andRRLyraeinstabilitystripwouldintersect the that itisjustinthismassrangetheextensionof the of massduringearly-main-sequenceevolution.Theypointout predict nodepletionduringmain-sequence evolution.Onthe presolar lithiumandberyllium hadbeencompletelydestroyed: relatively rapidrotation.Guzik,Willson,&Brunish(1987) their modelspredictedanextreme overdepletionoftheseele- solar model,withinitialmassM*=2M,andfoundthat they applied suchmasslosstothemain-sequenceevolution of a other hand,thesolarobservations indicateadepletionby The AstrophysicalJournal,377:318-329,1991August10 1984). Theypointoutthatpulsationonthemainsequence © 1991.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. 0 0 0 0 It hasbeensuggestedbyWillson,Bowen,&Struck-Marcell 7 7 9 6 6 79 10_ 79 convection. WehaveconsideredinitialmassesofMi=1.8,1.5,1.2,and1.1M©,eachwithmass-losstime entire evolution,keepingcarefulaccountofthechangesinsurfacelithiumabundanceduetomasslossand lowed thenucleardestruction(andproduction)ofLithroughoutinteriorstructurestarduring from Lidepletion). mass-loss timescalesr=0.985,0.5,and0.16Gyr,respectively(amuchweakerconstraintthanthatobtained exceed theprimordialabundance);thisconstrainsmasslossAMtobelessthan0.7,1.0,and1.5-2Mfor presolar heliumabundance,thesolarmetallicity,mixing-lengthparameter,andpredictedneu- of themass-losstimescale.WiththismasslossonlyAM«0.1M,thereareminorchangesin scales ofz=0.16,0.5,and0.985Gyr. solar surface.Main-sequencemasslossreducestherequiredpresolarheliumabundance(whichmustalways layers whereBehasbeenburned,resultinginlittlesurfacedepletion,agreementwithsolarobserva- trino capturerates.Weestimatethatthismasslosswouldcausetheconvectiveenvelopetojustreach observed surfacelithiumdepletionofafactor100.Furthermore,thisisalmostindependent initial declineduringthemass-lossperiod(ratherthanasteadybrightening)butwithsmallertotalvariation. losing cases. 4 x10K,respectively.Thedifferenceisduetothedeclineofthesetemperaturescausedbymassloss.We 2.5 x10and3.3K,respectively,butrequireshigherinitialtemperaturesofapproximately3 tions. ThehigherinitialmassesareruledoutunlessthereissomemechanismtoproduceLi(andBe)atthe would havehadasignificanteffectonthesolarplanetarysystem,whichshouldbeexploredfurther. This, togetherwiththelargerinitialgravityandsolarwind[^=(1-7)x10“Myr], have developedasimpleprescriptiontoestimatetheseinitialtemperaturesforLiandBeburninginmass- Subject headings:nucleosynthesis—:abundancesinterior m 0 G m 0 79 OUR SUN.II.EARLYMASSLOSSOF0.1MANDTHECASEMISSINGLITHIUM 0 We havecomputeddetailedsolarmodels,takingearly-main-sequencemasslossintoaccount.fol- A solarmodelofinitialmass«1.1Mcansolvethecasemissinglithium,producing The luminosityhistoryofourM,=1.1Mcaseisdifferentfromthatthestandardsolarmodel,withan For mass-losingcases,LiandBeburningdoesnottakeplaceatusuallyassumedtemperaturesof © American Astronomical Society • Provided by the NASA Astrophysics Data System 0 0 Canadian InstituteforTheoreticalAstrophysics,UniversityofToronto,60St.GeorgeStreet,Ontario,CanadaM5S1A1 1. INTRODUCTION W. K.KelloggRadiationLaboratory106-38,CaliforniaInstituteofTechnology,Pasadena,CA91125 I.-Juliana SackmannandWilliamA.Fowler Received 1990April3;accepted1991February13 Arnold I.Boothroyd ABSTRACT AND 318 7 7 79 7 example) withtheobservationsofHyadesstars(age0.6Gyr) of overshooting belowtheformal boundaryofenvelopeconvec- enhanced bysome“nonstandard ”assumptions,suchasstrong fairly largescatterinlithiumabundancecompared (for Cayrel etal.(1984). quite wellwithlithiumabundanceobservationsofPleiades main-sequence Lidepletion.Thesestandardmodelsagree licity (Proffitt&Michaud1989;Swenson,Stringfellow, & factor ofabout2forstandardmodelssolarmassandmetal- tion (D’Antona&Mazzitelli 1984;Kiziloglu&Eryurt-Ezer Duncan &Jones(1983)atloweffectivetemperatureshave a (age<0.1Gyr),althoughtherelevantobservations of fairly large,standardmodelsdoexhibitsomeLidepletion (Boesgaard &Steigman1985). factor ofabout100forLiandlittleornodepletionBe Faulkner 1990);highermassstarsencounterevenless pre- However, thispre-main-sequenceLidepletionisonly by a track (Bodenheimer1965),whichshouldbetakenintoaccount. during pre-main-sequencecontractiondowntheHayashi 1985). Swensonetal.(1990) showthatanotunreasonable 7 Pre-main-sequence Lidepletion couldbesignificantly Swenson (1990)pointedoutthat,unlesstheinitialmassis 1991ApJ. . .377. .318B 7 7 7 7 7 7 7 7 7 7 7 enhancement inthecontributionofmetalstointerior depletion ofHyadesstarswithoutrequiringanyfurthermain- opacities canresultingreatlyincreasedpre-main-sequence current Sunafterchangingtheiropacities,butSwenson(1990) abundance Yandmixing-lengthparameteraagaintothe sequence Lidepletion.Theydidnot“recalibrate”thehelium (1990) canbeusedtoestimatetheeffectofchanginginte- notes thatevenalargechangeintheinterioropacitiesresults model whoseeffectivetemperatureisconstrained:their40% only averysmallchangeina.ThusFigure2ofSwensonetal. Li depletion,perhapsevenallowingonetomatchthe more thanafactorof10increaseintheamountLideple- (which theyrequiretomatchtheHyades)resultsinslightly increase inthecontributionofmetalstointerioropacities rior opacitiesofthepre-main-sequenceLidepletiona model. Lithiumabundanceobservationsofstarsinanumber factor of2Lidepletionastandardpre-main-sequencesolar