aa r X i v : . [ a s t r o - ph ] J un Astronomy & Astrophysics manuscript no. 7256 c (cid:13)
ESO 2018November 5, 2018
On the robustness of H-deficient post-AGB tracks
M. M. Miller Bertolami , ⋆ , L. G. Althaus , ⋆⋆ Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) LaPlata, Argentina. Instituto de Astrof´ısica La Plata, IALP, CONICETe-mail: mmiller,[email protected]
November 5, 2018
ABSTRACT
Aims.
We analyze the robustness of H–deficient post–AGB tracks regarding previous evolution of their progenitor starsand the constitutive physics of the remnants. Our motivation is a recent suggestion of Werner & Herwig (2006) thatprevious evolution should be important in shaping the final post–AGB track and the persisting discrepancy betweenasteroseismological and spectroscopical mass determinations. This work is thus complementary to our previous work(Miller Bertolami & Althaus 2006) and intends to shed some light on the uncertainty behind the evolutionary trackspresented there.
Methods.
We compute full evolutionary models for PG1159 stars taking into account different extramixing (overshooting)efficiencies and lifetimes on the TP-AGB during the progenitor evolution. We also assess the effect of possible differencesin the opacities and equation of state by artificially changing them before the PG1159 stage. Also comparisons are madewith the few H-deficient post–AGB tracks available in the literature.
Results.
Contrary to our expectations, we found that previous evolution is not a main factor in shaping H–deficientpost–AGB tracks. Interestingly enough, we find that only an increase of ∼
50% in the intershell opacities at higheffective temperatures may affect the tracks as to reconcile spectroscopic and asteroseismologic mass determinations.This forces us to conclude that our previous tracks (Miller Bertolami & Althaus 2006) are robust enough as to beused for spectroscopic mass determinations, unless opacities in the intershell region are substantially different. Ourresults, then, call for an analysis of possible systematics in the usually adopted asteroseismological mass determinationmethods.
Key words. stars: evolution — stars: PG1159
1. Introduction
Post Asymptotic Giant Branch (AGB) stars constitute ashort–lived transition stage between AGB stars and whitedwarf stars. Among them a minority show H–deficient com-positions and are suppossed to be the main progenitors ofH–deficient white dwarfs, which account for about 15% ofthe white dwarf population (Eisenstein et al. 2006). Thegroup of H–deficient post–AGB stars displays a wide vari-ety of surface chemical compositions ranging from almostpure helium envelopes to the helium– (He), carbon– (C) andoxygen– (O) rich surface composition of the Wolf Rayet cen-tral stars of planetary nebulae ([WC]) and the PG1159 typestars; see Werner & Herwig (2006), from now on WH06.The surface composition of the last group resembles the in-tershell region chemistry of AGB star models when someovershooting at the base of the pulse driven convective zone(PDCZ) is allowed during the thermal pulses (Herwig et al.1997). For this reason, and also due to the fact that the oc-currence of late (i.e. post-AGB) thermal pulses is statisti-cally unavoidable in single stellar evolution modeling (Ibenet al. 1983), a late helium shell flash is the most accepted
Send offprint requests to : M. M. Miller Bertolami ⋆ Fellow of CONICET, Argentina. ⋆⋆ Member of the Carrera del Investigador Cient´ıfico yTecnol´ogico, CONICET, Argentina. mechanism for the formation of these stars (see, however,De Marco 2002). In this scenario, the remaining thin H–richenvelope is either burnt in a very late helium flash (VLTP)that occurs on the hot white dwarf cooling branch after Hburning has almost ceased, or diluted in a late helium flash(LTP) that develops when the H burning shell is still ac-tive during the horizontal evolution of the stars in the HRdiagram (Herwig 2001).Roughly a third of PG1159–type stars exhibit multiperi-odic luminosity variations caused by non–radial g–modepulsations. This has allowed researchers to derive struc-tural parameters — in particular the mass of these stars— of individual pulsators by means of asteroseismologicalstudies i.e. by comparing adiabatic pulsation periods withthe observed ones — e.g. Kawaler & Bradley (1994) andmore recently C´orsico & Althaus (2006). It is important tomention that for applications requiring accurate values ofadiabatic pulsation periods full evolutionary models with arealistic thermal structure should be used. Stellar massesof PG1159 stars can also be derived by comparing the val-ues of log g and log T eff coming from the fitting of line–blanketed non–LTE model atmospheres to the measuredspectra (Werner et al. 1991) with tracks coming from stellarevolution modeling. These two different approaches enableus to compare the derived stellar masses. Although previousspectroscopical mass determinations, based on old H–richpost–AGB models, show relatively good agreement with Miller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks asteroseismological masses (to about 5%, WH06, roughly0.03 M ⊙ ), the development of a new generation of stel-lar evolution sequences that account for the C– and O–rich surface abundances expected in PG1159 stars (Herwiget al. 1999) has changed the situation. As mentioned byWH06 the new post–AGB tracks are systematically hot-ter than the old ones, which leads to lower spectroscopi-cal masses. The new mean spectroscopical mass becomes0.573 M ⊙ , this is 0.044 M ⊙ lower than previous values; seeMiller Bertolami & Althaus (2006), from now on MA06.This is at variance with asteroseismological predictions. Infact from Table 3 of WH06 and Table 2 of MA06 the as-teroseismological masses are usually 10% higher than theirspectroscopical counterparts, except for the hottest knownpulsating PG1159 star RX J2117.1+3412, the spectroscop-ical mass of which is more than 20% higher than the as-teroseismological one; see C´orsico et al. (2007) for a recentand detailed study of this object. The difference in derivedmasses is a clear indication of the uncertainties weightingupon the mass determination methods.In this context, WH06 have recently compared new andold tracks and claimed that the previous evolution on thethermally pulsing AGB (TP-AGB) — particularly the thirddredge-up (3DUP) efficiency and mass–loss rates — playsa decisive role in the location of the tracks in the HR andlog T eff − log g diagrams during the post–AGB evolution.Specifically, as shown by Herwig et al. (1998), a strong3DUP changes the evolution of the core mass without al-tering the evolution of its radius. Consequently the mass–radius relation of the remnants will depend on the previ-ous TP-AGB evolution and, if we accept in the “predic-tion” of shell homology relations ( L shell ∼ M R core − ,for M core ∼ < . M ⊙ , Herwig et al. 1998), the post–AGBtracks would be accordingly altered. WH06 also point outthat mass loss can produce a similar effect as remnants ofsimilar mass may come out with very different degrees ofdegeneracy depending on the previous evolution. This beingthe case, as both mass–loss rates and 3DUP efficiency arepoorly known, the location of theoretical post–AGB tracks,and thus mass determination, would be highly model de-pendent and uncertain. These issues call for the need ofan analysis of the robustness of existing H-deficient post–AGB tracks and for a way of solving the mentioned massdiscrepanciesHowever, no calculation of the importance of these ef-fects was actually presented neither in WH06 nor in Herwiget al. (2006). The lack of consistent calculations to assess towhat a degree the location of the post–AGB tracks dependson the prior AGB evolution has motivated us to undertakethe present investigation. In the following sections we willelaborate on these issues. In this sense the present workis complementary to that of MA06 where H-deficient post–AGB tracks were presented but no analysis of its robustnesswas performed. In Sect. 2 we analyze how evolution pre-vious to the PG1159 stage affects PG1159 tracks in lightof the suggestion presented by WH06. Then in Sect. 3 weexplore to what an extent the constitutive physics of themodels at the PG1159 stage may affect the tracks. In Sect.4 we compare with other H-deficient tracks available in theliterature and also compare the location of tracks comingfrom LTP and VLTP events. Finally Sect. 5 is devoted tothe discussion of the results and making some final remarks.
