A double-modulation effect detected in a double-mode high-amplitude δ Scuti star: KIC 10284901
AA Double-modulation Effect Detected in a Double-mode High-amplitude δ Scuti Star:KIC 10284901
Tao-Zhi Yang and Ali Esamdin Xinjiang Astronomical Observatory, Chinese Academy of Science, Urumqi 830011, People ʼ s Republic of China; [email protected] University of Chinese Academy of Sciences, Beijing 100049, People ʼ s Republic of China; [email protected] Received 2018 November 4; revised 2019 May 21; accepted 2019 May 21; published 2019 July 5
Abstract
In this paper, we present an analysis of the pulsating behavior of
Kepler target KIC 10284901. The Fouriertransform of the high-precision light curve reveals seven independent frequencies for its light variations. Amongthem, the fi rst two frequencies are main pulsation modes: F0 = ( ) day − and F1 = ( ) day − ; the ratio F0 / F1 = fi es this star as a double-mode high-amplitude δ Scuti ( HADS ) star; anothertwo frequencies, f m = − and f m = − , are detected directly, and the modulations of f m and f m to F0 and F1 modes ( seen as quintuplet structures centered on these two modes in the frequency spectrum ) are also found. This is the fi rst detection of a double-modulation effect in the HADS stars. The features of thefrequency pattern and the ratio ( f m / f m ≈ ) , as well as the cyclic variation of amplitude of the two dominantpulsation modes, which seem to be similar to that in Blazhko RR Lyrae stars, indicate this modulation might berelated to the Blazhko effect. A preliminary analysis suggests that KIC 10284901 is in the bottom of the HADSinstability strip and situated in the main sequence. Key words: stars: oscillations – stars: variables: delta Scuti
1. Introduction
The
Kepler Space Telescope is designed to search forterrestrial planets orbiting the solar-type stars by the transitsmethod ( Borucki et al. 2010; Koch et al. 2010 ) . As a supportingprogram, the Kepler asteroseismolog program possesses anintrinsic important role in the core planet search project ( Gilliland et al. 2010 ) . Owing to the ultra-high photometricprecision data at the level of μ mag, the Kepler mission hassigni fi cantly improved our understanding of different types ofvariable stars ( e.g., Bedding et al. 2011; Giammichele et al.2018 ) ; and the continuous observations spanning about 4 yrprovide an excellent opportunity to monitor the amplitudemodulation of different pulsators ( e.g., Benkó et al. 2014;Bowman & Kurtz 2014; Bowman et al. 2016 ) . At present, the Kepler mission has found at least 2000 δ Sct stars ( Balona &Dziembowski 2011; Balona 2014; Bowman et al. 2016 ) .Among them, some stars show amplitude modulation ofpulsation modes caused for different reasons, e.g., beating,mode coupling, and rotation ( e.g., Bowman & Kurtz 2014;Bowman et al. 2016; Yang et al. 2018b ) . These targets areexcellent for asteroseismic study, as they could improve ourknowledge of the stellar structure and evolution for stars.High-amplitude δ Sct ( HADS ) stars are usually considered asubclass of δ Sct stars, with amplitudes of peak-to-peak lightvariations larger than 0.3 mag ( Breger 2000 ) . From the relation-ship between the amplitude and the measured rotational velocity ( v sin i ) provided by Breger ( ) , HADS stars are typical slowrotators with v sin i
