The born again (VLTP) scenario revisited: The mass of the remnants and implications for V4334 Sgr
aa r X i v : . [ a s t r o - ph ] J un Mon. Not. R. Astron. Soc. , 1– ?? (2007) Printed 27 October 2018 (MN L A TEX style file v2.2)
The born again (VLTP) scenario revisited: The mass of theremnants and implications for V4334 Sgr
M. M. Miller Bertolami , ⋆ and L. G. Althaus , † Facultad de Ciencias Astron´omicas y Geofisicas, UNLP, Paseo del Bosque s/n, La Plata, B1900FWA, Argentina Instituto de Astrof´ısica La Plata, CONICET-UNLP, Paseo del Bosque s/n, La Plata, B1900FWA, Argentina
ABSTRACT
We present 1-D numerical simulations of the very late thermal pulse (VLTP) sce-nario for a wide range of remnant masses. We show that by taking into account the dif-ferent possible remnant masses, the observed evolution of V4334 Sgr (a.k.a. Sakurai’sObject) can be reproduced within the standard 1D-MLT stellar evolutionary modelswithout the inclusion of any ad − hoc reduced mixing efficiency. Our simulations hintat a consistent picture with present observations of V4334 Sgr. From energetics, andwithin the standard MLT approach, we show that low mass remnants ( M ∼ < . M ⊙ )are expected to behave markedly different than higher mass remnants ( M ∼ > . M ⊙ )in the sense that the latter are not expected to expand significantly as a result of theviolent H-burning that takes place during the VLTP. We also assess the discrepancyin the born again times obtained by different authors by comparing the energy thatcan be liberated by H-burning during the VLTP event. Key words: stars:evolution, stars:AGB and post-AGB, stars: individual: V4334 Sgr
Hydrogen(H)-deficient Post-Asymptotic Giant Branch(AGB) stars display a wide variety of surface abundances,ranging from the almost pure helium atmospheres of O(He)stars to the helium(He)- carbon(C)- and oxygen(O)- richsurfaces of WR-CSPN and PG1159 stars (see Werner& Herwig 2006 for a review). In particular the surfacecomposition of the last group resembles the intershell regionchemistry of AGB star models when some overshooting inthe pulse driven convection zone (PDCZ) is allowed duringthe thermal pulses (Herwig et al. 1997). For this reason, anddue to the fact that the occurrence of late (i.e. post-AGB)helium flashes is statistically unavoidable in single stellarevolution modeling (Iben et al. 1983), a late helium flashis the most accepted mechanism for the formation of thesestars (see, however, De Marco 2002). In particular duringa very late helium flash (VLTP; Herwig 2001b), as theH-burning shell is almost extinguished, the PDCZ can reachthe H-rich envelope. As a consequence H-rich material iscarried into the hot C-rich interior, and violently burned(see Miller Bertolami et al. 2006, from now on Metal06, fora detailed description of the event). As was already notedby Iben et al. (1983), the timescale in which H is burnedis similar to that of convective motions and consequently ⋆ E-mail: [email protected] † E-mail: [email protected] the usually adopted instantaneous mixing approach is notvalid. For this reason only few numerical simulations of theVLTP exist in the literature: Iben & MacDonald (1995),Herwig et al. (1999), Herwig (2001a), Lawlor & MacDonald(2002), Lawlor & MacDonald (2003) and more recentlyMetal06.The identification of V4334 Sgr (a.k.a. Sakurai’s Ob-ject) as a star undergoing a VLTP event (Duerbeck &Benetti 1996) has renewed the interest in this particular kindof late helium flash. V4334 Sgr has shown a very fast evo-lution in the HR diagram of only a few years (Duerbeck etal. 1997, Asplund 1999, Hajduk et al. 2005). In this contextit is worth mentioning that the theoretical born again times(i.e. the time it takes the star to cross the HR diagram froma white dwarf configuration to a giant star one) is a con-troversial issue: whilst Iben & MacDonald (1995), Lawlor &MacDonald (2002) and Metal06 obtain born again times ofthe order of one or two decades, Herwig et al. (1999) obtaintimescales of the order of centuries.The difference between the theoretical born again timesof Herwig et al. (1999) and the observed timescale of V4334Sgr (and also V605 Aql, Duerbeck et al. 2002), promptedHerwig (2001a) to propose a reduction in the mixing effi-ciency during the conditions of the violent proton burning(by about a factor of 100 for their 0.604 M ⊙ sequence) inorder to match the observed timescale. However, in view ofthe lack of hydrodynamical simulations of the violent burn- c (cid:13) M. M. Miller Bertolami and L. G. Althaus ing and mixing process during the VLTP, the idea has animportant drawback: it introduces a free parameter (i.e. themixing efficiency) that can only be calibrated with the situa-tion that one wants to study, thus losing its predictive power.Aside from this philosophical aspect, current reduced mix-ing efficiency models (Herwig 2001a, Lawlor & MacDonald2003 and Hajduk et al. 2005) suffer from an internal in-consistency. Indeed, contrary to what is stated in Herwig(2001a) and Lawlor & MacDonald (2003), changes in con-vective velocities are expected to affect convective energytransport. In fact, as shown in appendix A, reducing con-vective velocities is completely equivalent to reducing themixing length. But more importantly, the reproduction ofthe born again timescale does not necessarily make modelscompletely consistent with observations. In fact, althoughHajduk et al. (2005) claim that models with a reduction inthe mixing efficiency reproduce “The real-time stellar evolu-tion of Sakurai’s object”, a closer inspection shows that sucha claim should be taken with a pinch of salt. In particularthe effective temperature (and also the cooling rate) of themodel in its (first) return to the AGB contradicts the in-ferred effective temperature (with 2 different methods) dur-ing 1996-1998 (Duerbeck et al. 1997, Asplund et al. 1999).In the same line, the extremely high luminosity of their the-oretical model in its first return to the AGB ( ∼