2. Influence of previous evolution
As was mentioned, uncertainties in mass–loss rates are ex-pected to affect the duration of the TP-AGB phase and tolead to remnants with different degrees of degeneracy andmass–luminosity relation. Also the initial-final mass rela-tion is expected to be altered by different mass–loss rates.By altering the intensity of mass loss we can get the samefinal remnant mass from progenitors of initially very differ-ent mass, and consequently very different previous evolu-tions (e.g., that have or have not undergone a helium coreflash at the tip of the RGB). We will address these issues inSect. 2.1. In Sect. 2.2 we elaborate on the effects of differ-ent 3DUP efficiencies on the TP-AGB, which is the otherpoint mentioned in WH06 as a possible cause for shiftsin post–AGB tracks. The main effect of 3DUP efficiencyis to change the initial-final mass relation. Indeed, strongdredge up events on the TP-AGB lead to lower final rem-nant masses for the same initial mass. In this context weanalyze sequences with different 3DUP efficiencies and, todisentangle this effect from the one studied in Sect 2.1, withthe same TP-AGB lifetime. Finally, Sect 2.3 is intendedto clarify the reason of the difference between MA06 andBl¨ocker (1995a,b) tracks and to study the extreme limitingcase for which no overshooting (OV) is allowed to operateat any convective boundary during the whole evolution. Inall the sequences presented in this section, mass loss hasbeen arbitrarily set during the departure from the AGBas to get a VLTP and the subsequent PG1159-like surfacecomposition.To visualize and quantify the change introduced by thevariations in the parameters of each sequence we will referand compare our sequences with those of MA06 and C´orsicoet al. (2006) which are assumed as standard in the presentwork. These sequences were calculated with an overshoot-ing efficiency of f = 0 .
016 at all convective boundaries; seeHerwig et al. (1997) for a definition of f . To quantify thechange that a variation in T eff and g for a sequence of agiven mass — caused by different physical assumptions inthe calculations — would produce in spectroscopical massdeterminations, we estimate a mass value for the sequencefrom its location relative to MA06 sequences — this is whatis called “mass derived from comparison” in Table 1 — andcompare that mass with the actual value of the mass. Thedifference between both values gives the shift expected inspectroscopical masses if tracks with a different physicalassumption are used in their derivations.It is worth noting that most of our article deals withpost-VLTP sequences. However, within the late heliumflash scenario for explaining the origin of PG1159 stars,these objects could also be the offspring of LTP events.In fact some PG1159 stars are known to be N-deficient,a fact usually asociated with post-LTP objects. In thesecases some H will be present but hidden below the detec-tion limit. If systematic differences exist between post-LTPand post-VLTP tracks, then this will introduce a system-atic effect in spectroscopic mass determiations. Althoughfrom figure 1 of Herwig one is tempted to conclude thatthis is not the case, it is worth noting that the presence ofH should be more important in the low mass region as thesestars display thicker H-envelopes. We will discuss this issuein section 4.3, where detailed comparisons between post-LTP and post-VLTP tracks will be made for a wide rangeof masses and various surface H-abundances. . iller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks 3Sequence Final Mass Mass derivedfrom comparisonNALT 0.607 0.612LALT 0.6035 0.614SALT 0.6033 0.5982.2MSALT 0.5157 0.524TPA008 0.617 0.621TPA004 0.633 0.6373 M ⊙ w/NOV 0.626 0.623 Table 1.
Values of the final masses of the sequences of thiswork and the masses derived from the comparison with the“standard” ones (MA06). Stellar masses are in M ⊙ . SeeSect. 2.1, 2.2, and 2.3 for definition of the sequences. As stated in WH06, for a similar core mass, a remnantthat spend more time on the TP-AGB will finish with amore compact and degenerate core. Then, different mass–loss prescriptions can lead to remnants with the same coremass but different radius and, by virtue of shell homologyrelations — that “predict” L shell ∼ M R − , Herwig etal. (1998) — different luminosities. This is supported bythe work of Herwig et al. (1998) that shows that, becauseof the continuous shrinking of the H-free core (HFC), theluminosity at the TP-AGB keeps increasing, despite theend of the effective core mass growth as consequence ofstrong dredge up events. In addition, Bl¨ocker (1995b) hasalready shown that a more compact remnant is more lumi-nous than a less compact one of similar mass. To analyzethe possible shift in the H-deficient post-AGB tracks result-ing from uncertainties in TP-AGB mass loss — and hencein different TP-AGB lifetimes — we have calculated the fullevolution of three sequences with the same prescriptions asin MA06 but changing mass loss at the TP-AGB to get dif-ferent TP-AGB lifetimes. These sequences are: NALT witha normal mass–loss prescription, SALT with a short TP-AGB lifetime and LALT with a longer TP-AGB. All thesesequences come from the same pre TP-AGB evolution of aninitially 3- M ⊙ ZAMS star. While NALT underwent 12 ther-mal pulses, SALT and LALT sequences experienced 6 and18 pulses, respectively. SALT (LALT) sequence has a TP-AGB lifetime a factor 2 shorter (1.5 longer) than NALT.Thus the sequences considered here take into account possi-ble uncertainties in TP-AGB lifetimes up to a factor three.This is more than what is expected from different mass–loss prescriptions (Kitsikis & Weiss 2007). Due to the highdredge up efficiency during the last thermal pulses the fi-nal remnant mass of all these sequences is very similar (seeTable 1), thus allowing a direct comparison of the effect ofdifferent TP-AGB lifetimes on the location of H-deficientpost–AGB tracks of similar mass. We mention that all ofthese sequences have been followed through an additionalpost-AGB thermal pulse (the VLTP) where the H-rich en-velope is violently burned.In some agreement with Bl¨ocker (1995b) we find a shiftin post–AGB tracks as a consequence of different TP-AGBlifetimes. However the effect is not very important. In fact,comparing SALT and LALT sequences (both with the samefinal mass) we see that a factor 3 in TP-AGB lifetimes leadsto a maximum shift of 0.03 dex in log T eff . A shorter TP-AGB leads to cooler tracks that would imply ∼ . M ⊙ larger spectroscopical masses. It is also worth noting that sun ]99.059.19.159.29.259.3 L og ( R ) [ c m ] Locus of MA06 sequences at the VLTPNALTTPA008TPA004LALTSALT3M sun , without overshooting
Fig. 1.
