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Astronomy&Astrophysicsmanuscriptno.aa1280˙lc˙arxiv February5,2008 (DOI:willbeinsertedbyhandlater) ⋆,⋆⋆,⋆⋆⋆ The mass loss of C-rich giants J.BergeatandL.Chevallier CentredeRechercheAstronomiquedeLyon(UMR5574duCNRS),ObservatoiredeLyon, 9avenueCharlesAndre´,69561St-Genis-Lavalcedex,France 6 Received12May2004/Accepted23July2004 0 0 Abstract. Themasslossrates,expansionvelocitiesanddust-to-gasdensityratiosfrommillimetricobservationsof119carbon- 2 rich giants are compared, as functions of stellar parameters, to the predictions of recent hydrodynamical models. Distances n andluminositiespreviouslyestimatedfromHIPPARCOSdata,massesfrompulsationsandC/Oabundanceratiosfromspec- a troscopy, andeffectivetemperaturesfromanewhomogeneous scale,areused.Predictedandobservedmasslossratesagree J fairlywell,asfunctionsofeffectivetemperature.ThesignatureofthemassrangeM≤4M⊙ofmostcarbon-richAGBstarsis 7 seenasaflatportioninthediagramofmasslossratevs.effectivetemperature.Itisflankedbytworegionsofmasslossrates 1 increasingwithdecreasingeffectivetemperatureatnearlyconstantstellarmass.Fourstarswithdetachedshells,i.e.episodic strong mass loss, and five cool infrared carbon-rich stars with optically-thick dust shells, have mass loss rates much larger 1 thanpredictedvalues.Thelatter(includingCWLeo)couldbestarsofsmallermasses(M≃1.5−2.5M⊙)whileM≃4M⊙ v isindicatedformostofthecoolestobjects.Amongthecarbonstarswithdetachedshells,RSclreturnedtoapredictedlevel 6 6 (16timeslower)accordingtorecentmeasurementsofthecentralsource.Theobservedexpansionvelocitiesareinagreement 3 withthepredictedvelocitiesatinfinityinadiagramofvelocitiesvs.effectivetemperature,providedthecarbontooxygenabun- 1 danceratiois1≤ǫ /ǫ ≤2,i.e.therangededucedfromspectraandmodelatmospheresofthosecoolvariables.Fivestarswith C O 0 detachedshellsdisplayexpansionvelocitiesabouttwicethatpredictedattheireffectivetemperature.Mirasandnon-Mirasdo 6 populatethesamelocusinbothdiagramsatthepresentaccuracy.Thepredicteddust-to-gasdensityratiosarehoweverabout 0 2.2timessmallerthanthevaluesestimatedfromobservations.Recentdriftmodelscancontributetominimizethediscrepancy / h sincetheyincludemoredust.Simpleapproximateformulaeareproposed. p - Keywords.stars:AGBandpost-AGB–stars:carbon–stars:massloss o r t s a 1. Introduction tio. For instance a model for the photodissociationof the cir- : cumstellar CO by Mamon et al. (1988) is usually applied to v TheobservationsofCOandHCNlinesinthemillimetricrange i radial brightness distributions with fair success, but clear de- X together with infrared fluxes, especially from the IRAS cata- viations were noted in a few cases (e.g. Schoïer & Olofsson r logue (1988), provide estimates of parameters of the circum- 2001). Predominantspecies, like hydrogenor helium, are not a stellarenvelopesofcarbon-richgiants.TheyareessentiallyTP- tracedoutandtheadoptedabundancesinfluencetheestimates AGBstars,i.e.latestagesofstellarevolutionoflowandinter- ofthetotalmasslossrate.Theestimateddistancesarealsoof mediate mass stars, experiencing substantial mass loss. This paramountimportancesincetheratecalculatedfromobserva- latter phenomenonmay terminatetheir evolutionon the AGB tionsincreasesasthesquareddistance.Afulldiscussionisout- beforerapidnucleosynthesisforcesthemtoleavethatbranch. side the scope of the presentpaper and we referthe readerto Masslossalsocontributestothereplenishmentoftheinterstel- thepaperscitedinSect.2. lar medium in carbonaceous material (dust and gas), mostly It has become clear that the interaction of pulsation and from extreme (“infrared”)carbon stars. Models and simplify- dust formationplaysa key role in the mass loss phenomenon ingassumptionsarerequiredtoderivetheestimatesofthemass (e.g. Wallerstein & Knapp 1998 for a review). The effects of loss rate M˙ in solar mass per year, the expansion velocity v e stellar pulsationare simulatedindynamicalmodelsbyapply- in kilometer per second, and the dust-to-gasmass density ra- ing a piston at the inner boundary. Consistent models of cir- cumstellardustshellsaroundcarbon-richlongperiodvariables Sendoffprintrequeststo:J.Bergeat (LPV)includetime-dependenthydrodynamics,adetailedtreat- ⋆ This research has made use of the Simbad database operated at mentoftheprocessesofformation,growthandevaporationof CDS ⋆⋆ Partially based on data from the ESA HIPPARCOS astrometry dust grains,a carbon-richchemistry,and radiativetransfer (e. satellite g. Fleischer et al. 1992, Fleischer 1994, Winters et al. 1994b, ⋆⋆⋆ Table3isonlyavailableinelectronicformattheCDSviaanony- 1995, Ho¨fner et al. 1995, 1996). The empirical Reimers rela- mousftp130.79.128.5 tionforRGB(Reimers1975)proveditsinadequacyforAGB. 2 J.BergeatandL.Chevallier:ThemasslossofC-richgiants Table 1. Mean distances (pc) of carbon-rich giants and dispersions as obtained from data of various authors: O93 stands for Olofssonetal.(1993a),G99forGroenewegenetal. (1999),SO1forSchoïer&Olofsson(2001),GO2forGroenewegenetal. (2002b),andL93forLoupetal.(1993).TheyarecomparedwithastrometricdatafromBergeatetal.(2002a,2002b=BKR)for (n)starsincommon.RatiostoBKRmeansarethenquotedinthelastline(Seetextforadiscussion). n O93 BKR n G99 BKR n O93 SO1 BKR n G02 BKR n L93 BKR 104 707 708 11 1141 1161 54 567 427 614 25 1535 748 21 721 536 ±218 ±258 ±459 ±452 ±171 ±166 ±218 ±927 ±357 ±424 ±182 1.00 0.98 0.92 0.70 2.05 1.35 Alternative relations were derived from observational results availablepredictionsfromhydrodynamicalmodelsaresumma- and/or theoretical models of mass loss (e.g. Volk & Kwok rized in Sect. 3 and equationsfitted on a grid from the litera- 1988,Vassiliadis&Wood1993,Blo¨cker1995).Themassloss ture are adopted.Valuesof stellar parametersfromBergeatet relation of Dominik et al. (1990) was the first one based on al. (2001, 2002b, 2002c) were compiled for stars in common detailed calculations of dust formation and growth (Gail & with millimetric studies and mean values collected for eight Sedlmayr1988),withratherextremestellarparametersadapted photometricgroups(Sect.4)tobereplacedintheformulaeof totheverylatestagesofAGBevolution.Approximateformu- Arndtetal.(1997).Then,thecomparisonofobservedandpre- laeformasslossrate,expansionvelocityanddusttogasden- dictedmasslossratesisperformedinadiagramagainsteffec- sityratiowerethencalculatedbyArndtetal.