12 500 L ⊙ )implies an extremely large distance of more than 8 Kpc (bycomparing with the values in Duerbeck et al. 1997). Sur-prisingly enough, this value is inconsistent with the 2 Kpcadopted by Hajduk et al. (2005) as well as with independentdistance estimations which place V4334 below 4.5 Kpc andpreferentially between 1.5 and 3 Kpc (Kimeswenger 2002).Hence, even if the reduction in the mixing efficiency by afactor 60 leads to born again times (Hajduk et al. 2005)similar to those displayed by V4334 Sgr, that model failsto match other well established observed properties. On theother hand, whilst the reduced mixing efficiency models ofLawlor & MacDonald (2003) do not suffer from this inconsis-tencies, their models show high H-abundances which are notconsistent with observations of both V4334 Sgr or PG1159type stars.In this context, it is worth noting that a strong reduc-tion in the mixing efficiency does not seem necessary in theVLTP models of Iben & MacDonald (1995) and Metal06to reproduce observations. Most of the existing computa-tions of the VLTP have been performed for masses in avery narrow range around the canonical mass ∼ . M ⊙ . Inthis context, we feel that an exploration of the (neglected)importance of the mass of the remnant for the born againtimescale is needed. This is precisely the aim if this arti-cle. Indeed some confusion seems to exist in the literaturewith respect to this issue: whilst it is usually stated thatlow mass/luminosity models evolve slower after a VLTP(Pollaco 1999, Kimeswenger 2002), it is clear from Herwig’s(2001a) 0.535 M ⊙ model that lower luminosities/masses leadto faster born again evolutions. In the present work we performed numerical simulations ofthe VLTP scenario for several different remnant masses. The -1-0.500.511.522.533.544.5 L og ( L / L s un ) sun sun sun -1-0.500.511.522.533.544.5 L og ( L / L s un ) sun sun sun eff )-1-0.500.511.522.533.544.5 L og ( L / L s un ) sun sun sun sun Start ofVLTP Back in the AGBStart ofVLTP the AGBBack in VLTPStart of Back in the AGB
Figure 1.
HR diagrams of the VLTP evolution for the sequencespresented in this work during their first return to the AGB. simulations have been performed with the LPCODE (Al-thaus et al. 2005) by adopting the Sugimoto (1970) schemefor the structure equations as described in the Appendix Aof Metal06. During the present work we have adopted thestandard MLT with a mixing length parameter α = 1 . D = 13 l v MLT = α / H P (cid:20) c gκ ρ (1 − β ) ∇ ad ( ∇ rad − ∇ ) (cid:21) / (1)as can be deduced from Cox & Giuli (1968) . As shown inMetal06, no difference would arise if the expression of Langeret al. (1985) was to be adopted. Diffusive overshooting wasallowed at every convective boundary with a value f = 0 . f ). By adoptingthe standard MLT we are ignoring the effect of chemical gra-dients during the violent proton burning that can certainlyinfluence the results (Metal06). We choose not to includethis effect for consistency, as our overshooting prescriptiondoes not include the effect of chemical gradients, and also fornumerical simplicity. Also, as will be clear in the followingsections, we do not intend to reproduce the exact evolution Note that there are two typos in the expression for D in footnote6 of Metal06. c (cid:13) , 1–, 1–
HR diagrams of the VLTP evolution for the sequencespresented in this work during their first return to the AGB. simulations have been performed with the LPCODE (Al-thaus et al. 2005) by adopting the Sugimoto (1970) schemefor the structure equations as described in the Appendix Aof Metal06. During the present work we have adopted thestandard MLT with a mixing length parameter α = 1 . D = 13 l v MLT = α / H P (cid:20) c gκ ρ (1 − β ) ∇ ad ( ∇ rad − ∇ ) (cid:21) / (1)as can be deduced from Cox & Giuli (1968) . As shown inMetal06, no difference would arise if the expression of Langeret al. (1985) was to be adopted. Diffusive overshooting wasallowed at every convective boundary with a value f = 0 . f ). By adoptingthe standard MLT we are ignoring the effect of chemical gra-dients during the violent proton burning that can certainlyinfluence the results (Metal06). We choose not to includethis effect for consistency, as our overshooting prescriptiondoes not include the effect of chemical gradients, and also fornumerical simplicity. Also, as will be clear in the followingsections, we do not intend to reproduce the exact evolution Note that there are two typos in the expression for D in footnote6 of Metal06. c (cid:13) , 1–, 1– ?? he born again (VLTP) scenario revisited Remnant Initial Ref. pre-flash t BA He-DCZ H-DCZ H-DCZ Loc.Mass Mass M H τ - τ τ - τ τ glo C C C N H-burn peak[ M ⊙ ] [ M ⊙ ] [ M ⊙ ] [yr] [sec] [sec] [sec] [ M ⊙ ]0.515 1 d 2 . × −
14 2183—2687 121—154 1931 6.6 13.4 0.4861120.530 1 b 2 . × − . × − . × − . × − . × − . × − † . × − † . × − † . × − — ∗ ∗ ∗ Table 1.
