The effects of thermohaline mixing on low-metallicity asymptotic giant branch stars
aa r X i v : . [ a s t r o - ph . S R ] D ec Mon. Not. R. Astron. Soc. , 1–12 (0000) Printed 6 September 2018 (MN L A TEX style file v2.2)
The effects of thermohaline mixing on low-metallicityasymptotic giant branch stars
Richard J. Stancliffe ⋆ Centre for Stellar and Planetary Astrophysics, Monash University, VIC 3800, Australia
Accepted 0000 December 00. Received 0000 December 00; in original form 0000 October 00
ABSTRACT
We examine the effects of thermohaline mixing on the composition of the envelopes oflow-metallicity asymptotic giant branch (AGB) stars. We have evolved models of 1, 1.5and 2 M ⊙ from the pre-main sequence to the end of the thermally pulsing asymptoticgiant branch with thermohaline mixing applied throughout the simulations. In agree-ment with other authors, we find that thermohaline mixing substantially reduces theabundance of He on the upper part of the red giant branch in our lowest mass model.However, the small amount of He that remains is enough to drive thermohaline mixingon the AGB. We find that thermohaline mixing is most efficient in the early thermalpulses and its efficiency drops from pulse to pulse. Nitrogen is not substantially af-fected by the process, but we do see substantial changes in C. The C/ C ratio issubstantially lowered during the early thermal pulses but the efficacy of the process isseen to diminish rapidly. As the process stops after a few pulses, the C/ C ratio isstill able to reach values of 10 − , which is inconsistent with the values measuredin carbon-enhanced metal-poor stars. We also note a surprising increase in the Liabundance, with log ǫ ( Li) reaching values of over 2.5 in the 1.5 M ⊙ model. It isthus possible to get stars which are both C- and Li-rich at the same time. We compareour models to measurements of carbon and lithium in carbon-enhanced metal-poorstars which have not yet reached the giant branch. These models can simultaneouslyreproduced the observed C and Li abundances of carbon-enhanced metal-poor turn-offstars that are Li-rich, but the observed nitrogen abundances still cannot be matched. Key words: stars: evolution, stars: AGB and post-AGB, stars: Population II, stars:carbon
It has long been known that models of asymptotic giantbranch (AGB) stars that only include mixing in convec-tive regions are incomplete. These canonical models cannotaccount for observations such as: the low C/ C ratiosin low-mass AGB stars (Abia & Isern 1997; Lebzelter et al.2008), Li and C-rich stars in our Galaxy (Abia & Isern 1997;Uttenthaler et al. 2007), isotopic ratios measured in pre-solar grains (e.g. Nollett et al. 2003, and references therein).It has therefore been suggested that material might circulatebelow the base of the convective envelope and into regionswhere nuclear burning can happen. This process is often re-ferred to as ‘cool bottom processing’.The carbon-enhanced metal-poor (CEMP) stars alsoshow abundance trends that are difficult to explain in thecontext of AGB models including only convective mixing. Inparticular, those CEMP stars that are rich in s -process el- ⋆ E-mail: Richard.Stancliff[email protected] ements (which we can, with some confidence, assume comefrom mass transfer in binary systems that once containedan AGB star) show C/ C ratios no greater than 100(e.g. Sivarani et al. 2006), even for stars at the Main Se-quence turn-off. AGB models predict C/ C ratios in ex-cess of 10 (Stancliffe et al. 2009). In addition, the mod-els predict that low-mass AGB stars should be rich in car-bon, but nitrogen-poor. Yet there is a tendency for thesestars to show nitrogen-enhancement alongside their carbon-enhancement . Furthermore, there have been detections oflithium in CEMP stars (e.g. Thompson et al. 2008). Canon-ical AGB models produce Li via the Cameron-Fowler mech-anism (Cameron & Fowler 1971), which involves the pro-duction of beryllium deep in the hydrogen burning shell viathe reaction He( He, γ ) Be and the immediate transport of However, Masseron et al. (2009) note that this trend only holdswhen considering CEMP stars as a whole. For each of the indi-vidual CEMP sub-classes, this trend does not hold.c (cid:13)
R. J. Stancliffe this to cooler regions of the star where the Li that forms(once the beryllium has undergone electron capture) is sta-ble against proton captures. This takes place in the moremassive stars which undergo hot bottom burning (HBB,where the base of the convective envelope lies in the topof the hydrogen-burning shell). Such stars would be rich innitrogen, not carbon. These observations suggest that some-thing is missing from the AGB models and an extra mixingmechanism must be at work.Lithium is a particularly important element fromthe point of view of mixing processes in CEMP stars.Stancliffe et al. (2007) pointed out that material accreted onto a low-mass companion does not just remain at the surfaceof the star and thermohaline mixing could efficiently mixthis material deep into the stellar interior. However, detec-tion of lithium in the CEMP binary system CS 22964-161 ledThompson et al. (2008) to suggest that the mixing efficiencycould not be so high. Li is a fragile element and is easily de-stroyed at temperatures in excess of about 2 . × K. Evena modest depth of mixing can lead to efficient Li-depletion(Stancliffe 2009) and hence the measurement of Li in CEMPstars could be a good test of the efficiency of thermohalinemixing. It is therefore crucial that we understand the originof this element.Despite the apparent need for extra mixing on both thegiant branches, the physical nature of the mechanism (ormechanisms) has proved illusive. Recently, Eggleton et al.(2006) showed that the lowering of the mean molecularweight by the reaction He( He,2p) He could lead to mix-ing in red giants via the thermohaline instability. Thiscan potentially explain the change in abundances seenin giants above the luminosity bump (Charbonnel & Zahn2007; Eggleton et al. 2008). The effect of this mecha-nism has been investigated beyond the first giant branch(Cantiello & Langer 2008) and in super-AGB stars (Siess2009). In this work, we wish to examine what the con-sequences of thermohaline mixing are for low-mass, low-metallicity AGB stars.
