Asteroseismology of the Double-Mode High-Amplitude δ Scuti Star VX Hydrae
Hui-Fang Xue, Jian-Ning Fu, L. Fox-Machado, Jian-Rong Shi, Yu-Tao Zhou, Jun-Bo Zhang, R. Michel, Hong-Liang Yan, Jia-Shu Niu, Wei-Kai Zong, Jie Su, A. Castro, C. Ayala-Loera, Altamirano-Dévora L
DDraft version July 12, 2018
Typeset using L A TEX preprint style in AASTeX61
ASTEROSEISMOLOGY OF THE DOUBLE-MODE HIGH-AMPLITUDE δ SCUTI STAR VXHYDRAE
Hui-Fang Xue, Jian-Ning Fu, L. Fox-Machado, Jian-Rong Shi,
3, 4
Yu-Tao Zhou,
3, 4
Jun-Bo Zhang, R. Michel, Hong-Liang Yan, Jia-Shu Niu,
5, 6, 7
Wei-Kai Zong, Jie Su, A. Castro,
2, 9
C. Ayala-Loera, and Altamirano-D´evora L. Department of Astronomy, Beijing Normal University, Beijing 100875, People’s Republic of China Observatorio Astron´omico Nacional, Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, Ap. P.877, Ensenada, BC22860, M´exico Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing100012, People’s Republic of China University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China Institute of Theoretical Physics, Shanxi University, Taiyuan, 030006, People’s Republic of China Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing,100190, People’s Republic of China School of Physical Sciences, University of Chinese Academy of Sciences, No.19A Yuquan Road, Beijing 100049,People’s Republic of China Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China Physics and Astronomy, University of Southampton, Southampton S017 1BJ, UK
ABSTRACTBi-site time-series photometric and high-resolution spectroscopic observations were made for thedouble-mode high-amplitude δ Scuti star VX Hya. The fundamental frequency f = 4 . − ,the first overtone f = 5 . − and 23 harmonics and linear combinations of f and f are detected by pulsation analysis. From the spectroscopic data, we get [Fe / H] = − . ± . /P )( dP /dt ) = (1 . ± . × − yr − . With these resultsfrom the observations, we perform theoretical explorations with the stellar evolution code MESA,and constrain the models by fitting f , f , and (1 /P )( dP /dt ) within 3 σ deviations. The results showthat the period change of VX Hya could be ascribed to the evolutionary effect. The stellar parametersof VX Hya could be derived as: the mass 2 . ± . M (cid:12) , the luminosity log( L/L (cid:12) ) = 1 . ± . . ± . × years. VX Hya is found to locate at the post-main-sequence stagewith a helium core and a hydrogen-burning shell on the H − R diagram.
Keywords: stars: individual (VX Hya) - stars: oscillations - stars: variables: deltaScuti - techniques: photometric - techniques: spectroscopic
Corresponding author: Jian-Ning [email protected] a r X i v : . [ a s t r o - ph . S R ] J u l Xue et al. INTRODUCTION δ Scuti stars are a class of short-period pulsating variable stars with periods between 0.0125 and0.25 days and amplitudes from milli-magnitude to tenths of a magnitude (Breger 2000; Casey et al.2013). They locate on either the pre-main sequence, or the main sequence, or the post-main sequenceevolutionary stages, lying at the bottom of the classical Cepheid instability strip with spectral classesof A-F. The excitation mechanism of δ Scuti stars is the κ -mechanism which is the same as that ofthe Cepheid and the RR Lyrae stars (Baker & Kippenhahn 1962, 1965; Zhevakin 1963).High-amplitude δ Scuti stars (hereafter HADS) are a subgroup of δ Scuti stars showing largeamplitudes (∆ V ≥ . m
3) with single or double radial pulsation modes (see, e.g., Poretti 2003;Niu et al. 2013, 2017). Another subgroup, the SX Phoenicis variables, contains δ Scuti stars ofPop.II, being the old disk population showing large amplitudes like HADS. As the HADS generallyrotate slowly with v sin i ≤
30 km/s, Breger (2000) indicated that the slow rotation seems to be aprecondition for high amplitudes and possibly even for radial pulsations. Petersen & Christensen-Dalsgaard (1996) pointed out that the observed period ratios and positions on the H − R diagramof double-mode HADS are in agreement with the assumption that these variables are normal starsfollowing standard evolution.Theoretically, stellar evolution induces changes of periods in the pulsating stars. The secularobservations of some variable stars show that the pulsation periods change in long time intervals.Whether the observed period change could be ascribed to the stellar evolution is an open question(Rodr´ıguez et al. 1995). In principle, if the observation-determined period change rate of a variablefalls into an interval derived from the model calculation, the observed period change is consideredto be due to the stellar evolution effect. For Classical Cepheids, this assumption helped Neilsonestimate the mass loss and evolutionary stage of Polaris (Neilson et al. 2012a; Neilson 2014). Inaddition, Neilson et al. (2012b) provided the first evidence that the enhanced mass loss must be aubiquitous property of Classical Cepheids. For δ Scuti stars, Breger & Pamyatnykh (1998) predictedthe values of (1 /P )( dP/dt ) from 10 − yr − on the main sequence to 10 − yr − on the post-mainsequence stages. With the same assumption, Yang et al. (2012) confirmed that the observed periodchange of the SX Phoenicis star XX Cyg could be successfully explained by the stellar evolutioneffect. Moreover, Niu et al. (2017) showed that the observed period change rate of the HADS AEUMa could not only be interpreted by the evolutionary effect, but also be regarded as an importantobservable that helps one perform successful asteroseismology on such kinds of stars.VX Hydrae ( α = 09 h m s , δ = − ◦ (cid:48) (cid:48)(cid:48) ) is a Pop.I HADS, discovered by Hoffmeister (1931)and then observed by Lause (1932) and Oosterhoff (1938). Fitch (1966) derived the periods of thefundamental mode P = 0 . P = 0 . uvby and β filters, measur-ing the variation ranges of the surface gravity log g ∈ [3 . , .
