1 Key words.Stars:pulsationStars:heliumburningphase- Variables: low-mass Cepheids 7 0 0 2 n a J 9 1 v 6 5 2 1 0 7 0 / h p - o r t s a : v i X r a Astronomy & Astrophysics manuscript no. sissa February 5, 2008 (DOI: will be inserted by hand later) Updated pulsation models for BL Herculis stars 1 12 Marconi, M. & Di Criscienzo, M. 1 INAF-Osservatorio Astronomico di Capodimonte, via Moiariello, 16,I-0080131, Naples, Italy 2 Universit´a diRoma “Tor Vergata” via della Ricerca Scientifica, 1, Rome, Italy Received ????; accepted ??? Abstract.ContextPopulationIIpulsatingvariablesplayarelevantrolebothasdistanceindicatorsandastracers of theproperties of old stellar populations. Aims In this paper we present an updated and homogeneous pulsational scenario for a wide range of stellar parameterstypicalofBLHerstarsi.e.,PopulationIICepheidswithperiodsshorterthan8days.MethodsTothis purpose,weadoptanonlinearconvectivehydrodinamicalcodeevaluatethestabilityandfullamplitudebehaviour ofanextensivesetofBLHerpulsationmodels.Variousassumptionsofmass,luminosityandmetallicityconsistent with the most recent evolutionary prescriptions, are adopted. Results We obtain the theoretical instability strip for both fundamental and first overtone pulsators and present a detailed atlas of light/radial-velocity curves. Some relations for the boundaries of the instability strip and for the dependence of the absolute magnitude on period,massandcolor,aswellasthefundamentalperiod-amplituderelationsarederived.Finally,weprovidethe theoreticalperiod-radiusrelationforBLHerandfindthatitisinexcellentagreementwiththeempiricalrelation by Burki& Meylan and consistent with theone holding for shorter periods for RRLyrae stars. 1. Introduction Buchler (1994) also evaluated the pulsation properties of first overtone (FO) models and found that the first over- The BL Her variables form a small but interesting group tone blue edge (FOBE) was very close (≤ 100 K) to the of radially pulsating Population II stars with period P in fundamental (F) one, producing a very narrow regime of the range 0.8-3 days. They are observed in globular clus- FO-only pulsation. Moreover, FO models were found to ters with few RR Lyrae stars and blue horizontal branch show very low pulsation amplitudes but it seemed to be (HB) morphology and they are brighter than RR Lyrae hard to discriminate the pulsation mode on the basis of but fainter than Anomalous Cepheids, for a fixed period. their Fourier coefficients. By relying on a nonlinear con- They are currently interpreted as central He-burning low vective pulsation analysis, Bono et al. (1997a) found a mass (M ≤ 0.6M⊙) stars which, after their main core good agreement between the predicted and the observed He-burning phase spent in the blue side of the Zero Age BL Her instability strip boundaries and suggested that HorizontalBranch(ZAHB),evolvetowardtheAsymptotic thesepulsatorsarepulsatingintheFmodewithatypical Giant Branch (AGB) crossing the instability strip at lu- mass of 0.52-0.59 M⊙. Moreover these authors predicted minosities higher than the RR Lyrae level (see Gingold a Period-Luminosity-Amplitude (PLA) relation in the B 1985; Bono et al. 1997a,Caputo 1998). band. The properties of these objects have recently been re- However the Bono et al. (1997a)nonlinear convective ap- viewed by Wallerstein (2002) and Sandage & Tammann proach was limited to a quite restricted range of stellar (2006). From the theoretical point of view, Buchler & parameters and adopted an old input physics both in the Moskalik (1992) and Buchler & Buchler (1994) presented evolutionary and the pulsational computations (see Bono a pulsational study of models with P ≤ 3 days based on & Stellingwerf 1994 and Bono et al. 1997 for details). a linear and nonlinear radiative analysis. These authors In this paper we present an updated and homogeneous found evidence for a resonance between the fundamental pulsational scenario for a wide range of stellar parame- mode and the second overtone, similar to the one often ters typical of BL Her stars. In an accompanying paper invokedasthe explanationoftheHertzsprung progression (Di Criscienzo et al. 2006) we will discuss the connection phenomenon in Classical Cepheids (see Bono, Marconi, between the pulsational scenario and the predictions of Stellingwerf 2000 for a detailed discussion). Buchler & updated evolutionary models in order to compare theory with