Description of the sequences analysed in this work. Fourth column shows the total amount of H ( M H ) left at the moment ofthe VLTP. Note the strong dependence of M H on the mass of the remnant. t BA stands for the time elapsed from the maximum of protonburning to the moment when the sequence reaches log T eff =3.8. Mean values of the local convective turnover timescales estimated withdifferent prescriptions are shown for both He and H-driven convective zones (He-DCZ and H-DCZ respectively). The global estimationof the turnover timescale for the H-DCZ is also shown. Last column shows the approximate mass location of the maximum of protonburning at the moment of its maximum. † In these sequences the return to the AGB is mainly powered by the He-shell flash. ∗ The 0.870 M ⊙ sequence was stopped at high effective temperatures due to numerous convergence problems. References are: a- Metal06, b- MillerBertolami & Althaus (2006), c- C´orsico et al. (2006), d- Althaus et al. (2007) and e- Unpublished. of born again stars (something that would require a muchmore sophisticated treatment of convection) but instead toshow the importance of the remnant mass for the subsequentevolution.A detailed description of the sequences considered inthe present work is listed in Table 1. HR diagram evolutionof the sequences during their first return to the AGB dur-ing the VLTP is shown in Fig. 1. We mention that for moremassive sequences this will be their only return to the AGBas they only experience the He-driven expansion. With ex-ception of the sequence 0.561 M ⊙ , the prior evolution of ourVLTP sequences has been presented in our previous works(Metal06, Miller Bertolami & Althaus 2006, Corsico et al.2006 and Althaus et al. 2007). All of them are the result offull and consistent evolutionary calculations from the ZAMSto the post-AGB stage. In all the cases the initial metallic-ity was taken as Z=0.02. One of the most remarkable fea-tures displayed by Table 1 is the strong dependence of thetotal amount of H at the moment of the VLTP ( M H ) onthe mass of the remnant. This is important in view of thediscussion presented in Section 4. Note however that, asidefrom this strong dependence on the mass of the remnant, M H will also depend on the previous evolution and on theexact moment at which the VLTP takes place. This is whythe 0.584 (0.530) M ⊙ model has a higher M H value than the0.565 (0.515) M ⊙ model. As was already mentioned in Metal06, an extremely hightime resolution ( ∼ − yr) during the violent proton burn-ing is needed in order to avoid an underestimation of theenergy liberated by proton burning . Such small time steps In fact this can be one of the reasons for the discrepancy in theborn again times of different authors. are close to the timescale needed to reach the steady statedescribed by the MLT (see Herwig 2001a). To be consistentwith the treatment of convection the time step should bekept above this value.The convective turnover times can be estimated withdifferent prescriptions which can lead to significantly dif-ferent values. In order to have a feeling of what is reallyhappening we have adopted three different estimations: τ , τ and τ glo (two local and one global timescale), which aredefined as: τ = 1 | N | = r kTµ m p g |∇ − ∇ ad | (2) τ = α H P v MLT (3) τ glo = Z R top R base drv MLT (4)The definition of the last two expressions is evident and thefirst is the inverse of the Br¨unt-V¨ais¨al¨a frequency and pro-vides the timescale for the growth of convective velocitiesin a convectively unstable region (see Hansen & Kawaler1994 for a derivation). In Table 1 we list typical values ofthese timescales. Values at the He-driven convection zone(He-DCZ) correspond to the moment just before the vio-lent proton ingestion (second stage of proton burning, asdescribed in Metal06) whilst values at the convective zonedriven by the violent proton burning (H-DCZ) correspondto the moment of maximum energy release by proton burn-ing (which is also the moment at which important amountsof H start to be burned). For the local estimations the av-eraged value over the whole convective zone is displayed.It is worth mentioning that whilst global and local turnover times coincide for the He-DCZ, the global estimationis, roughly, one order of magnitude larger in the H-DCZ.From the values in Table 1 one would be tempted to statethat by adopting a minimum allowed timestep of ∼ − yr ( ∼ c (cid:13) , 1– ?? M. M. Miller Bertolami and L. G. Althaus burning, we are allowing convective motions to develop and,thus, being consistent with the steady state assumption ofthe MLT. However, a point should be