Calculations in this work have been carried out using the stars stellar evolution code which was originally developedby Eggleton (1971) and has subsequently been updated bymany authors (e.g. Pols et al. 1995; Stancliffe & Eldridge2009). The version used here includes the nucleosynthe-sis routines of Stancliffe et al. (2005) and Stancliffe (2005),which follow the nucleosynthesis of 40 isotopes from Dto S and important iron group elements. The code usesthe opacity routines of Eldridge & Tout (2004), which em-ploy interpolation in the OPAL tables (Iglesias & Rogers1996) and which account for the variation in opacity asthe C and O content of the material varies. An approxi-mation of the contribution to the molecular opacities is in-cluded via the method of Marigo (2002) and is described inStancliffe & Glebbeek (2008) . The modifications for follow- Recently, low-temperature opacity tables for variable com-positions have become available (Lederer & Aringer 2009;Marigo & Aringer 2009). These will eventually replace the lessaccurate approximations used here. ing the evolution through the thermally-pulsing AGB (TP-AGB) phase are described in Stancliffe, Tout & Pols (2004).Thermohaline mixing is included throughoutall the evolutionary phases via the prescription ofKippenhahn, Ruschenplatt & Thomas (1980), with themixing coefficient being multiplied by a factor of 100 assuggested by the work of Charbonnel & Zahn (2007). Theseauthors find that with a factor of this magnitude they areable to reproduced the abundance trends observed towardsthe tip of the red giant branch. Stancliffe et al. (2009)also showed that a coefficient of this magnitude couldreproduced the observed mixing trends in both low-massmetal-poor stars and carbon-enhanced metal-poor stars onthe upper part of the first giant branch.We evolve stars of 1, 1.5 and 2 M ⊙ from the pre-main se-quence to the end of the thermally pulsing asymptotic giantbranch (TP-AGB) using 999 mesh points. Reimers (1975)mass-loss prescription, with η = 0 .
4, is used from the MainSequence up to the TP-AGB; the Vassiliadis & Wood (1993)mass-loss law is employed during the TP-AGB. A mixinglength parameter of α = 2 . Z = 10 − ([Fe/H] ≈ − .
3) and the ini-tial abundances are assumed to be solar-scaled accordingto Anders & Grevesse (1989), with the exception of Li, forwhich we adopt a value of X Li = 1 . × − which is equiva-lent to the Spite plateau value. It would be more appropriateto adopt an α -enhanced composition for these models as themetal-poor stars are observed to have [ α /Fe] ≈ .
4. However,opacity tables for an α -enhanced mixture that have also beencomputed with variable C and O abundances do not exist.We therefore choose to have our models self-consistent bykeeping our initial compositions solar-scaled like our opac-ity tables. The evolution of He, log ǫ ( Li) and the C/ C ratio atthe surface of each of the models, up to the beginning ofthe first thermal pulse in each model, is displayed in Fig. 1.The behaviour of each model is qualitatively similar. The He abundance initially increases before dropping slightlyand then levelling off. At the same time, the lithium abun-dance and the C/ C ratio both drop. These changes areall caused by the onset of first dredge-up – the deepeningof the convective envelope as the star ascends the red giantbranch. In each of the models, thermohaline mixing beginsto affect the surface abundances at around 10 years beforethe first thermal pulse. The 1 M ⊙ model is most stronglyaffected and its surface He abundance (by mass fraction)falls from around 9 × − to about 1 . × − . At the sametime, the remaining Li is efficiently destroyed by the mixingand the C/ C ratio drops from its post-dredge-up valueof around 30 to around 5.The two higher mass models bring less He to the sur-face during first dredge-up. In addition, their surfaces areless affected by thermohaline mixing on the giant branch.There are only minor reductions in the surface abundances [A/B] = log ( N A /N B ) − log ( N A /N B ) ⊙ , where N i is thenumber abundance of species i .c (cid:13) , 1–12 hermohaline mixing in low-Z AGB stars Figure 1.
Evolution of the surface abundances of various chem-ical species for the 1 M ⊙ model (dotted line), 1.5 M ⊙ model(dashed line) and 2 M ⊙ model (solid line). of He, Li and the C/ C ratio. The mean molecularweight inversion that formed during the giant branch is notcompletely removed by the time the star reaches the tip ofthe giant branch and consequently thermohaline mixing con-tinues to act throughout the core He-burning phase (as wasnoted by Cantiello & Langer 2008). The surface changes itcauses are not dramatic and only the lithium shows a no-ticeable level of depletion. Eventually, as the convective en-velope deepens when the star ascends the early AGB, thesurface abundances of He, Li and the C/ C ratio all fallslightly as the envelope once again dredges up material thathas undergone nuclear burning.Each of the models is evolved to the onset of the super-wind phase at which point numerical problems occur and theruns were terminated. As it is clear that the effects we areinterested in only occur at the beginning of the TP-AGB,we need not be concerned with this failure to complete thefull thermally pulsing phase. In addition, because each of themodels has entered the superwind, mass loss would quicklystrip off the remaining envelope and we should not havemissed more than a couple of pulse cycles. These missingpulses will not substantially affect the composition of theejecta. Details of the models are given in Table 1.The occurrence of mixing by thermohaline convectiondriven by the burning of He is seen to occur in each of themodels. Mixing is able to affect the abundances of He, Liand C. Heavier elements are not affected as their proton-burning reactions are activated at much higher temperaturesthan the mixing mechanism is able to reach. We discuss indetail the case of the 1.5 M ⊙ model, the salient propertiesof which are displayed in Fig. 2. The other models behavein much the same way (see section 3.3 for a discussion ofthe differences between them). The mixing is at its mostefficient in the earliest pulses. After the first thermal pulsewith dredge-up, the C/ C ratio falls dramatically in the interpulse. Just after the end of third dredge-up (TDUP)the C/ C ratio is around 600 while by the end of theinterpulse it has fallen to just 120. At the same time the Heabundances drops from 2 . × − to 2 . × − and the Liabundances increases from zero to 1 . × − (equivalentto log ǫ ( Li) = 1 . ). On the next pulse, the effect isdiminished with the C/ C ratio being 540 at the end ofTDUP and by the end of the interpulse it has dropped to350. The He abundances falls to 2 . × − and log ǫ ( Li)reaches 2.14.Why does the effect fall off so rapidly? The point atwhich the µ -minimum is located is critical. What deter-mines this is the competition between two things: the rateat which the mean molecular weight can be lowered via He( He,2p) He and the rate at which it is increased byother reactions producing He. In the absence of efficientCNO cycling in the H-shell the µ -minimum can lay deeperin the star as the pp-chains dominate. When dredge-uphappens, the enhanced C-abundance increases the efficiencyof the CNO cycle. This forces the µ -minimum outward tocooler temperatures and higher He abundances, with themagnitude of the µ -dip being reduced. The effect of this istwo-fold. First, the reduction in the size of the µ -dip makesthe mixing less efficient, as the µ -gradient (which the mix-ing rate is proportional to) is also reduced. As the locationof the µ -dip occurs at progressively higher He abundancesfrom pulse to pulse (the He abundance at the µ -minimumis around 6 × − during the second interpulse, 1 . × − during the third interpulse and 2 × − during the fourthinterpulse), the rate of He depletion falls off because mate-rial that is less depleted in He is now being mixed into theenvelope. Secondly, the µ -minimum is pushed out to a re-gion of cooler temperature. In this region, the C(p , γ ) Nreaction is less efficient and the material that is mixed to thesurface has a smaller fraction of C (formed from the decayof N). This, coupled with the reduction in the mixing effi-ciency, means that the C/ C ratio falls more slowly witheach subsequent pulse.