75] and of the effective temperature T eff ∈ [6500 , g = 3 .
47 and the mean effective temperature T eff = 7000 K were also calculated. After re-analyzing the data of Breger (1977), McNamara (1997)determined the mean effective temperature T eff = 7100 K, the metallicity [Fe / H] = 0 .
05, and thesurface gravity log g ∈ [3 . , . f = 4 . − and of the first overtone f = 5 . − . From the above frequency analysis results, one may note that (i) the frequencyof fundamental mode of VX Hya (4.4765 c days − ) is smaller than that of most HADS (see McNa- steroseismology of VX Hya Table 1.
Journal of the Photometric Observations for VX Hya.Observatory Longitude Latitude Telescope Filter Date Nights FramesYAO 102 . ◦
78E 25 . ◦
03N 101.6 cm V . ◦
46W 31 . ◦
04N 84 cm V Table 2.
The Properties of VX Hya, the Comparison and the Check Star.
Star name α (2000) δ (2000) B (mag) V (mag) B − V (mag)Target = VX Hya 09 h m . s − ◦ (cid:48) . (cid:48)(cid:48) . ± .
309 10 . ± .
205 0 . ± . h m . s − ◦ (cid:48) . (cid:48)(cid:48) . ± .
009 12 . ± .
022 0 . ± . h m . s − ◦ (cid:48) . (cid:48)(cid:48) . ± .
009 12 . ± .
021 0 . ± . B and V data were taken from AAVSO Photometric All Sky Survey (APASS) catalog (Henden et al. 2016). mara 2000, Table 1); (ii) the amplitude ratio of the fundamental and the first overtone modes of VXHya A f /A f ∼ . ∼ .
5, Niu et al. 2017; for RV Ari ∼ .
2, P´ocs et al. 2002; etc.). These characteristics induced ourinterest to study VX Hya further based on our new observations.The aim of this paper is to derive the stellar parameters and the evolutionary stage of VX Hyaby means of asteroseismology. The paper is organized as follows. In Section 2, we report newphotometric and spectroscopic observations and corresponding data reduction process. Section 3presents the pulsation analysis and the period change rate calculation of VX Hya. In Section 4, weconstruct stellar evolution models and make pulsation frequency fitting. A brief discussion and theconclusions are presented in Sections 5 and 6, respectively. OBSERVATIONS AND DATA REDUCTION2.1.
Photometry
VX Hya was observed in Johnson V with the 101.6 cm telescope at Yunnan Astronomy Observatory(YAO) in China (hereafter 101T) and the 84 cm telescope at Observatorio Astron´ o mico Nacional atSierra San Pedro M´ a rtir (OAN-SPM) in Mexico (hereafter 84T), from 2015 January to February.101T and 84T were equipped with an Andor DW436 CCD camera and an E2V-4240 CCD camera,respectively. Both cameras had 2048 × µ m/pixel.The effective fields of view were 7 . (cid:48) × . (cid:48) and 7 . (cid:48) × . (cid:48) for 101T and 84T, respectively. Table 1 liststhe journal of the photometric observations. 2122 frames were collected over 10 nights with 101T,while 1534 frames were collected over 5 nights with 84T.Two stars on the same CCD image close to the target star VX Hya were used as the comparison star(TYC 5482-1225-1) and the check star (2MASS 09454598-1201204), respectively. Their brightnessesare similar to that of the target and stable during our observations (see Table 2 and Figure 1). Figure1 shows a CCD image taken with the 101T with the target star, the comparison and the check star Xue et al.
Figure 1.
