A rapidly evolving high-amplitude δ Scuti star crossing the Hertzsprung Gap
DDraft version February 23, 2021
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A rapidly evolving high-amplitude δ Scuti star crossing the Hertzsprung Gap
Jia-Shu Niu and Hui-Fang Xue
2, 3 Institute of Theoretical Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan030006, China Department of Physics, Taiyuan Normal University, Jinzhong, 030619, China Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong 030619, China
ABSTRACTPeople cannot witness the stellar evolution process of a single star obviously in most cases becauseof its extremely secular time-scale, except for some special time nodes in it (such as the supernovaexplosion). But in some specific evolutionary phases, we have the chances to witness such processgradually on human times-scales. When a star evolved leaving from the main sequence, the hydrogennuclei fusion in its core is gradually transferring into the shell. In the Hertzsprung–Russell diagram,its evolutionary phase falls into the Hertzsprung gap, which is one of the most rapidly evolving phasesin the life of a star. Here we report a discovery of a rapidly evolving high-amplitude δ Scuti starKIC6382916 (J19480292+4146558) which is crossing the Hertzsprung gap. According to the analysisof the archival data, we find three independent pulsation modes of it, whose amplitudes and frequenciesare variating distinctly in 4 years. The period variation rates of the three pulsation modes are one or twoorders larger than the best seismic model constructed by the standard evolution theory, which indicatesthe current theory cannot precisely describe the evolution process in this rapidly evolving phase andneeds further upgrades. Moreover, the newly introduced Interaction Diagram can help us to find theinteractions between the three independent pulsation modes and their harmonics/combinations, whichopens a new window to the future asteroseismology.MAIN BODY δ Scuti stars are a class of short-period pulsating vari-able stars with period between 0.02 and 0.25 day and thespectral classes A-F, which locate on the main sequenceor post main sequence evolutionary stage at the bot-tom of the classical Cepheid instability strip and excitedby the κ mechanism (Breger 2000). High-amplitude δ Scuti stars (hereafter HADS) are a subclass of δ Scutistars, who always have larger amplitudes (∆ V ≥ m. /P )( d P/ d t )) of HADS causedby the evolution should have a value from 10 − yr − on the main sequence to 10 − yr − on the post-mainsequence phase (Breger & Pamyatnykh 1998). Accord-ing to the seismic models, the observed period variation Corresponding author: Jia-Shu [email protected] rates of some HADS (such as XX Cyg (Yang et al. 2012),AE UMa (Niu et al. 2017), and VX Hya (Xue et al.2018)) can be interpreted by the evolutionary effect self-consistently, and each of these stars should be located inthe Hertzsprung gap in the H-R diagram with a heliumcore and a hydrogen-burning shell. In these researches,the period variation rates are obtained from the groundbased time-series photometric data accumulated in sev-eral decades, but only that of the fundamental pulsa-tion modes have a sufficient precision to be confirmedand tested because of the quality of the data. The highprecision photometric data from
Kepler space telescope lasted for about 4 years could provide us excellent oppor-tunity to determine the period and amplitude variationrates not only for the fundamental mode, but also forthe other pulsation modes. These observed quantitieswould tell us more secrets about the stellar evolution inthis special rapidly evolving phase.KIC6382916 (J19480292+4146558, also known asGSC 03144-595) is a HADS to be monitored extensivelyby the
Kepler space telescope for its pulsation proper-ties, which shows three independent frequencies (pul-sation modes) in its light curves (Ulusoy et al. 2013).These three pulsation modes have been confirmed as the a r X i v : . [ a s t r o - ph . S R ] F e b Niu & Xue fundamental, the first overtone and the second overtoneradial p-modes (Mow et al. 2016), and T eff = 6950 ± g = 3 . ± . v sin i = 50 ±