30 km s − . They seem to concentrate in thecenter part of the δ Sct instability region, and occupy a relatively narrow strip with a width of 300 K in temperature ( McNamara2000 ) . As seen with ground-based observations, HADS stars areusually pulsating with only one or two radial modes ( e.g., YZBoo: Yang et al. 2018a; KIC 5950759: Yang et al. 2018b, etc ) . Inrecent decades, some stars also been found to exhibit nonradialmodes with low amplitude, owing to extensive photometriccampaigns. With the advent of space missions, especially the Kepler mission, more interesting phenomena have been dis-covered, including low-amplitude pulsation modes and long-termvariations in pulsating stars ( Balona et al. 2012; Bowman &Kurtz 2014 ) . Several stars observed by Kepler show triplet orquintuplet structures in their frequency spectra ( Kolenberg et al.2011; Benkó et al. 2014 ) . For instance, in HADS star KIC5950759, a pair of triplet structures centered on the mainfrequency were detected in its frequency spectra, and the causeof the triplet structure is inferred to be the amplitude modulationof stellar rotation, 0.3193 day − ( v sin i ≈
33 km s − ) ( Yanget al. 2018b ) . These low-amplitude multiplet structures mightimprove our knowledge of the HADS stars and offer new cluesfor probing the stellar interior and physical processes.KIC 10284901 ( α = h m
46 4, δ = + ° ′
32 8,2MASS: J19434637 + ) was found to be a δ Scuti star inthe RApid Temporal Survey of the
Kepler fi eld ( RATS-
Kepler ) byRamsay et al. ( ) . In that survey, Ramsay et al. ( ) reportedKIC 10284901 was a mid-late A type star and it might be a HADSstar due to its high-amplitude light variations. Some basicproperties of this star from that survey and the Kepler
InputCatalog ( KIC; Brown et al. 2011 ) are listed in Table 1. This starwas also selected as a target in the Kepler
Guest Observer programand continuously observed for more than 10 months in both longcadence ( LC ) with 29.4 minute effective integrations and shortcadence ( SC ) with 58.8 s effective integrations ( Gilliland et al.2010 ) . Due to the strong effect of signal averaging in LC data, weonly use the SC data in this work. The unique and high-precisionphotometric data make KIC 10284901 an ideal source to deeplyinvestigate its pulsation behavior. The Astrophysical Journal, ( ))
Guest Observer programand continuously observed for more than 10 months in both longcadence ( LC ) with 29.4 minute effective integrations and shortcadence ( SC ) with 58.8 s effective integrations ( Gilliland et al.2010 ) . Due to the strong effect of signal averaging in LC data, weonly use the SC data in this work. The unique and high-precisionphotometric data make KIC 10284901 an ideal source to deeplyinvestigate its pulsation behavior. The Astrophysical Journal, ( )) , 2019 July 1 https: // doi.org / / / ab23f3 © 2019. The American Astronomical Society. Corresponding author.Original content from this work may be used under the termsof the Creative Commons Attribution 3.0 licence. Any furtherdistribution of this work must maintain attribution to the author ( s ) and the titleof the work, journal citation and DOI. . Observations and Data Reduction KIC 10284901 was observed continuously from BJD2456107.13 to 2456390.97 ( Q14.1: 34,809 SC observations;Q14.3: 49,957 SC observations; Q16.1: 7622 SC observations;,Q16.3: 46,461 SC observations; 138,849 SC observations intotal ) spanning 283.84 days. The Kepler Asteroseismic ScienceOperations Center ( KASOC ) database ( Kjeldsen et al. 2010 ) provides the SC photometric fl ux data of KIC 10284901 as twotypes: one is the “ raw ” data, which in fact has been reduced bythe NASA Kepler
Science pipeline, and the other is the fl uxdata corrected by KASOC Working Group 4 ( WG δ Scutitargets ) . We use the corrected fl ux and perform corrections,including eliminating outliers, as well as the possible lineartrends in some quarters. The fl ux data are converted to themagnitude scale, then the mean value of each quarter issubtracted, and the recti fi ed time series is obtained. A portionof the recti fi ed SC light curve is shown in Figure 1. It is clearthat the light amplitude of KIC 10284901 is larger than 0.3 magin SC data. Inspection of the light curve indicates that theamplitude has a repeat of about 2.5 days.