(1997).Theycon- tivetemperature(Sect.5).Inasimilarapproach,theobserved sideredsixstellarparametersasindependentones,namelythe andpredictedexpansionvelocitiesarecomparedbyplottingvs. temperature, the luminosity, the mass, the abundance ratio of effectivetemperature(Sect.6).Thediscrepancynotedbetween carbonto oxygen,the pulsationalperiodandthe velocityam- observedandpredicteddust-to-gasdensityratios,isestimated plitudeofthepulsation.Theythenderivedapproximateequa- and discussed (Sect. 7). A few conclusionsare givenanddis- tions on the basis of 48 dynamicalmodels, exploringvarious cussed(Sect.8)andsimpleapproximateformulasareproposed rangesforthesixparameters.Theyalsocomparedtheirresults inAppendixA. to thoseof 22analogousmodelsfromHo¨fner& Dorfi(1997) whoprovedtobewellapproximatedbytheirsetofequations. 2. Thedatafrommillimetricobservations ThepredictionsofArndtetal.arecomparedtothedatade- Mass loss rates, expansion velocities and dust-to-gas density ducedfrom observationsas publishedby variousauthors.We ratios were extracted from the literature for a large sample use the effective temperaturesof Bergeatet al. (2001) as stel- of carbon-rich giants. The early millimetric data and results lartemperatures.Usinganewhomogeneousscaleconstructed are summarizedin a cataloguebyLoupet al. (1993). Schoïer from model atmospheres, observed angular radii and spec- & Olofsson (2001) revised a set of data from Olofsson et al. tral energydistributions, they have provedefficientin solving (1993a,1993b).Themasslossrate,forstarsincommon,ison various problems. We also adopt the distances and luminosi- average 3.1± 3.6 times larger in the 2001 paper than it was ties derived by Bergeat et al. (2002b) from the HIPPARCOS inthoseof1993.ValuesinGroenewegenetal.(2002a)for18 data (1997) by taking into account three distinct biases (their stars in common with Olofsson et al. are on average 2 times Sect. 2.3; see also Knapik et al. 1998 and Bergeat et al. larger.Amean2.9ratiowasfoundfromtwostarsincommon 2002a). Pulsation masses were derived for carbon-rich LPVs withWintersetal.(2003).Admittedly,recentacceptedvalues by Bergeat et al. (2002c). The carbon to oxygen (C/O) ra- formasslossratesaretypically2-3timeshigherthanthepre- tios were taken from Lambert et al. (1986), Olofsson et al. viousonesfromOlofssonetal. (1993a, 1993b).We alsocon- (1993b)andAbia&Isern(1996).Theperiodsofpulsationused sidered data for a few stars from Groenewegen et al. (1999), arethephotometricperiodstakenfromtheGeneralCatalogue Schoïer & Olofsson (2000) and Le Bertre et al. (2001, 2003; of Variable Stars (GCVS; Kholopov et al. 1985) or from the seealsoMeixneretal.1998forRWLMi=CIT6andSkinneret HIPPARCOS data(1997). The valuesare those of fundamen- al.1998forCWLeo=IRC+10216). tal mode pulsators from the study of Bergeat et al. (2002c). Usually,thedistancesadoptedbytheauthorssubstantially Considering the complexities of those pulsating atmospheres differandthisisafundamentalproblemsincethemasslossrate andtheuncertaintiesontransferintheirmodels,weadoptthe (forbothgasanddust)increasesasthesquareddistance.Large referencevalue∆u = 2kms−1ofArndtetal.(1997)fortheve- discrepanciesmayexistbetweendistancesestimatedfromvar- locityamplitudeofthepulsation.Weshallseethatitsinfluence ious methods: mean absolute magnitude adopted, say in the isquitelimited. K-band, mean bolometric magnitude with bolometric correc- We re-scaled the mass loss rates from the literature to the tion or mean luminosity, absolute magnitude vs. period for distances of Bergeatet al. (2002b) and a f-correctionwas ap- long period variables, kinematical methods ... The astromet- pliedtoconvertthemtothescaleofSchoïer&Olofsson(2001) ric parallaxes of those bright distant giants are also affected who properly dealt with radiative transfer (Sect. 2). In addi- by substantial errors.First of all, we calculated the mean dis- tion, the expansion velocities and dust-to-gas density ratios tances and dispersions for the various samples of the above- werecompiledforasampleofcarbon-richgiants(Sect.2).The mentionedreferencesandcomparedthemtotheircounterparts J.BergeatandL.Chevallier:ThemasslossofC-richgiants 3 Table 2. The data for 119 carbon-richgiants (Sect. 2) with variable star names (Columns1 and 10, except for CGCS1006 in Stephenson1989andAlksnisetal.2001)shownwithboldfacecharactersfor19Miras.DatafromSEDsandspectraaregiven:the photometricgroupsinColumns2and11withanadditionalJfor13C−richstars,theeffectivetemperatures(Kelvin)inColumns 3and12,thecarbontooxygenabundanceratio(C/O)oftheatmosphereinColumns4and13,theabsolutebolometricmagnitude M inColumns5 and14,thatcanbetranslatedintoluminosityinsolarunitsfromEq.(3),andthedistanceD in parsecin b BKR Columns6 and 15. The data from millimetric observationsare displayedin Columns7 and 16 (mass loss rate as M˙′ =108M˙ whereM˙ insolarmassperyear),Columns8and17(expansionorterminalvelocityv inkms−1,andColumn9and18(r′ =103r e wherer=M˙ /M˙ isthedust-to-gasratioofmasslossrates).Additionalnotesdealwithranges:a HC5-CV4&3525-2775K,b d g HC5-CV6&3520-2645K&M = −3.72;−4.36,c 2290-2245K&M = −4.86;−4.53,d 2920-2840K&M = −5.22;−4.80, b b b e M = −5.81;−5.48,f 3345-2825K&M = −6.15;−5.67,g 2405-2525K&M = −5.89;−6.10,hIRC+102161915-2105K, b b b iCIT62425-2465K,j2100-1945K&M =−5.23;−4.60,k2865-2680K&M =−5.51;−5.00,lCV5-CV6&2650-2530K& b b M =−4.94;−5.06,m1885-1875K&M =−6.28;−5.93,n2945-2865K&M =−4.84;−5.61,o1975-1875K,p2240-2095K b b b &M =−5.42;−5.22,qCV1-CV5&3245-2605K&M =−3.01;−3.69,r2735-2655K&M =−4.78;−4.58,sCIT5,tIRAS b b b 05352+2247,u AFGL971,v AFGL2067=IRC-10396,w AFGL3116=IRC+40540,x AFGL1235,y AFGL1961(erroneously identifiedasV522Oph).NotshowninFig.1:accordingtoOlofssonetal.(2000),TTCyg(whichissurroundedbyathinshell) ispresentlylosingmassatamodestrateof310−8M⊙yr−1 whichisperhapstobemultipliedbythe3.9correctionfactorofEq. (2);accordingtoIzumiuraetal.(1996),YCVn**exhibiteda7−2010−6M⊙yr−1rateattheformationofitsthinshell. Name G Teff C/O Mb D M˙′ ve r′ Name G Teff C/O Mb D M˙′ ve r′ VXAnd CV6J 2520 1.76 -4.47 560 14 11.5 4.7 AQAnd CV5 2660 -5.24 1015 43 ZPsc CV2 3095 1.014 -4.65 465 6.1 3.5 2.9 RScl* CV4 2625 1.34 -5.48 475 640 16.9 0.86 RFor CV7 2000 -5.82 865 500 16.5 0.87 V623Cas CV1J 3360 -4.49 515 11 TWHor CV2 2950 -5.28 485 20 5.5 2.5 V384Pers CV7 1820 (-5.43) 720 590 15.0 0.25 YPer HC5a 3150a -4.09 975 29 7.2 0: UCam* CV4 2695 1.30 -5.09 525 200 20.6 1.0 V466Per CV4 2775 (-4.66) 530 24 9.0 2.3 SYPer CV5 2705 -5.86 1430 150 17.5 0.79 STCam CV4 2805 1.14 -6.07 800 110 9.0 0.89 TTTau CV2 3090 -4.88 585 20 5.0 0.42 V346Aur SCV 2880 -5.49 1110 20 AUAur CV5 2665 -5.01 1470 57 ROri HC5b 3083b -3.72b 1700 61 10.0 0.67 RLep CV6 2290c 1.030 -4.86c 335 160 18.0 1.4 ELAur CV4 2730 -4.60 650 28 WOri CV5 2625 1.16 -5.19 410 38 11.0 2.5 V431Ori CV6 2540 -3.78 540 22 UVAur CV3 2920d -5.22d 1090 160 10: SAur CV7 1940 1.014 (-5.43) 1130 420 25.5 1.0 RTOri CV4 2870 -4.31 675 22 CGCS1006t CV6 2300 (-5.05) 1870 43 TUTau CV3 2850 -5.81e 1055 57 YTau CV4 2735 1.040 -6.00 735 160 11.0 1.6 WPic CV6 2530 -4.85 665 51 16.0 1.7 FUAur CV2 3035 -5.56 