mentioned. Hydro-dynamical simulations of “standard” (i.e. without protoningestion) helium shell flashes (Herwig et al. 2006) showthat the steady state is not achieved in only one turn overtime. In fact Herwig et al. (2006) find that, at a standard( ǫ = 2 × erg s − gr − ) heating rate, about 10 turnover times are necessary. They also find, however, that atan enhanced heating rate (30 times larger) steady state isachieved about 2-3 times faster. Then, as the heating rate atthe base of the H-DCZ during the violent proton burning isabout ǫ = 10 —10 erg s − gr − (1000—5000 times largerthan the one at the base of the He-DCZ) one would expectthat the steady state will probably be reached in no muchmore than one turn over time. Consequently it seems rea-sonable to choose a minimum allowed time step of 10 − yr,which accordingly to our previous studies is needed to avoidan underestimation of the energy liberated by proton burn-ing . We have also checked that convective velocities remainsubsonic during the stage of maximum proton burning, be-ing in all the cases v MLT < . v sound . This is importantbecause during the maximum of proton burning convectivevelocities are high and the MLT is derived within the as-sumption of subsonic convective motions. In our view themain drawback of present simulations may come from thefact that convective mixing differs from a diffusion process.In fact, during a VLTP, H needs to be mixed with C inorder to be burned. It is probable that, initially, H may betransported downwards in the form of plumes and thus noimportant amounts of H would be (initially) mixed with C.This situation would be very different from the picture ofdiffusive mixing with or without reduced mixing efficiency.How far our results are from reality will depend on how farthe place at which most H is burned differs from the onepredicted by diffusive mixing within the standard (withoutreduced mixing efficiency) MLT.
As mentioned in the introduction, born again times from dif-ferent authors can differ by almost two orders of magnitude.Also, as found in Metal06 for ∼ . M ⊙ models, numericalissues can change the born again times completely and evensuppress the H-driven expansion (if energy from H-burningis underestimated by a factor ∼ M ⊙ models. Whilst the 0.584 M ⊙ model shows a H-driven ex-pansion and consequently short born again times (8.9 yr),the 0.609 M ⊙ model does not, displaying born again timesof 157 yr. In this context we feel it interesting to analysethe energetics of the VLTP. To this end, we estimate theenergy that can be liberated by proton burning if the wholeH-content of the star is burned ( E H ). This is shown in Fig. Time steps of ∼ − yr are only needed during the (short)runaway burning of protons where most of the H is burned (seeMetal06) and the energy liberated by proton burning is L H ∼ − L ⊙ . M remnant /M sun E n e r gy [ e r g ] Figure 2.
Comparison of the total amount of energy that can beliberated during the violent proton burning ( E H ; shaded region)with the energy necessary to expand the envelope above the pointof maximum proton burning ( | E tot | ; solid line). Broken line showsthe gravitational binding energy of that envelope ( | E g | ). C+p → N+ γ → C+e + + ν e (which lib-erates 3.4573 MeV per proton burned) and then through C+p → N+ γ → C+e + + ν e and C+p → N+ γ work-ing at the same rate (a process that liberates 5.504 MeV per proton burned, Metal06). These two values give rise tothe lower and upper boundary in Fig. 2. We also calculatethe gravitational binding energy E g , just before the violentstage of proton burning, of the zone above the peak of protonburning. From numerical models we know this is the zonethat expands due to the energy liberated by the H-burning(and will be denoted as envelope in the following) E g = − Z envelope Gmr dm (5)We also estimate the internal energy of the envelope E i as E i = Z envelope T c v dm (6)From these values we compute the total energy of the en-velope E tot (which is negative for a gravitationally boundsystem) as E tot = E g + E i (7) | E tot | is then the energy needed to expand the envelope toinfinity. Its value is lower than | E g | because (the Virial the-orem enforces) as the envelope expands it cools and internalenergy is released, helping the expansion. By comparing E H and | E tot | we can resonably decide if the burning of H in agiven remnant can drive an expansion of its envelope. Theresult is surprising, as inferred from Fig. 2. Note that formodels above 0.6 M ⊙ the energy that can be liberated byproton burning is not enough to expand the envelope. Thisresult is completely consistent with what is