model, whilePleiadesobservationsareconsistentwiththe “nonstandard” pre-main-sequenceLidepletionofasolar tion forsuchamodel.ThusHyadesobservationssuggestan absolute upperlimitofafactorabout20evenfor least uptoanageof~5Gyr.Wewillassumethatpre-main- increase (forfixedstellarmass)withincreasingclusterage,at that mostLidepletiontakesplaceonthemainsequencefor of openclusters(see,e.g.,Hobbs&Pilachowski1988)suggest mass lossofAM=1.0Mresultsintotaldepletion.Itwasour loss (astandardsolarmodel)resultsinnodepletion,whilea produce anyamountofmain-sequenceLidepletion:nomass stars of1Mandabove:theamountdepletiontendsto intent toexploretheconsequencesofdifferentamountsmass sequence solarLidepletionwasrelativelyslight. mine justhowmuchmasslosswasnecessarytoaccountforthe loss, andofdifferentmass-losstimescales.Wewishedtodeter- circulation, rotation-inducedturbulentmixing,convective . assume thatnoothermechanism(suchasdiffusion,meridional observed solarLidepletion.Throughoutthiswork,we overshoot, orvariablemixinglength)actstodepletesurface mass lossofAM=1.0Mrequiredapresolarheliumabun- dance Ylowerby0.04thantheirstandardmodel.Wewished to explorefurtherthisreductioninthepresolarheliumabun- 0 0 mass loss,determinedbytherequirementthatpresolar dance. Specifically,wewishedtoobtainanupperlimitonthe mass lossfromthatobtainedtheLidepletion.Wehave bang value.Thisisacompletelyindependentconstrainton the contentmustalwaysbelargerthantheprimordial big 0.239 +0.015,asobtainedbyobservationsofGalactic and taken theprimordialbigbangheliumabundancetobe = extragalactic Hnregions(Boesgaard&Steigman1985). Galactic chemicalevolutionwouldincreasetheexpected pre- ment as11.0M,(1) Our opacitieswerethoseofLAOL,whichincludedtheopac- 0 m =\r m 2.1. MassLossandSurfaceLithiumDepletion t,Zm Í0.99 M+{Mi-0.99)e~ [l.O Motherwise, qQ 0 © American Astronomical Society • Provided by the NASA Astrophysics Data System k f A*,-O.99M 0 U d{U) _Li- e=b dM M ce otherwise . M(t) >1.0Mq, BOOTHROYD, SACKMANN,&FOWLER (2) fc)fc1) 74 7 9 3 34 4 1 9 121316 10 31 171618 farther in,particularlyifthemasslayersarecloseenough is itevennecessarilyconstant:thestrongtemperaturedepen- where convective envelopeisconstantoveratimestep.Underthe ent. Itisalsoagoodapproximationtosaythatthemassof values Li£atmasslayerMandLi|,~_ dence oftheLi(p,

J No. 1, 1991 OUR SUN. II. 323 r"

Time t (Gyr) Time t (Gyr) Fig. 2a Fig. 2b

standard (constant-mass) case. Thus the = 1.1 M0 cases encounter less luminosity change between the ZAMS and the present than does the standard case, particularly for long mass- loss time scales. This different history of the luminosity and gravitational attraction of the Sun, together with the much larger , could have a significant impact on the . 3.3. The Presolar Helium Abundance Figure 3 illustrates how mass loss reduces the required pre- solar helium content Y. For the short mass-loss time scale (rm = 0.16 Gyr), the presolar value of Y remains nearly con- stant for different values of the initial mass M¿. However, for the longer mass-loss time scales, there is a large reduction in the presolar Y for the larger initial masses: for = 1.8 M0 with im = 0.5 Gyr, the presolar Y is reduced by 0.03, namely, to 7 = 0.25 (from Y = 0.28 for the ); for M = Time t (Gyr) t 1.8 Mq with Tm = 0.985 Gyr (our longest time scale case), Fig. 2c the presolar Y is reduced by nearly 0.06, to 7 = 0.224. Fig. 2.—Luminosity history on the main sequence for mass-losing cases The reason for this behavior of 7 as a function of Mj- and im and the standard Sun. is clear. If the mass-loss time scale tm is short, even a large Figures lb, 1c, and Id show an expanded view of the approach in the H-R diagram of the mass-losing cases to the 1.0 M0 case, for each of the mass-loss time scales Tm. Let us ignore the pre-main-sequence evolution. The mass-losing cases move down the main sequence until mass loss termi- nates; at this point they intercept the position in the H-R diagram of the 1.0 M0 (constant-mass) case at the correspond- ing time. For a short mass-loss time scale, the mass-losing cases intercept the 1.0 M0 track earlier in its evolution than for a long mass-loss time scale. After mass loss terminates, all models follow the same path toward the present Sun. 3.2. The Luminosity History Figures 2a, 2b, and 2c illustrate the history of the luminosity (up to the present) of our mass-losing solar models, as com- pared with the standard case. The cases with high initial masses naturally start out with initial very much larger than that of the present Sun, as much as a factor of 10 for the =1.8 Mq case; note, however, that the = 1.1 M0 case starts out with a luminosity only very slightly higher than Fig. 3.—Presolar helium abundance Y required for mass-losing solar that of the present Sun, or about 50% brighter than the initial models, of various initial masses and mass loss time scales.