1295 28 TUGem CV4 2715 1.12 (-4.66) 515 44 11.5 1.7 FUMon SCV 3085f -6.15f 1450 26 VAur CV4 2820 -5.47 1760 360 20 0.13 BNMon CV6 2410 (-5.05) 1280 420 24.2 0.73 ZZGem CV6 2530 (-5.05) 1835 59 6.9 0.22 BLOri CV2 3035 1.039 -5.13 595 25 9.0 1.2 CRGem CV2 2960 -5.60 920 84 14.9 1.3 UUAur CV4 2760 1.063 -5.78 415 76 11.0 0.16 NPPup CV2 3090 -4.54 510 19 9.5 1.0 RVMon CV3 2910 -4.90 670 17 V614Mon CV1 3320 -3.47 485 1.5 RYMon CV6 2440 1.19 -4.90 685 59 11.0 1.1 WCMa CV3 2975 1.046 -5.42 785 61 10.5 0.63 V406Pup CV3 2875 -4.15 610 11 RUPup CV3 2680 -3.33 455 13 V346Pupx CV7 1875 (-5.43) 1120 3310 20.7 0.50 RPyx CV6 2440 -4.52 1175 68 8.8 1.2 UZPyx CV1 3325 1.30 -4.63 815 26 XCnc CV5 2645 1.14 -5.77 710 62 7.0 1.3 TCnc CV6 2405g -5.89g 1085 97 CWLeoh CV7 1915h 1.4 -5.83 150 3300 14.5 0.66 YHya CV5 2645 1.52 -4.89 485 37 9.0 XVel CV5 2700 1.16 -5.46 620 63 10.0 0.68 SZCar CV4 2810 (-4.66) 365 46 14.0 0.30 RWLMii CV6 2445i (-5.05) 410 650 17.0 4.5 XZVel CV6 2430 1.22 (-5.05) 760 94 14.0 0.90 CZHya CV5 2525 1.02 (-4.82) 990 125 12.0 0.65 UAnt* CV4 2810 1.44 -4.93 320 200 21.8 UHya CV3 2965 1.043 -3.93 175 21 7.0 0.98 VYUMa CV2 2930 1.060 -4.67 445 14 6.0 0.97 VHya CV6 2160 1.050 -5.88 495 750 24.2 1.1 RRMus CV2 3090 1.010 -4.34 690 27 SSVir CV6 2560 1.080 (-5.05) 560 36 12.5 1.4 YCVn** CV5J 2760 1.087 -4.64 260 14 8.5 2.0 DYDru CV6 2420 1.150 (-5.05) 900 47 RUVir CV6 2100j -5.23j 675 230 18.4 RXCru CV6 2650 (-5.05) 845 62 RYDra CV5J 2810 1.18 -5.28 550 44 10.0 0.27 RVCen CV3 2865k 1.030 -5.51k 940 78 12.6 1.3 NSV6507 CV3 2920 (-4.39) 560 18 6.5 3.1 V996Cen CV4 2695 1.53 -5.81 830 44 11.0 1.1 ZLup CV5 2655 1.12 -5.05 1105 20 XTrA CV5 2710 1.17 -5.71 460 18 9.1 0.81 ASCir CV6 2420 (-5.05) 935 78 UAps CV5 2695 -4.68 825 28 VCrB CV7 2090 -4.80 845 130 7.5 1.3 VOph CV3 3010 -4.25 580 14 7.8 0.64 SUSco CV5 2655 (-4.82) 635 22 V1079Sco CV6 2510 1.10 -5.33 920 120 V2309Ophy CV6 2425 1.10 (-5.05) 1025 34 TWOph CV6 2440 1.20 -5.12 495 23 8.6 0.15 TTSco CV6 2430 (-5.05) 865 57 VPav CV6 2545 -4.74 430 66 16.0 1.1 SXSco CV4 2785 -5.02 830 38 TDra CV6 1850 -5.83 1425 820 13.5 1.3 ESSer CV6 2500 (-5.05) 905 30 TYOph CV5 2680 (-4.82) 845 32 TLyr CV6J 2310 1.29 -5.43 645 80 11.5 2.2 HKLyr CV3 2945 -4.66 730 26 HKLyr CV5 2620 -4.39 730 26 DRSer CV5 2650 -5.40 1295 100 20.1 3.9 SSct* CV4 2755 1.069 -5.18 580 560 17.3 UVAql CV5 2700 1.11 -5.01 910 30 VAql CV6 2525 1.25 -5.65 560 66 8.5 1.0 CGVul CV5 2685 (-4.82) 805 33 V1942Sgr CV2 2960 1.12 -4.60 550 19 10.0 1.1 UXDra CV2 3090 1.046 -5.53 545 37 4.0 0.49 AQSgr CV4 2790 1.033 -5.66 790 77 10.0 0.63 V391Aql CV6 2585 -5.05 1950 38 TTCyg* CV4 2825 -3.98 620 (12:) 13.5 AXCyg CV5 2655 -4.03 520 23 XSge CV5 2630 -4.60 860 24 SVCyg CV5 2600 -4.22 700 19 RSCyg CV2 3100 -5.08 655 20 RTCap CV6 2480 1.10 -4.95 560 23 8.0 1.8 UCyg CV5l 2650l -4.94l 700 100 13.0 1.5 VCyg CV7 1880m -6.28m 740 630 11.5 0.72 TInd CV2 2990 -5.36 645 17 6.0 2.0 YPav CV3 2945n -4.84n 420 23 8.0 1.4 V1426Cyg CV7 1975o (-5.43) 820 540 14.0 1.3 SCep CV6 2240p -5.42p 495 290 22.0 1.2 V460Cyg CV2 2950 1.062 -5.81 635 45 10.0 0.78 RVCyg CV5 2675 1.20 -5.57 640 140 13.5 1.2 LWCyg CV5 2580 -5.45 1095 35 RZPeg CV3q 2925q -3.3q 605 10 12.6 0.59 DGCep CV2 2985 -4.71 800 23 TXPsc CV2 3125 1.027 -5.09 315 32 10.7 0.72 WZCas CV2 3095 1.010 -5.48 650 13 3.8 1.0 V688Monu CV7 1670 (-5.43) 1370 2400 13.4 4.1 FXServ CV7 2050 (-5.43) 1230 3100 28.4 0.58 LPAndw CV7 2040 (-5.43) 840 2100 14.0 1.7 4 J.BergeatandL.Chevallier:ThemasslossofC-richgiants fromBergeatetal. (2002b, BKR) whose valuesare astromet- butwewereunabletosplitthesamples.ForLoupetal.(1993, ricinessenceanddeducedfromHIPPARCOSdata(Knapiket 7starsand9%),weobtainedf=1.2±0.5whilef=3.4±2.1 al.1998andBergeatetal.2002a),withnophotometriccriteria and f=3.1±1.7 were deduced for Le Bertre et al. (2001, 5 used.TheresultsareshowninTable1.Thereisapparentlyan stars) and (2003, 4 stars) respectively.Theselater five correc- excellentagreementforthe104starsofOlofssonetal.(1993a) tionfactorsandthoseinEq.(2)wereappliedtotheM˙ −data BKR incommon,with707-708pcasmeanvalues.Adetailedanal- derivedfromEq.(1).WerejectedthedataavailableinWinters ysis (Bergeat 2004) however proved this is fortuitous since a etal.(2003)sincethecorrectionfactorcuriouslyamountedto Malmquist-typebias occurred in the Olofsson et al. distances 0.4fromonlytwostarsincommonwithSchoïer&Olofsson. inthissampleofmagnitudesK≤2,adoptingM ≃ −8.1(ac- Both the obtained mean gas mass loss rates and the K tuallythereisa3-magnituderangewhosecentralvalueisclose D −distancesaregiveninTable2togetherwithphotometric BKR to -7.50). Conversely, no such bias appeared when compar- groups and effective temperaturesfrom Bergeat et al. (2001). ing the distances deduced from the M −magnitudes derived The luminosities in solar units can be derived from the abso- K by Mennessier et al. (2001) through a maximum likelihood lute bolometric magnitudes of Bergeat et al. (2002b) making methodfor124carbongiantsincommonwithBKR. Thereis useof alsoagoodagreementwithBKRinTable1forasmallsample L/L⊙ = 10−0.4(Mb−4.73) . (3) of 11 stars from Groenewegenet al. (1999). Theyare located farther away at 1141-1161 pc on average, that is 1.63 times Additionalstaridentifiersanddetailedbibliographicsourcesof moredistant.Wealsoconsideredasampleof54starsincom- the mass loss data for each star are providedin Table 3, only montoOlofssonetal.(1993a)andSchoïer&Olofsson(2001). availableinelectronicform.TheC/OratiosquotedinTable2 There is a 1.33 ratio (i.e. 1.76 on mass loss rates), in the di- weretakenfromLambertetal.(1986),Olofssonetal.(1993b) rectionoppositetothatofthe3.1meanratioofmasslossrates andAbia&Isern(1996).Alsogivenaretheexpansionveloc- mentionedabove.Itistobecomparedto0.92whenreferredto ities fromthe same five referencesmentionedabovefor mass BKR. The Groenewegenet al. (2002b) mean distance is 2.05 lossrates.Boththedustandgasmasslossratesareproportional timeslargerthantheBKR-valuefor25starsincommon,lead- to the adoptedsquareddistanceand no correspondingcorrec- ingtotheirmasslossratesbeingsystematicallylarger.Thefac- tions need be applied to the dust-to-gasratios. The dust mass torisonly1.35betweenLoupetal.(1993)andBKR. lossrateisderivedfromtheIRAS(60µm)makinguseofafor- Finally, for the sake of consistency, we transformed the mulagivenbySopkaetal.(1985;seee.g.Eq.(8)inOlofsson ratesM˙ givenatdistancesDinthementionedpapers,accord- et al. (1993a). The dust rate is proportionalto L−0.5. We thus ingtotherelation scaled the values from Olofsson et al. (1993b), Groenewegen etal. (1999) andGroenewegenetal. (2002b)to the luminosi- M˙ = M˙ (D /D)2 (1) BKR BKR tiesquotedinTable2,accordingtotheindividualluminosities whereD standsforthedistancesofBergeatetal.