10 km s − havebeen estimated by the high-dispersion spectra (Ulusoyet al. 2013). Using the long-cadence (LC) photomet-ric data of KIC6382916 lasted from BJD 2454953 to2456424 (Quarter 0-17), we extract the first 23 frequen-cies with largest amplitudes (which have a cut-off of 1.3mmag, see in Table 2 for more details) to study the vari-ations and interactions of them. Theses 23 frequenciesare composed as that of the fundamental ( f ), the firstovertone ( f ), and the second overtone ( f ) modes, to-gether with their harmonics/combinations. The IDs ofthese frequencies are also identified as the labels of thecorresponding pulsation modes in this work.In order to extract the variation information of theamplitudes and frequencies over time, we use the Short-Time Fourier Transformation to deal with the LC photo-metric data. The pre-whiting process in a time windowof 150 days is performed when the window is movingfrom the start to the end time of the LC data, with astep of 30 days. In each step, the frequencies are fixedas that have been extracted in the complete LC data(see in Table 2), while the amplitudes and phases areobtained by the non-linear least square fitting. Then,we get the amplitude and phase variations of the 23frequencies. Using the Fourier-Phase Diagram method(Paparo et al. 1998; Bowman et al. 2016; Xue et al.2018), the variations of the frequencies can be derivedfrom the variations of the phases.The amplitude and phase variations of f , f , and f are presented in Figure 1, and each one of them is per-formed by a quadratic fitting. In Figure 1, we findthe amplitudes of f ( A f ) is relatively stable, that of f ( A f ) is slightly decreasing while A f is distinctlyincreasing by about 24% in 4 years. On the otherhand, the phases (subtracted by its average) of f , f ,and f variate with a same trend but different levels,which is corresponding to the increasing periods of themwith different period variation rates: (1 /P )( d P / d t ) =(3 . ± . × − yr − , (1 /P )( d P / d t ) = (3 . ± . × − yr − , and (1 /P )( d P / d t ) = (2 . ± . × − yr − .Noting the decrement of A f and the increment of A f ,it is quit interesting to study the amplitude interactionsbetween the 23 pulsation modes, which also indicate theenergy transformation between them. Here, we intro-duce the Interaction Diagram (see Figure 2) to show the The variations of all the 23 pulsation modes are listed in Figure5, 6, and 7. amplitude interactions between the pulsation modes. InFigure 2, the color of a small square represents the corre-lation coefficient between the amplitudes of the labeledpulsation modes whose column and row intersect at thesquare. The dendrograms on the upper and left repre-sent the Agglomerative Hierarchical Clustering (AHC)process (Murtagh & Contreras 2012) in the amplitudeinteraction space which is spanned by the vectors (cor-responding to each of the pulsation modes) consisting ofthe correlation coefficients between a specific pulsationmode to all of them. In general, the AHC process re-orders the pulsation modes in order to cluster those whohave similar behaviors in the interaction space together.At first glance of Figure 2, the dominating interac-tions occur between the pulsation modes including f and f . In detail, all the pulsation modes including f are clustered together and have increasing amplitudes,while the pulsation modes of f , 2 f , 2 f + f , f + 2 f ,and f +3 f are anti-correlated with the above f relatedmodes and have decreasing amplitudes. What’s more in-teresting is that the 3 f , 3 f , − f + 2 f , and − f + 2 f modes, which are not f related, but have increasing am-plitudes like that of the f related modes. All these re-lations and structures shown in the Interaction Diagramof amplitudes indicate that there exists interactions orenergy transformations between the independent pulsa-tion modes and their harmonics/combinations, and thisopens a new window for exploring the interior of thestars in future researches. The single star evolutionary models are constructedby using different initial masses with three groups of[Fe/H] and two different values of equatorial rotationvelocities ( v eq ), from the pre-main sequence to the redgiant branch. At each of the steps on the evolutionarytracks, the pulsation frequencies are calculated basedon the corresponding stellar structure. Figure 3 showsthe best-fit seismic models of the observed independentfrequencies ( f , f , and f ) together with the corre-sponding evolutionary tracks for specific combinationsof ([Fe/H], v eq ). The detailed information of the best-fitseismic models is collected in Table 1.The results of the seismic models re-confirm the con-clusion that f , f , and f are the fundamental, firstovertone, and second overtone radial p-modes, respec- Here we use the Spearman’s rank correlation coefficient, whichis a measure of how well the relationship between two variablescan be described by a monotonic function. Comparing with the complicated interactions shown in Figure 2,the Interactions Diagram of the phases (see in Figure 8) showsthat almost all the pulsation modes have the same variation trendexcept the − f + 2 f , f + f − f , and 3 f modes, which leadsto the decreasing periods of these three modes. n HADS crossing the Hertzsprung Gap