Li production takes place in three stages and begins withthe deepening of the convective envelope at TDUP. This ho-mogenises the convective envelope, resulting in a constantmean molecular weight above the soon to ignite H-burningshell. Without this flattening of the µ -profile, the burning of He would not be able to produce a large enough change inthe mean molecular weight to allow thermohaline mixing totake place. After TDUP has occurred, the convective enve-lope retreats, hydrogen burning reignites and the He + Hereaction leads to the production of Be (top panels of Fig. 3).At the same time, the He( He , He reaction leads to areduction of the mean molecular weight. Once a µ -inversiondevelops, thermohaline mixing can take place and the beryl-lium is brought up to regions of cooler temperature wherethe Li that is formed from the beryllium is able to sur-vive for longer than it can closer to the burning shell (top log ǫ (X) = log ǫ ( N X /N H ) + 12, where N X is the numberabundance of species X and N H is the number abundance of hy-drogen.c (cid:13) , 1–12 R. J. Stancliffe ⊙ TP M ∗ M H τ ip log ( L maxHe / L ⊙ ) ∆M H ∆M DUP λ [C/Fe] log ǫ ( Li) C/ C( M ⊙ ) ( M ⊙ ) (10 yr) ( M ⊙ ) ( M ⊙ )1 0.868 0.537 ... 4.650 0.00828 ... ... -0.67 ... 5.72 0.868 0.546 17.58 6.71 0.00475 ... ... -0.67 ... 5.73 0.868 0.550 29.39 7.37 0.00806 0.00017 0.021 0.61 0.32 62.94 0.868 0.558 26.03 7.55 0.00922 0.00142 0.154 1.80 0.55 8125 0.868 0.566 17.90 7.45 0.00884 0.00095 0.107 1.98 0.65 11646 0.868 0.574 12.65 7.11 0.00714 ... ... 1.98 0.72 11067 0.867 0.581 13.91 7.52 0.00887 0.00105 0.118 2.17 0.78 16048 0.866 0.589 12.34 7.51 0.00864 0.00058 0.067 2.23 1.03 16499 0.840 0.597 11.50 7.62 0.00883 ... ... ... ... ...1.5 M ⊙ TP M ∗ M H τ ip log ( L maxHe / L ⊙ ) ∆M H ∆M DUP λ [C/Fe] log ǫ ( Li) C/ C( M ⊙ ) ( M ⊙ ) (10 yr) ( M ⊙ ) ( M ⊙ )1 1.436 0.586 ... 5.63 0.01152 ... ... -0.32 ... 16.92 1.436 0.598 12.64 7.17 0.00552 0.00168 0.304 1.24 1.56 1223 1.436 0.601 12.05 7.45 0.00709 0.00371 0.523 1.88 2.13 3504 1.436 0.605 10.65 7.61 0.00835 0.00544 0.651 2.25 2.33 7445 1.436 0.608 10.10 7.80 0.00959 0.00637 0.664 2.47 2.42 12296 1.436 0.611 9.65 7.97 0.01028 0.00686 0.667 2.63 2.46 17357 1.435 0.614 9.16 8.05 0.01061 0.00709 0.668 2.74 2.49 22218 1.427 0.618 8.63 8.13 0.01068 0.00706 0.661 2.83 2.51 26729 0.830 0.621 8.11 8.17 0.01060 ... ... ... ... ...2 M ⊙ TP M ∗ M H τ ip log ( L maxHe / L ⊙ ) ∆M H ∆M DUP λ [C/Fe] log ǫ ( Li) C/ C( M ⊙ ) ( M ⊙ ) (10 yr) ( M ⊙ ) ( M ⊙ )1 1.960 0.41469 ... 5.94 0.23905 ... ... -0.40 0.64 14.12 1.960 0.653 6.66 7.08 0.00429 0.00314 0.732 1.39 1.13 5973 1.960 0.654 6.59 7.50 0.00633 0.00568 0.897 1.94 1.29 20654 1.960 0.655 6.78 7.90 0.00823 0.00778 0.945 2.26 1.38 42195 1.960 0.656 7.02 8.22 0.00980 0.00924 0.943 2.46 1.43 65526 1.960 0.656 7.13 8.45 0.01087 0.01009 0.928 2.60 1.44 87287 1.960 0.657 7.13 8.63 0.01157 0.01065 0.920 2.71 1.44 106848 1.959 0.658 6.98 8.72 0.01188 0.01079 0.908 2.79 1.43 123649 1.956 0.659 6.77 8.78 0.01197 0.01080 0.902 2.86 1.40 1393610 1.946 0.660 6.53 8.83 0.01195 0.01074 0.899 2.92 1.38 1527811 1.859 0.661 6.00 8.80 0.01156 ... ... ... ... ... Table 1.