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1991ApJ. . .377. .318B interior abundanceprofileisstillverysimilartothatofthe 324 initial massisreducedveryquicklyto1.0M,andthe develop. Theseresultinrelativelylargechangesthestellar mass-loss timescaleislongandtheinitialmasslarge, cal evolutionwillproduceaheliumenrichmentAYprobably structure atthesolarage,requiringrelativelylargechangesin differences intheinteriorabundanceprofilehavetimeto remainsatahighermassforlongtime,andappreciable lower boundoneachofthesevalues,namely,Y=0.239 between 0.015and0.05,asdiscussedin§1.Eventakingthe the presolarheliumabundanceYinordertomatch standard (constant-mass)caseatthesameage.However,if namely, Yæ0.24.ThustheM=1.8casewithi0.985 observed solarluminosity. increase inaismuchlargerforlongmass-losstimescalest match thepresentsolareffectivetemperature;required extrapolation ofourresultsallowsustoestimatethatthis total amountofmain-sequencemassloss: below avalueof0.24allowsustoplaceanupperlimitonthe requirement thatthepresolarheliumabundanceYcannotfall Gyr, whichhasapresolarY=0.224,isnotallowed.The solar heliumcontentcanhardlyfallbelowthesumofthese, than forshortones.ForM*=1.8Mwiththelongesttime M, requirelargermixing-lengthparametersainorderto upper limittoliesbetween2.5and3M.In§3.7,weshow the standardsolarmodel.However,forM<1.2, scale (r=0.985Gyr),therequireda-valueisdoublethatof For theshortesttimescale(r=0.16Gyr),aconsiderable for muchthesamereason,exceptthataisdeterminedfrom is muchthesameasthatofpresolarheliumabundanceY, standard solarmodel(i.e.,within10%).Ifthemass-losstime required valueofaisonlyslightlydifferentfromthatthe tighter limitsonthemain-sequencemassloss. that thesurfacelithiumabundanceallowsustoplacemuch much furtherthanforthestandard case.Forthecase=1.8 high-Mj- caseswithlargethas centralhydrogenbeendepleted causes thebehaviorofYand adescribedabove.Onlyforthe solar ageillustratesthechangeinabundanceprofilewhich follows thebehaviorofmixing-lengthparametera (see matching thesolareffectivetemperatureatage. scale isshort(r=0.16Gyr),evenalargeinitialmassofM; case, M=1.8witht0.985Gyr,theconvective Table 1).Itremainsatapproximatelythesamevaluefor < left atthecenterasforstandard case. envelope massatthesolarageismorethandoublethatof the 0 Mq withT=0.985Gyr,there isonlyhalfasmuchhydrogen standard solarmodel. p 1.8 Mrequiresonlyasmall(10%)increaseina.Thisbehavior t0m m — 0.015=0.224andAY0.015,theminimumpossiblepre- + 0.015(Boesgaard&Steigman1985),whileGalacticchemi- 1.2 Mqorforr=0.16Gyr.However,themostextreme 0 0 t0 m m m m f0m m q m r) Note thattheprimordialheliumabundanceis1^,=0.239 Table 1demonstratesthatmodelswithlargerinitialmasses The valueofthecentralhydrogenabundanceA[°at The massoftheconvectiveenvelopeatsolarage L7Mf0r0985 Gr © American Astronomical Society • Provided by the NASA Astrophysics Data System M i5) U 3.5. TheCentralHydrogenDepletion ■' ~jioMforT=0.5Gyr. 0m 3.4. TheMixingLength BOOTHROYD, SACKMANN,&FOWLER 3-1 78198 3-1 71115 37 37 71115 71 7 capture rateislargerthanthatforthestandardcase,although capture rateswouldbereducedby0.6SNU(seePaperI). for themorerecentvalueofL=3.86x10ergss,these capture ratesforLi,Br,andMotargets(47,28,19 function ofMandr,asshowninTable1,issimilartothat rateswerecalculatedforL=3.90x10ergss; compared to8.3SNUforthestandardcase).Notethatthese this increaseissmallforthe=1.1Mcase(8.5-9.0SNU, marily toneutrinosfromthep-preaction,wouldbarely for targetssuchasGaandIn,whicharesensitivepri- the quantitiesdescribedinsectionsabove.Predicted respectively). increased onlybyfactorsof1.5and1.2,respectively,overtheir increase atall.Evenforthemostextremecase(M,=1.8M to thepredictedcapturerateforC1.However,rates The observedlowC1capturerateeliminatesanyhopeof escapethecenterofSunandthusbedetectable. values forthestandardmodel(whichare130and630SNU, and r=0.985Gyr),theGaIncaptureratesare resolving differentmass-losscasesvianeutrinoobservations. been majormassloss.However,thiswouldrequirethatall and theGaexperimentcouldhelpdecidewhethertherehas SNU, respectively,forthestandardcase)behaveverysimilarly below; thusnoneofthepredictedneutrinocapturerateswould be significantlydifferentfromthoseofthestandardmodel. an initialsolarmassmuchabove1.1M,asdescribedin§3.7 Furthermore, theobservedsolarlithiumabundanceprecludes lan &Fowler(1988),asdescribedin§2.Figure4illustratesthe account throughouttheinteriorstructureandtimeevolution profile ofLiforthecase=1.1Mwith0.16Gyr, of oursolarmodels,usingthenuclearreactionratesCaugh- near theZAMS(att=0.013Gyr),abouthalfwaythrough at threedifferenttimesduringthemain-sequenceevolution: 0 tm 0 0 the mass-lossepisode,andatsolar age. solar modelatthreedifferenttimes,namely, neartheZAMS,partwaythrough 0 m 0 0m 37 3 7 37 7 7 For allmass-losingcases,thepredictedC1neutrino The behaviorofthepredictedClneutrinocapturerateasa In principle,combiningtheresultsofC1experiment We havetakenLidestruction(andproduction)into Fig. 4.—ProfileoftheLi/Hnumber ratiointheinteriorofamass-losing 3.7. TheCaseoftheMissingLithium 3.7.1. LithiuminDetailedModels 3.6. SolarNeutrinos Vol. 377 ooPQ r" No. 1, 1991 OUR SUN. II. 325 oor- mass-loss phase (at i = 0.178 Gyr), and at the solar age (at observed meteoritic value of Li/H = 2.6 x 10" 9 (Boesgaard & ft i = 4.55 Gyr). One sees how mass loss eats beyond the steep Steigman 1985). This would shift the steep left-hand portion of decline of the lithium profile, and how the convective envelope the curves in Figure 5 upward by a factor of 2.6, and would acts as a reservoir for lithium. imply a somewhat larger mass loss of Figure 5 illustrates one of the key results of this paper, 7 0.128 Mq for T = 0.16 Gyr , namely, the surface Li abundance at the solar age as a func- m AM : 0.122 Mq for tion of initial mass Mf and mass-loss time scale Tm. One sees Tm = 0.5 Gyr , (7) that for a high initial mass (Mt = 1.8, 1.5, and even 1.2 M0), 0.116 Mq for Tm = 0.985 Gyr . 