A CCD image of VX Hya collected with 101T. The field of view is 7.3 (cid:48) × (cid:48) . North is up andeast is to the left. VX Hya, the comparison star, and the check star are marked. marked. Table 2 lists their properties. The CCDRED routine of IRAF was used to subtract thebias and dark, and divide the flat for all the frames. After that, we performed aperture photometryfor all images by utilizing the APPHOT routine of IRAF. Then, the magnitude differences betweenVX Hya and the comparison star were calculated together with those between the check star and thecomparison star. The standard deviations of the differential magnitudes between the check star andthe comparison star yield the estimations of the photometric precisions, with typical values of 0 . m . m
02 in good and poor observational conditions, respectively. The zero-point differences amongindividual nights with typical values of 0.001-0.004 mag, which might be caused by the transparencyinstability of the atmosphere during the observation campaign, are compensated. Figure 2 shows thelight curves of VX Hya in V band in 2015.2.2. Spectroscopy and Atmospheric Parameter Calculation
A high-resolution, high-signal-to-noise ratio (S/N) spectrum of VX Hya was obtained with the ARC´echelle Spectrograph (ARCES) mounted on the 3.5 m telescope located at Apache Point Observatory(APO) on 2015 December 08. The wavelength range of the instrument was 3200-10000 ˚A with theresolution of R ∼ Image Reduction and Analysis Facility is developed and distributed by the National Optical Astronomy Obser-vatories, which is operated by the Association of Universities for Research in Astronomy under operative agreementwith the National Science Foundation. Based on observations obtained with the Apache Point Observatory 3.5-meter telescope, which is owned andoperated by the Astrophysical Research Consortium. steroseismology of VX Hya Figure 2.
Light curves of VX Hya in V band in 2015. The top panels in each subfigure give the magnitudedifferences between VX Hya and the comparison star, while the bottom panels show the magnitude differencesbetween the check star and the comparison star. The orange points come from the observations with 101Tand the green ones from those with 84T. The solid curves represent the fitting with the 25-frequency solutionlisted in Table 5. Xue et al.
Table 3.
Characteristics of the Spectro-scopic Observations of VX Hya and Calcu-lated Stellar Parameters.Property ValueMagnitudes ( V ) a ∼ R π (mas) b . ± . c − . A V d . / H] − . ± . T eff spec (K) 7193 ± T eff photometric (K) 7155 ± spec (cm s − ) 3 . ± . trigonometric (cm s − ) 3 . ± . ξ t (km s − ) 2 . ± . v sin i (km s − ) 6 . ± . rad (km s − ) − . M (cid:12) . ± . Note —The R and S/N indicate the spec-trum resolution and S/N, respectively. π is the parallax. ξ t is the micro-turbulencevelocity. a Magnitudes was taken from APASS cata-log (Henden et al. 2016). b Parallax was taken from Gaia DR1 (GaiaCollaboration et al. 2016). c Bolometric correction was derived fromAlonso et al. (1999). d Extinction in V magnitude was taken fromSchlafly & Finkbeiner (2011). We employed the one-dimensional (1D) plane-parallel MAFAGS-OS atmospheric model to thespectral analysis, which was described in detail by Grupp et al. (2009). In our analysis, the spectralsynthesis method was applied to derive the abundances with Spectrum Investigation Utility (SIU).This code was based on the IDL program (Reetz 1991). The spectroscopic method was adopted to steroseismology of VX Hya Wavelength ( Å ) N o r m a li ze d F l ux Figure 3.
Spectral region covering the H α line profile of VX Hya. The observed spectrum is shown by theblack dots, while the synthetic spectrum with the adopted atmospheric parameters is presented by the reddashed line. Wavelength ( Å ) N o r m a li ze d F l ux vsini = 2.0 km s − vsini = 5.0 km s − vsini = 7.0 km s − vsini = 10.0 km s − Figure 4.
Spectral synthesis of Fe I line at 5991 ˚A with individual values of v sin i . determine the stellar parameters. The effective temperature was derived from the excitation equilib-rium of Fe I lines with the excitation energy higher than 2.0 eV. The surface gravity was approachedby requiring the same Fe abundances obtained from Fe I and Fe II lines. The micro-turbulence veloc-ity ξ t was determined when the Fe abundance derived from the individual Fe I lines does not dependon the equivalent widths. During our analysis, the nonlocal thermodynamic equilibrium (NLTE)effects on the Fe I lines had been considered. We also derived the photometric T eff from the relationbetween the intrinsic color V − K and T eff for the giant (Alonso et al. 1999), which is consistent withthe spectroscopic one. As the H α line wing is sensitive to the T eff variation, it is a good indicator to Xue et al. examine the effective temperature. Figure 3 shows the comparison between the observed spectrumand the synthetic spectrum calculated with the adopted atmospheric parameters. The H α profilecan be reproduced by the adopted T eff .We note that Gaia DR1 (Gaia Collaboration et al. 2016) provided the parallax value of this object;thus, we also calculated the trigonometric log g with the equation:log g = log MM (cid:12) + log g (cid:12) + 4 log T eff T eff (cid:12) + 0 . M bol − M bol (cid:12) ) . (1)Here, M stands for the stellar mass, which was estimated with the Bayesian method (da Silvaet al. 2006) by using the PARSEC isochrones (Bressan et al. 2012). The method requests a setof spectro-photometric parameters (i.e., T eff , [Fe/H], V mag , parallax) to estimate the stellar mass,age, surface gravity, radius, etc. By comparing with the theoretical isochrones, the probabilitydistribution functions (PDF) of each stellar parameter were computed over the stellar isochrones.Then, the parameters could be determined when the PDFs present a single well-defined peak. Theabsolute bolometric magnitude M bol is defined as M bol = V mag + BC + 5 log π + 5 − A v . (2)where the parallax was taken from the Gaia DR1 (Gaia Collaboration et al. 2016). The bolometriccorrection was derived from the empirical calibration of Alonso et al. (1999), and the extinction in V magnitude was estimated from the Galactic dust extinction (Schlafly & Finkbeiner 2011). Withinthe uncertainties, the surface gravities derived from both methods are in a good agreement with eachother.The metallicity [Fe/H] was finally adopted until the interactive process converged, and the stellarparameters with the corresponding uncertainties are presented in Table 3. The isolated 29 Fe I and8 Fe II lines had been adopted, which are listed in Table 4 in the Appendix, for the determination ofthe stellar parameters. The atomic line data were obtained from the Vienna Atomic Line Database(VALD; Kupka et al. 1999), and the oscillator strength (log gf ) values were derived by fitting thesolar spectrum.In order to determine the projected rotational velocity ( v sin i ), four relatively isolated iron lines at5198, 5576, 5991, and 6065 ˚A were adopted from the lines listed in Table 4 in the Appendix. Whenfitting the line profile, the broadenings caused by the instrumental broadening, the macroturbulenceand v sin i were included. The instrumental broadening was estimated from the Th-Ar lines withthe Gaussian profile. The macroturbulent velocity was assumed to be 5 km s − (Fekel 1997, forF subgiants). We kept the instrumental broadening and macroturbulence fixed, and the v sin i wasderived until the best match between the synthetic and observed spectra was achieved. The syntheticspectra with the individual v sin i are illustrated in Figure 4. PULSATION ANALYSIS3.1.
Frequency Analysis
The software Period04 (Lenz & Breger 2005) was applied to make frequency analysis based on theFourier transformations. Figure 5 shows the spectral window and amplitude spectra of the frequencypre-whitenning process for the light curves of VX Hya in V band in 2015.The solution of 25 frequencies with S/Ns larger than 4.0 (Breger et al. 1993) is listed in Table 5,including f = 4 . − , f = 5 . − and 23 harmonics or linear combinations of steroseismology of VX Hya Figure 5.
Spectral window and Fourier amplitude spectra of the frequency pre-whitenning process for thelight curves of VX Hya in V band in 2015. Xue et al.
Table 5.
Multi-frequency Solution of the Light Curves of VX Hya in V Band in 2015.No. Identification Frequency (c days − ) Amplitude (mmag) S/NF1 f f f + f f f − f f f + 2 f f − f f + f f f − f f f + 3 f f + f f + 2 f f + 3 f f − f f f + f f + 2 f f + 4 f f f + 3 f f + 2 f f + 4 f f and f . The errors of the frequencies and amplitudes in Table 5 were estimated with the MonteCarlo simulations, which were based on the simulated light curves produced by an addition of theobserved data and a Gaussian distribution random variable obeying N (0 , σ obs ) (see Sect. 3 of Fuet al. (2013)). The solid curves in Figure 2 show the fits with the multi-frequency solution. Here, σ obs indicates the observation error. steroseismology of VX Hya Table 6.
Journal of Different Groups of Light Curves.No. Year Number of nights Number of data points Number of hours References1 1954 5 166 17.5 Fitch (1966)2 1956 9 459 36.0 Fitch (1966)3 1957 5 297 21.1 Fitch (1966)4 1958 6 358 25.4 Fitch (1966)5 2005 5 236 30.0 AAVSO6 2006 19 3389 133.4 AAVSO7 2008 7 345 48.2 AAVSO8 2009 9 267 36.7 AAVSO9 2010 17 3831 66.5 AAVSO10 2011 5 313 20.9 AAVSO11 2012 4 244 17.8 AAVSO12 2013 6 325 21.1 AAVSO13 2014 11 656 43.2 AAVSO14 2015 15 3656 99.8 this workNote. “References” indicates the souces of the data.
The period ratio P /P = f /f = 0 . f is the frequency of the fundamental mode and f the first overtone (Petersen & Christensen-Dalsgaard 1996).3.2. Period Change Rate
Considering the similar amplitudes of f and f from the Fourier analysis (see Table 5), the tradi-tional O − C method (see, e.g., Yang et al. (2012); Niu et al. (2017)) could not be used to determinethe period change of f which is effective only when the amplitude of the dominant frequency is muchlarger than that of the second strongest frequency. Consequently, we employed the Fourier-phase di-agram method developed by Paparo et al. (1998) to calculate the period change rate of VX Hya.This method can help study not only the period change of f , but also that of f . P´ocs et al. (2002)analyzed the period change of the double-mode HADS star RV Ari with this method and obtaineda result that was consistent with that from the O − C method.The Fourier decomposition can be presented by the formula, m = m + (cid:88) i A i sin [2 π ( f i t + φ i )] = m + (cid:88) i A i sin [2 π Φ i ] , (3)where A i is the amplitude, f i the frequency, and φ i the corresponding initial phase. Here, we defineΦ i = f i t + φ i . If one wants to investigate the linear variation of f i , one should add a term in Φ i ,Φ i = f i t + 12 ˙ f i t + φ i . (4)2 Xue et al.