Details of the three models evolved. The columns are: TP – thermal pulse number; M ∗ – total stellar mass in solar masses;M H – hydrogen-exhausted core mass in solar masses; τ ip – interpulse period in 10 years; logarithm of the maximum helium luminosityreached; ∆M H – growth of the H-exhausted core mass in solar masses; ∆M DUP – mass of material dredged-up; λ – dredge-up efficiency(which is defined as ∆M DUP / ∆M H ). The remaining columns give the abundances of carbon, lithium and the C/ C ratio measuredat the end of the interpulse phase following the thermal pulse. Each model run encountered numerical problems at the peak of a thermalpulse and hence no information for ∆M
DUP , λ and abundances could be given for the last pulse. right panel of Fig. 3). The profiles of Be and Li are dueto the equilibrium between the rate at which these isotopesare transported up from the He burning regions (by the ac-tion of thermohaline mixing) and the rate at which they aredestroyed. A pocket of Li is formed. As the µ -inversion re-duces, the efficiency of mixing is reduced and Be is no longermixed outwards as far. All this takes place very shortly afterthe cessation of TDUP. At no point is the efficiency of themixing enough to get the Li into the convective envelopebefore it is destroyed.After the H-shell re-ignites the convective envelopemoves back inward in mass (bottom left panel of Fig. 3)and connects with the upper edge of the pocket of berylliumand lithium. This inward motion of the convective envelopetakes place during the interpulse period and it is at thispoint that the Li enhancement is first seen. The envelope does not move in to as great a depth as it did during TDUPbut it brings the base of the envelope close enough to thehydrogen-burning shell that Li can be transported up intothe envelope before it is destroyed (bottom right panel ofFig 3).
The production of C is much more straightforward. Abun-dance profiles for the 1.5 M ⊙ model during its second inter-pulse phase are shown in Fig. 4, with the isotopes He, C, C and N being displayed. The newly-dredged up C isconverted into C, which is then carried to the convectiveenvelope by the thermohaline mixing. Once lifted from theH-burning shell, C undergoes no further reactions and sowe see a steady increase in this isotope throughout the in- c (cid:13) , 1–12 hermohaline mixing in low-Z AGB stars Figure 3.
Abundance profiles of the 1.5 M ⊙ model after its second thermal pulse. In each panel, the abundances are: He (dot-short dashline), Be (dot-long dash line) and Li (dashed line). The mean molecular weight, µ is displayed by a solid line. The grey shading indicatesconvective regions. Top left:
Just after the end of third dredge-up.
Top right:
Retreat of the convective envelope. Thermohaline mixingleads to the formation of a pocket of beryllium and lithium.
Bottom left: . The hydrogen shell moves outward and the envelope comesback in again.
Bottom right
Just before the beginning of the next thermal pulse. terpulse (unlike Li, which still undergoes proton capturesas it is being transported outward). Thus the C/ C ra-tio drops throughout the interpulse. There is a concomitantslight increase in the surface N abundance as the temper-ature down to which mixing occurs is sufficiently high forincomplete CN-cycling. At the top of the H-burning shell, ni-trogen is enhanced to a few times its surface value after thirddredge-up, whereas C is over an order of magnitude moreabundant at the top of the H-shell than it is at the surface.Hence the N-enhancement caused by thermohaline mixingis insignificant (and becomes even less significant with eachpulse).During the third interpulse phase (see Fig. 5), the C/ C declines by much less. The dredge-up of carbonforces the µ -minimum out to cooler temperatures. As a con-sequence of this, the CN cycle is less efficient and less Cis produced: while the C abundance in the envelope hasincreased from 3 . × − to 1 . × − , the C abundanceat the µ -minimum is roughly constant at around 3 × − .In addition, the magnitude of the mixing coefficient is abouta factor of 2 lower in the third interpulse than it was in thesecond, owing to the much smaller µ -dip. To compound all this, the interpulse is also 2 × yrs shorter than the pre-ceding one. All this results in the C/ C ratio sufferingless of a reduction during the interpulse.The reduction in the efficacy of mixing continues frompulse to pulse and very quickly thermohaline mixing is nolonger able to affect the C/ C ratio. After three pulsesthere is barely a discernible change in it during the inter-pulse. The C/ C ratio keeps increasing with each episodeof third dredge-up and values of over 10 are reached by theend of the TP-AGB. The 2 M ⊙ model shows less of an effect on its surface abun-dances than the 1.5 M ⊙ model (see Fig. 6). For the C/ Cratio, there is less time for circulation of material close tothe H-burning shell because the interpulse periods are con-siderably shorter (by about a factor of 2) than in the 1.5 M ⊙ model. In addition, the lower He abundance that this modelstarts the TP-AGB with also reduces the efficiency of mix-ing. The peak Li abundance that is reached is also lowerthan in the 1.5 M ⊙ model due to the reduced availability of c (cid:13) , 1–12 R. J. Stancliffe
Figure 4.
Abundance profiles of the 1.5 M ⊙ model after its second thermal pulse. In each panel, the abundances are: He (dot-shortdash line), C (dotted line), C (short-dashed line) and N (long-dashed line). The mean molecular weight, µ is displayed by a solidline. The grey shading indicates convective regions. Top left:
Just after the end of third dredge-up.
Top right:
Retreat of the convectiveenvelope.
Bottom left:
The hydrogen shell moves outward and the envelope comes back in again.
Bottom right
Just before thebeginning of the next thermal pulse. He and the reduced mixing efficiency. In addition, we alsosee a decrease in the Li abundance towards the end of theTP-AGB because the temperature at the base of the convec-tive envelope increases sufficiently to allow Li destruction.We also note that the Li abundance increases directly af-ter the first thermal pulse, even though there is no thirddredge-up. This occurs because the convective envelope stillmoves inward after this thermal pulse but does not reachdeep enough to penetrate the C-rich layers of the intershell.The envelope reaches to the H-burning shell and erases theexisting mean molecular weight gradient. It is this flatten-ing of the mean molecular weight gradient by the convectiveenvelope that allows the He burning to form a mean molec-ular weight inversion and thus allows thermohaline mixingto occur.The 1 M ⊙ model also shows less of an effect on its sur-face abundances than the 1.5 M ⊙ model. Again, we attributethis to the lower He abundance (the model starts the TP-AGB with X He = 1 . × − ) which reduces the mixingefficiency. Unlike the 2 M ⊙ model, the first efficient episodeof thermohaline mixing happens after the first episode of third dredge-up. We thus get a substantial reduction of the C/ C ratio during the next interpulse phase.