(2002b). adopted by these authors. The obtained values were then di- BKR The gas mass loss rates were derived from CO observations videdbythecorrectionfactorsofEq.(2)orby3.3or1.9,i.e. byaformulaestablishedbyKnapp&Morris(1985)forunre- thef-factorsforthegasmasslossratesfromthosethreerefer- solved optically thick envelopes.From a more detailed radia- ences. As discussed in Sect. 7, the latter correction strongly tive transfer analysis, Schoïer & Olofsson (2001) found that shifts observations toward model predictions. Two estimates the rates resulting from the formulaare systematically under- areprovidedbyOlofssonetal.(1993a),namelyaloweroneas- estimatedwhencomparedtothenewvalues(theirFig.8).As sumingnodriftvelocityforthedust,andanupperoneadopting expected,thediscrepancyincreaseswithdecreasingratei.e.de- an upper limit for drift velocities (up to 20 km/s, i.e. eventu- creasing optical depth. From the comparison of M˙ for 53 allyhigherthanthegasexpansionvelocity)obtainedbyequat- BKR stars in common we obtained a mean correction factor (f) of ing the radiation force on the grains and the drag force due 2.5±1.7toscaletheratesfromOlofssonetal.(1993a)tothe to gas-grain collisions in the limit of supersonic motion (e.g. Schoïer & Olofsson (2001) level.We adoptedmean valuesin Kwok1975).TheestimatesfromGroenewegenetal.makeuse fourranges ofmoderatedriftvelocities,usually2-3km/s.Theirmeanval- ues, (1.2±1.3) 10−3 for Groenewegenet al. (1999, n=8) and 0.94±0.34≃1 if M˙BKR >410−6 (1.45±1.1) 10−3 for Groenewegen et al. (2002b, n=24) are 2.2±1.0 if 410−7 <M˙ <410−6 consistent with that of the “lower” values ((1.3±1.0) 10−3, BKR 2.5±1.4 if 710−8 <M˙ <410−7 n=55)ofOlofssonetal.(1993a),whilethe“upper”valuesyield BKR 3.9±3.1 if M˙ <710−8 (2) a significantly larger mean ((3.2±2.5) 10−3, n=55). Average BKR valuesoftheavailabledataarequotedinTable2,ignoringthe where quoted rates refer to values from Olofsson et al. “upper”onesinOlofssonetal.(1993a).Theythuscorrespond (1993a). This is in fair agreement with Fig. 8 in Schoïer tothelowdriftvelocitiesadoptedhere. & Olofsson (2001). About 70% of the data we used were The sample of Table 2 has 119 carbon-richgiants with at taken from those two references. From 17 stars in common, least one of the three quantities from millimetric data given. we found f=1.9±1.4 for Groenewegen et al. (2002b, 8%) Many“optical”carbonstarsoflowtomoderatemasslossrates while f=3.3±0.3 was deduced from the seven values of areincludedwhilethe“infrared”carbonstarswithhigherrates Groenewegenet al. (1999, 2.7%). Here too, there is some in- and often optically thick dust shells are fewer. This is due to dicationof f tendingto unityforobjectswith verylargerates our requirementthat stellar data be available from Bergeat et J.BergeatandL.Chevallier:ThemasslossofC-richgiants 5 al. (2001, 2002b, 2002c) that, in turn, necessitated sufficient We recall here the equations (1), (3) and (5) from Arndt photometricdatainthevisiblerange.Thisis veryoftenmiss- etal.(1997)whosepredictionsweintendtocomparewiththe ingfor“infrared”carbonstars.Thisisratherunfortunatesince observeddataascollectedinSect.2.Forthemasslossratein 330“infrared”carbonstarswerestudiedbyGroenewegenetal. solarmassperyeartheyderived (2002a,2002b). logM˙ =−4.95−9.45log(T/2600)+1.65log(cid:16)L/104L⊙(cid:17) −2.86log M/M⊙ +0.47log(ǫ /1.8ǫ ) (cid:0) (cid:1) C O 3. Thepredictionsfromhydrodynamicalmodels −0.146log(P/650)+0.449log(∆u/2.0) (4) As mentioned in Sect. 1, both pulsation and dust formation wheretheparametersofmajorinfluencearetheeffectivetem- are involved in the mass loss phenomenonon the asymptotic peratureTinKelvin,theluminosityLandthemassM.Themi- branch.ThemasslossrelationofDominiketal.(1990)isbased nor parametersare the carbonto oxygenabundanceratio, the on calculated models and was the first one available. It de- periodoftheLPVindays,andthepistonvelocityamplitudein scribes the stationary outflows of carbon-rich low mass stars. kms−1.Forthedust-to-gasdensityratio,theyobtained The mass loss rate increases with increasing luminosity and decreasingeffectivetemperature.Thestrongdependenceupon log(cid:16)ρd/ρg(cid:17)=2.74−1.62logT−0.248log(cid:0)L/L⊙(cid:1) theeffectivetemperatureisduetothefactthatdustalwayscon- −0.658log M/M⊙ +2.45log(ǫ /ǫ ) densesataboutthesametemperature.Forstarswithhigheref- (cid:0) (cid:1) C O +0.230logP−4.9310−3log(∆u) (5) fectivetemperatures,thiscondensationtemperatureisreached farther out in the shell where dust formation is less effective. with the same units. In addition to T, L and M, the carbon The mass loss rate increases as the stellar mass decreases. to oxygen abundance ratio is here a major parameter. Again, Otherthingsbeingequal,increasedgravityduetolargermasses the period and piston velocity amplitude have little influence. inhibitsthewind.Thedependenceofthemasslossrateonthe Finally,theygivefortheexpansion(oroutflow)velocity carbontooxygenratioismuchlessmarked,especiallyforval- ues larger than 1.5. Dominik et al. proposed an analytical fit, logv∞ =0.730−0.148logT+0.107log(cid:0)L/L⊙(cid:1) their formula (7), which is presumably accurate for extreme −6.4010−2log M/M⊙ +1.74log(ǫ /ǫ ) masslossrates,i.e.forM˙ ≥10−6M⊙yr−1.Theterminalveloc- +9.5610−2lo(cid:0)gP−7.(cid:1)0710−4log(∆Cu)O (6) ityrangesfrom5to40kms−1.Itincreaseswithdecreasingef- fectivetemperatureandwithincreasingluminosityorincreas- wheretheeffectivetemperature,theluminosityandthecarbon ingC/Oratio.TheauthorsproposedalinearfitastheirEq.(8). to oxygen abundance ratio prove to be the main parameters. Thedust-to-gasdensityratio isclosely relatedto the finalde- Thestellarmass,theperiodofpulsationandthepistonvelocity gree ofcondensation,as shownby formula(9) of Dominiket amplitudehavelittleinfluencehere. al.,anditrangesfrom610−4to510−3. Making use of the models of Fleischer et al. (1992) and 4. Therealmofcarbon-richAGBstars Fleischer (1994), Arndt et al. (1997) derived a grid of 48 dy- namical models usually with less extreme mass loss than in The adopted values of the six parametersin the modelsmen- the cases explored by Dominik et al. (1990). They are real- tionedinSect.3werevariedmoreorlessfreelybytheauthors. istic models of circumstellar dust shells around carbon-rich The resulting Eqs. (4), (5)and (6) are givenfor rangesof pa- long-period variables. They include the calculation of time- rameter values that are those of the used (satisfactory) mod- dependent hydrodynamics as well as a detailed treatment of els. Each parameteris supposedto vary independentlywhich dustformation,growthandevaporation.Themaximumlikeli- is actually notthe case for existing carbon-richgiantson the hoodmethodwasappliedtothegridwithsixparametersthey AGB. We briefly