7 there is almost no Li left at the surface; on the other hand, an Note that Guzik et al. (1987) concluded that a solar model initial mass of Mf = 1.1 M0, i-e.9 a mass loss ofOA M0, repro- 7 9 7 with Mf = 2 Mq would result in total surface Li (and Be) duces almost exactly the observed Li abundance at the solar depletion; they did not use this to attempt to place limits on surface, namely, a main-sequence depletion by a factor of 100 9 main-sequence mass loss, but instead suggested that the from a ZAMS value of Li/H = 10“ . (For such models, Proffitt 7 9 7 observed solar abundances of Li and Be could perhaps be & Michaud 1989 predict a pre-main-sequence Li depletion fac- achieved via spallation reactions in solar flares. Schramm, tor of 1.5, for a total depletion factor of 150 from the presolar Steigman, & Dearborn (1990) point out two strong arguments 7Li abundance.) Another key result displayed in Figure 5 is 7 against this production mechanism. First, spallation reactions that the surface Li depletion is almost independent of the mass- would badly underproduce 7Li relative to 9Be. (The problem of loss time scale xm. To be precise, to match a main-sequence 6 7 7 spallation overproduction of Li relative to Li could possibly solar Li depletion factor of 100 requires mass loss of be avoided by 6Li burning at the base of the convective 7 9 io.112 M0 for Tm = 0.16 Gyr , envelope, but neither Li nor Be abundances would be AM ~ \ 0.106 M for T = 0.5 Gyr , (6) affected : the convective envelope does not reach down to high q m enough to burn either of these.) Second, the [0.100 Mq for T = 0.985 Gyr . m energy available for spallation in solar flares is totally insuffi- (These total mass-loss values were obtained by taking the cient to produce the required amount of7 Li, provided that the Mf =1.1 Mq cases and using the slope of Fig. 5 to estimate present rate of flare activity was typical of the Sun’s past also. how much further mass loss would be required to obtain a Guzik et al. (1987) argue that observations of X-ray fluxes of F- main-sequence depletion factor of 100.) The fact that the 7 Li and G-type stars of ages younger than ~ 1.0 Gyr show that in depletion is not a monotonie function of rm for the Mf = 1.2 such stars there would be enough energy to produce the neces- Mq cases (see Fig. 5) may be a numerical artifact, due perhaps sary amount of 7Li and 9Be over a time scale of ~ 1.0 Gyr (note to the slight inaccuracy of the match to the current Sun. that these stars would resemble the early Sun, for a mass-losing However, it may well be a real effect, since the greatest depth case). In any case, the first objection remains: the observed reached by envelope convection switches from being an solar 7Li/9Be ratio is far from what one would expect from increasing function of im to a decreasing function of im at this spallation reactions. initial mass, and Mce and MLi are also functions of xm : one of 3.7.2. The Lithium Depletion-Mass Loss Formula these working against the other could result in nonmonotonic 7 Using the same approach as Weymann & Sears (1965), behavior of the Li depletion as a function of Tm. We have used a number ratio of Li/H = 1.0 x 10“9 for the Hobbs et al. (1989) presented a formula for the amount of mass ZAMS 7Li abundance, corresponding to a presolar value of loss, AM, required to match any observed lithium depletion Li/H = 1.5 x 10-9. An upper limit for the main-sequence mass / = Lif/Lii, namely, loss may be obtained by taking the largest reasonable initial AM = (MU-Mce)-Mceln/, (8) 7Li abundance, i.e., a ZAMS 7Li abundance equal to the where Mce is the mass of the convective envelope and Mu is the depth in mass below the initial mass surface at which lithium is burned. Note that the above formula assumes that the convective envelope mass Mce remains constant while the mass loss is taking place, and that the lithium discontinuity at 7 MLi is sharp; the depletion factor /is that of main-sequence Li depletion. The simplest approach is to use values obtained from a standard solar model at the solar age. From Table 1, our standard solar model at the solar age has a convective ( } envelope mass of M c® = 0.017 M0, and lithium has burned to 2) half its initial abundance at a depth M[V = 0.029 M0; for a lithium depletion /= 1/100, equation (8) would then yield a mass loss of AM = 0.090 M0. This is in fairly good agreement with the results (shown in eq. [6]) of our detailed calculations. Use of the values from Table 1 for the cases Mf = 1.1 M0, ( /2) namely, M ¿i ä 0.05 M0 and Mce æ 0.019 M0, results in a value of AM æ 0.118 M0, nearly as far off in the other direc- tion. Using equation (8), Hobbs et al. (1989) obtained a mass Initial Mass Mi (M©) loss AM = 0.041 M0: this resulted from their use of an older Fig. 5.—Predicted surface 7Li abundance for our mass-losing solar models solar model (Iben 1967), which had a much smaller convective for various initial masses and mass-loss time scales. (The positions of the envelope mass (of M^ = 0.0045 M0) than is obtained by breaks in the curves are approximate.) most authors for recent solar models (see Paper I). They also