There are many uncertainties associated with thermohalinemixing as a mechanism, most notable of which is the effi-ciency of the mixing. The prescriptions of Ulrich (1972) andKippenhahn et al. (1980) give diffusion coefficients of thesame functional form, but with a dimensionless constant (re-lated to the geometry of the mixing) that differs by orders ofmagnitude (see Charbonnel & Zahn 2007, for details). Thereare also differences in the way that one applies the mixing.Eggleton et al. (2008) use a mixing coefficient that dependson the difference between µ at a given point in the starand the minimum value of µ . This is a non-local prescrip-tion, whereas Stancliffe et al. (2007) use a local prescriptionwhere the mixing efficiency depends on local differences in µ (i.e. the µ -gradient). Denissenkov & Pinsonneault (2008b)also include overshooting in their prescription so that theyactually mix to below the location of their µ -minimum. c (cid:13) , 1–12 hermohaline mixing in low-Z AGB stars Figure 5.
Abundance profiles of the 1.5 M ⊙ model after its second thermal pulse. In each panel, the abundances are: He (dot-shortdash line), C (dotted line), C (short-dashed line) and N (long-dashed line). The mean molecular weight, µ is displayed by a solidline. The grey shading indicates convective regions. Top left:
Just after the end of third dredge-up.
Top right:
Retreat of the convectiveenvelope.
Bottom left:
The hydrogen shell moves outward and the envelope comes back in again.
Bottom right
Just before thebeginning of the next thermal pulse.
In light of these uncertainties, we make the followingtests on our 1.5 M ⊙ model to determine the robustness ofour results. We have run two model sets in which we changethe diffusion coefficient of the existing prescription. In one,we increase the diffusion coefficient by a factor of 5 (modelD5) and in the other we decrease it by a factor of 10 (modelD0.1). We also implement a prescription similar to that ofEggleton et al. (2008) (hereafter EDL08), where the diffu-sion coefficient is given by D th = C ( µ − µ min ) (1)for regions above the point of minimum µ . Here, D th is thediffusion coefficient, C is a free parameter, µ is the meanmolecular weight and µ min is the minimum mean molecularweight in the model. The free parameter C is set to 10 asthis gives approximately the same carbon depletion on theupper giant branch as is obtained in our standard model. Us-ing this prescription, the mixing is made global rather thanlocal (model EDL). We note that the EDL08 prescription An increase of a factor of 10 proved to be unstable and themodel could not be converged. gives a very different functional form for the mixing coeffi-cient to our standard implementation. The EDL08 prescrip-tion gives faster mixing further away from the µ -minimumbecause this is where µ − µ min is greatest, whereas our stan-dard prescription gives most efficient mixing close to thepoint of µ -minimum because this is where the µ -gradient isat its steepest.Each of these models is run from the end of first dredge-up until the superwind phase of the TP-AGB. Table 2 showshow the He & Li surface abundances and the C/ C ratioat various evolutionary stages change with the varying of theparameters.If we increase the diffusion coefficient by a factor of 5,we get substantially more processing of material on the firstgiant branch. The drop in the He during the mixing episodeon the upper giant branch is about 6 times as great as forthe standard diffusion coefficient. The lithium abundancedrops from log ǫ ( Li) = 0 .
41 to log ǫ ( Li) = − . C/ C ratio drops to just 10.1, compared to 21.2 in thestandard case. Despite the depletion in the helium-3 reser-voir, the model is still able to undergo thermohaline mixingon the AGB. The total Li production is greater than in the c (cid:13) , 1–12 R. J. Stancliffe
Figure 2.
Properties of the 1.5 M ⊙ model as a function of timesince the first thermal pulse. Top panel: The evolution of thehelium-burning luminosity. Middle panel: The evolution of thesurface C/ C ratio (solid line) and the surface Li abundance(dashed line). Bottom panel: The evolution of the surface Heabundance.
Figure 6.
Properties of the 2 M ⊙ model as a function of timesince the first thermal pulse. Top panel: The evolution of thehelium-burning luminosity. Middle panel: The evolution of thesurface C/ C ratio (solid line) and the surface Li abundance(dashed line). Bottom panel: The evolution of the surface Heabundance. Model He log ǫ ( Li) C/ CPost 1 st dredge-up 3 . × − . × − . × − -0.30 10.1D0.1 3 . × − . × − . × − -1.40 11.7D5 1 . × − -1.82 8.11D0.1 3 . × − -0.48 15.3EDL 1 . × − -2.57 6.02Final modelStandard 2 . × − . × − . × − . × − Table 2.
Surface abundances by mass fraction of isotopes affectedby thermohaline mixing and how they vary with parameters inthe 1.5 M ⊙ model, together with the C/ C ratio, at variousevolutionary stages. The models are all started from the samepoint after the end of first dredge-up. standard case by over a factor of 2, with the final modelreaching a surface abundance of log ǫ ( Li) = 2 .
89. Thefinal C/ C ratio reached is 2317, which is only slightlylower than the standard case. This is because thermohalinemixing is only effective at reducing this ratio in the earlypulses and this does not change with changes in the diffu-sion coefficient.Reducing the diffusion coefficient by an order of magni-tude does not substantially affect the amount of processingthat takes place on the first giant branch. The surface abun-dances of He, Li and the C/ C ratio are almost identicalin the two models. The slower diffusion coefficient meansthere is less of a reduction in the He, Li and C/ Cratio between the tip of the giant branch and the begin-ning of the TP-AGB. Along the TP-AGB, the reduction ofthe diffusion coefficient shows a much more significant ef-fect. The reduction in the efficiency of mixing means thatless Li is brought into the envelope and the model reachesonly log ǫ ( Li) = 0 .