displayed byour detailed modeling of the process. Although this extremeconsistency with such rough estimation can be just a coin-cidence, the different behaviour of E tot and | E H | with themass of the remnant is worth emphasising. Whilst the for-mer decreases only slightly with the stellar mass, the later c (cid:13) , 1– ?? he born again (VLTP) scenario revisited shows a steeper behaviour. This is expected because the to-tal H-amount of a remnant is a steep function of the stellarmass. This fact yields a transition at a certain stellar massvalue below which the H-driven expansion is possible andabove which the H content of the remnant is not enough forH-burning to trigger an expansion.This energetic point of view not only holds the clueto understand the distinct behaviour of our sequences butalso helps to understand the differences in previous works.Note that models with masses close to the ∼ . M ⊙ transi-tion value will be very sensitive to numerical issues that caneventually alter the release of H-burning energy. This couldexplain why an underestimation by factor 2 in the H-burningenergy reported in Metal06 strongly affects the born againtimes. It also helps to explain why Althaus et al. (2005), inwhich no extreme care of the time step during the violentproton burning was taken, reported longer born again times(20-40 yr) than Metal06 for the same initial model . In thisview, the short born again times found by Lawlor & Mac-Donald (2002) are probably due to the low mass of theirmodels (0.56-0.61 M ⊙ ). Also the difference in born againtimes between Iben & MacDonald (1995) and Herwig et al.(1999) simulations can probably be understood in this con-text. In fact, whilst Iben & MacDonald (1995) and Herwiget al. (1999) sequences have similar remnant masses (0.6 M ⊙ and 0.604 M ⊙ , respectively) the total amount of H ismarkedly different in both cases, being ∼ . × − M ⊙ and ∼ . × − M ⊙ , respectively. This leads to energies( E H ) of 3.29—5.23 × erg for Herwig et al. (1999) (sim-ilar to the values of our own 0.609 M ⊙ sequence, see Fig.2) and 1.56—2.48 × erg for Iben & MacDonald (1995).This difference of almost a factor of five in the energy re-leased by proton burning is very probably the reason whyIben & MacDonald (1995) find a H-driven expansion whilstHerwig et al. (1999) do not.On the basis of these arguments, the finding of Herwig(2001a) about that a reduction in the mixing efficiency canlead to shorter born again times can be understood as fol-lows. By reducing the mixing efficiency the point at whichthe energy is liberated by proton burning is moved outwards.Then the value of | E tot | is lowered whilst the total amountof H remains the same and thus no change in E H will hap-pen. Then, changing the mixing efficiency moves the solidline in Fig. 2 up and down, which alters the transition massvalue at which H-driven expansion begins to be possible . In Fig. 3 we compare the evolution of luminosity and ef-fective temperature of our sequences with that observed atV4334 Sgr (Duerbeck et al. 1997, Asplund et al. 1999). Alsothe preoutburst location of V4334 Sgr and V605 Aql (fromKerber et al. 1999 and Herwig 2001a) is compared with thelocation of the sequences at the VLTP in Fig. 4. Luminosities This was also due to a difference in the definition of the diffusioncoefficient by a factor 3 (see Metal06). There is also a secondary effect which comes from the fact thatthe closer to the surface the energy is released, the shorter it takesthe liberated energy to reach the surface of the star. -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Time [yr]3.63.844.24.44.64.8 L og ( T e ff ) sun sun sun sun sun sun Hajduk et al. L og ( L / L s un ) X H Figure 3.
Bottom and middle panel show the evolution of lumi-nosity and effective temperature of the models after the VLTP(set at 0 yr) compared with the observations of V4334 Sgr (Duer-beck et al. 1997, Asplund 1999; + and × signs respectively). Theevolution of luminosity and effective temperature of Hajduk etal. (2005), extracted from Fig. 2 of that work, is shown for com-parison. The zero point in the x-axis of the observations was ar-bitrarily set to allow comparison with the models. Upper pannelshows the evolution of the H abundance at the outermost layerof the models (which should be close to the surface abundance)compared with the observed abundances at V4334 Sgr (Asplund1999). eff )11.522.533.5 L og ( L / L s un ) P r e - ou t bu r s t ( ) [ K p c ] P r e - ou t bu r s t ( ) [ K p c ] P r e - ou t bu r s t ( ) [ K p c ] sun sun sun sun sun sun sun sun sun sun V605 Aql [3.1 Kpc] V4334 Sgr [5.5 Kpc]V4334 Sgr [1.5 Kpc]
Figure 4.