© American Astronomical Society • Provided by the NASA Astrophysics Data System 326 BOOTHROYD, SACKMANN, & FOWLER Vol. 377 have a smaller value of MLi = 0.025 M0 than our value of same result that Mce is not constant; in addition, pre-main- 7 0.029 M0. The former value was obtained by Hobbs et al. sequence Li depletion can be significant for such stars. (1989) by taking the commonly used value for lithium burning Recently, Swenson & Faulkner (1990) computed some of T æ 2.5 x 106 K: at this , the 7 Li-burning time stellar models with main-sequence mass loss, evolving them to scale is 7 Gyr, so that about half of the initial lithium would be an age of 0.6 Gyr (the age of the Hyades). Their models indi- burned after 4.55 Gyr. Thus for the standard solar model, the cated that, if main-sequence mass loss was assumed to be the 7 definition of Mu used by Hobbs et al. (1989) is the same as our cause of the observed Hyades Li depletion, then most Hyades definition, and the slight difference between their value of 0.025 G dwarfs must descend from an extremely small range of initial 7 M0 and ours of 0.029 M0 is due to a difference in the structure masses (e.g., 1.08 ± 0.01 M0), with the different Li abun- of the two solar models. dances and effective temperatures being due to different mass- From Table 1, one can check the assumption made in equa- loss rates; and any stars with lower initial masses would have 7 tion (8) that the convective envelope mass Mce remains con- to suffer sufficient mass loss to deplete Li below the observa- stant from the moment when the convective envelope reaches ble threshold, while higher initial masses encountered very the lithium discontinuity to the moment mass loss terminates. little mass loss. Such a sharp-edged (nearly step-function) dis- The quantity is the mass of the convective envelope at the tribution of mass-loss rates as a function of initial stellar mass moment when surface lithium has declined by a factor of 2, i.e., seems improbable, suggesting that main-sequence mass loss is shortly after the convective envelope encountered the lithium unlikely to be the predominant explanation for main-sequence discontinuity. The quantity M£ax is the mass of the convective 7Li depletion. envelope at its maximum extent, which occurs at the moment that mass loss terminates. The difference between these shows 3.7.3. The Lithium-burning Temperatures the extent to which the convective envelope mass deviates from Figures 6a and 6b show the temperature history of the constancy. For the Mt = 1.1 M0 cases, this deviation from deepest mass layer ever reached by envelope convection, for constancy is about 0.005 M©, which is about a quarter of the our shortest and longest mass-loss time scales. For the convective envelope mass. For a depletion /= 1/100, a change constant-mass 1.0 M0 case, the deepest convective envelope of this size in the value used for Mce would result in a change of occurs at the beginning of the main-sequence evolution, and nearly 20% in the mass loss predicted by equation (8). The retreats thereafter. The temperature for that mass layer is situation grows rapidly worse for larger amounts of mass loss : shown in Figures 6a and 6b : it remains very nearly constant at the mass of the convective envelope changes by a factor of 3 for 2.5 x 106 K. Lithium at this layer is depleted by a factor of j at the Mj = 1.2 M0 cases. the solar age, but even if the convective envelope had not Swenson (1990) has noted that equation (8) has limited valid- quickly retreated from this layer, there still would not have ity; in a forthcoming paper, he and Faulkner intend to discuss been any significant surface lithium depletion, since the mass of its drawbacks and possible improvements. It is clear that equa- the burning layer is much smaller than the mass of the convec- tion (8) will work less well for stars either more massive or less tive envelope. (Note that, in fact, the convective envelope has massive than the Sun, even if the most appropriate values of retreated to a temperature of 2 x 106 K by the solar age.) Mce and Mu are used (rather than the solar values, which Figures 6a and 6b show that the convective envelope for the could be rather poor approximations). For stars more massive M, = 1.1 Mq cases reaches down to layers that have experi- than the Sun, the convective envelope mass declines increas- enced temperatures of 4 x 106 K, and that the temperature at ingly quickly as a function of stellar mass, and the assumption this layer declines with a time scale tt, which is 2-3 times as that Mce is constant during mass loss is violated. For stars less long as the mass-loss time scale im. (Note that tt is not the massive than the Sun, the large observed amount of 7Li deple- same for all layers of the star.) More relevant to the surface tion requires relatively large amounts of mass loss, with the lithium depletion is the behavior which determines the position 7.0