200 400 600 800 1000 1200 140079.079.580.080.581.0 A m p li t u d e ( mm a g ) best fit A f
200 400 600 800 1000 1200 1400Time (BJD - 2454833)505 σ e ff
200 400 600 800 1000 1200 14000.0050.0000.005 φ i − < φ i > best fit φ f − <φ f >
200 400 600 800 1000 1200 1400Time (BJD - 2454833)505 σ e ff
200 400 600 800 1000 1200 140077.077.578.078.579.0 A m p li t u d e ( mm a g ) best fit A f
200 400 600 800 1000 1200 1400Time (BJD - 2454833)505 σ e ff
200 400 600 800 1000 1200 14000.0050.0000.005 φ i − < φ i > best fit φ f − <φ f >
200 400 600 800 1000 1200 1400Time (BJD - 2454833)505 σ e ff
200 400 600 800 1000 1200 14001112131415 A m p li t u d e ( mm a g ) best fit A f
200 400 600 800 1000 1200 1400Time (BJD - 2454833)505 σ e ff
200 400 600 800 1000 1200 14000.020.000.02 φ i − < φ i > best fit φ f − <φ f >
200 400 600 800 1000 1200 1400Time (BJD - 2454833)505 σ e ff Figure 1.
Variation of the amplitudes and phases (subtracted by their averages) of f , f , and f . In each of the subfigures, thedata points are fitted by a quadratic fitting, the best-fit result is presented by a solid black line, and the corresponding residualsare plotted in the lower panel. The 2 σ (deep red) and 3 σ (light red) bounds of the fitting are also shown in the subfigures. Inthe lower panel of each subfigures, the σ eff is defined as ( Q obs − Q cal ) /σ , where Q obs and Q cal are the values which come fromthe observation and model calculation, respectively; σ is the uncertainty of the observed points. tively (Mow et al. 2016). Moreover, it shows thatthis star locates in the later evolutionary phase of theHertzsprung gap, which is a more rapidly evolving phasecompared with those stars in the earlier evolutionaryphase of the Hertzsprung gap (such as AE UMa (Niuet al. 2017) and VX Hya (Xue et al. 2018)). The most in-credible thing is that the observed period variation ratesof f and f are an order of magnitude larger than thetheoretical predicted ones, and two orders of magnitudelarger for the case of f . If we ascribe these observed pe- riod variation rates to the stellar evolution (which seemsto be the most reasonable case), it indicates that thestar evolving more rapidly than theoretical predictionand the standard stellar evolution theory cannot pre-cisely describe a single star’s evolution process in thisrapidly evolving phase.For KIC6382916, following the current variation rates,the amplitude of f will exceed that of f at about BJD2460331 (Jan, 2024), and exceed that of f at aboutBJD 2460343 (Feb, 2024). This prediction can be tested Niu & Xue f + f f f f + f f + f f − f f + f f f − f + f f + f f + f f f − f + f − f + f − f + f + f − f + f f + f f f + f − f + f f + f − f ID f + 3 f f f f + f f + 2 f f − f f + f f f − f + f f + 2 f f + f f f − f + 2 f − f + 2 f − f + f + f − f + f f + f f f + f − f + f f + f − f I D Figure 2.
Interaction diagram of amplitudes of the 23 pulsation modes. in the near future by the following photometric obser-vations, which also could provide us an opportunity towitness the stellar evolution process of a single star grad-ually in this special evolutionary phase. Moreover, thecurrent continuous photometric data from
Kepler cansufficiently support us to carry out the researches onthe interactions between the pulsation modes via theInteraction Diagram, which could not only be used asa new tool to classify the different pulsation modes ofa pulsating star or classify the different kinds of pulsat- ing stars, but also opens a new window for the futureasteroseismology.ACKNOWLEDGMENTSJ.S.N. acknowledges support from the National Natu-ral Science Foundation of China (NSFC) (No. 11947125and No. 12005124) and the Applied Basic Research Pro-grams of Natural Science Foundation of Shanxi Province(No. 201901D111043). H.F.X. acknowledges support bythe Scientific and Technological Innovation Programs of n HADS crossing the Hertzsprung Gap log T eff l og ( L / L fl ) M = 1 . M fl , [Fe / H] = 0 . , v eq = 40 km s − M = 1 . M fl , [Fe / H] = 0 . , v eq = 120 km s − M = 1 . M fl , [Fe / H] = 0 . , v eq = 40 km s − M = 1 . M fl , [Fe / H] = 0 . , v eq = 120 km s − M = 1 . M fl , [Fe / H] = 0 . , v eq = 40 km s − M = 1 . M fl , [Fe / H] = 0 . , v eq = 120 km s − Figure 3.