Evolution of the HFC (mass and radius) duringthe TP-AGB (solid lines, filled circles) and at the VLTP(dashed lines, empty circles) for selected sequences (valuesare taken just before each thermal pulse). Also the locus ofthe standard models at the moment just before the VLTPis shown for comparison. Note that, due to the turn off ofthe H-burning shell, compression before the VLTP does notfollow the trend in the AGB. [color figure only available inthe electronic version]tracks for LALT and NALT sequences are almost identi-cal regardless the difference in TP–AGB lifetime of 50%. Itseems that, while shortening the TP–AGB does change thepost–VLTP tracks, prolonging it does not produce a size-able effect. To understand this, we show in Figs. 1 and 2 themass-radius relations of our sequence for both the H– andHe– free cores — HFC and HeFC, respectively. The evo-lution of the HFC is in agreement with that presented byHerwig et al. (1998) which shows that the HFC continues tocontract even when the core mass growth is stopped by ef-ficient 3DUP events. Because this behaviour is the basis ofthe argument of WH06 the following should be noted. Firstthe radius of the HFC at the moment of the VLTP does notfollow the trend during the TP-AGB. This is a result of theaccelerated compression of the intershell caused by the de-cline of the H-burning shell when the star approaches thewhite dwarf cooling track. Second, and more importantly,the post–VLTP sequences are powered by the He-burningshell and consequently, if shell homology are to be usedin the analyzes, the relevant values should be the
HeFCmass and radius . Note that the HeFC (Fig. 2) seems toconverge to a certain locus in the core mass-radius diagramfaster than the HFC. In fact while in all the sequences theHFC radius gets smaller with each thermal pulse, the HeFCends its compression after about ∼
10 thermal pulses. Thishelps to understand why there is almost no difference be-tween NALT and LALT sequences. The 6 “extra” thermalpulses of LALT sequence do not introduce any significantchange in the mass-radius relation and thus, according toshell homology relations, their post-AGB luminosity shouldbe similar.As mentioned early, different mass–loss rates can alsochange the initial-final mass relation of the sequences, lead-ing to final remnants with very different previous evolutionbut similar final mass. In this connection, we have com-puted the evolution of an initially 2.2- M ⊙ sequence by as- Miller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks sun ]8.878.888.898.98.918.928.938.948.958.968.978.98 L og ( R ) [ c m ] Locus of MA06 sequences at the VLTPNALTTPA008TPA004LALTSALT3M sun , without overshooting2.2MSALT at the VLTP
Fig. 2.
Same as Fig. 1 but for the HeFC. Note that thelocation of the HeFC on this diagram seems to converge,after not many thermal pulses, to a certain locus. [colorfigure only available in the electronic version]suming an extreme mass–loss rate during the whole AGB(sequence 2.2MSALT). As a result, this sequence underwentonly 5 thermal pulses on the AGB — as compared with the15 AGB pulses of the 2.2- M ⊙ sequence in MA06. The finalmass of the remnant is of 0.516 M ⊙ , much lower than the0.565 M ⊙ quoted in MA06. The track for this sequence inthe log T eff − log g plane is shown in Fig. 3 together with theother sequences of this work and those of MA06. Note thatthe 2.2MSALT track is more luminous and hotter than thatof the standard sequence of similar mass (the 0.512 M ⊙ se-quence in MA06). Note that in this case, the shift in the M - g - T eff relation of the remnants would imply a decrease of ∼ . M ⊙ in spectroscopical masses. This value is un-expectedly low in view of the fact that the two standardsequences in the same region of the log T eff − log g diagramhave a very different previous evolution. Indeed, the 0.512and 0.53 M ⊙ sequences in MA06 have been calculated froman initially 1- M ⊙ progenitor that went through the heliumcore flash. Also, the 0.512 M ⊙ has a very different intershelland surface composition with only 2%, by mass, of oxygen;see MA06 for a description of this sequence. Again, it is in-teresting to look at the structure of the HeFC to understandthis change. As can be seen in Fig. 2 (black star symbol),although the mass and radius of this model fall almost inthe standard locus (the thick grey line in figure 2), its HeFCmass ( ∼ . M ⊙ ) is significantly higher than that of thestandard 0.512 M ⊙ sequence ( ∼ . M ⊙ ) and thus shouldbe, again from shell homology arguments, more luminousthan the standard sequence. Indeed that is what actuallyhappens. Even more, the 2.2MSALT sequence has a HeFCmass that falls almost in the middle of that of the 0.512 and0.53 M ⊙ MA06 sequences and its track in the log T eff − log g diagram does exactly the same. These considerations seemto support the idea that is the HeFC structure — and notthe HFC — which is important to understand H-deficientpost–AGB tracks.So, although the findings of this section confirm thatdifferent TP-AGB lifetimes may result in changes in thepost–AGB tracks, we find that this effect is not enough toaccount for the mass discrepancy mentioned in the intro-duction. Indeed, we find that the PG1159 spectroscopical eff )5.566.577.58 L og ( g ) NALTTPA008TPA004LALTSALT2.2MSALT3M sun , NOV
Fig. 3.
PG1159 tracks of this work as compared withthose of MA06. Thin solid lines correspond to the stan-dard ( f =0.016 at all convective borders) tracks of MA06with stellar masses of (from right to left) 0.512, 0.53, 0.542,0.565, 0.585, 0.609, 0.664 M ⊙ .masses inferred from the MA06 post-AGB tracks would behigher by at most ∼ . M ⊙ (for stars close to the 0.6 M ⊙ tracks) if in their calculations MA06 had consideredmuch shorter TP-AGB lifetimes during the progenitor evo-lution of their PG1159 sequences. On the other hand wefind impossible to get a similar shift for stars close to the0.512 M ⊙ track. This is so because the lack of 3DUP in lowmass stars. To explore the role of the 3DUP efficiency during the TP-AGB in the location of post–AGB tracks, we have followedthe TP-AGB evolution for three different values of the over-shooting efficiency ( f ) at the pulse driven convection zone(PDCZ) that develops during each He-shell flash. As shownin Herwig (2000), higher f values at the bottom of thePDCZ lead to more intense helium shell flashes and more in-tense third dredge up events, while the value of f at the baseof the convective envelope only plays a secondary role in de-termining the 3DUP efficiency (the reasons for this are ex-tensively discussed and shown in sections 4 and 5 of Herwig2000). We have, thus, calculated three different sequencesfor a 3- M ⊙ progenitor by adopting values of f =0.016, 0.008,0.004 at both convective borders of the PDCZ, from nowon sequences NALT, TPA008 and TPA004; sequence NALTcorresponds to that previously described. At any other con-vective zone — for example the AGB convective envelopeand the core burning regions in the previous evolution —the “standard” value of f =0.016 has been used. We stressthat the “standard” value f = 0 .