Location of the models in the HR diagram at themoment of the VLTP. The lines show the possible 1976 detectionof V4334 (taken from Herwig 2001a and rederived for differentassumed distances). Also inferred pre-flash location of V4334 Sgrand V605 Aql (Kerber et al. 1999, Lechner & Kimeswenger 2004)are shown for comparison. Note the strong dependence of the T eff on the mass of the remnant.c (cid:13) , 1– ?? M. M. Miller Bertolami and L. G. Althaus have been rederived for different distances to allow compari-son. This was done by making use of the fact that interstellarextintion is supposed to be constant in that direction of thesky for distances above 2 Kpc. For remnants that display aH-driven expansion ( ∼ < . M ⊙ ) the main feature can bedescribed as follows. Whilst the preoutburst location of leastmassive remnants ( ∼ < . M ⊙ ) is compatible with higherdistances ( > < M ⊙ models, preoutburst locations are compatiblewith a distance of ∼ M ⊙ displaycooling rates which are compatible with that observed inV4334 Sgr. Also, their born again times range from 5 to 11yr not far from that observed at V4334 Sgr (or V605 Aql,Duerbeck et al. 2002).It is particularly interesting to note the behaviour ofour 0.561 M ⊙ model. It reaches the effective temperatureat which V4334 Sgr was discovered in about ∼ C/ C ∼ C/ C atV4334 Sgr (Asplund et al. 1999). Our models also predicthigh N abundances at their first return to the AGB (seeTable 1), at variance with Herwig (2001b) but in agreementwith observations of V4334 Sgr (Asplund et al. 1999).In Fig. 3 the evolution of the reduced mixing efficiencymodel of Hajduk et al. (2005) is shown for comparison. Al-though this model evolves initially faster than our sequencesthe behaviour of its luminosity and temperature are not con-sistent with those observed at V4334 Sgr. In particular allthe effective temperature determinations of V4334 Sgr in itsfirst return to the AGB lie beyond the minimum effectivetemperature attained by that sequence. Also its luminos-ity at low effective temperatures implies a distance of d > M ⊙ sequences —those which are both com-patible with pre- and post- outburst inferences for the tem-perature and luminosity of V4334 Sgr — also reproducequalitatively the drop at low effective temperature in the Habundance of V4334 Sgr observed by Asplund et al. (1999).Note however that there is a quantitative disagreement bymore than one order of magnitude between observations andmodel prediction. This may be either due to an intrinsic fail-ure of the models or because the quantities plotted are notexactly the surface abundances of the models but, instead,the H-abundance at the outermost shell of the models (whichcan differ from the actual surface value).As was already noted by Metal06, models without re-duced mixing efficiency fail to reproduce the fast reheatingof V4334 Sgr (as reported by Hajduk et al. 2005). One mayargue that this can be due to the fact that mass loss was ig-nored in the models, whilst V4334 Sgr displayed strong mass loss episodes once its temperature dropped below ∼ × − M ⊙ /yr, Hajduk et al. 2005)we find that the 0.561 M ⊙ model reheats (reaches temper-atures greater than 10000 K) in only 27 yr and the 0.584 M ⊙ model does it in 24 yr. This is certainly faster than inthe absence of mass loss but still a factor of ∼ − ad − hoc reduction of the mixing efficiency if differ-ent remnant masses are allowed. Even more, this approach(aside from not introducing a free parameter) leads to mod-els which are more consistent with the observations of V4334Sgr than models with a reduced mixing efficiency (Herwig2001a, Hajduk et al. 2005). In the present work we have presented 1-D hydrostatic evo-lutionary sequences of the VLTP scenario for different rem-nant masses. In Section 4 we have made an analysis of theenergetics of the VLTP that shows the importance of themass of the remnant for the born again timescale. In partic-ular, that argument shows that, within the standard MLTwith no reduction of the mixing efficiency, it is expected thatthe H-driven expansion that leads to short born again timeswill only be present in VLTPs of low mass remnants. Thetransition remnant mass value below which the H-driven ex-pansion (i.e. a short born again timescale) takes place is closeto the canonical mass value ∼ . M ⊙ . The precise value ofthis transition mass depends on the exact location at whichmost H is burned, and thus on a detailed description of themixing and burning process. A more accurate value of thetransition mass will have to wait until hydrodynamical simu-lations of the violent H-ingestion became available. We havealso shown that the energetic point of view discussed in Sec-tion 4 can help to understand the differences in the calcu-lated born again times by different authors.We have also compared our predictions with the obser-vations of real VLTP objects (mainly V4334 Sgr.). As wasnoted early in this work we do not expect from such simpli-fied models and treatment of convection (as those presentedin this and all previous works) to reproduce the exact evo-lution observed at real VLTP stars (like V4334 Sgr or V605Aql). In fact this is one of the reasons why inferences suchas the need of a