rm = 0.985 Gyr: Mr = 0.9769 M©, Mj = 1.0 Mq Mr = 0.9796 M0, Mj - 1.1 Mq 6.8 Mr = 0.9784 M0, Mj = 1.2 Mq Mr = 0.9719 Mg> Mj = 1.5 Mq Mr = 0.9584 Mg, Mj = 1.8 Mq

6.5

L' ' log T(Mr= 1.04M0) —log T(Mr=1.137M0) 6.3 2 Time t (Gyr) Fig. 6a Fig. 6b Fig. 6.—Temperature as a function of time for the deepest mass layer in the star ever reached by surface convection. For some models, the temperature as a function of time for a mass layer near the lithium discontinuity is also given (see text).

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1991ApJ. . .377. .318B i0) /2) (2) 1/100) 6 2) i0)/2) 6 /2) 6 6 6 v No. 1,1991 for theM=1.2casewithlongmass-losstimescale, mass layersatwhichburningreducesthelithiumabundance of thelithiumdiscontinuity.ForM=1.1cases,and (see Fig.6a),thismasslayerisatM=1.025,i.e.,a by afactorof100.FortheM,=1.1M,r0.16Gyrcase Figures 6aand6bshowthebehavioroftemperaturefor mass oftheconvectiveenvelope.Thisisanothersourceerror depth ofM[y=0.075Mbelowtheinitialsurface.(Note in themass-lossformulaofeq.[8].)Similarly,forM=1.1 this illustratesthefactthatlithiumdiscontinuityisnot that thisissomewhatdeeperthanM[l=0.061forcase: 0.06 M©(cf.MV=0.049fromTable1).Forthe in Figure6bforthemasslayerM=1.04,i.e.,M^ sharp, butisspreadoutoveramassrangecomparabletothe lithium discontinuitylaybetween3.2x10and3.3K. initial temperaturesoftheselayersnearthebottom lithium profile.)FromFigures6aand6b,onecanseethatthe M, T=0.985Gyrcase,thetemperaturebehaviorisshown The correspondinginitialtemperaturesfortheMjylayers M =1.137M©,i.e.,M[y0.063M©(cf.M^0.052 range between2.95x10and3.05K;forthepurposes M© fromTable1).(Again,theseillustratethespreadin cases, theinitialtemperatureatmasslayeroflithium the lithium“discontinuity.”Inotherwords,formass-losing of equation(8),M[lgivesabetterestimatetheposition indeed 2.5x10K.Thedifferenceliesinthefactthatfor the constant-masscasecorrespondingtemperatureis discontinuity isnot2.5x10K(aswouldbecommonly f0 constant-mass casethetemperatureatthislayerremainsprac- assumed) butrather3x10K.Recall,asnotedabove,thatfor initial temperatureforlithiumburningcanbederivedanalyti- tically constantwithtime,whileforthemass-losingcases t0 cally, assumingaknowledgeoft. temperature dropsfairlysteeplywithtime.Theappropriate 1.2 M©,T=0.985Gyrcase,thetemperatureisshownfor r0 0m nentially, i.e.,T=Tiexp(—t/xf),andthatthenuclearburning 0 where T(T)isthenuclearburningtimescaleattemperature at thislayerdropsas rate isproportionaltoT,thentheabundanceXofanelement f Lt r0i 0m r t m nuc (LX _X/TV Assuming thatthetemperatureatagivenlayerdropsexpo- dt T(T)(7;)Vv nuc © American Astronomical Society • Provided by the NASA Astrophysics Data System 7 b initial masses0.9M<<1.2 ;theexpectedivaluesareforaLdepletionof/= given densitiesp,whichshouldbewithin 10%ofthecorrectvaluesforZAMSmodelswith G0 r 2.5 0.2452.67-176.920.297 2.7 0.3171.23-161.1619.615.8 2.6 0.2795.85-172.7819.938 2.9 0.4104.93-160.22419.23.0 2.8 0.3602.51-160.5019.46.7 3.2 0.5973.15-150.024118.50.31 3.1 0.5281.74-150.04918.70.64 3.0 0.4659.41-160.10318.91.36 3.5 0.8531.62-140.003317.9 0.041 3.4 0.7589.57-150.006218.1 0.078 3.3 0.6735.55-150.012118.3 0.154 3 b -17 6-31 Thenucleartimescaleiscomputed forahydrogenabundanceX=0.7bymassandthe Powerof10notation: 2.67—17=x10. (10 K)(gem)[cm(moles)](Gyr)Indexv (Gyr) 7 3 T pReactionRateiNuclearExpected x nuc T Lí-burning TimeScalesforAppropriateTemperaturesandDensities OUR SUN.II. TABLE 2 2) 6 6 6 time t: T. ThisequationcanbesolvedtoyieldXasafunctionof i.e. The timescaletwithwhichthetemperaturedropsdepends initial mass-lossrateM.Usingthisempiricalrelationfort where equation(2)hasbeenusedtogiveanexpressionforthe estimate thatforinitialmassesnottoofarfrom1.0M©the stant throughoutthestellarinterior.Lookingatappropri- on theinitialmassandmass-lossrate;itisalsonotcon- depletion/= Xf/Xiinamass-losingmodel: time scaletforlayersrelevanttothelithiumdiscontinuityis ate layers(i.e,atM^y)inourdetailedmodels,wecan and thedepletiongivenbyequation(10),onemayestimate initial nuclearburningtimescalerequiredtoobtainagiven lithium-burning layer.Thisispreciselythevalueobtainedfrom would implyaninitialtemperatureof~2.95x10Kforthe depletion f=\.Forthe1.1M©,t0.985Gyrcase, values ofthenuclearpowerindexvandtimescalerfora temperatures isgiveninTable2,alongwiththecorresponding models, aperiodoforder10Myrduringwhichthetem- models of3.06x10K.Partthisdiscrepancyfortheshort provide suchaperfectestimate:fortheA/=1.1M©,T=0.16 Mj/I M|=9.85Gyr,implyingx~2Gyr;fromTable2,this For lithiumburning,thenucleartimescaleTforseveral perature atthelithium-burninglayerisapproximatelycon- Gyr case,theformulagivesaninitialtemperatureof stant atitsinitialZAMSvalue.Ineffect,thetimescaleifor time scalecomesfromthepre-main-sequenceevolutionofour our detailedmodel.However,thisformuladoesnotalways t ft t m r m tT nuc ~3.2 x10K,ascomparedwiththevaluefromourdetailed r ^nuc(lî) =- X —X¡exp f X =exp t V In/5v\Mj~5v(M-M)‘ f %T ~5\mJ5(M,.-M)’ f T 1/MA TM IL„uc(7¡) V mi (e - Xt/XT for t>—.