Table 7.
Fourier Phases and Their Errors.HJD − ϕ σ ( ϕ ) ϕ σ ( ϕ )5158.30025 0.096 0.007 0.066 0.0085715.26675 0.148 0.006 0.112 0.0076198.87175 0.132 0.005 0.141 0.0086596.83505 0.111 0.004 0.050 0.00523444.92019 0.176 0.005 0.060 0.00623809.51435 0.189 0.002 0.094 0.00224507.54468 0.166 0.005 0.134 0.00624870.01302 0.134 0.006 0.106 0.00725277.19806 0.140 0.001 0.118 0.00225629.94859 0.133 0.006 0.137 0.00625989.91636 0.127 0.007 0.119 0.00726340.96562 0.143 0.005 0.090 0.00626696.99443 0.128 0.004 0.097 0.00527054.08968 0.116 0.001 0.096 0.002 Note —HJD is the middle heliocentric Juliandate between the first and the last data pointin each data group. ϕ and ϕ are the Fourierphases of f and f , respectively. σ ( ϕ ) and σ ( ϕ ) are the corresponding uncertainties. With the Fourier-phase diagram method, one may fix the amplitudes and frequencies in Eq. (3)and use the variations in φ i to reflect the change of f i . In that case, one has,Φ i = ˆ f i t + ˜ φ i = ˆ f i t + a i + b i t + c i t . (5)Here, ˆ indicates fixed values, and ˜ φ i the variation term in the method, which is represented by aquadratic polynomial ( ˜ φ i = a i + b i t + c i t ).Comparing Eq. (4) and (5), one gets the meanings of a i , b i , and c i : a i represents a constant valueof phase; b i a correction on the value of ˆ f i , while ˆ f i + b i is the corrected value of the frequency afterfitting; 2 c i the value of ˙ f i in Eq. (4). Moreover, the standard deviation of b i and 2 c i can be regardedas the standard deviation of f i and ˙ f i .In order to apply this method, we collected light curves from Fitch (1966), the American Associationof Variable Star Observers (AAVSO), and our observations, and divided them into 14 groups (seeTable 6). We used Period04 to extract the first two frequencies ( f and f ), which have the largest All of these data have been observed in V band. steroseismology of VX Hya P h a s e R e s i d u a l s (a) P h a s e R e s i d u a l s (b) P h a s e R e s i d u a l s (c) P h a s e R e s i d u a l s (d) Figure 6.
The top panels of subfigure (a) and (c) show the linear and the quadratic fitting results of˜ φ , respectively. The top panels of (b) and (d) show the linear and the quadratic fitting results of ˜ φ ,respectively. The bottom panels of each subfigure show the corresponding fitting residuals. The 2 σ (deepred) and 3 σ (light red) bounds are also shown in the panels. amplitudes for the whole data sets. Then, for each group, we fixed the amplitudes and frequencies,and let the phases be a free parameter to fit the light curves. The nonlinear least-square method wasused to calculate the best-fit values of the phases. The obtained phases ( ˜ φ and ˜ φ ) and their errorsfor each group are listed in Table 6.For ˜ φ and ˜ φ , we considered both the linear fitting ( ˜ φ i = d i + e i t ) and the quadratic fitting( ˜ φ i = a i + b i t + c i t ). The Markov Chain Monte Carlo method (MCMC; see Sharma 2017 for review)was used to determine the best-fit values of the parameters and their errors. The best-fitting resultsand the corresponding residuals of ˜ φ and ˜ φ are shown in Figure 6. The mean values and thestandard deviations of the parameters are listed in Table 8, where a i , b i , c i are the quadratic fittingparameters and d i , e i the linear fitting parameters of ˜ φ i . With the values of the parameters a i , b i , c i in Table 8, we derived f = 4 . ± . − , ˙ f = − . ± . × − c days − , The errors of the phases were estimated by Monte Carlo simulations, similar to the method used in the frequencyanalysis. Xue et al.
Table 8.