Best-fit seismic models for KIC6382916 along with the corresponding evolutionary tracks from the main-sequenceto the red giant branch. The solid lines and the dash lines represent the evolutionary tracks of v eq = 40 km s − and v eq =120 km s − , respectively. The different colors represent different value of [Fe/H]. The colored pluses and crosses represent thebest-fit seismic models of different value of v eq . Table 1.
Frequencies and period variation rates of the three independent pulsation modes from the best-fit seismic models andobservations (which are listed in the last column). f f f (1 /P )( d P / d t ) (1 /P )( d P / d t ) (1 /P )( d P / d t ) Mass [Fe/H] Rotation(c days − ) (c days − ) (c days − ) (yr − ) (yr − ) (yr − ) M (cid:12) dex km s − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . ± . × − (3 . ± . × − (2 . ± . × − – – – Higher Education Institutions in Shanxi (STIP) (No.2020L0528).
Software: emcee (Foreman-Mackey et al. 2013),MESA (Paxton et al. 2019), GYRE (Townsend & Teitler2013)REFERENCES
Aerts, C. 2021, Rev. Mod. Phys., 93, 015001,doi: 10.1103/RevModPhys.93.015001Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009,ARA&A, 47, 481,doi: 10.1146/annurev.astro.46.060407.145222Bai, Y., Liu, J., Bai, Z., Wang, S., & Fan, D. 2019, AJ, 158,93, doi: 10.3847/1538-3881/ab3048 Berger, T. A., Huber, D., Gaidos, E., & van Saders, J. L.2018, ApJ, 866, 99, doi: 10.3847/1538-4357/aada83Bowman, D. M., Kurtz, D. W., Breger, M., Murphy, S. J.,& Holdsworth, D. L. 2016, MNRAS, 460, 1970,doi: 10.1093/mnras/stw1153Breger, M. 2000, in Astronomical Society of the PacificConference Series, Vol. 210, Delta Scuti and RelatedStars, ed. M. Breger & M. Montgomery, 3
Niu & Xue
Breger, M., & Pamyatnykh, A. A. 1998, A&A, 332, 958Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman,J. 2013, PASP, 125, 306, doi: 10.1086/670067Luo, A. L., Zhao, Y. H., Zhao, G., & et al. 2019, VizieROnline Data Catalog, V/164Mathur, S., Huber, D., Batalha, N. M., et al. 2017, ApJS,229, 30, doi: 10.3847/1538-4365/229/2/30Montgomery, M. H., & Odonoghue, D. 1999, Delta ScutiStar Newsletter, 13, 28Mow, B., Reinhart, E., Nhim, S., & Watkins, R. 2016, AJ,152, 17, doi: 10.3847/0004-6256/152/1/17Mowlavi, N., Meynet, G., Maeder, A., Schaerer, D., &Charbonnel, C. 1998, A&A, 335, 573Murtagh, F., & Contreras, P. 2012, WIREs Data Miningand Knowledge Discovery, 2, 86,doi: https://doi.org/10.1002/widm.53Niu, J.-S., Fu, J.-N., & Zong, W.-K. 2013, Res. Astron.Astrophys. , 13, 1181, doi: 10.1088/1674-4527/13/10/004Niu, J.-S., Fu, J.-N., Li, Y., et al. 2017, MNRAS,doi: 10.1093/mnras/stx125Paparo, M., Saad, S. M., Szeidl, B., et al. 1998, A&A, 332,102Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192,3, doi: 10.1088/0067-0049/192/1/3Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208,4, doi: 10.1088/0067-0049/208/1/4Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS,220, 15, doi: 10.1088/0067-0049/220/1/15Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS,234, 34, doi: 10.3847/1538-4365/aaa5a8Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243,10, doi: 