(10b) V (10a) (12) (11) 327 1991ApJ. . .377. .318B 6 2) /2) 6 /2) 9 (2 96 79 6 9 9 7 /2) 91 -9 1 6 328 mass lossnecessarytomatchtheobservedsolarlithiumabun- mated tobe~4.09x10K;fromourmodels,thesewould implies M[V=0.029M. imply massesM^=0.102and0.131M,respectively.The mass-loss timescale(r=0.16Gyr),thetemperatureisesti- yields alithium-burningtemperatureof2.5x10K,which factor off=j,thisimpliesT6.6Gyr;fromTable2, decline increaseslater. for theshorttimescalecasewouldpresumablyresultina values. Forthe=1.1Mcasewithlongmass-losstime nuclear burningtimescaleT=—t/\nf.Foradepletion the temperaturedeclineislongerinitially;speedof scales, respectively(seeTable1).Itthusreacheslayers0.120 tional 0.020and0.022M,forthelongshorttime The convectiveenvelopereachesfartherdownwardbyanaddi- correspondingly slightlylargermassofM^=0.132M.) dance wasAM=0.100and0.112M,respectively(seeeq. scale (r=0.985Gyr),thetemperatureisestimatedtobe temperature forBeburning.Table3givesthecorresponding abundance, whichshowslittleornodepletioncomparedwith not inconsistentwiththeobservationsofsolarberyllium the slightinaccuracyofequation(8),thismodestdepletionis tively. ConsideringtheerrorsinourestimatesofM£\and surface, butnotmuch:invertingequation(8)forthecaseof some beryllium-depletedmaterialwouldbemixedtothe and 0.134M,respectively,beneaththeoriginalsurface.Thus above, theseyieldBe-burningtemperaturesof~3.83x10 time scales,respectively(seeeq.[7]).Usingthesamemethodas AM =0.116and0.128Mforthelongshortmass-loss dance, i.e.,Li/H=2.6x10“,thenonerequiresmasslosses and ~4.11x10K,respectively;fromourdetailedmodels, the cosmicabundance:Boesgaard(1976)givessolarBe/H= Be, oneobtainsdepletionfactorsof/=0.4and0.9,respec- [6]): thesevalueswereobtainedusingZAMSLi/H= these implyvaluesforM^ofabout0.104and0.139M, Be/H =(1.31±0.36)x10“,i.e.,0.7;$/<1. 1.0 x10).(TheslightlylargerinitialmassofM=1.112 1.13 x10“,andestimatesthecosmicabundancetobe 0 e0 m nuc ~3.81 x10K,whilefortheM=1.1casewithshort 0 nuc0 0 e0 0 m 0 0 e0 t0 t0 One canusethesameapproachtodetermineinitial Note thatforconstant-massmodelsadepletion/requires If oneusesthemeteoriticvalueforZAMSlithiumabun- © American Astronomical Society • Provided by the NASA Astrophysics Data System 3.7.4. BerylliumDepletion 4.2. 4.1. 4.0. 4.4. 4.3. 4.5. 3.9. 3.8. 3.5. 3.4. 3.3. 3.7. 3.6. b 17 a 6 Powerof10notation:1.19—17= x10~. SeenoteatoTable2. (10 K) 9 T Be-BURNiNG TimeScalesforAppropriateTemperaturesandDensities“ BOOTHROYD, SACKMANN,&FOWLER (g cm 0.957 0.853 0.758 0.673 2.39 2.20 1.47 1.19 1.97 1.80 1.63 1.33 1.07 P _ 31 [cm (moles)"] Reaction Rate b 4.63-16 2.66-16 4.43-17 2.33- 17 9.06-15 2.20-15 7.92-16 3.58-15 8.23-17 5.73-15 1.33- 15 1.19 —17 1.49-16 TABLE 3 69101X 91 9 9 7 9 79 7 79 /2) 9 7 61/2) 69 (slightly highervaluesoff)ifthemodelhadbeenmoreaccu- initial modelsontheHayashitrack,ratherthannearmain program toincludeLi,Be,,and(andalsoallow presolar value. missing solarlithium.Themasslossis0.10-0.11Mifone paper, weseethatthedepletionwouldhavebeenslightlyless abundance ofBe/H=2.5x10“(Cameron1973)forthe solar Bedepletionof/«0.5,whenoneusesthemeteoritic respectively. Weinvertequation(8)togetberylliumdepletions medium) value. depletion, andthuscanbeconsideredtoupperlimits.Note presume thatthereisnoothermechanismforsurfacelithium 2.6 x10“,themeteoriticvalue.(Thesemass-lossvalues loss is0.12-0.13MifonetakestheZAMSLi/Hratiotobe value observedinPopulationImain-sequencestars.Themass time scaleofT=—t/\nftogetadepletion/atthesolar tion resultagreesfairlywellwiththeaboveestimates. rately matchedtothepresentSun.ThispreliminaryBedeple- by afactor/=0.32;comparingwiththemodelsofpresent sequence Lidepletionbyafactor/=1/125andBe M andmass-losstimescaler=0.5Gyrresultedinmain- sequence). Apreliminary“nearlysolar”testrunwithMf=1.1 of/ =0.2and0.6,respectively.Thisagreeswiththeobserved (Proffitt &Michaud1989):theZAMSLiabundancewillhave main-sequence lithiumdepletion.)Notethatforinitialmasses takes theZAMSLi/Hratiotobe1.0x10“,maximum estimated M^=0.07M.) then yieldsaBe-burningtemperatureofslightlybelow age; foradepletionf=j,thisimpliesT6.6Gyr.Table3 been reducedbythisfactorfromitspresolar(interstellar depletion (ontheHayashitrack)ofafactorabout1.5 M ~1.1,standardmodelsshowpre-main-sequenceLi this mass-lossmechanismbeingprimarilyresponsiblefor that recentworkbySwenson&Faulkner1990arguesagainst of themass-losstimescale. 3.3 x10K,givingM^=0.065M.(NotethatHobbsetal. 1989 usedatemperatureof3.7x10KforBeburning,and 0 0 nucQ 0m e0 nuc t0 e0 We areintheprocessofmodifyingourstellarevolution 2. Therequiredamountofmasslossisnearlyindependent For aconstant-masscase,onerequiresnuclearburning 1. Early-main-sequencemasslosscansolvethecaseof 0.039 0.074 0.143 0.0036 0.0064 0.0114 0.0208 0.28 0.57 2.57 0.0021 5.6 1.20 (Gyr) Nuclear Index v 20.8 21.0 21.3 21.5 22.3 20.4 20.6 21.7 21.8 22.1 22.5 20.1 20.3 4. CONCLUSIONS Expected z T 40 88 18.4 (Gyr) 0.163 0.30 0.57 0.029 0.050 0.091 2.13 4.3 8.7 1.09 Vol. 377 1991ApJ. . .377. .318B 6 6 79 6 v / ataget(asisthecaseforconstant-masscase),weshow No. 1.1991OURSUN.II.329 estimate theamountofmasslossAMrequiredbyanobserved lithium (orberyllium)depletion/,namely,AM=(M—M) lithium (orberyllium)isburned.Thismass-lossformulaworks is thedepthinmass(belowinitialsurface)atwhich mass (i.e.,<0.1M),providedthatoneobtainsgoodvaluesfor fairly wellforstarsverycloseto1Mwhichdonotlosemuch mass-losing models,wheretemperaturesdeclinewhilethemass loss takesplace,higherinitialtemperaturesarerequired.We temperatures of2.5x10and3.3K,respectively.In higher orlowermass. compensate forthetemperaturedecline,namely,T= have developedaprescriptiontoestimatethesetemperatures M and.Itisexpectedtobelessaccurateforstarsof t isthetimescaleoftemperaturedecline.