016 comes from the fittingof the width of the main sequence (Herwig et al. 1997),and thus is appropriate for the core H-burning zone. But itmay be unrealistic for the conditions at the PDCZ (Herwig2004). All of these sequences have similar TP-AGB life-times. This enables us to disentagle the 3DUP effect fromthe one coming from different TP-AGB lifetimes, whichwas studied in the previous section. Also, for these threesequences, mass loss during the last interpulse phase has iller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks 5 been artificially set in order to obtain a VLTP and conse-quently a H-deficient post–AGB remnant. M H fr ee c o r e NALTTPA008TPA004
Fig. 4.
HFC evolution during the TP-AGB for three se-quences with different f values at the PDCZ (masses in M ⊙ ).Note from Fig. 4 that different values of f yields differ-ent evolution of the HFC. For models with higher 3DUPefficiencies the “effective” growth of the HFC is stopped.This is because the increase in the HFC induced by the H-burning shell is compensated for by a decrease during the3DUP events. Not only the HFC mass is altered but also,as expected, the surface and intershell abundances — inparticular the O intershell abundance; see Herwig (2000).As a result of the different adopted 3DUP efficiencies, thefinal remnant masses are different, being 0.607, 0.617 and0.633 M ⊙ for NALT, TPA008 and TPA004 respectively. Itis worth noting that TPA004 hardly undergoes any 3DUPevents. So this sequence should be representative of the casein which no overshooting is considered at the PDCZ.Our results suggest that different 3DUP efficiencies donot seem to lead to an important shift in the location ofthe theoretical post–AGB models in the M - g - T eff space.Indeed, sequences TPA008 and TPA004 are located in thezone of the log T eff − log g diagram corresponding to rem-nants of similar mass of the standard sequences; see Fig. 3.A quantitative measure of the possible shift in the tracksrelative to the standard MA06 ones is given Table 1. Notethat there is a small shift of 0.005 M ⊙ in the derived massfor the NALT sequence as compared to the actual one — weremind that NALT sequence has the same overshooting pre-scription than that assumed in MA06. This is probably dueto a combined effect of a different number of thermal pulsesand slightly different envelope composition — which leadsto different intershell opacities, see Sect. 3. Because thethree sequences have similar TP-AGB lifetimes, this smallshift should be taken as the level of uncertainties in thesecomparisons. Keeping this in mind, the masses derived forTPA008 and TPA004 are practically similar to the actualmasses of these sequences. This shows that, at least around ∼ . M ⊙ , the theoretical M - g - T eff relation of the MA06 H-deficient post-VLTP sequences does not seem to depend onthe intensity of 3DUP events. This can be understood fromFig. 2. Note that HeFC mass-radius values of sequencesTPA008 and TPA004 at the moment of the VLTP lie on the same locus than the standard MA06 sequences of sim-ilar masses. Thus, according to shell homology relations,the He-shell luminosity-mass relation for these sequencesshould be similar to the MA06 ones — which do experienceefficient 3DUP events. Finally, we mention that the centralvalues of density and temperature ( T c , ρ c ) show that theHeFC readjusts its structure to the new mass after eachthermal pulse. At the end of the TP-AGB the T c , ρ c valuesof TPA004 are within those of NALT — of final HeFC mass0.572 M ⊙ — and those of the 3.5 M ⊙ sequence of MA06 —of final HeFC mass 0.638 M ⊙ —, a fact which is consistentwith the final HeFC mass of 0.601 M ⊙ that characterizessequence TPA004. We explore now the effect of overshooting efficiency dur-ing both the early AGB and the core He-burning phaseon the location of the post–AGB tracks. This bears alsosome relevance on the fact that, as inferred from the twoprevious sections, neither the TP-AGB lifetime nor the3DUP efficiency are the reasons for the fact that the MA06tracks are markedly hotter than the older H-rich tracks(Bl¨ocker 1995b). To assess these issues, we have calculatedthe evolution of an initially 3- M ⊙ progenitor but without overshooting mixing at any convective border of the starduring its whole evolution. After 19 thermal pulses, a H-deficient post–VLTP sequence of 0.626 M ⊙ is obtained —early AGB and TP-AGB lifetimes are ∼ . × yr and ∼ . × yr, respectively. This is similar to one of thesequences of Bl¨ocker (1995b) that consisted of an initially3- M ⊙ model that after 20 thermal pulses ends its post–AGB evolution as a 0.625 M ⊙ remnant — with early AGBand TP-AGB lifetimes of ∼ . × yr and ∼ . × yr,respectively — and will allow us for comparison. The mainevolutionary difference between both sequences is the oc-currence of a VLTP in the post-AGB evolution of our se-quence.The resulting H-deficient post–VLTP track is very sim-ilar to the MA06 one and thus much hotter than the old,H-rich, Bl¨ocker’s 0.626- M ⊙ track. In fact if we estimate itsmean mass from comparison with the standard MA06 se-quences we get almost the actual mass (see Table 1). Themayor difference is that the model is slightly cooler at theknee — a shift that would affect spectroscopical masses lessthan ∼ . M ⊙ . From Fig. 2 we can see that the evolutionof the HeFC mass-radius relation is different from that ofthe standard sequences. But even in this case the differ-ence in the radius of the He-free core at the moment of theVLTP amounts to only a 4% as compared with the stan-dard sequences of similar mass. Consequently, it should notbe surprising that the tracks are similar.This shows that the M - g - T eff relation for the post–AGBtracks is not significantly affected by the previous evolution.Thus, differences in the previous evolution do not seem toprovide a possible solution to discrepancy between aster-oseismological and spectroscopical masses nor an explana-tion to the difference with older tracks. Miller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks
3. The role of microphysics and composition in thelocation of post–VLTP tracks