10.3847/1538-4365/ab2241 Sharma, S. 2017, ARA&A, 55, 213,doi: 10.1146/annurev-astro-082214-122339Smith, J. C., Stumpe, M. C., Van Cleve, J. E., et al. 2012,PASP, 124, 1000, doi: 10.1086/667697Stumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012,PASP, 124, 985, doi: 10.1086/667698Townsend, R. H. D., & Teitler, S. A. 2013, MNRAS, 435,3406, doi: 10.1093/mnras/stt1533Ulusoy, C., Ulas , , B., G¨ulmez, T., et al. 2013, MNRAS, 433,394, doi: 10.1093/mnras/stt731VanderPlas, J. T. 2018, ApJS, 236, 16,doi: 10.3847/1538-4365/aab766Wils, P., Rozakis, I., Kleidis, S., Hambsch, F. J., &Bernhard, K. 2008, A&A, 478, 865,doi: 10.1051/0004-6361:20078992Xiang, M., Ting, Y.-S., Rix, H.-W., et al. 2019, ApJS, 245,34, doi: 10.3847/1538-4365/ab5364Xue, H.-F., & Niu, J.-S. 2020, ApJ, 904, 5,doi: 10.3847/1538-4357/abbc12Xue, H.-F., Fu, J.-N., Fox-Machado, L., et al. 2018, ApJ,861, 96, doi: 10.3847/1538-4357/aac9c5Yang, X. H., Fu, J. N., & Zha, Q. 2012, AJ, 144, 92,doi: 10.1088/0004-6256/144/4/92Zong, W., Charpinet, S., Fu, J.-N., et al. 2018, ApJ, 853,98, doi: 10.3847/1538-4357/aaa548Zong, W., Charpinet, S., & Vauclair, G. 2016a, A&A, 594,A46, doi: 10.1051/0004-6361/201629132Zong, W., Charpinet, S., Vauclair, G., Giammichele, N., &Van Grootel, V. 2016b, A&A, 585, A22,doi: 10.1051/0004-6361/201526300 n HADS crossing the Hertzsprung Gap A. PHOTOMETRIC DATA REDUCTIONThe long-cadence (LC) photometric data of KIC6382916 from the
Kepler space telescope were used in this work,which covers from BJD 2454953 to 2456424 (Quarter 0-17) (publicly available PDC data Stumpe et al. (2012); Smithet al. (2012)). We downloaded the light curves (in the format of reduced BJD and magnitudes) of KIC6382916 fromMikulski Archive for Space Telescope (MAST) , which were then normalized to be zero in the mean for each quarter.An overview of all the normalized LC data in time domain and frequency domain are shown in Figure 4.
200 400 600 800 1000 1200 1400 1600Time (BJD - 2454833)4002000200400600800 N o r m a li z e d K e p l e r m a g n i t u d e ( mm g )
765 770 775 780 785 790 795 8003002001000100200 5 10 15 20Frequency (c/d)020406080100 A m p li t u d e ( mm g ) f f f Figure 4.
Overview of the normalized LC data in time domain and frequency domain of KIC6382916.
All the above normalized LC data were pre-whiten to extract the frequencies, amplitudes, and phases of the pulsationmodes. This process was cut-off until the amplitude smaller than 1.3 mmg and we totally got 23 pulsation modes(see in Table 2), which is much higher than the typical noise level in
Kepler data of this star ( ∼ . ≤ f ≤ . − were consideredin this work, which was determined by the LC Nyquist frequency (Bowman et al. 2016).The Short-Time Fourier Transformation (Bowman et al. 2016; Zong et al. 2016a,b, 2018) was then performed to thenormalized LC data to get the variation of the amplitudes and phases. In this process, a time window of 150 days wasmoving from the start to the end time of the LC data, with a step of 30 days. In each step, the pre-whiting process http://archive.stsci.edu/kepler Niu & Xue
Table 2.