3. Frequency Analysis
We performed a Fourier transform for the recti fi ed SC lightcurve using PERIOD04 ( Lenz & Breger 2005 ) . The light curveis fi tted with the following formula: ( ( )) ( ) p f = + S + m m A f t sin 2 , 1 i i i where m denotes the zero-point, A i is the amplitude, f i is thefrequency, and f i is the corresponding phase.To detect more signi fi cant frequencies, a frequency range of0 < ν <
50 day − , which covers the typical pulsation fre-quency of the δ Sct stars, was chosen in this work. In the extraction of signi fi cant frequencies, the highest peak in thefrequency spectrum was considered as a signi fi cant frequency,then a multi-frequency least-squares fi t using formula 1 wasconducted for the light curve with all the signi fi cant frequenciesdetected, resulting in the solutions of all the signi fi cantfrequencies. Next, all the frequencies of combination signalsare fi xed to the exact values they are supposed to be, and onlyleave the independent frequencies, all amplitudes, and allphases as free parameters to be improved. A constructed lightcurve using the above solutions was subtracted from the data,and the residual was obtained to search for a signi fi cant term innext step. The above steps were repeated until there was nosigni fi cant peak in the residual. The criterion ( S / N > ) suggested by Breger et al. ( ) was adopted to judge thesigni fi cant peaks. The uncertainties of frequencies werecalculated following Montgomery & Odonoghue ( ) .A total of 151 signi fi cant frequencies were extracted in thiswork, and a full list of the extracted frequencies ( f S to f S ) ,with their corresponding amplitude and identi fi cations, is givenin Table 2. After pre-whitening of the 151 frequencies, theamplitude spectrum of the residual is shown in Figure 2. Nopeak is statistically signi fi cant in the residual and the overalldistribution of the residual has typical noise.Seven high-amplitude independent frequencies were detected, fi ve of which were found in the range of 18 –
25 day − , and theother two ( i.e., f S and f S ) were in the range of 0 – − .Among these independent frequencies, f S and f S give a ratioof 0.7805, which is in the typical period ratio range of the fi rstovertone and the fundamental mode for the double-modeHADS star. If the highest frequency f S is assumed to be thefundamental mode, then KIC 10284901 is classi fi ed as a double-mode HADS star. We thus marked the frequencies f S and f S with “ F0 ” and “ F1 ” in the last column of Table 2, respectively.Petersen & Christensen-Dalsgaard ( ) presented a detaileddiagram of double-mode HADS stars of different metallicities intheir Figure
3. Their study showed that higher values of theP ( F1 ) / P ( F0 ) ratio are found for metal-poor stars. For KIC10284901, the higher period ratio ( ) indicates that it maybelong to Population II stars. f S and f S are interesting, as they are out of the typicalfrequency range of the δ Scuti stars, and they are notcombinations of F0 and F1. Considering a repeating withabout 2.5 days in the light curve, the frequency f S was markedwith f m , and f S with f m in Table 2, respectively. Three otherfrequencies, f S , f S , and f S , were also considered asindependent frequencies, as they are neither any combinationsnor harmonics of other frequencies. Thus, we marked thesethree frequencies with “ independent ” in the last column ofTable 2. Another eight frequencies ( marked with “ T1 ” and “ T2 ” in the last column of Table 2 ) can be divided into twogroups: one group consists of the frequencies marked with “ T1, ” which are combinations of f m and the main frequencies ( i.e., F0 and F1 ) ; the other group is composed of the frequenciesmarked with “ T2, ” which are combinations of f m and the mainfrequencies.