61 by the onset of the superwind. Theproduction of C is substantially reduced because of theweaker mixing and the model has a final C/ C ratio thatis a factor of about 4 higher than in the standard case.The EDL prescription gives similar changes in the abun-dances between the end of first dredge-up and the tip of thegiant branch than those obtained in the standard case. Thisis not so surprising: we have chosen the free parameter ofthis prescription such that it gives roughly the same level ofcarbon depletion as the standard case, so we would expectthe other species to behave in a similar way. However, theEDL prescription leads to significant depletion of He and Li at the beginning of the core helium burning phase. Bythe time the star reaches the AGB, the surface He abun- c (cid:13)000
61 by the onset of the superwind. Theproduction of C is substantially reduced because of theweaker mixing and the model has a final C/ C ratio thatis a factor of about 4 higher than in the standard case.The EDL prescription gives similar changes in the abun-dances between the end of first dredge-up and the tip of thegiant branch than those obtained in the standard case. Thisis not so surprising: we have chosen the free parameter ofthis prescription such that it gives roughly the same level ofcarbon depletion as the standard case, so we would expectthe other species to behave in a similar way. However, theEDL prescription leads to significant depletion of He and Li at the beginning of the core helium burning phase. Bythe time the star reaches the AGB, the surface He abun- c (cid:13)000 , 1–12 hermohaline mixing in low-Z AGB stars dance has dropped to just 1 . × − , nearly a factor of 3less than the standard case. This severely limits the action ofthermohaline mixing along the AGB. With less He avail-able, the mixing efficiency is substantially reduced and sothe drop in the He along the TP-AGB is much less than inthe standard case. In addition, less C can be transportedup from the burning shell and the model reaches the super-wind phase with a very high C/ C ratio. Li productionsuffers two-fold. There is less He available from which toproduce Be and the reduction in the transport efficiencymeans more beryllium and lithium is destroyed in the stel-lar interior before it can reach the safety of the convectiveenvelope. The final surface lithium abundance is over 2 dexlower than in the standard case, with log ǫ ( Li) being just0.28.There are also uncertainties that we are unable to as-sess at present. Foremost among these is the suggestion thatthermohaline mixing can be inhibited by the presence of ro-tation (Denissenkov & Pinsonneault 2008a). Until hydrody-namic simulations of the interactions of these two mecha-nisms are made, this point cannot be addressed. It is alsoprobable that other effects (e.g. the presence of magneticfields) will also influence the degree of mixing.
The action of thermohaline mixing during the TP-AGB doesreduce the C/ C ratio, so that the ejecta will have aratio of a few thousand rather than the 10 predicted bycanonical models (e.g. Stancliffe & Glebbeek 2008). Whilethis is an improvement it still leaves the models predictinghigher C/ C ratios than are observed in CEMP turn-offstars. While more evolved CEMP stars will undergo mix-ing of CN-cycled material on the first giant branch (eitherat first dredge-up or through extra mixing above the lumi-nosity bump) which will lower their surface C/ C ratios,the turn-off objects are not evolved enough for this to havehappened. Another mechanism for reducing this ratio mustclearly be sought. It is also unable to affect the N abun-dance and so cannot account for the correlation of C- andN-enhancement observed in the CEMP stars . However, ifmixing occurs to regions below the point of minimum µ dueto some sort of overshooting (Denissenkov & Pinsonneault2008b), it is possible that substantial enhancements of Cand N could result.The surprising outcome of these simulations is the highLi abundances that can be produced. It is usually supposedthat only the higher mass AGB stars which undergo hot bot-tom burning are able to produce Li via the Cameron-Fowlermechanism (Cameron & Fowler 1971). This work shows thatit is possible that low-mass AGB stars could be producersof lithium-7 . We also note that the action of thermohalinemixing on the AGB is subtly different from its action of theRGB. On the RGB, it leads to a depletion of Li with the Masseron et al. (2009) suggest that the apparent C and N cor-relation does not hold when considering the individual CEMPsubclasses. Therefore, the failure of the model to produce a Cand N correlation may not be so serious. Object [Fe/H] [C/Fe] log ǫ (Li) log g Refs.CS 22964-161A -2.41 1.35 2.09 3.7 1CS 22964-161B -2.39 1.15 2.09 4.1 1HE 0024-2523 -2.7 2.6 1.5 4.3 2CS 31080-095 -2.85 2.69 1.73 4.5 3CS 31062-012 -2.53 2.14 2 . Table 3.
Properties of CEMP turn-off stars with measured Li-abundances, as extracted from the SAGA database. References:1 – Thompson et al. (2008), 2 – Lucatello et al. (2003), 3 –Sivarani et al. (2006), 4 – Aoki et al. (2008) star leaving the RGB with virtually no lithium left. How-ever, on the AGB thermohaline mixing can substantially in-crease the Li abundance up to values above the Spite plateauvalue. It is therefore possible that low-mass AGB stars havecontributed to the Galaxy’s Li budget. The effect of a pop-ulation of low-mass, low-metallicity lithium producers onGalactic chemical evolution models should be investigated.These models do improve the agreement of the AGBmodels with observations of Li in carbon-enhanced metal-poor stars. We have extracted from the Stellar Abundancesfor Galactic Archaeology (SAGA) database (Suda et al.2008) those CEMP stars that are both C-rich and havemeasured Li-abundances, and also that are still close to themain sequence turn-off (because first dredge-up will signif-icantly reduce the surface Li abundance). We select onlythose stars in the metallicity range − < [Fe/H] < − s subclass, i.e.those stars which have [Ba/Fe] > > r/s subclass of CEMP stars, which have0 < [Ba/Eu] < .
5, is currently unknown. It is possible thattheir s -process enrichment has come from a binary masstransfer event, in which case the models presented hereinwould apply, but the enrichment may have another sourceentirely (see e.g. Lugaro et al. 2009, for a possible alternativeformation scenario). This should be borne in mind through-out the following discussion. Thompson et al. (2008) reported abundances for this in-triguing system. It is a double CEMP binary, with bothcomponents being turn-off objects. It is presumed that thissystem is actually a hierarchical triple system, in which anAGB star polluted the tight inner binary which we now ob-serve today.As both objects in the system have [C/Fe]=1.21 we are c (cid:13) , 1–12 R. J. Stancliffe forced to assume that some mixing of accreted material hastaken place in order to get the surface [C/Fe] low enough , asthe three AGB models presented here all give [C/Fe] > ad hoc one.Models of the secondary involving these three processes(see Stancliffe 2009, for details) were run, accreting 0.001-0.1 M ⊙ of material from companions of 1, 1.5 and 2 M ⊙ ,in each case producing a star of 0.8 M ⊙ . The evolution ofthese models through the [C/Fe]-log ǫ (Li) plane are shownin the top panel of Fig. 7. We find that we can reproducethe observed properties of CS 22964-161 if around 0.002 M ⊙ of material was accreted from the 1.5 M ⊙ companion, withstar A accreting slightly more mass than star B. In contrast,models including thermohaline mixing alone deplete far toomuch of their lithium to match the observations (lower panelof Fig. 7), as was noted by Stancliffe (2009). Detailed abundance analyses for this object have beencarried out by Carretta et al. (2002) and Lucatello et al.(2003). These authors report [Fe/H] for this object as be-ing -2.62 and -2.72 respectively. Both studies agree that thesurface gravity of the object is log g = 4 .