Frequency solution of the first 23 frequencies with largest amplitudes.ID Frequency Amplitude Phase(c days − ) (mmag) (rad / π ) f ± ± − . ± . f ± ± − . ± . f + f ± ± . ± . − f + f ± ± . ± . f ± ± − . ± . f ± ± − . ± . f ± ± − . ± . f + f ± ± . ± . − f + 2 f ± ± − . ± . f − f ± ± . ± . f + 2 f ± ± . ± . f + f ± ± − . ± . f + 2 f ± ± . ± . f + f ± ± − . ± . − f + f ± ± − . ± . − f + f ± ± − . ± . f + f − f ± ± − . ± . f ± ± − . ± . − f + f + f ± ± − . ± . f ± ± − . ± . f + 3 f ± ± . ± . f + f ± ± − . ± . − f + 2 f ± ± . ± . was performed to extract the amplitudes and phases of the specific 23 pulsation modes, while the frequencies werefixed as the values obtained in the complete LC data in Table 2.At last, the amplitude and phase (subtracted by its average value) for each of the 23 pulsation modes in each of themoving window were collected (with the times which were defined as the midpoints of the window). After removingthe periodic signals between 350 to 400 days which is related to the 372.5 day Earth-trailing orbit of the Kepler SpaceTelescope , the variations of the amplitudes and phases of the 23 pulsation modes were presented in Figure 5, 6, and 7.In this work, the pre-whiting process was performed by the Fourier decomposition which can be presented by theformula m = m + (cid:88) A i sin [2 π ( f i t + φ i )] , (A1)where m is the shifted value, A i is the amplitude, f i is the frequency and φ i is the corresponding phase.The uncertainties all through the work were estimated by the following expressions (Montgomery & Odonoghue1999; Aerts 2021): σ A = D (cid:114) N σ N , σ f = D √ σ N π √ N AT , σ φ = Dσ N π √ N A . (A2)In these expressions, σ A , σ f , and σ φ are the uncertainties of the amplitude, frequency, and phase, respectively; σ N can be approximated by the standard deviation of the final residual light curve; A is the amplitude; N and T are thetotal number of data points and the total time base-line employing in the pre-whiting process, respectively; D can beestimated as the square-root of the average number of consecutive data points of the same sign in the final residuallight curves. In this work, the residual light curve were conservatively considered as the light curves in which thespecific 23 pulsating signals had been removed. n HADS crossing the Hertzsprung Gap B. THEORETICAL MODEL CALCULATIONIn order to determine the stellar mass and evolutionary stage based on the single star evolutionary models, theopen source 1D stellar evolution code Modules for Experiments in Stellar Astrophysics (MESA (Paxton et al. 2011,2013, 2015, 2018, 2019)) was used to construct the structural and evolutionary models. At each step along with theevolutionary tracks, the pulsation frequencies of the specific structure were calculated by the stellar oscillation codeGYRE (Townsend & Teitler 2013) (see, e.g., Niu et al. (2017); Xue et al. (2018); Xue & Niu (2020)).The initial parameters used to construct pre-main sequence evolutionary models of KIC6382916 were configured asfollows. Different metallicity [Fe/H] with values of 0 . . .
040 dex were considered as the initial metallicityof the evolutionary models (see Table 3 for more details).
Table 3.
Observed stellar parameters of KIC6382916.Parameter Value Reference[Fe/H] (dex) 0 . ± .
021 Luo et al. (2019)0 . ± .
021 Xiang et al. (2019)0 . +0 . − . Mathur et al. (2017) T eff (K) 7075 ±
281 Bai et al. (2019)6950 ±
100 Ulusoy et al. (2013)6923 . ± .
99 Luo et al. (2019)6737 . ± .
29 Xiang et al. (2019)6548 +148 − Mathur et al. (2017)6557 ±
229 Berger et al. (2018) v sin i (km s − ) 50 ±
10 Ulusoy et al. (2013)
The following formulas were used to calculate the initial heavy element abundance Z and initial hydrogen abundance X : [Fe / H] = log ZX − log Z (cid:12) X (cid:12) , (B3) Y = 0 .
24 + 3 Z, (B4) X + Y + Z = 1 , (B5)where X (cid:12) = 0 . Z (cid:12) = 0 . / H], we got ( X = 0 . Z = 0 . X = 0 . Z = 0 . X = 0 . Z = 0 . (cid:12) to 2.5 M (cid:12) with a step of 0 .
01 M (cid:12) , covering the typical mass range of HADS. The rotation of the starhad also been considered in the model calculation. Because Ulusoy et al. (2013) provided us the projected rotationalvelocity v sin i = 50 ±
10 km s − from high-resolution spectroscopic observation, the equatorial rotation velocities v eq = 40 km s − and v eq = 120 km s − were set to be the inputs in the model calculation, which covered a reasonablerange of v eq . The mixing-length parameter was set to be a value of α MLT = 1 .