observations. Send offprint requests to: Marconi, M. Marconi & Di Criscienzo: Updated pulsation models for BL Herculis stars 3 Table 1.InputparametersofthecomputedBLHermod- strip, which depends on the intrinsic parameters of pulsators. els. An helium abundance Y=0.24 has been adopted. For each given metallicity, mass and luminosity, we have cal- culated themaximum and minimum effectivetemperature for Z M/M⊙ LogL/L⊙ FOBE FBE FORE FREthe onset of either fundamental or first-overtone pulsation, as 0.0001 0.60 1.95 - 6850 - 5750 reported in the last four columns of Table 1, where FOBE 2.05 - 6750 - 5650 (FORE)andFBE(FRE)aretheblue(red)limitforfirstover- 2.15 - 6750 - 5550 toneandfundamentalpulsatorrespectively.Itisofimportance 0.65 1.91 6950 6850 6100 5750 to note that our calculations confirm the earlier suggestion 2.01 6750 6850 6300 5750 by Bono et al. 1997(a,b,c) that, for a given helium content 2.11 - 6750 - 5550 and mass, there exists a ”intersection” luminosity LIP where 0.001 0.50 2.11 - 6650 - 5450 FOBE=FBE and that above this luminosity only the funda- 2.41 - 6350 - 5150 mental mode has a stable nonlinear limit cycle. As shown by 0.55 1.81 6875 6850 6400 5650 Di Criscienzo et al. 2006, the evolutionary models show that 1.91 - 6850 - 5550 BL Her variables occur at luminosity higher than this inter- 2.01 - 6750 - 5450 section point: therefore, in the following we will consider only 0.65 1.81 7050 6750 6650 5750 fundamental models. As for the red boundary of the funda- 1.91 6850 6750 6150 5650 mentalmode,it isworthmentioningthatit isexpectedtode- 2.01 6650 6850 6350 5650 pend on theefficiency of convection and, in particular, on the 0.004 0.55 1.81 - 6950 - 5750 valueofthemixinglengthparameter l/Hp weassumetoclose the nonlinear system of dynamical and convective equations 1.91 - 6850 - 5650 (see Stellingwerf 1982 and Bono & Stellingwerf 1992, 1994 for 2.01 - 6750 - 5450 details on the treatment of convection in the adopted hydro- dynamical code). To test this occurrence, we have computed The organizationofthe presentpaperis the following: additional models with l/Hp increased from 1.5 to 2.0 and we in Sect.2 we present the new computed models. In Sect. confirmthegeneraltrendshownbyRRLyrae(DiCriscienzoet 3 we deal with predicted light/velocity curves and visual al.2004)andClassicalCepheidmodels(Fiorentinoetal.2006, amplitudes.SameimportantrelationsaregiveninSection inpreparation)inthattheFREmovestowardhighereffective 4, whereas the conclusions close the paper. temperatures as l/Hp increases. In addition, we find that this effect is less important with increasing theluminosity. In par- ticular,for M =0.50M⊙ andZ=0.001 theFREblueshiftisof 2. The stellar pulsation models about 200 K at logL/L⊙ = 2.01 but reduces to less than 100 K for logL/L⊙ = 2.41. This occurrence is due to the reduced To study BL Her stars, we have adopted the same pul- density values, and in turn to the increase of the required su- sation code and the same physical and numerical as- peradiabaticity, in the outer layers of stellar structures with sumptions used for RR Lyrae (Bono & Stellingwerf 1994, low mass and veryhigh luminosity. Bono, Castellani & Marconi 2000, Marconi et al.2003, Di Criscienzo et al. 2004) and Anomalous Cepheid models 3. Light curves and pulsation amplitudes (Marconi et al. 2006). Model sequences have been com- puted as one parameter families with constant luminos- For each investigated model, the non linear analysis provides ity, mass and chemical composition, by varying the effec- the variation of relevant parameters, namely luminosity, ra- tive temperature with a step of 100 K. The various as- dius, radial velocity, effective temperature and gravity, along sumptions of mass, luminosity and metallicity are consis- a pulsational cycle. A subsample of bolometric light curves tent with the most recent evolutionary prescriptions (see (Z=0.001andM =0.65M⊙ andthelabeledluminositylevels) is shown in Figure 1, while Figure 2 exhibits the correspond- Pietrinfernietal.2004,2006)andarereportedinthefirst ing radial velocity curvesas a function of pulsation phase. All three columns of Table 1. these curves show a large variety of shapes, which is perhaps Linear regression through the model’s parameters the most striking feature of BL Her models as already sug- allows us to derive analytical relations connecting the gested by Moskalik and Buchler (1994). In