reduction in the mixing efficiency comingfrom the fitting of the born again times of 1D hydrostatic se-quences for a single remnant mass should be taken with care.Also the born again times given in Table 1 should be takenwith care, as for example a reduction by a factor of three inthe mixing efficiencies leads to a reduction of a factor of twoin the born again times of one of the sequences presented c (cid:13) , 1– ?? he born again (VLTP) scenario revisited by Metal06. However, as argued in Section 2, we have somereasons to believe that present models may not be that farfrom reality. In this context the comparison with observa-tions in Section 5 shows that it is possible to roughly repro-duce the observed behaviour of V4334 if a mass of ∼ . M ⊙ for the remnant is assumed and a distance of ∼ ∼ . − . M ⊙ sequences also reproduce qualitatively the drop in theH abundance observed in V4334 Sgr at low effective tem-peratures. As this late drop in H-abundance is partially dueto dilution of the outer layers of the envelope into deeperlayers of the star (and also due to a deepening of the shellat which τ / is located), it is expectable that this drop inH would be accompanied by a raise in s-process elementsabundances, just as observed in V4334 Sgr.The main drawback of our models is that they fail toreproduce the fast reheating of V4334 Sgr (Hajduk et al.2005) by about a factor of 4. However as discussed in Section5 this is not a surprise as one of the main hypothesis ofthe modeling (that of hydrostatic equilibrium) is explicitlybroken in the outer layers of the models once the VLTP staris back to the AGB at very low tempetures. Thus, thereis no reason to expect that the sequences of this work willaccurately reproduce reality at that course of evolution.We conclude then, that V4334 Sgr is not “incomprehen-sible” (Herwig 2001a) within the standard MLT approachand that there is “apriori” no need for a reduction of themixing efficiency. Needless to say, this statement does notimply that mixing efficiency is not reduced during the protoningestion in the VLTP nor that the MLT approach is correctduring the VLTP —something that will only be known oncehydrodynamical simulations of the H ingestion and burn-ing become available—, but only shows that it is possibleto roughly reproduce the observations within the standardMLT approach.In any case, the main conclusion of the present work isthat different remnant masses have to be considered whencomparing theoretical expectations with real VLTP objects. ACKNOWLEDGMENTS
This research was supported by the Instituto de Astrof´ısicaLa Plata and by PIP 6521 grant from CONICET. M3Bwants to thank the Max Planck Institut f¨ur Astrophysikin Garching and the European Assocciation for Research inAstronomy for an EARA-EST fellowship during which thecentral part of this work was conceived. We warmly thankA. Serenelli, K. Werner, M. Asplund and an anonymous ref-eree for a careful reading of the manuscript and also forcomments and suggestions which have improved the finalversion of the article. M3B wants to thank A. Weiss for use-ful discussions about convection. We also thank H. Viturroand R. Martinez for technical support.
REFERENCES
Althaus L. G., Serenelli A. M., C´orsico A. H., MontgomeryM. H., 2003, A&A, 404, 593Althaus L. G., Serenelli A. M., Panei J. A., et al., 2005,A&A, 435, 631Althaus L. G., C´orsico A. H., Miller Bertolami M. M., 2007,A&A, 467,1175Asplund M., Lambert D., Pollaco D., Shetrone M., 1999,A&A, 343, 507C´orsico A. H., Althaus L. G., Miller Bertolami M. M., 2006,A&A, 458, 259Cox J., Giuli T., 1968, Principles of Stellar Structure (NewYork, Gordon and Breach)De Marco O., 2002, ApSS, 279, 157Duerbeck H. W., Benetti S., 1996, ApJ, 468, L111Duerbeck H. W., Hazen M. L., Misch A. A., Seitter W. C.,2002, ApSS, 279, 183Duerbeck H. W., Benetti S., Gautschy A., et al., 1997, AJ,114, 1657Hajduk M., Ziljstra A., Herwig F., et al., 2005, Science,308, 231Hansen C. J., Kawaler S. D., 1994, Stellar Interiors(Springer)Herwig F., 2001a, ApJ, 554, L71Herwig F., 2001b, ApSS, 275, 15Herwig F., 2002, ApSS, 279, 103Herwig F., Bl¨ocker T., Langer N., Driebe T., 1999, A&A,349, L5Herwig F., Bl¨ocker T., Sch¨onberner D., El Eid M., 1997,A&A, 324, L81Herwig F., Freytag B., Hueckstaedt R., Timmes F., 2006ApJ, 642, 1057Iben I., MacDonald J., 1995, Lecture Notes in Physics(Berlin: Springer Verlag), 443, 48Iben I., Kaler J. B., Truran J. W., Renzini A., 1983, ApJ,264, 605Kerber F., K¨oppen J., Roth M., Trager S., 1999, A&A, 344,L79Kimeswenger S., 2002, ApSS, 279, 79Kippenhahn R., Weigert A., 1990, Stellar Structure andEvolution, Springer-VerlagLanger N., El Eid M. F., Fricke K. J., 1985, A&A, 145, 179Lawlor T. M., MacDonald J., 2002, ApSS, 279, 123Lawlor T. M., MacDonald J., 2003, ApJ, 583, 913Lechner M. F. M., Kimeswenger S., 2004, A&A, 426, 145Miller Bertolami M. M., Althaus L. G. 2006, A&A, 454,845Miller Bertolami M. M., Althaus L. G., Serenelli A. M.,Panei J. A., 2006, A&A, 449, 313 [Metal06]Pollaco D., 1999, MNRAS, 304, 127Schlattl H., Cassissi S., Salaris M., Weiss A., 2001, ApJ,559, 1082Sugimoto D., 1970, ApJ, 159, 619Werner K., Herwig F., 2006, PASP, 118, 183
APPENDIX A: CONSISTENT TREATMENT OFREDUCED CONVECTIVE VELOCITIES.