89 (Yang et al. 2012). All the evolutionarytracks were calculated from the pre-main sequence to red giant branch.Based on the pulsation frequencies calculated in every step along with the evolutionary tracks, we got the best-fitseismic models (which have the smallest χ with respect to the observed values of f , f , and f ) with the differentcombinations of ([Fe / H] , v eq ) (see in Figure 3 and Table 1).The possibility that f , f , and f could be the non-radial pulsation modes was also tested in our calculation. Itcan be excluded based on two reasons: (i) we explored the frequency spectrum carefully and did not find any hintsof rotation splits of the identified frequencies; (ii) according to the model calculation, the non-radial pulsation modesrepresented negative period variation rates in post-main sequence phase, which was opposite to the observed values.0 Niu & Xue C. FITTING ON THE THREE INDEPENDENT PULSATION MODESThe amplitudes and phases of the three independent pulsation modes ( f , f , and f ) were fitted by the quadraticpolynomial a + b · t + c · t . The Markov Chain Monte Carlo (MCMC) algorithm (Sharma 2017) was used to determinethe coefficients and their uncertainties in above expression, and the fitted coefficients are listed in Table 4. Table 4.
Quadratic fitting results of the amplitudes and phases of f , f , and f .Variable a b c χ / d . o . f .A f . ± .
1) 7( ± × − − ± × − A f . ± . − ± × − − . ± . × − A f . ± . − . ± . × − . ± . × − φ f − < φ f > − . ± . × − . ± . × − − . ± . × − φ f − < φ f > − . ± . × − − . ± . × − − . ± . × − φ f − < φ f > − . ± . × − . ± . × − − . ± . × − Based on the Fourier Phase Diagram method (Xue et al. 2018), the period variation rates can be directly derivedvia the fitted coefficients of the quadratic term of the phases. D. FIGURES AND TABLES n HADS crossing the Hertzsprung Gap
200 400 600 800 1000 1200 140079.580.0 A m p li t u d e ( mm a g ) f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 140077.077.578.078.5 A m p li t u d e ( mm a g ) f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 140028.028.5 A m p li t u d e ( mm a g ) f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 140016.016.517.0 A m p li t u d e ( mm a g ) − f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 140015.516.016.5 A m p li t u d e ( mm a g ) f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14001214 A m p li t u d e ( mm a g ) f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 140011.011.512.0 A m p li t u d e ( mm a g ) f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14008.08.59.0 A m p li t u d e ( mm a g ) f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14006.57.0 A m p li t u d e ( mm a g ) − f +2 f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14005.05.5 A m p li t u d e ( mm a g ) f − f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i > Figure 5.
Variation of the amplitudes and phases of the 23 pulsation modes, Part I. Niu & Xue
200 400 600 800 1000 1200 14004.04.5 A m p li t u d e ( mm a g ) f +2 f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14003.03.54.0 A m p li t u d e ( mm a g ) f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14003.03.5 A m p li t u d e ( mm a g ) f +2 f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14003.03.54.0 A m p li t u d e ( mm a g ) f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14002.53.03.5 A m p li t u d e ( mm a g ) − f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14002.02.53.0 A m p li t u d e ( mm a g ) − f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14001.52.02.53.0 A m p li t u d e ( mm a g ) f + f − f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14001.82.02.22.4 A m p li t u d e ( mm a g ) f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14001.52.02.5 A m p li t u d e ( mm a g ) − f + f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14001.52.0 A m p li t u d e ( mm a g ) f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i > Figure 6.
Variation of the amplitudes and phases of the 23 pulsation modes, Part II. n HADS crossing the Hertzsprung Gap
200 400 600 800 1000 1200 14001.52.0 A m p li t u d e ( mm a g ) f +3 f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14001.01.5 A m p li t u d e ( mm a g ) f + f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i >
200 400 600 800 1000 1200 14001.01.52.0 A m p li t u d e ( mm a g ) − f +2 f
200 400 600 800 1000 1200 1400
Time (BJD - 2454833) φ i − < φ i > Figure 7.