Period-colour (PC) relations may be used to study the interaction of the stellar photosphere and the hydrogen ionisation front (HIF). RR Lyrae (RRL) and long period classical Cepheids (P>10d) have been found to exhibit different PC behavior at minimum and maximum light which can be explained by the HIF-photosphere interaction based on their location on the HR diagram. In this work, we extend the study to include type II Cepheids (T2Cs) with an aim to test the HIF-photosphere interaction theory across a broad spectrum of variable star types. We find W Vir stars and BL Her stars to have similar PC relations as those from long period and short period classical Cepheids, respectively. We also use MESA to compute RRL, BL Her, and classical Cepheid models to study the theoretical HIF-photosphere distance and find the results to be fairly consistent with the HIF-photosphere interaction theory

Bellinger, E. P.

Bhardwaj, A.

Das, Susmita

Kanbur, S. M.

Singh, H. P.

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arXiv:2010.06624v1  [astro-ph.SR]  13 Oct 2020A study of the stellar photosphere-hydrogen ionisation frontinteraction in pulsating variables using period-colour relationsSusmita Das,1 Shashi M. Kanbur,2 Earl P. Bellinger,3 Anupam Bhardwaj,4 andHarinder P. Singh,51Department of Physics & Astrophysics, University of Delhi, Delhi 110007,India; susmitadas130@gmail.com2Department of Physics, State University of New York Oswego, Oswego, NY13126, USA; shashi.kanbur@oswego.edu3Stellar Astrophysics Centre, Department of Physics and Astronomy, AarhusUniversity, Aarhus 8000, Denmark;School of Physics, UNSW Sydney, NSW 2052, Australia; bellinger@phys.au.dk4Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He YuanLu 5, Hai Dian District, Beijing 100871, China; abhardwaj@pku.edu.cn5Department of Physics & Astrophysics, University of Delhi, Delhi 110007,India; hpsingh@physics.du.ac.inAbstract. Period-colour (PC) relations may be used to study the interaction of thestellar photosphere and the hydrogen ionisation front (HIF). RR Lyrae (RRL) and longperiod classical Cepheids (P > 10d) have been found to exhibit different PC behaviorat minimum and maximum light which can be explained by the HIF-photosphere inter-action based on their location on the HR diagram. In this work, we extend the studyto include type II Cepheids (T2Cs) with an aim to test the HIF-photosphere interactiontheory across a broad spectrum of variable star types. We find W Vir stars and BL Herstars to have similar PC relations as those from long period and short period classicalCepheids, respectively. We also use MESA to compute RRL, BL Her, and classicalCepheid models to study the theoretical HIF-photosphere distance and find the resultsto be fairly consistent with the HIF-photosphere interaction theory.1. IntroductionClassical pulsators like RRLs, T2Cs, and classical Cepheids are important astrophysi-cal objects. They are used as tracers of stellar populations and extragalactic distances,thanks to their well-defined period-luminosity relations (Muraveva et al. 2015; Bhard-waj et al. 2016). Bhardwaj et al. (2014); Das et al. (2018), and references therein)studied the PC relations to gain insight into the structure of the outer envelopes ofRRLs and classical Cepheids. The PC results remained consistent with different obser-vational datasets: RRLs have flat PC slope at minimum light with a significantly greaterPC slope at maximum light while long-period classical Cepheids (P > 10 d) have a flatPCmax slope with a significant PCmin slope. These results may be explained by theinteraction of the stellar photosphere and the HIF (Kanbur & Phillips 1996). The dif-12 S. Das, S. M. Kanbur, E. P. Bellinger, A. Bhardwaj, and H. P. Singh0 1 2log(P)−1012(V−I)minBL HerW VirpW VirRV Tau−1012(V−I)maxBulge0 1 2log(P)LMC0 1 2log(P)SMCFigure 1. The period-colour relations for the different subclasses of type IICepheids in the Bulge and the Magellanic clouds at maximum and minimum light.The best fit linear regression for T2Cs as a whole with the different sub-classes com-bined is represented by the dashed line, after recursively removing the 3σ outliers(shown in smaller symbols).ference in the behaviour of PC relations between RRLs and classical Cepheids may beattributed to their different locations on the Hertzsprung-Russell (HR) diagram- RRLsare hotter and less luminous than classical Cepheids. A lower L/M ratio and/or a hottereffective temperature indicates smaller distance between the HIF and the photosphereor equivalently, a situation with “engaged” HIF-photosphere.Conversely, according to the HIF-photosphere interaction theory, we expect starsoccupying similar regions on the HR diagram to show similar behaviour in their PCrelations. With an aim to analyse this, we extend our study to include the PC relationsof the different subclasses of T2Cs. We use the classification as provided by Soszyńskiet al. (2018): BL Hers are those with periods between 1 − 4 days, W Vir stars haveperiods between 4−20 days and RV Tau stars have P > 20d. We also compute a samplegrid of RRL, classical Cepheid, and BL Her models using the radial stellar pulsationcodes in Modules for Experiments in Stellar Astrophysics (MESA, Paxton et al. 2019)to study the distance between the HIF and the photosphere theoretically.2. Data and MethodologyThe optical (VI) light curves of the T2Cs in the Galactic bulge (Soszyński et al. 2017)and the Magellanic Clouds (Soszyński et al. 2018) are obtained from the OGLE-IVcatalogue. We choose stars with well-sampled light curves for our analysis, havingmore than 30 epochs of observation in both V and I bands. The light curves are fittedwith the Fourier sine series (Das et al. 2018): m = m0 +∑Nk=1 Ak sin(2πkx + φk), whereHIF–Photosphere Interaction 31.791.801.811.821.83log(L)0.91.01.11.2Period0 1000 2000 3000 4000 50000.0000.0050.0100.0150.020rsp_GREKMAge of star (days)Figure 2. An example of a full-amplitude stable pulsation BL Her model with theinput parameters Z = 0.004, X = 0.756,M = 0.55M⊙, L = 64.56L⊙, T = 5950K.x is the phase and N is the order of fit (4 ≤ N ≤ 8) obtained from the Bart’s criteria(Bart 1982). We define colour at maximum and minimum light as:(V − I)max = Vmax − Iphmax; (V − I)min = Vmin − Iphmin, (1)where Iphmax and Iphmin correspond to the I-mag at the same phase as that of Vmax andVmin, respectively.The PC relations for the T2Cs are obtained after correcting for extinction usingstandard methods. The colour excess for T2Cs in the Magellanic Clouds are obtainedfrom the reddening maps of Haschke et al. (2011) while those in the Galactic bulgeare obtained from Gonzalez et al. (2012). The extinction values are then estimated byadopting the reddening law of Cardelli et al. (1989).3. Results from the observational aspectThe PC relations for the T2Cs in the Galactic bulge and the Magellanic Clouds at min-imum and maximum light are displayed in Fig. 1. The PCmax slopes for W Vir starsin the Bulge (0.037 ± 0.141), LMC (−0.033 ± 0.086), and SMC (−0.169 ± 0.189) arenot significantly different from zero (i.e., flat) while those for PCmin are significantlysloped. This is similar to the nature of PC relations for long period classical Cepheids(P > 10d) in Galaxy, LMC, and SMC (Bhardwaj et al. 2014). The similarity of the PCrelations between W Vir and classical Cepheids may be explained by their occupationof the same regions on the HR diagram and thus, equivalent L/M ratios and effectivetemperatures. This observational evidence is thus in support of the HIF-stellar photo-sphere interaction theory. BL Her stars, on the other hand, have a positive PC slope inthe Bulge and the LMC and a flat (but negative) slope in the SMC at both minimum and4 S. Das, S. M. Kanbur, E. P. Bellinger, A. Bhardwaj, and H. P. Singhmaximum light. The nature of the sloped PC relations at both minimum and maximumlight for BL Her stars is similar to those for the short period classical Cepheids.4. Results from the theoretical aspectTo study the HIF-photosphere interaction from a theoretical perspective, we computeRRL, BL Her, and classical Cepheid models using the radial stellar pulsation code inMESA (Paxton et al. 2019). We choose to use the convection parameter set A from Ta-ble 4 of Paxton et al. (2019) which corresponds to the simplest convection model. Forour analysis, we proceed with the models that have full-amplitude stable pulsations inthe fundamental mode, an example of which is shown in Fig. 2. The stellar photosphereis defined as the zone with optical depth of 2/3 while the HIF is the zone with the steep-est gradient in temperature. The HIF-photosphere distance, ∆ is then defined in termsof Q = log(1 − Mr/M) (Kanbur et al. 2004). Fig. 3 displays the HIF-photosphere dis-tance as a function of log(P) for the computed models at minimum and maximum light.For a particular model, we find that the HIF-photosphere distance is always smaller atmaximum light. The top panel of Fig. 3 shows the HIF-photosphere distance of thedifferent variable star types to be almost constant at maximum light. However, from thelower panel, we may clearly observe the smaller