Linear and Quadratic Fitting Results for ˜ φ and ˜ φ .Parameters Fitting Results χ / dof a − ± × − b . ± . × − c − . ± . × − d . ± . × − e . ± . × − a ± × − b ± × − c − ± × − d ± × − e . ± . × − Note — a i , b i , c i are the quadratic fitting pa-rameters and d i , e i are the linear fitting pa-rameters of ˜ φ i . f = 5 . ± . − , and ˙ f = − ± × − c days − . Hence, the period changerates of f and f can be derived as follows,1 P d P d t = − f d f d t = (1 . ± . × − yr − , (6)1 P d P d t = − f d f d t = (5 . ± . × − yr − . (7)Comparing the χ / dof between the linear and quadratic fitting results, one can conclude that (i)for the fundamental mode, a quadratic fitting is necessary and a reliable value of ˙ f can be obtained;(ii) for the first overtone mode, it is not absolutely necessary to employ a quadratic fitting. As aresult, we used only ˙ f as an effective constraint in our substantial stellar model calculation. CONSTRAINTS FROM THEORETICAL MODELSWe used Modules for Experiments in Stellar Astrophysics (MESA Paxton et al. 2011, 2013, 2015),a suite of open source for computation in stellar astrophysics, to construct the theoretical models.The stellar evolution module MESA star combines a number of numerical and physical modules forsimulations of stellar evolution. The ρ − T tables are based on the updated OPAL EOS tables in2015 (Rogers & Nayfonov 2002) and extended to lower temperatures and densities by the SCVHtables (Saumon et al. 1995). HELM (Timmes & Swesty 2000) and PC (Potekhin & Chabrier 2010) Here, χ = ( O th − O obs ) σ obs , where O th and O obs are the theoretically calculated and the observed values of anobservable, respectively; σ obs is the observed error. dof implies the degree of freedom, which is defined as the numberof data points minus the number of free parameters in the fitting. steroseismology of VX Hya Table 9.
Pulsation Parameters of VX Hya. f (c days − ) f (c days − ) (1 /P )( dP /dt ) (yr − )4 . ± . . ± . . ± .
09 (10 − ) tables are used at the outside region of OPAL and SCVH. MESA opacity tables are composed ofOPAL Type 1 and 2 tables (Iglesias & Rogers 1993, 1996) for the high temperature region, tablesof Ferguson et al. (2005) for the low temperature region, and tables from OP (Seaton 2005) as thetable format is identical. The standard mixing-length theory of convection of Cox & Giuli (1968,chap.14) and the modified MLT of Henyey et al. (1965) are used in MESA. The overshooting mixingdiffusion coefficient presented by Herwig (2000) is adopted in MESA star. The adiabatic pulsationcode ADIPLS (Christensen-Dalsgaard 2008) in MESA enables pulsation frequencies to be calculated.4.1. Pulsation Parameters
The frequencies of the fundamental mode f and of the first overtone f determined from theobservations in 2015 were utilized to restrict the theoretical models. In addition, as the value ofperiod change rate of the fundamental mode f is within the predicted values of δ Scuti stars (Breger& Pamyatnykh 1998), we assumed that the period change rate of VX Hya calculated in this paper isdue to the stellar evolutionary effect, which was applied to help constrain the models. These valueswere utilized as pulsation parameters of VX Hya, as listed in Table 9.4.2.
Setup of the Model Calculation
As pointed out by Poretti et al. (2005), the period ratios P /P of long-period HADS stars higherthan 0.770 could only exist for the models with the mass M larger than 2.00 M (cid:12) , while one notesthat P /P = 0 . ± . δ Scuti star. Conservatively, we constructed aseries of stellar models with the mass range from 1.80 M (cid:12) to 2.80 M (cid:12) with the step of 0.01 M (cid:12) . Theheavy-element mass fraction Z was determined as 0.010 from the metallicity value [Fe / H] = − . X was set as 0.7 as usual. The value of themixing-length parameter was taken as α MLT = 1 .
77 which has a slight effect on the models (c.f. Breger(2000); Yang et al. (2012); Niu et al. (2017)). As VX Hya is a slow rotator with v sin i ∼ . − (see Table 3), the effects of rotation were not considered in our calculations (Breger 2000; Petersen1998). Each model of the evolution sequence started from the ZAMS to the post-main-sequence stageby specifying the mass M and the initial chemical composition ( X , Y , Z ). From the stage of the mainsequence to the post-main sequence, the ADIPLS program was invoked to calculate the pulsationfrequencies of each evolutionary step. In our calculation, f and f were derived with the quantumnumbers of ( (cid:96) = 0 , n = 1) and ( (cid:96) = 0 , n = 2), respectively.4.3. Parameter Fitting
The evolutionary tracks from the main sequence to the end of the post-main sequence in the massrange of 1.8-2.8 M (cid:12) are shown in Figure 7. In all the three subfigures, the blue regions on thetracks indicate the models for which the calculated (1 /P )( dP /dt ) fit the observation-determined(1 /P )( dP /dt ). In subfigure (a), the black lines mark the models for which the calculated f fit theobservation-determined f ; in subfigure (b), the black lines mark the models for which the calculated6 Xue et al.
Table 11.
Physical Parameters of VX Hya Derived from the Fitted Models. M ( M (cid:12) ) 2 . ± . years) 4 . ± . T eff (K) 8015 ± L/L (cid:12) ) 1 . ± . g . ± . Table 12.