summarize here the results from Bergeat et consideredasindependent,namelythestellartemperature,the al. (2001, 2002b, 2002c) and references therein, on this spe- stellarluminosity,thestellarmass,theabundanceratioofcar- cific topic. The bright carbon-rich giants are located on the bon to oxygen,the pulsational period and the velocity ampli- asymptotic branch, with various initial masses and chemical tudeofthepulsation.Theyobtainedtheirequations(1),(3)and compositions. Consequently, those Galactic giants populate a (5)forthemasslossrate,the dust-to-gasratio andtheexpan- strip in the HR diagram with increasing luminosity for de- sion(outflow)velocity,respectively.Thetrendsareessentially creasing effective temperature (See Figs. 8 and 9 of Bergeat thesameasthoseobtainedbyDominiketal.andsummarized et al. 2002b). Mean values of both quantities were given in above.Keepingconstanttheparametersoflowinfluence,they their Table 3 for the HC-CV groups. This classification was also wrotethe simplified relationstheynumbered(2),(4)and showntobestronglycorrelatedwitheffectivetemperaturede- (6)forthesamethree“massloss”quantities. creasingalongthesequenceHC0toHC5andthenCV1toCV7 Applying a piston and a variable luminosity at the inner (Bergeatetal.2001).Influencedbytruedispersionanderrors boundaryinthe modelsofHo¨fneretal. (1995, 1996), Ho¨fner ondistanceestimates,thepopulatedstripisratherlargebutthe &Dorfi(1997)constructedasetof22carbon-richLPVmod- trend in mean values is obvious. A study of pulsation modes els.TherangesoftheparametersweresmallerbutArndtetal. resulted in the evaluation of pulsation masses in good agree- (1997) concludea goodagreementwith their work for values ment with evolution tracks in the HR diagram (Bergeat et al. incommon. 2002c) for long period variables. Mean pulsation masses and 6 J.BergeatandL.Chevallier:ThemasslossofC-richgiants periods increase with decreasing effective temperature along Table4.Themeanstellardataforcarbon-richgiants(Columns the sequence of photometric groups, as shown in their Table 2-6) from Bergeat et al. (2001, 2002b, 2002c) and the corre- 1.Inamass-luminositydiagram(theirFig.4),themeanlumi- sponding mass loss data (Columns 7-9) calculated from for- nosityincreaseswith increasingmean mass. For the coolcar- mulae (4), (5) and (6) taken from Arndt et al. (1997). For bonvariables(CV-stars),thecarbontooxygenabundanceratio photometric groups of Column 1, the mean values of effec- was plotted against effective temperature in Fig. 7 of Bergeat tive temperatures in Kelvin (Column 2), the mean luminosi- etal.(2002c).Themeanvaluesanddispersionsareincreasing ties l=<L/L⊙ > (Column 3) and masses m=<M/M⊙ > with decreasingeffectivetemperaturesalongthe CV1 to CV6 (Column 4) in solar units, the mean carbon to oxygen abun- sequence. They however decrease below 2500 K at the CV6- danceratioC/O(Column5)andthemeanpulsationperiodPin CV7junction,aresultpossiblyduetolargedustcondensation days(Column6).ThegroupCV5“und”correspondstoasmall (SiCandcarbonaceousgrains)intheatmosphereandresulting sample of underluminous CV5-stars which are not included. carbondepletionin the gasphase.The onlyparameterof for- Adoptingthepistonvelocityamplitude∆u=2kms−1 favored mulae (4) to (6) in Sect. 3 for which we have no indication, byArndtetal.,thecorrespondingmasslossratesM˙ ′ =107M˙ is the piston amplitude velocity.It is howeversupposed to be (Column7)whereM˙ isinsolarmassperyear,thedust-to-gas higher in Miras and semi-regularswith large amplitudes than abundance ratios ρ =104ρ /ρ (Column 8) and the expan- d/g d g it is in semi-regulars with low amplitudes and irregular vari- sionvelocitiesv inkms−1(Column9). e ables.Itsinfluenceishoweversmallonthethreequantitiescal- culatedfromEqs.(4)to(6).Wethusadoptedthe∆u=2kms−1 G Teff l m C/O P M˙ ′ ρd/g ve valueas favoredbyArndtet al. (1997).We havesummarized HC5 3470 1735 0.57 1.01 290 1.7 8.6 6.5 thevaluesinColumns7to9ofTable4ascalculatedfromthe ±0.20 ±0.01 ±70 CV1 3290 2265 0.55 1.005 299 5.4 9.2 6.7 meandataofColumns2to6,followingthesamesequenceof ±0.15 ±0.01 ±98 photometricgroups.Inthosecalculations,weeventuallywent CV2 3095 4130 1.0 1.01 339 5.2 6.3 7.1 ±0.2 ±0.01 ±71 beyondthevalidityrangesofformulae(4)to(6)strictosensu, CV3 2920 4885 1.6 1.03 355 2.7 5.0 7.3 asdefinedbytheparametervaluesthatArndtetal.introduced ±0.3 ±0.03 ±73 CV4 2775 5960 1.9 1.13 353 3.7 5.7 8.8 intheir48models.Discussionsarepostponedtothefollowing ±0.35 ±0.13 ±55 sections. CV5 2640 6530 2.6 1.2 389 2.7 5.8 9.8 ±0.5 ±0.2 ±40 CV5 2645 1600 0.5: 423: und 5. Masslossratevs.effectivetemperature CV6 2435 8635 4.2 1.4 448 2.5 6.9 13.4 ±0.8 ±0.4 ±53 ThemasslossratesM˙ asquotedinTable2(Columns7and16) CV7 1950 16445 8.2: 1.4: 444: 8.7 5.3 13.9 ±0.4 ±107 areplottedonalogarithmicscaleinFig.1asafunctionofthe CV7 1950 16445 4.2 1.4: 444: 62 8.4 14.5 effectivetemperaturesfromColumns3and12.Themaintrend of increasingmass loss rate with decreasingeffectivetemperature is observed as expected from theory (Sect. 3). Typical error bars on both coordinatesare also shown. The Miras are displayed as crosses while the filled circles stand for semi- 10−6M⊙yr−1 in the 2500-2800 K range are representative of regulars. Among the latter, four stars with detached shell(s) temporarily-enhanced mass loss leading to detached shell(s) areshownwith×-symbolsinthe2600-2800Krange.Denoted hundredsora fewthousandsofyearslater (see e.g.Lindqvist by asterisks in Table 2, they are U Ant, S Sct, R Scl and U etal1996forUCam,Izumiuraetal.1997forUAnt,Izumiura Cam. The observationof detachedshell(s) is indicativeof re- etal.1996andLeBertre&Ge´rard2004forYCVn). cent enhanced mass loss. As a consequence, those four stars At lower temperatures, we used symbols surrounded by do exhibit larger mass loss rates than the bulk of our sam- squaresforfivestarswithoptically-thickdustshellsasshown ple, in the same temperature range. Recent observations of a bylargeinfraredexcesses.TheyareCWLeo(IRC+10216),V HCNlineemissioninthecentralsourceofRScl(2625K)that 346PupandLPAnd(threeMiras),andRWLMi(CIT6)and was resolved were secured with a FWHM of 1” by Wong et FXSer(twosemi-regulars).Theyexhibitmasslossratesmuch al. (2004). The authors thus derive for the present mass loss larger than those from the carbon-rich giants with optically- rate 210−7M⊙yr−1 for an adopted distance of 360 pc, a rate thin dustshells in the same 2500-1800K range,that is larger we convertinto3.510−7M⊙yr−1 forour475pcdistance esti- than about610−6M⊙yr−1 for effective temperaturesless than mate (see Column 15of Table 2). Thislatter value,displayed 2500 K. A few intermediate objects are also noticed (V Hya, as a diamond-symbolin Fig. 1, is located in the main locus. BNMon). Thecircumstellarenvelopecharacteristicsof9car- This is about 16 times less than the earlier 5.510−6M⊙yr−1 bonstarswereprobedbyScho¨ıeretal.