Linear nonadiabatic pulsation models of the double-mode Cepheid V371 Per

TL;DR

This paper models the pulsation behavior of the unique double-mode Cepheid V371 Per, revealing it is likely crossing the instability strip for the first time and highlighting the importance of metallicity and temperature in its pulsation characteristics.

Contribution

It introduces linear nonadiabatic pulsation models tailored to V371 Per, providing insights into its evolutionary stage and the effects of metallicity on its pulsation properties.

Findings

V371 Per is likely crossing the instability strip for the first time.

Models require specific mass and Teff ranges that depend on metallicity.

Observed and model Teff ranges overlap marginally, needing further data.

Abstract

Recently found double-mode Cepheid with the shortest period in our galaxy and abnormal period ratio, V371 Per is investigated by linear nonadiabatic pulsation models. V371 Per is likely to be crossing the instability strip for the first time because mass derived from pulsation models is larger than the evolution mass for the second or higher crossing objects. This result seems to support the conclusion obtained from the spectroscopic observation. We also found that models with observed period and period ratio of V371 Per need to have mass and Teff in a narrow range which shifts as heavy element abundance Z changes. We have checked the agreement between Teff ranges estimated observationally and derived from pulsation models using observational Z. We found that those ranges overlap marginally. We need more spectroscopic estimations of the Teff and [Fe/H], and more photometric monitoring to estimate the evolutionally period change for confirmation of our result.

Tables

Table 1. Table 1: Summary of the Model Parameters.

Parameters
Luminosity (in L\sun)	420, 460, 500
Teff	7000 $\sim$ 5400
X	0.70
Z	0.001	0.004	0.01	0.02	0.03
Mass (in M\sun)	1.5 $\sim$ 6.0

Full text

1 Introduction

Double-mode cepheids are the subgroup of the classical Cepheids, which pulsate in two radial pulsation modes simultaneously (For a review article, see for example Petersen & Takeuti ([2001]) and references there in). These stars are checked for the difference from the single-mode Cepheids for many aspects, however no difference is reported up to now. We can interpret that double-mode cepheids are the normal classical Cepheids, which happen to pulsate in two pulsation modes. There are 22 clearly established galactic members of this subgroup. The period ratios of the two periods are from $0.6967$ to $0.7195$ with two exceptions with ratios around $0.80$ (CO Aur and HD304373), which are considered as the first overtone and the second overtone pulsators. The periods of the fundamental mode for other 20 stars are between $2.139$ and $6.293$ days.

Recently, V371 Per is reported as a new candidate member of this subgroup (Wils et al. [2010]). V371 Per was classified as irregular variable by Weber([1964]), but Schmidt et al.([1995]) suggested this star to be a double-mode cepheid from the detected period and excessive scatter in the light curve. Thus, Wils et al.([2010]) made new observations and indicated that this star is indeed a double-mode Cepheids with the shortest fundamental mode period of 1.738 days and unusually high period ratio of 0.731. They discussed that the high value of the period ratio may indicate that the metalicity of this star is much lower than the other Galactic double-mode cepheids from the relation between period and period ratio versus metalicity Z discussed by Buchler & Szabo([2007]). They also pointed out that this star is in the Thick Disk or the Halo of our Galaxy. Furthermore, Kovtyukh et al.([2012, 2016]) discussed that V371 Per can be considered as a first crossing object from its peculiar abundances, especially from richness of the lithium.

Wils et al.([2010]) further discussed on some properties of this star, however, the properties they deduced from pulsation models are not convincing because of the lack of computations with appropriate parameters due to its short period and other unusual properties. We would like to present here some results of linear nonadiabatic pulsation models with special interest to this unusual star.

2 Models

For calculating linear non-adiabatic period, we used the same code used in Ishida([1995]), namely, Castor([1971]) type procedure with OPAL opacity tables (Iglesias, et al. [1992]; Rogers & Iglesias [1992]). The effect of convection seems not efficient for our purpose, as the derived metalicity for V367 Sct and Y Car using linear non-adiabatic calculations in Ishida([1995]) is not far from recent observation (Andrievsky et al. [2002]). Therefore, the effect of convective energy transport is not included in all of the models.

The luminosity of the model is estimated from the Fouqué et al. ([2007])’s period-luminosity relation for Galactic Cepheids and Torres([2010])’s polynomial fits for the Flower([1996])’s bolometric correction. Luminotity of the V371 Per is estimated to be 460L\sun. Considering differences between distance derived from the period–luminosity relations in the different photometric bands into account, $L/L\sun=500$ and $420$ models are also examined.

The effective temperature of V371 Per is reported as; 6000K or lower from unreddened $B-V$ by Wils et al.([2010]), 6215K from average of low-resolution spectra by Schmidt([2011]), and 5950 $\sim$ 6213K from high resolution spectra by Kovtyukh et al.([2012]). Considering spectral classification as G0 by Bond([1978]) into account, we set wide parameter range for effective temperature from 7000 to 5400K with 100K difference.