We explore now the importance of microphysics and chemi-cal compositions. Specifically we assess the effects of chang-ing the equation of state (EoS), chemical composition ofthe C-O core and opacities — both radiative and conduc-tive. Here, we do not calculate new evolutionary sequencesfrom the ZAMS to the PG1159 stage; instead we considersome post–VLTP sequences of MA06 and alter their micro-physics before entering the PG1159 stage. We have checkedthat the models are already relaxed to the new physics wellbefore reaching the knee in the HR and log T eff − log g dia-grams. We check this by first doing the changes at differenttimes in the post–VLTP evolution. We find that the tracksdo not depend on the exact moment the changes are done,thus suggesting that the structure has already relaxed tothe new situation. Additionaly, we estimate the thermal re-laxation time of the envelope as τ ∼ R M ⋆ M e . b . c v T dm / L ⋆ —where M e . b . stands for the mass coordinate at the bottomof the envelope. We concentrate on the 0.53 and 0.584 M ⊙ remnants of MA06. For these sequences we find that τ isabout one order of magnitude lower than the time it takesthe remnants to evolve from log T eff ∼ . τ is about 2500 and 1600 yr for 0.53and 0.584 M ⊙ remnants respectively, as compared with the ∼ ∼ To analyze the importance of the C-O core composition andEoS we considered the 0.53 M ⊙ post–VLTP sequence fromMA06. With regard to the EoS we compare tracks result-ing from the use of the standard EoS of LPCODE — seeAlthaus et al. (2005) for references — with those comingfrom an updated version of Magni & Mazzitelli (1979) EoS.The latter is a more detailed equation of state which takesinto account non-ideal corrections such as the pressure ef-fects on ionization and includes Coulomb interactions alsoin the non-degenerate regime — our standard EoS only in-cludes Coulomb corrections in the degenerate regime. Toanalyze the role of the core composition we reset the abun-dances below ln(1 − m ( r ) /M ⋆ ) = − . T eff to exceed 0.01 dex, being generally much smaller.Consequently, neither the C-O core composition nor theEoS assumed in the computation of post–VLTP sequencesplay a role in the derivation of spectroscopical masses, andwe can discard these two factors as possible reasons for ashift in post–VLTP tracks. Because the outer structure of PG1159 stars is completelyruled by radiative transport of energy, changes in the opac-ities could yield differences in the tracks. This may be par-ticularly interesting as the sequences of MA06 have beencalculated for radiative opacities with a solar scaled metal-licity and PG1159 stars are known to present surface abun-dances rich in s-process elements and iron deficient (Miksaet al. 2002, WH06). In this regard, by how much the trans-formation of iron into heavier elements may alter the opac-ities in PG1159 is not known — for example, in a differentcontext, Jeffery & Saio (2006) find differences in the pulsa-tion properties of subdwarf stars depending on whether itis iron or nickel that it is enhanced. Also the exact value ofthe original metallicity of the progenitor star of PG1159 isnot known. We analyze the effect of changing both radia-tive and conductive opacities with very different results ineach case. Full calculations of the VLTP and post–VLTPby means of consistent opacities are out of the scope of thiswork, however we can try to get an idea of how much theopacities affect the post–AGB tracks by artificially chang-ing the opacities in the post–VLTP evolution by arbitraryfactors or by adopting different opacity tables.As a result of these experiments we find that for conduc-tive opacities even a change of 3 orders of magnitude do notproduce significant changes in the post–VLTP tracks. Quiteon the contrary, the tracks are more sensitive to radiativeopacity changes. In fact we find that — for both the 0.53and the 0.584 M ⊙ sequences — increasing the opacities bya factor 1.5 produces a reduction of ∼ .
04 dex in T eff and ∼ . L/L ⊙ . Similarly a reduction in the opacityby a factor 0.5 leads to increases of ∼ .
075 dex and ∼ . . The shift in the location of post–VLTP tracks due to changes in the opacities is displayed inTable 2, where we show the change in log T eff for differentvalues of g and for two different remnant masses (0.53 and0.607 M ⊙ ). Also the induced shift in the mass derived fromcomparison with the g T eff values of MA06 tracks is shown.Two things deserves comments. The effect of different ra-diative opacities is much larger for higher remnant massesand at larger luminosities (i.e. lower gravities). Indeed, notethat for the 0.53 M ⊙ remnant an increase in the opacity of50% would not produce a shift of more than 0.01 M ⊙ inspectroscopical mass determinations, and for the 0.607 M ⊙ remnant the increase in the spectroscopical mass becomesvery important, reaching up to 0.07 M ⊙ at high luminosi-ties. Note also that the shift in log T eff is almost the samefor the same change in κ regardless of the mass.Due to the importance of this effect we consider inter-esting to analyze if the effect is due to the value of theopacity at some speciffic region of the star — e.g. the He-burning shell. We proceed then to make localised changesin the opacity and found, against expectations, that it isnot the value of the opacity (per unit mass κ ) at someparticular region that is relevant but the total opacity ofthe envelope ( R envelope κ dm ). By looking at the models, we It is worth noting that we do not expect important changesin asteroseismological inferences due to changes in opacities.This is so because asteroseismological determinations are usu-ally based on adiabatic period studies, which are barely affectedby changes in the opacities.iller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks 7Sequence g = 5 . g = 6 g = 6 . g = 7 g = 7 . M ⊙ κ × .
5) (-0.1072) (-0.0687) (-0.045) (-0.0292) (-0.0188)0.53 M ⊙ -0.0462 -0.045 -0.044 -0.0432 -0.0365( κ × .
5) (0.0096) (0.0106) (0.0099) (0.0086) (0.0066)0.607 M ⊙ κ × .
5) (-0.1748) (-0.126) (-0.0888) (-0.0545) (-0.031)0.607 M ⊙ -0.0455 -0.0442 -0.0425 -0.0418 -0.0422( κ × .
5) (0.0684) (0.0484) (0.0366) (0.0243) (0.0158)
Table 2.