(2002),combiningradio valuequotedforRSclinColumn16ofTable2,aratherdrastic measurementsandinfraredobservationsfromISO.Theirradia- change.This is convincingevidencefor highly episodic mass tivetransferanalysisallowedustoputconstraintsonthemass loss,possiblycausedbyathermalpulse.Thedetachedshellof loss history of those objects, specially CW Leo=IRC+10216 RSclhasrecentlybeeninvestigatedforMgS,miningthemass- and RW LMi=CIT6 (studied here) and IRAS 15194-5115=II loss history of this object (Hony & Bouwman 1998; see also Lup=CGCS 3592 (not included). They found that those stars Olofsson et al. 2000 for a detailed study of circumstellar CO are notlikely to have experiencedany drastic long-termmass inTTCyg).Weconcludethatmasslossrateslargerthanabout lossratemodulations,atleastlessthanafactorofabout5,over J.BergeatandL.Chevallier:ThemasslossofC-richgiants 7 -4.5 -5 -5.5 dt) M/ -6 d g ( o l -6.5 -7 -7.5 3400 3200 3000 2800 2600 2400 2200 2000 1800 1600 Teff Fig.1. Logarithm of the mass loss rate in solar mass per year as a function of effective temperature in Kelvin, as taken from Table2(observations).Typicalerrorbarsaredisplayedatleft.Semi-regularsandMirasareshownasfilledcirclesandcrosses respectively with 4 non-Miras with detached shells (× and diamond symbol for present R Scl), and 5 objects with optically- thickdustshells(squaredsymbols).Forafewvariables,therearetwosymbolsfortemperaturesobtainedatdistinctphases.The predictionsfrommodels(Table4)areplottedasfilledsquaresconnectedbyfulllines.Additionalpredictionsforvariousmean massesareshownasfilledtrianglesconnectedwithdashedlines(seetext). thepastthousandsofyears.Thisisasituationincontrastwith rangeofeffectivetemperaturesandtheobservedflatportionis thecaseofthedetachedshellsmentionedabove. confirmedby the predictions.We shall identify its cause later Ifweremovetheninesymbolsofthosehighmasslossob- inthissection.Below2400K,theobservedlocusiswellrepro- jectsinFig.1,theremainingfilledcircles(semi-regulars)and ducedbypredictionsprovidedM=4.2M⊙ isadoptedinstead crosses(Miras)populatealocuswithaflatportioninthe2900- oftheuncertainM=8.2M⊙valueestimatedbyBergeatetal. 2400Krangeandforn=66stars, (2002c)fromafewextremeobjects.Thishighvaluewaspos- sibly induced by overestimating the “photospheric” radius of logM˙ ≃(−0.8±1.7)logT −(3.6±5.8) (7) eff very cool giants with dust shells. The right portion of Fig. 1 is an additional argument in favor of masses not exceeding withanaveragevalueof about 4M⊙ for a large majority of luminous carbon-rich gi- <M˙ >≃(5.8±4.4) 10−7 M⊙yr−1 (8) ants.Foreffectivetemperatureshigherthan3000K,thereisan increasingdisagreementbetweenobservationsandpredictions for this wide 10−7−210−6M⊙yr−1 nearly horizontal strip. It upto3400K.Hotvariablesexhibitanirregularpulsationalbe- isflankedbytwo rangesofincreasingmasslossratewith de- haviour with small amplitudes, resulting in dubiouspulsation creasingeffectivetemperature.ForTeff ≥2900Kweobtainfor modeandperiod.Uncertaintiesincreaseaccordingly.Itshould 32stars be notedthat, accordingto Arndtet al. (1997), one has to re- frainfromapplyingEqs.(4)to(6)beyond3000Ksincenone M˙ ≃4.511020T−7.83 M⊙yr−1 (9) eff of their 48 models lies there. In addition, none of their mod- andforT ≤2400Kwith17stars els is fainter than 5103L⊙ contraryto the observedaveraged eff meanluminositiesofHC5,CV1andCV2-starsinTable4.We M˙ ≃8.461024T−9.20 M⊙yr−1 . (10) however attempted to apply the <M>=1.6M⊙ mean value eff of the CV3-group in Table 4 to the other earlier groups CV1 Those three relations which are not satisfied by the 9 objects andCV2 with otherdataotherwiseunchanged.Togetherwith withstrongmasslossandpossiblyafewothers,arenotshown an additional calculation from Eq. (4) assuming for CV1 the inFig.1.InsteadthepredictionsofSect.4asquotedinTable4 <M>=1.0M⊙valuequotedforCV2inTable3,theyareplot- areplottedasfilledsquare-symbolsconnectedbyafullbroken tedasfilledtrianglesinFig.1.Thedashedlinesconnectingthe line. A very good agreementis observed in the 3000-2400K 8 J.BergeatandL.Chevallier:ThemasslossofC-richgiants 30 25 C/O = 2 20 e 15 v 10 5 C/O = 1 0 3400 3200 3000 2800 2600 2400 2200 2000 1800 Teff Fig.2.Outflow(expansion)velocitiesinkm/sasafunctionofeffectivetemperaturesinKelvin,astakenfromTable2(observations)andTable4(predictionsfrommodels).Typicalerrorbarsarealsodisplayedatleft.SamesymbolsusedasinFig.1.with 5non-Miraswithdetachedshells(×).Thepredictionsfromtheoryareplottedasfilledsquaresconnectedbyfulllines.Extreme linescorrespondingtoC/O=1andC/O=2respectivelyarealsoshown(seetext). CV3-square to the filled triangles for CV1 and CV2 and the standardmass loss in objects ofdiminished masses. Itshould initial segmented fullline reasonablywell encompassthe ob- benotedthatArndtetal.(1997)didnotcalculatemodelscooler servations.ThisresultsuggeststhattheslopeforT >2900K than2300K.Additionaldataandmodelsareclearlynecessary eff in Fig. 1 is indicative of a nearly uniform value of about one beforeafirmconclusioncanbereached. solar mass. A large correction factor (3.9 or 2.5) from Eq. We shall see in Sect. 6 that the four stars with detached (2) was applied to part of the contributing data. Without that shells mentioned above also have expansion velocities larger correction,theshiftofobservationsbelowthecontinuousline than average. Actually, it happens that velocities are little in- of predictionswould have been substantially larger. Similarly fluencedbyadoptedmasses(Sects.3and6).Inaddition,RScl the slope below 2400 K seems to correspondto a nearly uni- hasnowreturnedtolowmasslossrates(Wongetal.2004).For form value of about 4M⊙. The latter is predicted as the up- those stars with detached shells and intermediate (2400-2900 perlimitfromevolutionarycalculationspresumablyduetohot K) effective temperature, the above explanation of enhanced bottom-burningmost effective in AGB stars of larger masses masslossintermsofdiminishedmassesisnotacceptable.By (Bergeat et al. 2002b and references therein). We also calcu- continuity, we are lead to the conclusion that the 3000-2400 latedthemasslossrateoftheCV6andCV7-groups,assuming the<M>=2.6M⊙valuequotedforCV5inTable3,keeping K flat portioncorrespondsto the approximate1−4M⊙ mass range. Discussing their formula (1), i.e. our Eq. (4), Arndt et the otherparametersunchanged.They are shown as filled tri- al.(1997)concludedthattheabundanceratio,pulsationperiod anglesconnectedtotheCV5-squarebyabrokendashedline.A and piston amplitude velocity have only small influences on possibleexplanationofhighermasslossratemightbesmaller themasslossrate.Accordingly,wefindnocleardifferencebe- mass due to previous strong mass loss. Precisely, Scho¨ıer et tween the loci of Miras and semi-regulars. It remains to con- al.