Observational estimates of the iron abundance are; $[Fe/H]=-0.05$ to $-0.40$ from low-resolution spectra by Schmidt([2011]), and $[Fe/H]=-0.42$ from high-resolution spectra by Kovtyukh et al.([2012]). Thus, chemical compositions of the calculated models are $Z=0.001,0.004,0.01,0.02$ and $0.03$.

The mass of the models are changed from 1.5 to 6.0 $M\sun$ with 0.5$M\sun$ difference. Selected parameter range is summarized in Table 1.

3 Results of the linear nonadiabatic models

As a typical example, the results for $L=460L\sun$, $Z=0.004$ are summarized in Fig. 1.

The observed period 1.737 days and period ratio 0.731 is realized around $M=3.35M\sun$ and Teff=6020. In $L=420L\sun$ models, both of the instability edges move slightly to the higher-temperature direction, the constant period lines move to the lower-temperature direction, and the constant period ratio lines move to the lower-mass direction. Thus the observed period and period ratio are realized at the lower-mass and lower-temperature point than $L=460L\sun$ model. In $L=500L\sun$ models, changes are oposite directions and the observed period and period ratio are realized at the higher-mass and higher-temperature point than $L=460\sun$ models.

The results for the $L=460L\sun$ and $Z=0.02$ models are shown in Fig. 2. For comparison with Fig. 1, we need to remark that the presented mass range is changed from 1.5$\sim$5.0 $M\sun$ for $Z=0.004$ models to 4.0$\sim$6.0 $M\sun$ for $Z=0.02$ models. In the $Z=0.02$ models, both of the instability blue edges move to the lower temperature direction, the constant period lines move to the higher-temperature direction, and the constant period ratio lines move to the higher-mass direction, comparing to the $Z=0.004$ models. It is also worth noting that the slope of the constant period ratio lines is steeper in the $Z=0.02$ models. Thus the observed period and period ratio is realized at the higher-mass and lower-temperature point than the $Z=0.004$ model, namely, around $M=5.22M\sun$ and Teff=5600.

Derived mass and effective temperature to realize the observed period and period ratio is summarized in Fig. 3 and Fig. 4. As pointed out in the description of the Fig. 2, the Mass of the satisfactory model increases as Z increases, and the Teff of that decreases. We need to remark that both of the fundamental and the first overtone modes are pulsationally stable in the $Z=0.001$ model.

4 Discussion

As shown in Figs.3 and 4, the observed period and period ratio is realized with wide range of the chemical composition, however, the mass and effective temperature needs to change as chemical composition. Namely, the model with observed period and period ratio should be in a narrow band on the mass and Teff versus Z planes. Therefore, when we fix Z to the observationally estimated value, whether the observationally estimated Teff is in the range derived from the pulsation model or not will be a check of the agreement between observation and the pulsation theory. From 4 high-resolution spectra, Kovtyukh et al. ([2012]) derived $[Fe/H]=-0.42$ and Teff = 5950 $\sim$ 6213. The simple conversion from $[Fe/H]$ to $Z$ indicates $Z\sim 0.007$. From Fig.4, we can estimate Teff = 5700 $\sim$ 5900. Considering above observed Teff range is from only 4 spectra into account, we can consider that the observations and our estimation from pulsation models is marginally overlapping. More observational estimations of the Teff and chemical composition are needed to confirm whether our results agrees with the observations.

In the low Z models, helium content $Y$ used here is especially high. Therefore, additional calculation for $(X,Z)=(0.80,0.01)$ models are performed to confirm the effect of high $Y$ values. The process to search for the mass and Teff to satisfy the observed properties of V371 Per is the same as that used in the previous section. The obtained mass is 0.2M$\sun$ larger than that obtained for $(X,Z)=(0.70,0.01)$ models. The obtained Teff is almost unchanged. Thus, the result presented in the previous section will not be largely affected by using high $Y$ values.

The effect of inclusion of convection on the fundamental period and period ratio of the first overtone mode to the fundamental mode is investigated by Saio et al. ([1977]). Their result indicate that the fundamental period becomes slightly longer and the period ratio becomes smaller as the mixing length $l/H_{p}$ becomes larger. On the Fig.1 and 2, inclusion of the convection will shift the period constant lines slightly to the higher-mass direction and shift the period ratio constant lines to the lower-mass direction. Because the effect on the period ratio is larger than the effect on the period, the position of the model with observed period and period ratio will shitf to the lower-mass direction, namely, the derived mass will be smaller and derived effective temperature will be higher. Therefore, inclusion of convection will make better agreement between the observation and our estimation from pulsation models for Teff discussed in the previous paragraph.