Shifts in effective temperature ( δ log T eff ) induced by changes in κ for different values of g . The value betweenbrackets is the predicted induced shift in spectroscopical masses (in M ⊙ ).find that altering the radiative opacity produces almost nochange in the structure of the envelope. Then, as dT /dm isnot altered by changes in κ , varying κ leads to an oppositeand proportional change in the luminosity l ( m ) of the starvia the relation dTdm = − π ac × κlr T (1)A clue of why only l reacts to a change in κ can be obtainedfrom the following simple analytical argument. If we assumethat the envelopes of these objects are nearly homologicalto each other, then we have that under homology changes(with x = δr/r ) the change in the pressure and density ofa shell is (see Kippenhahn & Weigert 1990 for a deduction) δPP = − x, δρρ = − x. (2)Then if we assume an ideal gas equation of state for theenvelope — which is quite correct — we have the additionalrelation δTT = − δρρ + δPP = − x. (3)Using the equation of the temperature profile (Eq.1) byimposing an arbitrary change in the opacity δκ/κ and using δ (cid:18) dTdm (cid:19) = dTdm × δTT (4)we get δTT = δκκ + δll − δrr − δTT (5)and by using Eq. 2 and 3 we finally find δll = − δκκ , (6)which is quite similar to what it is observed in the numericalmodels — during the horizontal part of the tracks. Let usnote that this change in the luminosity does at first orderbalance (in Eq. 1) the change in opacity, leaving only secondorder effects on the factor κl to be balanced by the otherfactors in Eq. 1: κ , l , = κl (cid:18) δκκ (cid:19) (cid:18) δll (cid:19) = κl − (cid:18) δκκ (cid:19) ! . (7)Also, note that due to the high powers of r and T in Eq.1, small changes in these quantities should be enough tobalance the remaining second order effects. In fact, when looking at the numerical models all r ( m ) , P ( m ) , T ( m ) re-main almost unchanged by the change in the opacity, being l ( m ) the only structure variable that undergoes an impor-tant variation. The change in l ( m ) seems to be associatedwith a change in the energy liberated by the helium burn-ing shell —change which can be attained with almost nochange in T ( m ) due to the extremely high sensitivity oftriple alpha reaction rates with temperature— and by achange in the dS/dt term of the energy equation — thelower the opacity, the faster the evolution and contractionof the envelope. Sumarizing we can say that altering theradiative opacity of the envelope leads to a similar changein the l ( m ) profile of the star which balances (at first or-der) the effects of the opacity change in Eq. 1. It seems thatas consequence of this balance only minor changes appearin the other structure variables which remain almost un-changed — this probably reflects the fact that the run ofthese variables in the envelope of the star is forced by theradius and mass of the He-free core where most of the gravi-tational field is generated. Although this does not intend tobe a complete explanation, something which is impossiblenowadays due to the lack of an accepted explanation for thebehaviour of structures with burning shells , we think thatit sheds some light on what is happening on these models.Finally let us mention that as r ( m ) is not changed, then theradius of the star is almost the same, independently of thevalue of the opacity. Then, due to Steffan-Boltzmann’s lawwe have that the change in the opacity produces a variationin the effective temperature of δT eff T eff = 14 δL ⋆ L ⋆ − δR ⋆ R ⋆ ∼ δL ⋆ L ⋆ (8)In fact for the 0.584 M ⊙ models with normal and enhanced(for a factor 1.5, δκκ ∼ .
5) opacity we have that, in theknee, δLL ∼ .
34 and δT eff T eff ∼ .
09 and thus δT eff T eff ∼ δL ⋆ L ⋆ .Note however that δLL ∼ .
34 is far from the value 0.5 in δκκ ( δLL is close to 0.5 in the horizontal part of the track, i.e.log T eff < . As models seem to be sensitive to the value of the radia-tive opacity in the envelope, we have analyzed how muchopacities can change due to different adopted compositions.Firstly, we have assessed possible changes in the tracksdue to changes in the total amount of metals in the models In fact the problem is in some way related with the longstanding problem of why stars become red giants; see Faulkner(2005) for a recent discussion, and review, of this issue. Miller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks R o ss e l a nd O p ac it y [ c . g . s . ] OPALOP with OPAL compositionOP with correct NOP with correct N and NeOP with Ni instead of Fe
Fig. 5.
Value of the opacity for different adopted composi-tions.— with the exception of C and O which are always kept con-sistent with the envelope abundance. We did this by usingOPAL C- O- enhanced tables for Z = 0 .
01 and Z = 0 .
03— all of our previous sequences correspond to Z = 0 . Z = 0 .
01 and Z = 0 .
03 tracks differ ineffective temperature by only ∼ .
006 dex.Secondly, we have explored the use of opacities fullyconsistent with the abundances of the models. This is nota minor issue as Ne and N can be much larger than theirsolar scaled values — also Mg can reach values of 2%; seeWerner et al. 2004. Specifically, we have used the tool at OPproject website (Badnell et al. 2005) which allows to cal-culate opacities for arbitrary compositions. In this case wehave not made any track calculation but instead we havejust compared the opacities for a given model (i.e. for agiven T and ρ profile). We compare first OP and OPALopacities for the same imput composition. The result isshown in Fig. 5 Note that, for log T >
7, OP opacitiestend to be about 5 to 7 % larger. This would probably in-troduce a small change of about 0 .
01 dex in the tracks. Theinclusion of N and Ne — with abundances consistent withthose displayed by the stellar models — markedly increasesthe opacity values below log T ∼ .
3, but almost no changesare present at higher temperatures where most of the massof the envelope is stored, see Fig.5. As PG1159 stars aresupposed to be iron deficient due to s-process (WH06), wehave analyzed the extreme case for which all the iron waschanged into Ni. In this case, the opacity bump is locatedat larger T values, thus increasing opacity between log T =7.3 and 7.6. Although this change in the opacity is notenough to reconcile the discrepancy between spectroscopi-cal and asteroseimic masses of PG1159 stars, it is importantto note that modifying the heavy metal distribution doesintroduce a change in the opacities at high temperatures.Indeed, as opacities increase with the atomic number of theelements — due to the increase in the possible atomic tran-sitions, Roger & Iglesias (1994) — it remains to be seen towhat a degree an important increase in the content of veryheavy metals due to s-process (both in the AGB and atthe VLTP) can increase the opacity at the bottom of theenvelope. Note that because of the higher ionization po- tentials, those elements are expected to affect opacities atmuch higher temperatures than do Fe or Ni.
4. Other issues
We have compared our tracks with the few models of mod-ern H-deficient post–AGB sequences available in the liter-ature, particularly the 0.604 M ⊙ model of Herwig (2005)and the 0.632 M ⊙ model of Lawlor & Mac Donald (2006). eff )55.566.577.588.5 L og ( g ) Herwig’s 0.604 M sun
Lawlor’s 0.632 M sun
O’Brien’s 0.573 M sun
Fig. 6.