(2002)deducedthatmasslossratesremainedhighoverthe sidertheeffectivetemperature,theluminosityandthemassof past thousands of years in objects like CW Leo=IRC+10216 thestar.Since,onaverage,luminosityincreaseswithdecreas- and RW LMi=CIT6. From the comparison of theoretical nu- ingeffectivetemperature,bothparameterscontributetoanin- cleosynthesismodelsand measurementsofabundancesin the creaseofthemasslossrate.FollowingEq.(4),thatincreaseis circumstellarenvelope,Kahane et al. (2000) favor a mass not larger than 2M⊙ for CW Leo, well below the 4.2M⊙−limit inhibited in the 3000-2400K range by the increase of stellar massinthe1−4M⊙domain.The3000-2400Kflatportionin consideredforthestarsofEq.(10).Wintersetal.(1994a)pre- viouslyestimated M≤2M⊙ with C/O≃1.4for CW Leo. In Fig.1isafurtherargumentinfavorofthe1−4M⊙rangefor brightcarbon-richgiants(lessluminousobjectscanpopulate otherwords,theapparent(lasting)“superwinds”couldwellbe the0.5−1M⊙ domain),inadditiontotheuseofevolutionary J.BergeatandL.Chevallier:ThemasslossofC-richgiants 9 tracksintheHRdiagramandtothecalculationsofpresentpul- 7. Dust-to-gasratiovs.effectivetemperature sationmasses(Bergeatetal.2002b,2002c).Weconcludethat Thediagramofthedust-to-gasratioagainsteffectivetempera- themodelsandrelationsfromArndtetal.(1997)satisfactorily tureisnotshownhere.Thereisactuallynomarkedslopewhen reproducethemasslossratededucedfrommillimetricobserva- Eq.(5)andthecorrespondingpredictionsofTable4areused. tions.Largermeanmassthusimplies,otherthingsbeingequal, In the 2300-3000K validity domainof modelsfrom Arndtet alowermasslossrate. al.(1997),themeanvalueamountsto <ρ >=<ρ /ρ >≃(5.9±0.7) 10−4 (11) d/g d g 6. Outflowvelocityvs. effectivetemperature As described in Sect. 2, the mean observed values quoted in Theoutfloworexpansionvelocitiesv ,i.e.estimatesofveloc- Table 2are typically“lower”onesderivedassuminglow drift e ityatinfinityinkm/sasquotedinTable2(Columns8and17) velocities.For72documentedstars,weobtained areplottedinFig.2asafunctionoftheeffectivetemperatures M˙ /M˙ ≃(1.3±1.1) 10−3 (12) d g fromColumns3and12.Themain trendof increasingexpan- sionvelocitywithdecreasingeffectivetemperatureisobserved avalueabout2.2timeslargerthanthepredictedone.Thereis asexpectedfromtheory(Sect.3),butwithalargescatterhere. practically no slope in the diagram of “observed” dust-to-gas Typicalerrorbarsonbothcoordinatesarealsoshownandare ratios of rates against effective temperatures from Table 2, at much smaller. The same symbols as in Fig. 1 are used. The least within the large error bars. It must be emphasized here ×-symbols point to five stars with detached shells (U Ant, S thatthedatausedwascorrectedforthef-factorofSect.2and Sct,RScl,UCamandTTCyg).Thelatteralsoshowlargeve- forasmallerfactorforadoptedluminosity llaorcgiteiresth(a1n4-p2r1edkimct/esd)fvoarluTeesff.≃W2e6a0l0re−ad2y8f0o0uKndabinouSteactf.a5ctothra2t (cid:16)M˙d/M˙g(cid:17)BKR =(cid:16)M˙d/M˙g(cid:17)Lit (LBKR/LLit)−0.5 (13) theydisplayhighermasslossratesthanpredicted.Theissueis Without that f-correction, the above discrepancy ratio would dubiousfor the five coolextreme objectsof Sect. 5. Theyare have jumped from 2.2 to about 6. The mean “upper” value scatteredwith CWLeo=IRC+10216onthepredictedrelation of(3.2±2.5) 10−3 derivedinSect.2isabout5.4timeslarger andFXSerlocatedwellaboveit. than the Eq.(11)predictedvalue.Apparently,adoptinglarger Thepredictionsofv∞fromEq.(6)asquotedinTable4,are drift velocities could make the discrepancy between observa- showninFig.2asfilledsquare-symbolsconnectedbyabroken tions and predictions even worse, but predicted values would fullline.Thelatterfitsreasonablywellthe86%ofthestarsin alsobeincreased.Veryrecentmodelswiththeinfluenceofthe thesamplethatarelocatedinthelowerpartofthediagram.The grain drift velocity properly treated (Sandin & Ho¨fner 2004) modelsseem to correctlypredictthelow expansionvelocities can contribute to shift predictionscloser to observations. The whichroughlycorrespondtothelowmasslossratesofSect.5, derivationoftheratiofromtheobservationsmakesuseofdata themselveswell-predictedbythesamemodels.Thedecreasing from both infrared and millimetric ranges, with simpliflying effective temperature and increasing luminosity along the se- assumptions(like a uniquetemperatureforinstance).Itis not quenceofTable3bothcontributetotheincreaseoftheoutflow clearwhetherthesamelayersareinvolvedintheextendedat- velocitybuttheeffectissmallwhencomparedtotheinfluence mosphereandshell(frominnerdustcondensationzonetoouter of the ǫ /ǫ abundance ratio. This point is illustrated by the regionswithterminalvelocityreached). C O twodashedlinesinFig.2asobtainedkeepingtheratioto1or FollowingEq.(5)takenfromArndtetal.(1997),itappears 2insteadofthemeanvaluesofTable4.Withinerrors,theob- thatinadditiontodecreasingeffectivetemperatureandincreas- servedvaluesliebetweenthosetwolines.Thisispreciselythe ing luminosity whose influences counteract here, the stellar rangeoftheabundanceratioasderivedfromspectraandmodel massandtheǫC/ǫO abundanceratiodohaveanappreciablein- atmospheresofthecoolcarbon-richgiants(Lambertetal.1986 fluence.Thepredicteddust-to-gasdensityratioincreaseswith andreferencescitedinSect.2;seealsoFig.7ofBergeatetal. both decreasing stellar mass and increasing abundance ratio. 2002c). Adopting for instance M=1M⊙ and ǫC/ǫO ≥2 can rise the We alsonotethatabove2900KtheobservationsinFig.2 Eq.(11)valuehigherthanthatinEq.(12),reachingthelevelof tendtoconcentratetowardstheC/O=1line,below10-15km/s. thehighest“lower”valuesobserved.Itseemsoutsidetherange Thiscanbeattributedtothefactthatstarsinthehotterdomain ofacceptableparametervaluestakingintoaccounttheanalyses ofSects.5and6fromTable4data.However,keepingthemass ofTable 4 dohavemeanabundanceratiosclose to unitywith a small range apart. Below 2900 K (CV3 and later groups), rangeofTable4thatfitsnicelythegasmasslossratedata(Sect. the C/O mean values and ranges increase (Table 4). On av- 5),anincreaseofabout20%oftheC/OratiosfromLambertet erage, larger velocities are reached there. The influence of al.(1986)canhelp,whichremainscompatiblewiththeveloc- the C/O ratio is responsible for a triangular void in Fig. 2 at itycomparisoninFig.2.Theirabundanceratiosfor30carbon Teff ≥2800K that is inserted between the C/O=1 and C/O=2 starsleadto< (cid:16)12C+13C(cid:17)/H >≃ (5.5±1.6) 10−4innumbers lines. Predictionsare consistentwith millimetric observations ofatomsand(2.7±1.0) 10−3inmassforagasprincipallyofH andvisibleandinfraredspectroscopywith1≤ǫ /ǫ ≤2,and andHe,iftheirC/Oratiosarekept.Theestimateisthenof22% C O with themeanvaluesanddispersionsasfunctionsofeffective ofcarbonlockedupingrainsifpredictionofEq.