all the sequences period of models to the intrinsic stellar parameters, (in particular for radial velocity curves), thepresence and the namely mass, luminosity and effective temperature, i.e: progression of the main bump is evident in analogy of what observed for Classical Cepheids with periods around 10 days L M (the well known Hertzsprung progression). A detailed investi- logP = 11.579(±0.015)+0.893log −0.89log L⊙ M⊙ gation of this issue, through the study of Fourier parameters, − 3.54logTeff (1) will be addressed in a future paper. In order to compare theoretical results with observations, for fundamental pulsators and: the bolometric lights curves have been transformed into the L M photometric bands UBVRIJK, using bolometric corrections logP = 10.784(±0.003)+0.806log −0.66log L⊙ M⊙ and temperature-color relations provided by Castelli, Gratton & Kurucz (1997a,b). In this way, light-curve amplitudes and − 3.31logTeff (2) mean absolute magnitudes, either intensity-weighted hMi or for FO ones. magnitude-weighted (M), are derived in various photometric It is well known that radial pulsation occurs in a quite bands.Afterhavingtestedthatintensity-weightedmagnitudes well-definedregionoftheHRdiagram,thesocalledinstability approximatebetterthestaticvaluesthanmagnitude-weighted 4 Marconi & Di Criscienzo: Updated pulsation models for BL Herculis stars PPhhaassee Fig.1. Theoretical bolometric light curves for a subsample (Z=0.001 and M=0.65 M⊙) of fundamental models.The luminosity levels are labeled. ones, similarly to what found for RR Lyrae (see e.g. Marconi minosityRRLyraestars).Howeverintherangeoflinearitywe et al. 2003) and Classical and Anomalous Cepheids (Caputo, obtain: Marconi,Ripepi1999,Marconietal.2004),inthefollowingwe M give the predicted relations with h Mi also because empirical logP = −0.033(±0.034)−1.15log −0.475log<MV > investigations usually providethis typeof mean values1. M⊙ In Figure 3, we show the behaviour of visual amplitudes − 0.195AV (3) as a function of the period for different values of metallicity, We notice that this dependence of the period on amplitude massandluminosity.AsforRRLyraemodels,wefindthatfor is, within the errors, in agreement with those found in Di fixedperiodtheamplitudeincreasesastheluminosityincreases Criscienzo et al. 2004. and as themass decreases (middle and bottom left panels, re- spectively), while it remains quite constant in the considered interval of metallicity, at fixed mass and luminosity (top left 4. Same relevant relations panel). We notice that part of the observed behaviour in the period-amplitude diagram is due to the dependence of period The linear regression through the data reported in Table 1 on mass and luminosity, as simply derived even from a sim- provides thefollowing analytical relations: ple linear adiabatic approach. The change in the pulsation in amplitude is mainly related to the distance from the FBE, as L M showninthetworightpanelsofthesamefigure.Thedeviation logTe(FBE)=3.912(±0.005)−0.035logL⊙ +0.048logM⊙(4) fromlinearityofthehighestluminositylevelmodels(toppanel L M ofFigure 3) isrelated tothecomplex couplingbetween pulsa- logTe(FRE)=3.925(±0.005)−0.075log +0.118log (5) L⊙ M⊙ tion and convection for these low density and cool structures. (see also Bono et al. 1997c for a similar behaviour in high lu- witharmsof0.005,validontheadoptedrangeofmetallicity and stellar mass. The location of these edges in the MV-logP 1 Similar relations involving magnitude-weighted mean val- planeispresentedinFigure4andthecorrespondinganalytical ues are available upon request to theauthors relations are: Marconi & Di Criscienzo: Updated pulsation models for BL Herculis stars 5 Fig.2. The same as Fig. 2 but for theoretical radial velocity curves. M − 1.10log (8) M⊙ <MV(FBE)> = −0.36(±0.04)−1.55log M −2.37logP hMVi = −2.08(±0.02)−2.88logP −4.13[hMVi−hMKi] M⊙ M − 2.14log (9) + 0.05(±0.03)logZ (6) M⊙ M <MV(FRE)> = 0.14(±0.03)−2.25logM⊙ −2.17logP with no dependence on metallicity. According to these re- + 0.05(±0.02)logZ (7) lations, for a sample of variables at the same distance and with the same reddening, e.g., for variables in a given globu- lar cluster or galaxy with no differential reddening, one could estimatethemassrangespannedbythevariablesfromtheob- FromthesameFigure,wenotethattheBLHerinstability served luminosity range, once periods and intrinsic colors are