In Fig.3, the mass for the calculated luminosity derived from the evolutional relations by Becker et al. ([1977]) is also plotted. The masses estimated by the mass-luminosity relation for the second crossing objects are shown by the dashed line and those for the first crossing objects are shown by the solid line. Our results indicate that the pulsationally derived mass is massive than the second crossing mass in almost all range of Z. It is even massive than the first crossing mass from about $Z=0.004$ to $0.02$. At the observed chemical composition $Z\sim 0.007$, the mass estimated from our results is $\sim 4.1M\sun$. Considering the effect of convection discussed in the previous paragraph into account, the excess over the first crossing mass will be reduced. Thus, our result from pulsation models seems to indicate that V371 Per is too massive as the second or higher crossing object, and is likely to be the first crossing object as disscussed by Kovtyukh et al. ([2012, 2016]) from spectroscopic observations.

More modern evolution calculation with OPAL opacity is presented by Bono et al. ([2000]) and they obtained a mass-luminosity relation for classical Cepheids. Although, their relation is for the second and the third crossing objects, and for the objects in the mass range $4$ – $15M\sun$, the estimated mass for $L=460L\sun$ with $(Y,Z)=(0.293,0.007)$ is about $3.34M\sun$ from their relation. This estimation is less massive than the mass estimated from our results in the previous paragraph, and seems to support that V371 Per is a first crossing object. It is also worth noting that the $4.0M\sun$ model in Bono et al. ([2000])’s table 7 with $(Y,Z)=(0.23,0.004)$ has similar first crossing luminosity $\log L/L\sun=2.620$ ($416L\sun$) to the present model $460L\sun$ and with shorter period $\log P=0.0171$ (1.040 days) at higher temperature $\log T=3.808$ (6426K). Our consequnce that V371 Per is likely to be a first crossing object seems also valid with state-of-the-art evolution calculation.

Wils et al. ([2010]) reported that both of the fundamental and the first overtone modes frequencies are decreased. Frequency decrease means period increase. Period increase is realized when the Cepheid is the 1st, 3rd, or 5th crossing of the instability strip. Effects of the chemical composition and the periods on the evolutionary period change of Cepheids is already investigated by Saitou([1989]) using Becker et al.([1977])’s evolution calculation. His results indidate that the evolutionary period changes for the 1st, 4th+5th crossing is larger than those for the 2nd and 3rd crossing, especially at the short period region. He also derived abundance dependent period change–period relations for each crossing. We can estimate the period change for V371 Per to be about $\log[\Delta P/P]_{100}=-3.11$ from Wils et al. ([2010])’s Figure 5. On the other hand, we can estimate from the Saitou([1989])’s fittings, that $\log[\Delta P/P]_{100}\sim-2.88$ for the first crossing and $\sim-4.67$ for the 3rd crossing with $Z=0.007$. Consequently, the observed period change seems to fit to the first crossing. Recently, evolutionary period change is also discussed by Pietrukowics([2003]) and Turner et al.([2006]). We can convert above period change to $\log{\dot{P}}\sim 0.063$ and ${\dot{P}}/P^{2}\sim 12.14$. Both values put V371 Per in the first crossing region of Fig.2 of Pietrukowics([2003]) and Fig.3 of Turner et al.([2006]). Therefore, the period change reported by Wils et al.([2010]) seems to support our result that V371 Per is a first crossing object derived from linear pulsation models. However, Wils et al.([2010]) reported that the period change of the V371 Per should be treated with caution, because this change highly depends on the photographic data. Thus, we need more photometric monitoring of this interesting and unusual Cepheid and more precise estimation of its period change.

5 Conclusions

We have investigated recently found double-mode Cepheid V371 Per by linear nonadiabatic pulsation models. Our result indicates that V371 Per is likely to be a first crossing object, because the mass derived from linear pulsation results is larger than the evolutionary mass for the second crossing object. Thus, our result obtained from pulsation models seems to support the conclusion from spectroscopic observation by Kovtyukh et al.([2012, 2016]). Decreases of the frequencies reported by Wils et al.([2010]) seems to support this consequence. We also found that models with observed period and period ratio of V371 Per need to have mass and Teff in a narrow range which shifts as heavy element abondance $Z$ changes. We have checked whether observational Teff is in the range derived from pulsation models using observational $Z$ or not. Two Teff ranges is marginally overlapping, and agreement between observation and our result seems to be marginal. We need more spectroscopic estimations of the Teff and $[Fe/H]$ to check our result, and more photometric monitoring to estimate the evolutionally period change and thus confirm evolutional status of V371 Per.

Acknowledgments

I would like to appreciate to my colleagues for their useful comments during this investigation. Computations are carried out mainly on the workstation prepared by the research grant of the University of the Hyogo.