H-deficient tracks of Herwig (2005) and Lawlor &Mac Donald (2006) compared with our post–VLTP tracks(with masses 0.87, 0.741, 0.664, 0.609, 0.585, 0.565, 0.542,0.530, and 0.515 M ⊙ ). Also a non–late helium flash (butH–deficient) track from O’Brien (2000) is shown for com-parison.Note from Fig. 6 that our models show a good agree-ment with the tracks of both authors. The agreement isremarkable despite the very different input physics and evo-lutionary history of progenitor stars considered by those au-thors. Indeed, Lawlor & Mac Donald (2006) models do notinclude any kind of overshooting prescription and Herwig’smodel is the result of an initially 2 M ⊙ star model and thuswith a distinct previous evolution than our 0.609 M ⊙ se-quence which comes from a 3 M ⊙ model. In addition, theEoS are different in all the cases. This supports the findingsin Sects 2 and 3. For a quantitative inference, we estimatemasses for those tracks by comparing their relative loca-tion with MA06 tracks. We derive masses of about 0.611 M ⊙ and 0.623 M ⊙ for Herwig and Lawlor & Mac Donaldsequences respectively — note that the resulting Herwig’strack becomes bluer than ours, leading to slightly lowerspectroscopical masses. In both cases the induced shift inspectroscopical masses would be lower than 0.01 M ⊙ , thusreinforcing the robustness of the MA06 post–AGB tracks. As mentioned early in this work H-burners old tracks arecooler than MA06 tracks, thus leading to a much betteragreement with asteroseismology (WH06). As we discussedin Sect 2, the difference in the location between old and new iller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks 9 tracks cannot be tracked back to a distinct previous evo-lution, with the exception of the VLTP event. It is worthnoting that old models (Bl¨ocker 1995a,b) are based on theCox & Stewart (1970) opacities, in contrast to new modelsthat use OPAL and molecular opacities. In this connec-tion we feel important to recall that already Dreizler &Heber (1998) noted a shift of 0.03 M ⊙ between old Wood& Faulkner (1986) and O’Brien & Kawaler (as they appearin Dreizler & Heber 1998, here shown in Fig. 6) helium–burning tracks, where the latter make use of modern OPALopacities. Appart from possible changes that could arisefrom the different opacities used in the calculations, the dif-ference in the tracks can be expected from the very fact thatthe Bl¨ocker’s tracks are H-burners while VLTP are heliumburners. In fact, we note from our sequences that there isa noticeable difference in H–burning-post–AGB and post–VLTP tracks for low remnant masses ( ∼ < . M ⊙ ). Thiscan be seen in Fig 7, where we show our post–AGB H–burning tracks (with H–rich surface compositions) of 0.517and 0.53 M ⊙ compared with the post–VLTP tracks of sim-ilar mass. Note that post–VLTP tracks are certainly bluerthan their H–rich counterparts and that they are more com-pact in the high gravity region of the diagram (log g > . M ⊙ track is very similar to Bl¨ocker’s 0.524 M ⊙ track,strongly suggesting that the difference with Bl¨ocker (1995b)tracks at (very) low masses is because MA06 tracks arepost–VLTP tracks. Using H-rich tracks to determine spec-troscopical masses for PG1159 stars will certainly influencethe result, in particular the stellar masses for high–gravityPG1159 stars would be largely overestimated. eff )44.555.566.577.58 L og ( g ) sun , H-burner0.530 M sun , H-burner0.515 M sun , post-VLTP0.530 M sun , post-VLTPBloecker’s 0.524 M sun Fig. 7.
Comparison between H-rich, H-burning tracks andH-deficient, He-burning tracks of low mass. It is clear fromthe figure that H-burners have lower T eff and are much sim-ilar to Bl¨ocker’s tracks. Using H-rich tracks to estimatePG1159 masses can lead to important overestimations inthe low mass range. As mentioned early in this work, N-deficient PG1159 ob-jects are probably the descendents of LTP events. Thus, apriori one should be careful about using post-VLTP tracksfor all PG1159 stars. In this context we now turn to an- alyze the question if there are systematic differences be-tween post-VLTP and post-LTP tracks. From figure 1 ofHerwig 2001 it seems that there is no differece once thestar enters the PG1159 stage. However we will now ana-lyze a wider range of masses. In Fig. 8 PG1159-tracks com-ing from VLTP and LTP are compared for similar rem-nant masses. In the upper panel LTP tracks with differentH-abundances are compared with VLTP tracks of similarmass. The ∼ . M ⊙ tracks correspond to the sequenceanalysed in Althaus et al. (2007). In these sequences two dif-ferent post-LTP evolutions have been considered. The firstin which the final surface H-abundance is normal and asecond in which due to mass loss episodes the whole H-richenvelope was eroded, exposing the He-rich intershell. Due tothe absence of the H-burning shell in the second case it liesvery close to the postVLTP tracks. The second experimentis also shown in the upper panel of Fig. 8 corresponds toan LTP sequence (0.543 M ⊙ ) in which the total H-contentof the star was artificially diluted to different depths, thusleading to different final surface H-abundances. As can beseen once the star reaches the PG1159 stage, the lower thesurface H-abundance the closer the LTP-tracks gets to theVLTP track of similar mass. Finally, in the lower panel ofFig. 8 post-LTP tracks of H abundances close to the usualdetection limit are compared with VLTP tracks of similarmass. From that plot is clear that for surface gravities abovelog g = 6, where almost all PG1159 stars lie, VLTP tracksand LTP tracks with low H-abundances are similar. Thenusing post-VLTP tracks for spectroscopic mass determina-tions of LTP objects with no detectable H does not seemto introduce any systematic effect on the mass determina-tion. On the contrary using post-VLTP tracks for hybridPG1159 stars may produce an important underestimationof the mass.
5. Discussion and final remarks
In the present work we have analyzed how uncertaintiesin the modeling of H-deficient post-VLTP remnants couldaffect spectroscopic mass determinations of PG1159 typestars. In Sect. 2, inspired by a suggestion in WH06 we haveanalyzed the importance of previous evolution. As the cal-culation of each full sequence is extremely time consum-ing (both computational and human, as at some stages themodels need hand interaction to converge them) we hadto restrict ourselves to a limited region of the parameterspace. Even in this case some conclusions can be drawn.Third dredge up alone does not seem to change the theoret-ical log T eff -log g - M locus and consequently its uncertaintiescan not affect spectroscopic mass determinations. On theother hand differences in mass loss rates (i.e. TP-AGB life-times) alter the location of the tracks, but only slightly. Infact our simulations show that even a reduction by a factorof 3 of TP-AGB lifetimes would not increase spectroscopicmass determinations by more than ∼ . M ⊙ . Going evenfurther we have shown that even extreme mass losses thatproduce low mass remnants ( ∼ . M ⊙ ) from very dif-ferent progenitors than those in MA06, does not introduceimportant shifts in spectroscopic mass determinations, be-ing only ∼ . M ⊙ . In an even more extreme limiting casewe have computed a sequence, in which no overshooting was In this sequence, due to the very low remnant mass, the lowintensity of the He-flash does not lead to any 3DUP.0 Miller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks eff ) 67 l og ( g ) sun , VLTP0.542 M sun , VLTP0.565 M sun , VLTP0.589 M sun , VLTP0.513 M sun , LTP, No H0.543 M sun , LTP, H=0.050.562 M sun , LTP, H=0.050.589 M sun , LTP, H=0.04 l og ( g ) sun , VLTP0.542 M sun , VLTP0.513 M sun , LTP, No H0.517 M sun , LTP, H=0.680.543 M sun , LTP, H=0.050.543 M sun , LTP, H=0.100.543 M sun , LTP, H=0.200.543 M sun , LTP, H=0.39 Fig. 8.