3. The bulk of theabundances for this object come from the latter work andit is from here that we quote the abundances [C/Fe]=2.6and log ǫ (Li) = 1 .
5. The barium abundance according toLucatello et al. (2003) is [Ba/Fe]=1.46, while Carretta et al.(2002) give [Ba/Fe] = 1.76. The object is therefore s -processrich. Only an upper limit for europium, [Eu/Fe] < .
16 isgiven by Lucatello et al. (2003) and so it cannot be ruledout that HE 0024-2523 belongs to the r/s CEMP sub-class.However, these authors do report that the star is extremelylead-rich, with [Pb/Fe] = 3.3.HE 0024-2523 could conceivably have accreted materialfrom a companion in a similar mass range to the models pre-sented here. The [C/Fe] and log ǫ (Li) values of the modelsare in the region of the values observed in this object. Wewould have to assume that no mixing of the accreted mate-rial took place, as any mixing would reduce the C- and Liabundances too much. Sivarani et al. (2006) have performed a detailed abundanceanalysis of this object. They find [Fe/H] = -2.85, [C/Fe] = Alternatively, the AGB models could be over-predicting theamount of C that is dredged-up. A simple back of the envelopecalculation suggests that to reach [C/Fe] = +1 about 5 × − M ⊙ of intershell material would have to be dredged-up, compared witharound 4 × − M ⊙ in the 1 M ⊙ model. Figure 7.
The evolution of log ǫ (Li) with [C/Fe] when accret-ing material from a 1.5 M ⊙ companion. The cases displayed arefor when 0.001 M ⊙ (solid line), 0.01 M ⊙ (dotted line) and 0.1 M ⊙ (dashed line) is accreted. In each case, the secondary is left witha total mass of 0.8 M ⊙ . Bold lines indicate where log g passesfrom 4.5 to 3.5 as the object evolves off the main sequence. Theerrorbars denote the locations of specific observed systems. Toppanel:
The secondary is modelled including thermohaline mix-ing, gravitational settling and an extra turbulent process.
Bot-tom panel:
The secondary is modelled with thermohaline mixingalone. ǫ (Li) = 1 .
73. A moderate barium enhance-ment of [Ba/Fe] = 0.77 is reported, but this is the heaviestelement for which they give an abundance.CS 31080-095 could conceivably come from a similarmass range to the models presented here. The C and Liabundances of this object lie between those of our 1 and1.5 M ⊙ models, so we would expect that accreting materialfrom a companion in this mass range would give appropriateabundances, provided the accreted material did not mix intothe secondary. CS 31062-012 has been studied by Norris et al. (1997),Aoki et al. (2001), Aoki et al. (2002a), Aoki et al. (2002b)and Aoki et al. (2008). These authors give [Fe/H] inthe range -2.53 to -2.74, with [C/Fe] in the range2.1 to 2.15. Aoki et al. (2008) give the lithium abun-dance as log ǫ (Li) = 2 .
3, but an earlier studyby Charbonnel & Primas (2005) gives a lower value oflog ǫ (Li) = 1 . s subclass.We find we are able to model this system using ejectafrom a 1.5 M ⊙ companion. If around 0.002 M ⊙ of material isaccreted on to the secondary, and both thermohaline mix- c (cid:13) , 1–12 hermohaline mixing in low-Z AGB stars Object [Fe/H] [C/Fe] log ǫ (Li) log g Refs.CS 29497-030 -2.7 2.38 < < . < < . < . Table 4.
Properties of CEMP turn-off stars with upper limits onthe Li abundance, as extracted from the SAGA database. Refer-ences: 1 – Sivarani et al. (2004), 2 – Aoki et al. (2008). ing and gravitational settling are included (but not the ad-ditional turbulent process), then the properties of the sys-tem can be reproduced. Dilution of the small quantity ofmaterial accreted allows the carbon abundance to drop tothe requisite level, while the gravitational settling createsa mean molecular weight barrier that prevents the mixinggoing too deep, allowing a substantial quantity of lithium tosurvive. Inclusion of the extra turbulent mixing process ofRichard et al. (2005) leads to a degree of carbon depletionthat is too great.It is rather unsatisfactory that for some objects we seemto require some degree of mixing of the accreted materialand for others we require that no mixing of this materialtakes place. Similar conclusions were reached by Stancliffe(2009) when examining the trends of various light elementsin CEMP stars. The physical reason why some stars mixtheir accreted material and others do not remains unknown.It is likely that the interaction of these mixing processeswith other physical processes not included here (e.g. rota-tion, magnetic fields) is responsible.
In addition to the above systems which have measurementsof the lithium abundances, SAGA lists five systems for whichan upper limit for lithium has been determined. These sys-tems are: CS 29497-030, CS 29526-110, SDSS 0924+40,SDSS 1707+58 and SDSS 2047+00. The salient details ofthese models are given in Table 4. The high upper limitsderived for the latter four systems do not severely restrictwhich models can be applied. For example, a model involvingaccretion of material from a 1 M ⊙ AGB star, with no mixingof the accreted material into the secondary would give sur-face abundances broadly consistent with the observations ofall but SDSS 0924+40. However, accretion of ejecta from amore massive AGB companion would also fit the data pro-viding some mixing of the accreted material took place in or-der to lower the surface carbon abundance of the secondary.Without better constrains on the lithium abundance, it isnot possible to determine which of these scenarios actuallytook place.The systems CS 29497-030 and SDSS 0924+40 are morerestrictive. The high [C/Fe] of SDSS 0924+40 rules out ac-cretion from a 1 M ⊙ companion because this model doesnot produce enough carbon. We can also rule out the caseof accretion of material from a 1.5 M ⊙ model. While thismodel produces roughly the correct amount of [C/Fe], its Liabundance is about 0.5 dex above the observed upper limit.Accretion from a more massive companion with some mixing of the accreted material is a viable scenario. For CS 29497-030, accretion from a 1 M ⊙ companion is consistent with theobserved abundances, provided no mixing of the accretedmaterial takes place. However, it is also possible the com-panion was more massive and that mixing of the accretedmaterial happened. We have investigated the effect that thermohaline mixinghas on the abundances of low-mass, low-metallicity AGBstars. We find that enough He remains after the first giantbranch that thermohaline mixing can still take place on theAGB. However, the effect is only felt during the very firstthermal pulses with dredge-up. Thermohaline mixing canlead to substantial production of Li – even up to valuesabove the Spite plateau value. Thus it is possible to recon-cile C- and Li-rich metal-poor stars with having come from abinary mass transfer scenario. We demonstrate that the Li-enrichments of five known Li-rich, turn-off CEMP stars canbe explained using the abundances of our models. Thermo-haline mixing does not reduce the C/ C ratio sufficientlyto reconcile the AGB models with observations of CEMPstars at the Main Sequence turn-off, nor does it raise thesurface nitrogen abundance sufficiently. The possibility thatlow-mass, low-metallicity stars could be producers of lithiumis intriguing and their role in Galactic chemical evolutionshould be investigated.Many uncertainties about the nature of thermohalinemixing still remain and we have shown that the Li abun-dance is particularly sensitive to the nature of the mixing.There is also the question of how other physical processes(e.g. rotation) interact with thermohaline mixing and theconsequences of this remain to be explored.