(Notethat that theinitialburningtimescalehastobeshorter,inorder as afunctionofinitialstellarmassMandmass-lossrate Allen, C.W.1963,AstrophysicalQuantities(2ded.;London:Athlone) peratures ofapproximately3x10Kforlithiumburningand t, namely,~Thisresultsininitialtem- detailed models,wehaveestimatedanempiricalformulafor v~20 forbothLiburningandBeburning.)Fromour Mj. InsteadofrequiringT=—t/\nftogetadepletionfactor Bahcall, J.N.,Huebner,W.F.,Lubow,S.H.,Parker,P.D.,&Ulrich,R.K. losing cases.(Notethattheprecisevaluesofthesetemperatures approximately 4x10Kforberylliumburningthemass- Bahcall, J.N.,&Ulrich,R.K.1988,Rev.Mod.Phys.,60,297 larger valuesforthesolarmetallicityZ,mixing-length depend ontheinitialmassandmass-lossrate.) Boesgaard, A.M.,&Steigman,G.1985,ARA&A,23,319 Boesgaard, A.M.1976,ApJ,210,466 Bodenheimer, P.1965,ApJ,142,451 ferences, however,arenegligibleifonehasamassloss presolar heliumabundanceYthanthestandardmodel,and Cameron, A.G.W.1973,SpaceSei.Rev.,15,121 Brown, A.,Vealé,Judge,P.,Bookbinder,J.,&Hubeny,I.1990,in Proc. Boothroyd, A.I.,&Sackmann,I.-J.1988,ApJ,328,653 parameter a,andthepredictedsolarneutrinoflux.Thesedif- Lice Caughlan, G.R.,&Fowler,W.A.1988,AtomicDataNucl.Tables, 40, AM ~0.1M,whichisthevaluerequiredtomatch Cayrel, R.,CayreldeStrobel,G.,Campbell,B.,&Däppen,W.1984,ApJ, 283, larger thantheprimordialheliumabundanceplacesmuchless — MIn/,whereistheconvectiveenvelopemassand 0 0 Grevesse, N.1984,Phys.Scripta,T8,49 Duncan, D.K.,&Jones,B.F.1983,ApJ,271,663 Dominy, J.F.,Wallerstein,G.,&Suntzeff,N.B.1986,ApJ,300,325 Dominy, J.F.,&Wallerstein,G.1987,ApJ,317,810 D’Antona, F,&Mazzitelli,1.1984,A&A,138,431 Cox, A.N.,&Stewart,J.N.1970,ApJS,19,243 stringent limitsonthemasslossthandoeslithiumdeple- observed solarlithiumdepletion. nuc ceLi — T/(vln/),wherevisthenuclearindex(i.e.,ocT~)and t t t nuc 0 ceLi rnuc G. Wallerstein(ASPConf.Ser.,Vol.9;SanFrancisco:Bookcrafters),183 Sixth CambridgeWorkshoponCoolStars,StellarSystems,andtheSun, ed. 283 205 1982, Rev.Mod.Phys.,54,767 4. Constant-massmodelsburnlithiumandberylliumat 3. Asimpleformula(Hobbsetal.1989;seeeq.[8])existsto 5. Mass-losingsolarmodelsrequiresmallervaluesforthe 6. Therequirementthatthepresolarheliumabundancebe © American Astronomical Society REFERENCES Provided bythe NASA Astrophysics Data System -1 14- increase asinthestandardcase.Themass-losingcaseshavea respectively. M formass-losstimescalesofx=0.985,0.5,and0.16Gyr, tion. ThemasslossAMmustbelessthan0.7,1.0,and1.5-2 extreme M=1.8case).However,forourpreferredM¿ higher initialluminosity(byasmuchafactorof15forthe different fromthatofthestandardcase,firstdecliningduring than thatofthestandardcase(barelyhigherpresent the periodofmasslossandthenfollowingsamesteady further. larger solarwind(M~10°Moyrratherthanthepresent history, thestrongerinitialgravitationalattraction,and solar luminosity),andthetotalvariationinluminosityis wishes inadditiontothankScottD.TremaineandPeterG. rate ofM~10“yr)>couldhavehadsignificantcon- Theoretical Astrophysics.WeareindebtedtoCharlesA. by theKelloggRadiationLaboratory.Oneofus(A.I.B.) sequences ontheplanetarysystemthatshouldbeexplored less thanforthestandardcase.Thisdifferentluminosity .1987,ApJ,319,957 mation. Oneofus(I.-J.S.)wishestoexpresshergratitude Martin forthesupportprovidedbyCanadianInstitute 1.1 Mcases,theinitialluminosityisonlyafactorof1.5higher Guzik, J.A.,Willson,L.&Brunish,W.M.1986,BAAS,18,664 Guenther, D.B.,Jaffe,A.,&Demarque,P.1989,ApJ,345,1022 mother, whodrovethousandsofmilesmanyatimetoassist with insightfuldiscussionsortasksofdailylife.Sheisalso Barnes forencouragementandenlighteningdiscussions,to effectively pursued.WewishtoacknowledgetheworkofKath- with thecareofchildrensothatresearchcouldbemore indebted totheirreplaceablesupportofMrs.LillyStelter,her Fritz J.Swenson(thereferee)forhelpfulcommentsandinfor- Iben, L,Jr.1967,ApJ,147,624 Hobbs, L.M.,&Pilachowski,C.1988,ApJ,334,734 Hobbs, L.M.,Iben,L,Jr.,&Pilachowski,C.1989,ApJ,347,817 leen E.Kraemer,whoasaCaltechsummerstudentin1988 Robert F.Christy,herhusband,forhisgenerosityinhelp,beit 0m Keady, J.1985,privatecommunication carried outmanycomputationswithanearlierversionofthe ¿0 Kiziloglu, N.,&Eryurt-Ezer,D.1985,A&A,146,384 This workwassupportedinpartbyagrantfromtheNatural Sackmann, I.-J.,Boothroyd,A.L,&Fowler,W.1990,ApJ,360,727(Paper Proffitt, C.R.,&Michaud,G.1989,ApJ,346,976 Maeder, A.1984,inStellarNucleosynthesis,ed.C.Chiosi&Renzini a grantfromtheNationalScienceFoundationPHY-8817296. stellar evolutionprogram.Thisworkwassupportedinpartby Schramm, D.N.,Steigman,G.,&Dearborn,S.P.1990,preprint Willson, L.A.,Bowen,G.H.,&Struck-Marcell,C.1987,Comm.Ap.,12,17 Willson, L.A.,&Bowen,G.H.1984,Nature,312,429 Weymann, R.,&Sears,R.L.1965,ApJ,142,174 Swenson, F.J.,Stringfellow,G.S.,&Faulkner,J.1990,ApJ,348,L33 Swenson, F.J.,&Faulkner,J.1990,BAAS,21,1101 Swenson, F.J.1990,privatecommunication Sciences andEngineeringResearchCouncilofCanada. 0 0 (Dordrecht: Reidel),115 I) FERMILAB-Pub-90/13-A 7. Theluminosityhistoryofourmass-losingcasesisquite We wishtothankStevenE.Kooninforthesupportsupplied

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