Observation-determined Fundamental Frequencies, Period Change Rates and Physical Parame-ters from the Best-fit Models of Five HADS Stars.star f (c days − ) (1 /P )( dP /dt ) (yr − ) M/M (cid:12)
Age (10 years) [Fe/H] ReferencesVX Hya 4.4763 1 . ± . × − . ± . × − . ± . × − . ± . × − f fit the observation-determined f ; in subfigure (c), the red lines mark the models for which thecalculated f and f fit simultaneously the observation-determined f and f . On these tracks, theobservation-determined f , f and (1 /P )( dP /dt ) are marked within 3 σ . More details about thefitted models can be found in Table 10 in the Appendix . The physical parameters of VX Hya derivedfrom the fitted models are listed in Table 11. DISCUSSIONIn Figure 7 (a) and (b), one can see that large numbers of models on the tracks have the observedvalues of f and f within uncertainties, respectively. However, as shown in Figure 7 (c), only afew models satisfy the constraints from both f and f . On the other hand, the fitting results showthat the period change of VX Hya can be successfully ascribed to the evolutionary effect as shownin Figure 8. In addition, our results also indicate that VX Hya as a HADS is a normal star evolvingin the post-main-sequence stage.As one may note, five HADS stars have been studied with the stellar masses and evolutionarystages determined by asteroseismology in our series publications. According to Yang et al. (2012),Niu et al. (2017), Li et al. (2018), Yang et al. (2018) and this work, we plot the positions of the fivestars on the H − R diagram in Figure 9. The observation-determined fundamental frequencies, theperiod change rates, and the physical parameters from the best-fit models are listed in Table 12. Asone can see, VX Hya has the largest mass and evolves further than the other four stars. One can usethe basic pulsation relation P (cid:112) ¯ ρ/ρ (cid:12) = Q to roughly compare the mean density ¯ ρ of the stars, where P is the period of pulsation and Q the pulsation constant. Compared with the other stars listed in Here, σ values are the standard deviations of the quantities determined from observations. Of course, the number of fitted models depends on the evolutionary steps and the initial mass steps that we havechosen. However, we think at present configurations, it is sufficient for us to get some valuable results. steroseismology of VX Hya (a) (b)(c) Figure 7.
The evolutionary tracks from the main sequence to the end of the post-main sequence in themass range of 1.8-2.8 M (cid:12) with the step of 0 . M (cid:12) . In all three subfigures, the blue regions on the tracksindicate the models for which the calculated (1 /P )( dP /dt ) fit the observation-determined (1 /P )( dP /dt ).In subfigure (a), the black lines mark the models for which the calculated f fit the observation-determined f ;in subfigure (b), the black lines mark the models for which the calculated f fit the observation-determined f ; in subfigure (c), the red lines mark the models for which the calculated f and f fit simultaneously theobservation-determined f and f . On these tracks, the observation-determined f , f and (1 /P )( dP /dt )are marked within 3 σ . Xue et al.
Figure 8.
The evolutionary tracks from the main sequence to the end of the post-main sequence in the massrange of 2.35-2.42 M (cid:12) . The black crosses mark the models for which the calculated f , f and (1 /P )( dP /dt )fit simultaneously the observation-determined f , f and (1 /P )( dP /dt ) within 3 σ . Figure 9.
Evolutionary tracks for the best-fit models of five HADS stars. See Table 12 for details. steroseismology of VX Hya P which leads to the smallest ¯ ρ , indicating that VX Hya evolvesinto the latest stage among the five HADS.From Figure 9 and Table 12, one can note that the four HADS stars that locate at the post-main-sequence stages with the hydrogen-burning shell show a tendency that the star with lowerfundamental frequency shows higher period change rate and evolves into later stage. Figure 5 ofBreger & Pamyatnykh (1998) shows the theoretically calculated periods of the radial fundamentalmodes and their changes during late main-sequence and post-main-sequence evolution of the 1.8 M (cid:12) model, which is consistent with the tendency derived from the observations of these four stars. Basedon our test, the tendency is insensitive to the mass and the metallicity of a δ Scuti star. Hence, onecan roughly deduce the evolutionary stage of a δ Scuti star by its observation-determined frequencyof the radial fundamental mode and period change rate. CONCLUSIONSBy analyzing the photometric data gathered during 15 nights in 2015 of VX Hya, we have de-tected 25 frequencies that include the two independent frequencies f = 4 . − and f = 5 . − and 23 harmonics and linear combinations. Based on the P /P ratio, f and f were found to be the