4. Discussion
The four pairs of side peaks around the main frequencies F0and F1 ( i.e., ( ) f S = − and f S = − ,centered on F0 with an interval of f m , ( ) f S = − and f S = − , centered on F0 with an interval of f m shown in the left panel of Figure 3; ( ) f S = − and Table 1
Basic Properties of KIC 10284901Parameters KIC 10284801 References K P P G Gaia g SDSS i SDSS z SDSS J H K g U − g g − r ( ) - E G G
BP RP A G T KIC ±
250 K a T GTC ±
180 K b T Gaia ±
324 K dlog g ± Note. ( a ) KIC Brown et al. ( ) . ( b ) Ramsay et al. ( ) . ( c ) Greiss et al. ( ) . ( d ) These parameters are available in the
Gaia
Archive ( http: // gea.esac.esa.int / archive / ) , and the uncertainty of the effective temperature is atypical value ( Andrae et al. 2018 ) . KASOC database: http: // kasoc.phys.au.dk. The Astrophysical Journal, ( ))
Archive ( http: // gea.esac.esa.int / archive / ) , and the uncertainty of the effective temperature is atypical value ( Andrae et al. 2018 ) . KASOC database: http: // kasoc.phys.au.dk. The Astrophysical Journal, ( )) , 2019 July 1 Yang & Esamdin S = − , centered on F1 with an interval of f m ; and ( ) f S = − and f S = − , centered onF1 with an interval of f m , shown in the right panel of Figure 3 ) ,are the most interesting features in the frequency spectrum of KIC10284901. Also listed in Table 2 are some other visible peaks thatare not labeled in Figure 3, f S and f S , which are the peaks to theleft of f S , and the three peaks f S , f S , and f S , which are to theleft of f S .The side peaks around F0 and F1 in the frequency spectra ofKIC 10284901 form four pairs of uniformly spaced tripletswith intervals of f m = − and f m = − .To investigate these side peaks, we fi rst checked the possibilitythat these triplets were from the instrumental effects of Kepler ,as performed by Yang et al. ( ) , and found that none of theknown frequencies of instrumental effects from Kepler wereequal to f m or f m . Hence, the side peaks detected in thefrequency spectra of this star are not caused by the knowninstrumental effects. has found that the multiplet structures can be shownin the frequency spectra of different types of pulsating variables ( e.g., Kolenberg et al. 2011; Benkó et al. 2014; Yang et al.2018b ) . In examining the δ Sct stars observed by
Kepler ,Breger et al. ( ) noted that in several stars the equallyspaced frequency components were presented in theirfrequency spectra and these multiplet structures were con-sidered to be from the stellar intrinsic variations. Withasteroseismology, these equally spaced frequencies mightprovide more information about the global properties. Hence,it is astrophysically interesting to explore the nature of thetriplet structures in the frequency spectra of KIC 10284901. For the equidistant or nearly equidistant frequency tripletsshown in the spectra of the pulsating stars, Breger & Kolenberg ( ) proposed an explanation called “ The CombinationMode Hypothesis. ” In this scenario, the frequency f S ( = − ) detected in the SC spectrum of KIC 10284901 can be regarded as the third mode ( F0 and F1 are thefundamental and the fi rst overtone mode, respectively ) , and f S ( = − ) can be considered the combination of 2F0and f S , i.e., f S = − f S . Some other frequencies aretherefore the combinations of F0, F1, and f S . Similarly, f S ( = − ) can be regard as the fourth mode and somefrequencies are also combinations of F0, F1, and f S . Thus,KIC 10284901 might be a multimode radial variable. However,the ratios of F0 / f S ( = ) and F0 / f S ( = ) are faraway from the typical ratio for the P O / P F ( ∼ ) and P O / P F ( ∼ )( P F is the period of the funda-mental radial mode, and P O and P O are the periods of thesecond and third radial overtones, respectively ) ( Stelling-werf 1979 ) . This seems to rule out the possibility that these twofrequencies belong to the radial modes.If f S or f S is assumed to be a nonradial mode, theequidistant triplet structure formed by the nonradial modes canoccur only when the stars rotate with an extremely lowvelocity. When the stars rotate only about a few km s − , therotationally split multiplet will not be symmetric, and theequidistant structure in the spectrum will also not be seen ( Pamyatnykh 2000 ) . Consequently, f S and f S are unlikely tobe nonradial modes of KIC 10284901. The side peaks might becaused by some modulation effects. Rotation plays an important role in stellar evolution andpulsations ( Pamyatnykh 2000 ) . Several δ Scuti stars found inthe
Kepler mission show equidistant multiplets in theirfrequency spectra that were caused by modulation of theamplitude with stellar rotation ( Breger et al. 2011 ) . A low-amplitude modulation frequency of 0.16 day − to the dominantfrequencies was detected in KIC 9700322 and it was con fi rmedto be the stellar rotation frequency by the high-dispersionspectral observations ( Breger et al. 2011 ) .Another example is KIC 11754974 ( Murphy et al. 2013 ) ,which is a metal-poor double-mode HADS star. The light curveof this star shows apparent amplitude modulation, similar tothat in KIC 10284901. The Fourier transform of its light curvealso reveals a high number of combination frequencies of the Figure 1.