(11)istrusted temperaturefrom Bergeatetal.(2001),asdisplayedinFig.7 against 48% for the mean observed ratio of Eq. (12). If con- ofBergeatetal.(2002c). firmed,such figureswouldmeanthat carbonmoleculescould 10 J.BergeatandL.Chevallier:ThemasslossofC-richgiants beappreciablydepressed.Veryhighdriftvelocitiescouldlead loweffectivetemperatures,Eq.(10)correspondstoaconstant tounreasonablyhighestimates. mass of about 4M⊙. At high temperatures, Eq. (9) describes carbon variables of about1M⊙ or less, but both observations andpredictionsare uncertainthere.Ninegiantsinoursample 8. Discussionandconclusions (at least) do have mass loss rates larger than predicted from Wehavecomparedtheresultsofmillimetric(andinfrared)ob- Eqs. (7) to (10). Four cases of detached shells are found in servations,namelygasmasslossrate,expansion(outflow)ve- the2600-2800Krange,i.e.objectshavingexperiencedhighly locityanddust-to-gasdensityratio,tothepredictionsfromhy- episodicmasslosspossiblycausedbyathermalpulse.Oneof drodynamicaltime-dependentmodelsofdust-drivenwindsfor them(RScl)hasreturnedtoamoderatemasslossrate(Wonget carbon-rich chemistry. Corrections were applied to the pub- al.2004),andthisiscertainlytruefortheothers(e.g.Olofsson lished “observed” values of the first (distance and radiative etal.2000forTTCyg).Thisisastrongargumentinfavorofa transfer)andthethirdquantity(luminosityandradiativetrans- specificmasslossmechanismatwork.Theotherfivestarsare fer) as described in Sect. 2. The prediction of models are ap- very cool objects with T ≤2500K and optically-thick dust eff pliedtolongperiodvariableswhosepulsationissimulatedbya shells shown by strong infrared excesses, a consequence of pistonofgivenamplitudevelocityandperiodactingatthebasis very high mass loss rates, like RW LMi (CIT6) or CW Leo oftheatmosphere.We madeuseoftheapproximateformulae (IRC+10216). Scho¨ıer et al. (2002) have shown that the gas of Arndt et al. (1997) based on 48 models exploring ranges masslossrateofthoseextremeobjectsdidnotvarymuchover in the six parameters they include. Among them, the pulsa- the last thousand of years. As argued in Sect. 5, they may be tion period and the piston amplitude velocity have only little stars with masses of about 1.5−2.5M⊙, in contrast with the influence(Sect. 3).We foundnomarkeddifferenceinthe ob- 4.2M⊙ favored for the cool objects with lower rates as given serveddatabetweenMirasandnon-Miras(heresemi-regulars byEq.(10).Iftrue,nospecificmasslossmechanismwouldbe withavailableperiods).Fourparametersremainwhoseranges requiredhere,contrarytothecaseofdetachedshells. may influence the results, namely effective temperature taken Agoodagreementisalsoobtainedbetweenpredictionsand asthestellartemperatureusedinthemodels,luminosity,mass observationsinthediagramofexpansion(outflow)velocityvs. andcarbontooxygenabundanceratio.Theobservationaldata effectivetemperature(Fig.2),asdescribedinSect.6.Theex- wassummarizedinTable2for119carbon-richgiantsformerly pansion (outflow)velocity increases with decreasingeffective studiedbyBergeatetal.(2001,2002band2002c),foreffective temperature,withabout86%oftheobjectswithintherangeof temperatures,luminosities,stellar massesand C/O abundance predictionsasquotedin Table4 fromEq.(6).Towithin error ratio.Theformulae(4),(5)and(6)astakenfromArndtetal. bars, the observations are located between the two lines ob- leadforthemeanvaluesofBergeatetal.totheestimatedmeans tained forǫ /ǫ =1 and ǫ /ǫ =2 respectively,the other pa- C O C O of Table 4 for the samples corresponding to the photometric rametersbeingkeptequaltotheirvaluesinTable4(including groupsHC5andCV1toCV7witharangeofeffectivetemper- themeanmassdistributionsuccessfullyusedformasslossrates atures from 3470 K down to 1950 K. This latter parameteris and described above). This is the range deduced from spec- considered as the most accurate one at present and it is used troscopyandmodelatmospheres(Lambertetal.1986;seealso forreference.Wealsoholdfixedtoreferencevaluesthestellar Fig. 7ofBergeatetal. 2002c).InFig. 2, itis remarkablethat massandtheC/Oabundanceratiokeepingunchangedtherun for effectivetemperatureshigherthan about2900K, the stars of effective temperaturesand luminosities of Table 4, namely concentratetowardstheǫ /ǫ =1line,below10-15km/s.The C O theTP-AGBofGalacticcarbon-richgiantsintheHRdiagram, maximumobservedvelocityrapidlydecreaseswithincreasing asobtainedbyBergeatetal.(2002c). temperature,leavingemptyalargetriangularareainFig.2.The AsdescribedinSect.5,weobtainedagoodagreementbe- resultofBergeatetal.(2001,Fig.7in2002b)isconfirmedthat tweenpredictionsandobservationsinthediagramofmassloss above2900K, theC/O ratioremainscloseto unitywith little rate vs. effective temperature (Fig. 1). For about 90% of our dispersion.Theǫ /ǫ abundanceratioisconfirmedhereasthe C O sample, the mechanism involving gas lifted by pulsation and thirdmajorparameterwitheffectivetemperatureandluminos- radiationpressureoncondenseddustisconfirmed.Threeequa- ity, while stellar mass has little influence in Fig. 2. Cases of tions(7) and thus(8),(9) and (10)were obtainedin the three superwind however blur the information from that figure. We rangesofeffectivetemperaturesdistinguished.Theusualstate- conclude that both mass loss rates and outflow velocities are ment(vanLoonetal.2003)thatM˙ ∝T−8isapproximatelytrue correctlymodeledfromrecenthydrodynamicalcalculationsof eff onlyabove2900K(Eq.(9))andbelow2400K(Eq.(10)),with dust-driven winds. Both the levels and the dependenceon ef- differentproportionalitycoefficientshowever. There is practi- fectivetemperaturearefairlywellreproduced. cally no variation over the 2400-2900 K range in Fig. 1 (in- Boththepredictedandobservedratiosofdust-to-gasmass cluding about 55% of our sample). Predictions and observa- loss rates show little variation with effective temperature if tions are in agreement provided the variations of luminosity any. As found in Sect. 7, the average observed value of ontheAGBintheHRdiagram(Bergeatetal.2002b)andthe <M˙ /M˙ >≃(1.3±1.1) 10−3 is a factor 2.2 larger than the d g meanmassesrangingfromabout1M⊙around3000Ktoabout mean predicted value <ρ >=<ρ /ρ >≃(5.9±0.7) 10−4 d/g d g 4M⊙around2400K(seeTable4),asdeducedfromtheoretical fromEq.(5).WehavediscussedinSect.7theinfluenceofthe tracksintheHRdiagramandfrompulsationsmasses(Bergeat stellar mass and C/O abundanceratio. It should be noted that etal.2002c).Heremassisthethirdimportantparameter,with theuncertaintiesareverylargehere.Amongthepossibleexpla- influencemuchlargerthanthatoftheC/Oabundanceratio.At nations is a moderate underestimate of abundance ratios (say