It has been proposed in some articles (Herwig 2001a, Schlattlet al. 2001) that convective mixing efficiency can be reduced c (cid:13) , 1– ?? M. M. Miller Bertolami and L. G. Althaus during violent H-flashes. Even more, Herwig (2001a) men-tions that convective material transport can be changed inmain sequence stellar models by orders of magnitude with-out any change in stellar parameters. Going even furtherLawlor & MacDonald (2003) conclude that, as reduced mix-ing efficiency does not produce significant changes in usualstellar evolutionary stages, reduced mixing velocities couldbe usual in stellar evolution. In what follows it is shown that,if a reduction of mixing velocities is considered in the fulltreatment of the MLT, the effect of changing the convectivevelocities is undistinguishable from changing the value of α (the mixing length to pressure scale height ratio). Then theprevious conclusions come from an inconsistent treatmentof the reduction in mixing velocities. A1 Reduced convective velocites and convectiveenergy flux
Following Kippenhahn & Weigert (1990) we can derive themean work done by buoyancy on the convective elements tobe:¯ W ( r ) = gδ ( ∇ − ∇ e ) l m H P , (A1)where the symbols have their usual meanings Then to pro-pose a reduced mixing velocity within the MLT, we proposethat only a fraction of the work done by buoyancy goes intothe kinetic energy of the convective elements. We can writethis as:¯ v = f v (cid:18) gδ ( ∇ − ∇ e )8 H P (cid:19) / l m . (A2)Here f v is a new (and, a priori, “free”) parameter giving thefactor by which the standard value of ¯ v is reduced. FollowingKippenhahn & Weigert (1990) and inserting this relation inthe equation for convective energy transport, F con = ρvc P DT, (A3)and replacing DT in terms of ( ∇ − ∇ e ) we get: F con = C P ρT √ gδ √ l m (cid:18) ( ∇ − ∇ e ) H P (cid:19) / f v . (A4)The convective flux is thus accordingly reduced by the factor f v . A2 Modified dimensionless equations
From the relation h ∇ e − ∇ ad ∇ − ∇ e i = 6 acT κρ C P l m ¯ v (A5)and defining the usual dimensionless quantities U and W : U = 3 acT c P ρ κl m r H P gδ (A6) W = ∇ rad − ∇ ad (A7)we get new equations (equivalent to equations 7.14 and 7.15in Kippenhahn & Weigert 1990) for U :( ∇ e − ∇ ad ) = 2 Uf v √∇ − ∇ e (A8) ( ∇ − ∇ e ) / = 89 Uf v ( ∇ rad − ∇ ) (A9)It seems natural then to define the quantity U ′ = Uf v . U ′ = 3 acT c P ρ κl m f v r H P gδ (A10)Then defining the quantity ζ = + √∇ − ∇ ad + U ′ we getthat the new dimensionless cubic equation for the MLT is:( ζ − U ′ ) + 8 U ′ (cid:0) ζ − U ′ − W (cid:1) = 0 (A11)This becomes the new equation to be solved in order toobtain the real value of ∇ . This is the same equation thanthe 7.18 one given in Kippenhahn & Weigert (1990), beingthe only difference the definition of U ′ . A3 The diffusion coefficient
In the context of diffusive convective mixing, material trans-port and mixing is ruled by the diffusion coefficient, whichis usually defined as D = l m v . Taking Eq. A2, replacing( ∇ − ∇ e ) with Eq. A9 and using the definiton of U ′ we getfor the mean velocity of convective motions the expression:¯ v = (cid:18) f v l m H P (cid:19) / (cid:20) cg ∇ ad (1 − β ) ρκ ( ∇ rad − ∇ ) (cid:21) / (A12)Introducing this into the definition of D we get (using l m = H P α ): D = 13 ( α f v ) / H P (cid:20) cg (1 − β ) ρκ ∇ ad ( ∇ rad − ∇ ) (cid:21) / (A13) A4 Conclusion As U ′ (and thus ∇ ) and D depend only on the product α f v (and not on both quantities independently) then changes inthe mixing length and changes in the mixing velocities areindistinguishable, making it unnecessary to consider a newfree parameter within the MLT. A reduction in the mixingefficiency should be regarded as equivalent to reducing themixing length. Then, as a consequence, changing the meanvelocity of convective motions would produce an apprecia-ble difference in any case in which non-adiabatic convectiontakes place (lower main sequence, RGB, AGB stellar mod-els). c (cid:13) , 1–, 1–