Comparison between post-LTP and post-VLTPtracks. Upper panel: Comparison between VLTP tracksand post-LTP tracks of similar mass but different Habundances. Lower panel: Comparison between post-VLTPtracks with post-LTP tracks that display surface hydrogenabundances, close to the detection limit.considered in the whole evolution, and found a very sim-ilar post-VLTP track than MA06. All these experimentssuggest, contrary to the argument in WH06, that previousevolution only plays a secondary (and not very important)role in determining the theoretical log T eff -log g - M locus.In light of these results we have discussed why the argu-ment presented in WH06 does not apply to post-VLTP se-quences. We showed that it is the HeFC (if any), and notthe HFC, mass and radius that is important for post-VLTPtracks. In particular we find that the HeFC converges fasterthan the HFC in the mass-radius diagram. However, shellhomology relations (as those used to derive the luminosity-mass-radius relation) should not be taken too seriously inthese models, as they neglect the importance of the enve-lope and only relate the luminosity of the burning shellsto the properties of matter in the burning shells and to thevalues of mass and radius of the core, since we find (in Sect.3.2) an important dependence of the shell luminosity withthe whole opacity of the envelope.We have roughly addressed the robustness of the tracksregarding EoS, C-O core composition, conductive andradiative opacities. We find that only radiative opacitymay affect the location of the tracks to some an extent.Specifycally we find that the luminosity of the post–VLTP sequences in the horizontal part of the HR diagram is verysensitive to the envelope opacity. In fact the luminosity ofthe He-burning shell turns to be sensitive to the total opac-ity of the envelope. We also present some analytical argu-ments to explain the shift induced by changes in radiativeopacitites. In this connection we explore how important theenvelope composition can be for the opacity of the envelope.We find that changes in light metals (Ne and Mg) can makeimportant changes in the opacities but only at low temper-atures ( T < × K) where no much mass of the envelopeis stored. Although this may be important for pulsationalstudies of PG1159 stars, it will certainly not change the T eff of the sequences. By contrast changing Fe into Ni inthe opacity calculations we find a more slightly importantchange. This particularly leaves open the question of howmuch opacities at the bottom of the envelope can change ifimportant amounts of Fe are transformed into very heavymetals by s-processes. We can conclude that, unless thereare important changes in the abundances of very heavy el-ements due to s-process, an increase in the opacity at high T is not expected to change more than 10%.All these arguments show that MA06 tracks are robustenough as to be used for spectroscopical mass determina-tions of PG1159-type stars (specially at high gravities; log g ∼ > ∼ < . M ⊙ inspectroscopic mass determinations) between those trackswith the other modern post–VLTP tracks available in theliterature (Herwig 2005 and Lawlor & Mac Donald 2006).We have also addressed in Sect. 4.3 if any systematic in themass determination may be due to the fact of some PG159stars being post-LTP objects with H-abundances below thedetection limit. We find that the resulting tracks in thePG1159 region of the T eff -g (log g >
6) diagram are verysimilar to post-VLTP tracks when surface H abundance isbelow ∼
5% by mass fraction. Thus, we conclude that thepost-VLTP tracks of MA06 are solid enough for spectro-scopic mass determinations of post-LTP objects with H-abundances below the detection limit and, thus, it seemsthat no systematic should be present due to this effect.On the contrary, we find that using post-VLTP tracks forPG1159 stars with important H-abundances (the so calledhybrid PG1159 stars) may lead to an important underesti-mation of the mass. Regarding the difference with Bl¨ocker’sH-rich post–AGB tracks we can say that, for low mass rem-nants ( ∼ < . M ⊙ ), the differences in the tracks seem to bemainly due to the fact that those tracks are H-burners sinceour own H-burner sequences are much colder than our post–VLTP ones. Other differences with older tracks may be dueto the difference in the opacities adopted for the He, C -rich intershell (note that older works make use of old Cox& Stewart opacities). These seems to be supported by thegood agreement between all H-deficient tracks that includemodern OPAL opacities —Herwig 2005, Lawlor & MacDonald 2006 and, more roughly ( ∼ . M ⊙ ), even withthe non- late helium flash 0.573 M ⊙ sequence of O’Brien2000.From the present work we judge that the systematicdiscrepancy between asteroseismological and spectroscopi-cal mass determination methods should not be attributedto uncertainties in post-AGB tracks. Whether the discrep-ancy comes from errors in asteroseismological or spectro-scopical determinations is not known, however some points iller Bertolami & Althaus: On the robustness of H–deficient post–AGB tracks 11 are worth emphasising. Although asteroseismology is usu-ally accepted as a more accurate method (very low errorbars are usually given), its robustness is not so clear. Infact recent works (C´orsico & Althaus 2006, C´orsico et al.2007a and C´orsico et al. 2007b) show the results of aster-oseismology to be method dependent. In this context it isworth emphasising that the asteroseismic mass of PG 1159-035 is reduced to ∼ . M ⊙ —only ∼ . M ⊙ higher thanits spectroscopical mass— when detailed evolutionary mod-els and averanged period spacing (instead of the usuallyadopted asymptotic period spacing) are used in the analy-sis, see C´orsico et al. (2006). Interestingly enough, duringthe referee stage of this article a new study of PG 0122+200(C´orsico et al. 2007b) which is based on our evolutionarymodels and a detailed period by period fitting procedure,reduces the mass discrepancy (with MA06 value) in thisstar to less than a 4%. This clearly shows the existence ofserious systematics in standard (i.e. based on asymptototicperiod spacing) asteroseismological determinations. In thiscontext is worth noting that a mean PG1159 mass of 0.573 M ⊙ like the one deduced from MA06 tracks, even if sensi-tively lower (0.044 M ⊙ ) than previously thought, is in goodagreement with that of their probable descendants, the DBwhite dwarfs (0.585 M ⊙ , Beauchamp et al. 1996) . Thenour results not only call for a revision of PG1159 modelatmospheres but, specially, for a revision of systematicsin usually adopted asteroseismological mass determinationmethods.Our full set of evolutionary tracks for post-VLTP objects is available at our web site at . Acknowledgements.
M3B wants to thank Achim Weiss, Agis Kitsikisand Alejandro C´orsico for useful and instructive discussions and theMax Planck Institut f¨ur Astrophysik in Garching and the EuropeanAssossiation for Research in Astronomy for and EARA-EST fellow-ship during which part of this work was done. This research was par-tially supported by the PIP 6521 grant from CONICET.
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