stripistheextensionathigherluminositiesofthefundamental firmly known. However, these relations have the disadvantage instabilityregionforRRLyraestars,whereasitdoesnotover- of depending on reddening uncertainties and for this reason, lap theoneforAnomalous Cepheids(ACs) sincetheACFRE as discussed in several earlier papers (e.g., Madore 1982, Di is hotter than the BL Her FBE (see also Caputo et al. 2004). Criscienzo et al. 2004, Udalski et al.1999), Period-Wesenheit This result, also related to the significant mass difference be- relations are widely used, with the Wesenheit functions being tween the two classes of pulsators, is fully consistent with the bydefinitionreddeninginsensitive.Thefullsetofthepredicted empirical evidence that ACs are brighter than BL Her stars, Period-Wesenheit and Period-Magnitude relations, including for a given period. the evolutionary properties of the BL Her stars, are reported The natural outcome of the period relation (Eq. 1) in the in Di Criscienzo et al. (2006). Figure 5 shows the behavior of observational plane is the Period-Magnitude-Color relation in thepredictedradiifor fundamentalpulsators, obtained byav- which the pulsation period is correlated with the pulsator ab- eragingthetheoreticalradiuscurve.Thesolidlineisthelinear solute magnitude and color, for a given mass. As a matter of regression through theentire set of models (PR relation): example, linear interpolations through theresults give: hMVi = −1.22(±0.05)−2.60logP −3.03[hMBi−hMVi] logR=0.87(±0.01)+0.529(±0.006)logP (10) 6 Marconi & Di Criscienzo: Updated pulsation models for BL Herculis stars 22 11..55 11 00..55 00 22 11..55 11 00..55 00 22 11..55 11 00..55 00 --00..22 --00..11 00 00..11 00..22 00..33 00 00..11 00..22 00..33 00..44 llooggPP Fig.3. (Right panels)Visual amplitudes versus periods for selected models varying the metallicity at fixed mass and luminosity (upper panel), varying the luminosity at fixed mass and metallicity (Z=0.001, intermediate panel) and varying the mass at fixed luminosity and metallicity (Z=0.001, lower panel).(Left panels) The same but visual amplitudes are now plotted versus ∆ log(P)=log (P)- log(PFOBE) where R is in solar units. This relation is in agreement with furtherclassofradialpulsatorsobservedinsimilarmetalpoor theempirical PR relation obtained byBurki& Meylan, 1986. stellarfields:PopulationIICepheids.Inparticular,wepresent In the same Figure we have also reported the theoretical thenewnon-linearandconvectivepulsationalmodelsforshort PRrelationrecentlyobtainedbyMarconietal.(2005) forRR period variables, generally named BL Her stars. Lyrae. This very good agreement between the theoretical PR We have investigated the instability strip and shown the relations obtained for RR Lyrae and BL Her stars supports dependence of the pulsation region on mass and luminosity, earlier suggestions by Burki & Meylan (1986) concerning the as well as the effect of varying the the mixing lenght parame- similarity of the PR relations for these two classes of Pop. II ter.Wehavepresentedadetailed atlas oflight/radial-velocity variables, and is consistent with the results by Caputo et al. curves. All these curves show a large variety of shapes, which (2004) concerning the behaviour of RR Lyrae and BL Her in isperhapsthemoststrikingfeatureofBLHerstarsasalready thePeriod-Magnitude diagram. suggested by Moskalik & Buchler (1994) on the basis of non- linear butradiative models. 5. Conclusion and final remarks Theoretical light curves have been transformed in the UBVRIJK photometric bands, using the bolometric correc- This work is part of a larger project that has the scope to tions and color-temperature relations provided by Castelli, study, from the theoretical point of view, Population II vari- Gratton &Kurucz(1997a,b). Onthisbasis, meanmagnitudes ables in Globular Clusters and dwarf galaxies. In the last andcolorshavebeenevaluatedandsomerelationsbothforthe few years we have provided the results for RR Lyrae stars boundaries of the instability strip and for the dependence of (Marconi et al 2003, DiCriscienzo et al.2004) and Anomalous theabsolutemagnitudeonperiod,massandcolorarederived. Cepheids (Marconi, Fiorentino & Caputo,2004, Caputo et al. 2004,Fiorentinoetal.2006).Thepresentpaper,togetherwith Finally, we have calculated the period-radius relation and the investigation by Di Criscienzo et al. 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