fundamental and the first overtone radial pulsation modes, respectively.From the results of the high-resolution spectroscopic observations, we conclude that VX Hya is aslow rotator with v sin i = 6 . ± . − , and derive the metallicity [Fe / H] = − . ± . /P )( dP /dt ) = (1 . ± . × − yr − while the constraint on the first overtone mode is weak.The stellar evolutionary models were constructed with the initial masses between 1.80 M (cid:12) and 2.80 M (cid:12) , and Z of 0.010. With the constraints from the observed values of f , f , and (1 /P )( dP /dt ) arewithin 3 σ deviations, the stellar parameters of VX Hya can be determined (more details are found inTable 11). Consequently, VX Hya is a HADS star lying after the second turn-off of the evolutionarytrack leaving the main sequence with a helium core and a hydrogen-burning shell.Moreover, we would like to point out that (i) for VX Hya, the period change can be successfullyinterpreted by the evolutionary effect; (ii) for VX Hya, the frequencies of the fundamental and thefirst-overtone modes could be used to determine the stellar parameters efficiently, which provided asuccessful example of asteroseismology on such kind of stars; (iii) the results provide a direct supportto the general consensus that HADS are probably normal stars evolving in either the main-sequenceor the post-main-sequence stages; (iv) the comparison of five HADS stars indicates that the frequencyof the radial fundamental mode and its period change rate are sensitive to the evolutionary stages ofthe HADS stars. ACKNOWLEDGMENTSJ.N.F. acknowledges the support from the National Natural Science Foundation of China (NSFC)through the grant 11673003 and the National Basic Research Program of China (973 Program2014CB845700). L.F.M. and R.M. acknowledge the financial support from the DGAPA, Univer-sidad Nacional Aut´onoma de M´exico (UNAM), under grant PAPIIT IN 100918. J.S. acknowledgesthe support from the China Postdoctoral Science Foundation (grant No. 2015M570960) and the foun-dation of Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy0 Xue et al. of Sciences (grant No. OP201406). J.S.N. acknowledges the support from the Projects 11475238and 11647601 supported by National Science Foundation of China, and by Key Research Program ofFrontier Sciences, Chinese Academy of Sciences. Special thanks are given to the technical staff andnight assistants at the Yunnan Astronomy Observatory, Sierra San Pedro M´artir observatory andthe Apache Point Observatory for facilitating and helping making the observations. We acknowledgewith thanks the variable star observations from the AAVSO International Database contributed byobservers worldwide and used in this research. We thank Michel Bonnardeau who as an observer ofAAVSO, provided us with the observations from 2011 to 2014.
Software:
IRAF (Tody 1986, 1993), PARSEC (Bressan et al. 2012), MAFAGS-OS Gruppet al. (2009), Period04 (Lenz & Breger 2005), MESA (Paxton et al. 2011, 2013, 2015), ADIPLS(Christensen-Dalsgaard 2008) APPENDIXDetailed Information about Atomic Parameters and Fitted Models.
Table 4 . Atomic Parameters of the Adopted Iron Linesand Equivalent Widths (EW) for VX Hya. λ (˚A) Species χ (eV) log gf EW (m˚A)5109.652 Fe I 4.30 -0.620 38.5485121.639 Fe I 4.28 -0.690 43.2145141.739 Fe I 2.42 -2.044 36.1585195.472 Fe I 4.22 -0.106 68.9885198.717 Fe I 2.22 -2.155 54.0765217.389 Fe I 3.21 -1.060 66.3825228.376 Fe I 4.22 -1.030 26.0035242.497 Fe I 3.63 -0.827 56.4515250.646 Fe I 2.20 -1.981 56.3525263.306 Fe I 3.27 -0.899 71.0025445.042 Fe I 4.39 0.040 87.6375576.096 Fe I 3.43 -0.870 74.3765624.542 Fe I 3.42 -0.755 89.2106024.058 Fe I 4.55 0.050 73.0646065.492 Fe I 2.61 -1.460 82.6896136.615 Fe I 2.45 -1.400 98.8706252.555 Fe I 2.40 -1.607 77.593
Table 4 continued on next page steroseismology of VX Hya Table 4 (continued) λ (˚A) Species χ (eV) log gf EW (m˚A)6393.612 Fe I 2.43 -1.430 79.1916411.649 Fe I 3.65 -0.600 80.1966677.987 Fe I 2.69 -1.318 87.1695983.690 Fe I 4.55 -0.538 26.6756003.002 Fe I 3.88 -0.990 37.2816056.010 Fe I 4.73 -0.360 39.2746078.490 Fe I 4.79 -0.171 35.7116191.571 Fe I 2.43 -1.417 99.6236246.327 Fe I 3.60 -0.773 64.7476301.508 Fe I 3.65 -0.718 71.3796302.499 Fe I 3.69 -0.973 55.0556430.856 Fe I 2.18 -1.966 70.7205132.669 Fe II 2.81 -4.100 35.7665264.808 Fe II 3.23 -3.050 90.4875425.257 Fe II 3.20 -3.280 70.6685991.380 Fe II 3.15 -3.600 56.8186084.111 Fe II 3.20 -3.831 35.4426416.919 Fe II 3.89 -2.977 66.3396432.683 Fe II 2.89 -3.610 71.0636516.080 Fe II 2.89 -3.272 106.256
Note — χ is the excitation energy. log gf is theoscillator strengths. Xue et al.
Table 10.
Models within 3 σ Deviations of f , f and (1 /P )( dP /dt ). M ( M (cid:12) ) Age (10 years) log T eff log( L/L (cid:12) ) log g f (c days − ) f (c days − ) P dP dt ( × − yr − ) χ REFERENCES
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