Portion of the SC light curve of KIC 10284901. Inspection of the light curve indicates that the amplitude has a repeat of about 2.5 days. The Astrophysical Journal, ( ))
Portion of the SC light curve of KIC 10284901. Inspection of the light curve indicates that the amplitude has a repeat of about 2.5 days. The Astrophysical Journal, ( )) , 2019 July 1 Yang & Esamdin able 2 All Frequencies Detected in SC Data ( Denoted by f Si ) f Si Frequency ( day − ) Amplitude ( mmag ) S / N Identi fi cationIndependent frequencies1 18.994054 ± ± ± f m ± f m ± ± ± ± + f m , “ T1 ” ± − f m , “ T1 ”
10 19.80665 ± + f m , “ T2 ”
11 18.18152 ± − f m , “ T2 ”
12 24.7766 ± + f m , “ T1 ”
13 23.8952 ± − f m , “ T1 ”
14 25.1483 ± + f m , “ T2 ”
15 23.5233 ± − f m , “ T2 ” Combination frequencies16 37.988110 ± ± − F018 43.329943 ± + F119 43.58978 ± + f S
20 48.67163 ± ± f S − F022 38.36385 ± + f S
23 0.37577 ± f S − F024 43.70589 ± f S + F125 38.73945 ± f S
26 42.1720 ± + f S
27 4.1839 ± f S − F028 48.9316 ± + f S
29 4.9661 ± − f S
30 0.2595 ± f S − F131 47.5139 ± + f S
32 22.7373 ± f S − f m
33 13.65236 ± − F134 38.42881 ± + f m
35 37.17565 ± − f m
36 29.67758 ± − F037 24.71221 ± − F0 + f S
38 4.90113 ± − F0 − f m
39 43.77050 ± + F1 + f m
40 13.39238 ± − f S
41 38.29677 ± f S − f m
42 5.7180 ± f S + F1 − ± f S − F044 18.6183 ± − f S
45 6.1543 ± − F0 + f m
46 42.8892 ± + F1 − f m
47 14.8101 ± − f S
48 37.5475 ± − f m
49 5.7824 ± − F0 + f m
50 42.5175 ± + F1 − f m
51 4.5292 ± − F0 − f m
52 38.8007 ± + f m
53 14.0281 ± f S − F1 + F054 0.7514 ± f S − ± f S − F156 3.7432 ± f S − F0 − f m
57 18.7342 ± + F1 − f S
58 23.9601 ± + F1 − f S
59 17.8352 ± f S − F1 + F060 19.2539 ± + f S − F161 44.0309 ± + f S + f m The Astrophysical Journal, ( ))
59 17.8352 ± f S − F1 + F060 19.2539 ± + f S − F161 44.0309 ± + f S + f m The Astrophysical Journal, ( )) , 2019 July 1 Yang & Esamdin able 2 ( Continued ) f Si Frequency ( day − ) Amplitude ( mmag ) S / N Identi fi cation62 20.2476 ± + f m + f m
63 29.9374 ± − F0 + f S
64 44.1422 ± + F1 + f m
65 4.9964 ± f S − F0 + f m
66 32.64640 ± − F167 19.30291 ± f S − F0 − f m
68 14.09292 ± − F1 + f m
69 10.6835 ± − ± − F1 − f m
71 32.3866 ± − f S
72 33.3978 ± + f S − F173 4.5903 ± − f S + F174 15.2508 ± − f S + f m
75 37.6118 ± − f S
76 24.2717 ± − F0 + f S − f m
77 13.9609 ± f S − F1 − f m
78 33.8049 ± − f S
79 33.0222 ± + f S − F180 29.2365 ± − F0 − f m
81 44.3413 ± f S + f S − F082 14.4647 ± − F1 + f m
83 28.5876 ± + f S + f m − f S
84 38.2478 ± + f S − F185 0.3089 ± f S − − f m
86 8.3105 ± − ± − F1 + f m