The anonymous referee is thanked for her/his useful com-ments that have helped to improve the clarity of thismanuscript. The author thanks Evert Glebbeek for readingthe manuscript prior to submission. RJS is funded by theAustralian Research Council’s Discovery Projects schemeunder grant DP0879472. This work was supported by theNCI National Facility at the ANU.
REFERENCES
Abia C., Isern J., 1997, MNRAS, 289, L11Anders E., Grevesse N., 1989, Geo.Cosmo.Acta, 53, 197Aoki W. et al., 2008, ApJ, 678, 1351Aoki W., Norris J. E., Ryan S. G., Beers T. C., Ando H.,2002a, PASJ, 54, 933Aoki W. et al., 2001, ApJ, 561, 346Aoki W., Ryan S. G., Norris J. E., Beers T. C., Ando H.,Tsangarides S., 2002b, ApJ, 580, 1149Beers T. C., Christlieb N., 2005, ARA&A, 43, 531Cameron A. G. W., Fowler W. A., 1971, ApJ, 164, 111Campbell S. W., Lattanzio J. C., 2008, A&A, 490, 769 c (cid:13) , 1–12 R. J. Stancliffe
Cantiello M., Langer N., 2008, in Deng L., Chan K. L., eds,IAU Symposium Vol. 252 of IAU Symposium, Thermoha-line mixing in low-mass giants. pp 103–109Carretta E., Gratton R., Cohen J. G., Beers T. C.,Christlieb N., 2002, AJ, 124, 481Charbonnel C., Primas F., 2005, A&A, 442, 961Charbonnel C., Zahn J.-P., 2007, A&A, 467, L15Denissenkov P. A., Pinsonneault M., 2008a, ApJ, 684, 626Denissenkov P. A., Pinsonneault M., 2008b, ApJ, 679, 1541Eggleton P. P., 1971, MNRAS, 151, 351Eggleton P. P., Dearborn D. S. P., Lattanzio J. C., 2006,Science, 314, 1580Eggleton P. P., Dearborn D. S. P., Lattanzio J. C., 2008,ApJ, 677, 581Eldridge J. J., Tout C. A., 2004, MNRAS, 348, 201Fujimoto M. Y., Iben I. J., Hollowell D., 1990, ApJ, 349,580Iglesias C. A., Rogers F. J., 1996, ApJ, 464, 943Kippenhahn R., Ruschenplatt G., Thomas H.-C., 1980,A&A, 91, 175Lau H. H. B., Stancliffe R. J., Tout C. A., 2009, MNRAS,396, 1046Lebzelter T., Lederer M. T., Cristallo S., Hinkle K. H.,Straniero O., Aringer B., 2008, A&A, 486, 511Lederer M. T., Aringer B., 2009, A&A, 494, 403Lucatello S., Gratton R., Cohen J. G., Beers T. C.,Christlieb N., Carretta E., Ram´ırez S., 2003, AJ, 125, 875Lugaro M., Campbell S. W., de Mink S. E., 2009, PASA,26, 322Marigo P., 2002, A&A, 387, 507Marigo P., Aringer B., 2009, ArXiv e-printsMasseron T., Johnson J. A., Plez B., Van Eck S., Pri-mas F., Goriely S., Jorissen A., 2009, A&A, in press.ArXiv:0901.4737Nollett K. M., Busso M., Wasserburg G. J., 2003, ApJ, 582,1036Norris J. E., Ryan S. G., Beers T. C., 1997, ApJ, 488, 350Pols O. R., Tout C. A., Eggleton P. P., Han Z., 1995, MN-RAS, 274, 964Reimers D., 1975, Memoires of the Societe Royale des Sci-ences de Liege, 8, 369Richard O., Michaud G., Richer J., 2005, ApJ, 619, 538Siess L., 2009, A&A, 497, 463Sivarani T. et al., 2006, A&A, 459, 125Sivarani T. et al., 2004, A&A, 413, 1073Stancliffe R. J., 2005, PhD thesis, University of CambridgeStancliffe R. J., 2009, MNRAS, 394, 1051Stancliffe R. J., Church R. P., Angelou G. C., LattanzioJ. C., 2009, MNRAS, 396, 2313Stancliffe R. J., Eldridge J. J., 2009, MNRAS, 396, 1699Stancliffe R. J., Glebbeek E., 2008, MNRAS, 389, 1828Stancliffe R. J., Glebbeek E., Izzard R. G., Pols O. R.,2007, A&A, 464, L57Stancliffe R. J., Lugaro M. A., Ugalde C., Tout C. A.,G¨orres J., Wiescher M., 2005, MNRAS, 360, 375Stancliffe R. J., Tout C. A., Pols O. R., 2004, MNRAS,352, 984Suda T. et al., 2008, PASJ, 60, 1159Thompson I. B. et al., 2008, ApJ, 677, 556Ulrich R. K., 1972, ApJ, 172, 165Uttenthaler S., Lebzelter T., Palmerini S., Busso M.,Aringer B., Lederer M. T., 2007, A&A, 471, L41 Vassiliadis E., Wood P. R., 1993, ApJ, 413, 641 c (cid:13)000