HIF-photosphere distance for the RRLmodels at minimum light (a situation of “engaged” HIF-photosphere) as opposed to themuch larger HIF-photosphere distance for classical Cepheid models. BL Her modelsshow an intermediate behaviour to those from RRL and classical Cepheid models.5. Discussion and ConclusionsWe have analysed the PC relations of the different subclasses of T2Cs in the Galacticbulge and the Magellanic Clouds using OGLE-IV data. We find the W Vir stars to havea flat PC slope at maximum light and a significantly sloped PCmin; a situation similarto long-period (P > 10d) classical Cepheids (Bhardwaj et al. 2014). On the other hand,BL Her stars have sloped PC relations at both minimum and maximum light, similar tothe short-period classical Cepheids. These results may be explained by the HIF-stellarphotosphere interaction theory (Kanbur & Phillips 1996). A lower L/M ratio and/ora hotter effective temperature indicates smaller HIF-photosphere distance. This is truefor W Vir stars and classical Cepheids at maximum light when the temperature of thephotosphere is the same as the temperature of the HIF, thereby resulting in a flat PCmax.RRLs always have an “engaged” HIF-photosphere. However, at minimum light, thisoccurs at a regime where Saha ionisation equilibrium is somewhat independent of thetemperature, thereby resulting in a flat PC relation at minimum light (Das et al. 2018).BL Her stars occupy a region intermediate to the RRLs and classical Cepheids on theHR diagram, and this may explain the intermediate nature of their PC relations.We have also computed RRL, BL Her, and classical Cepheid models using MESA(Paxton et al. 2019) to theoretically test out the HIF-photosphere interaction theory bylooking at their relative locations. We find that HIF-photosphere distance to be alwayssmaller at maximum light, for a particular model. At minimum light, we find the HIF-photosphere distances for the classical Cepheid models to be much larger than that forthe RRL models. The MESA computed models therefore support the HIF-photosphereinteraction theory to a reasonable extent.HIF–Photosphere Interaction 5−0.5 0.0 0.5 1.0 1.5 2.0log(P)0.00.2Δ(MIN)RRLBLHCep−0.20.00.2Δ(MΔX)Figure 3. The plots of distance (∆) between HIF and stellar photosphere as func-tion of log(P) at maximum and minimum light.Acknowledgments. SD acknowledges the INSPIRE SRF provided by DST, Govt. of India vide Sanc-tion Order No. DST/INSPIRE Fellowship/2016/IF160068 and the travel support provided by SERB, Govt.of India vide file number ITS/2019/004781 to attend the RR Lyrae/Cepheid 2019 Conference “Frontiersof Classical Pulsators: Theory and Observations”. HPS and SMK thank the IUSSTF for funding the Indo-US virtual joint networked centre on “Theoretical analyses of variable star light curves in the era of largesurveys”. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foun-dation (#DNRF106). AB acknowledges research grant #11850410434 awarded by the National NaturalScience Foundation of China through the Research Fund for International Young Scientists, China Post-doctoral General Grant (2018M640018), and Peking University Strategic Partnership Fund awarded toPeking-Tokyo Joint-collaboration on Research in Astronomy and Astrophysics. The authors acknowledgethe use of High Performance Computing facility Pegasus at IUCAA, Pune and the software MESA r11701.ReferencesBart M. L., 1982, IMA J. Num. Analysis, 2, 241Bhardwaj A., Kanbur S. M., Singh H. P., Ngeow C.-C., 2014, MNRAS, 445, 2655Bhardwaj A., Kanbur S. M., Macri L. M., Singh H. P., Ngeow C.-C., Wagner-Kaiser R., Sara-jedini A., 2016, AJ, 151, 88Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ, 345, 245Das S., Bhardwaj A., Kanbur S. M., Singh H. P., Marconi M., 2018, MNRAS, 481, 2000Gonzalez O. A., Rejkuba M., Zoccali M., Valenti E., Minniti D., Schultheis M., Tobar R., ChenB., 2012, A&A, 543, A13Haschke R., Grebel E. K., Duffau S., 2011, AJ, 141, 158Kanbur S. M., Phillips P. M., 1996, A&A, 314, 514Kanbur S. M., Ngeow C.-C., Buchler J. R., 2004, MNRAS, 354, 212Muraveva T., et al., 2015, ApJ, 807, 127Paxton B., et al., 2019, ApJS, 243, 10Soszyński I., et al., 2017, AcA, 67, 297Soszyński I., et al., 2018, AcA, 68, 89

A Study of the Stellar Photosphere-Hydrogen Ionisation Front Interaction in Pulsating Variables Using Period-Colour Relations

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A Study of the Stellar Photosphere-Hydrogen Ionisation Front Interaction in Pulsating Variables Using Period-Colour Relations

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