88 13.2759 ± − f S − F189 1.7424 ± f S − F0 − f m
90 25.3469 ± f S − + f S
91 44.7550 ± f S − F1 − f m
92 23.0762 ± + F1 − f m − f S
93 31.8340 ± − F1 − f m
94 5.0330 ± − f S + F0 + f m
95 5.2773 ± f S + F1 − − f m
96 16.6799 ± f S − + F097 43.6392 ± − F0 + f S − f m
98 18.2427 ± − f S
99 34.7890 ± + f S − f S
100 25.7609 ± f S − F0 − F1 − f m
101 18.6854 ± − f S + f m
102 27.3046 ± − ± f S − F0 + f S + f m + f m
104 24.0270 ± + F1 − f S + f m
105 35.6489 ± + f S − − f m + f m
106 13.7167 ± + f m − f S − F1107 32.2699 ± − f S − F1108 6.0934 ± − + f S
109 17.8683 ± − f S
110 16.6548 ± + f S − − f m + f m
111 6.7667 ± f S − − F1 − f m
112 9.3597 ± − − f m
113 24.6049 ± f S − F0 − − f m
114 12.2272 ± − f S + F1 + f m
115 36.8624 ± − f S
116 23.1281 ± + f S + f S + f m − f S − F1117 13.2396 ± + f S − − f m
118 22.7643 ± + F1 − f m − f S
119 23.0953 ± f S + − f S
120 5.7546 ± + f m − − f S
121 5.6111 ± f S − − − f m
122 37.9959 ± − f S + f S − − f m
123 24.7485 ± + f m − − f S
124 18.8870 ± f S + F0 − − f m − f S
125 28.7143 ± + F1 + f m + f S − f S The Astrophysical Journal, ( ))
125 28.7143 ± + F1 + f m + f S − f S The Astrophysical Journal, ( )) , 2019 July 1 Yang & Esamdin ndependent pulsation modes. Moreover, a quintuplet, which isassumed to be stellar rotationally split, is detected in thefrequency spectra, and its separation is nearly equal, with a mean separation of 0.218 day − ( Murphy et al. 2013 ) .However, it is different from that in KIC 10284901, as thequintuplet in this work includes two exact interval, i.e., Table 2 ( Continued ) f Si Frequency ( day − ) Amplitude ( mmag ) S / N Identi fi cation126 28.6591 ± f S + − − f m + f m − f S
127 43.7427 ± + f m − − f S
128 18.1570 ± − − f m + f m − f S
129 47.6534 ± + f S − − f m + f m − f S
130 29.7254 ± + f S − − f S − f m
131 20.3316 ± f S − − f m − f S + F1132 21.5457 ± f S − − f m − + f S
133 2.2550 ± f S − ± − + f m + f S + f S − f S
135 1.3374 ± f S − − f m − f S + F1136 37.1511 ± − − f m + f m − f S
137 2.5516 ± f S − − f m − + f S
138 20.0159 ± − − f S + f m − f m + f S
139 1.0216 ± − − f S + f m − f m + f S
140 17.6568 ± − f S + f m + f S − F1141 24.6291 ± + f S + f m − − f S − f S
142 16.4421 ± + + f m − f S − f S
143 17.9722 ± − + f S − f m + f m − f S
144 32.3149 ± − + f m + f S + f S − f S
145 5.2586 ± + f S + f m − − f S − f S
146 24.6680 ± f S − + − f m + f S
147 35.0051 ± + f S − − f m − f S − f m
148 16.0114 ± + f S − − f m − f S − f m
149 5.3288 ± + f S + f m + f m − − f S − f S
150 21.4558 ± f S − − f m − f S + F1151 2.9824 ± − − f S + f m + f S + f m Figure 2.
Residuals left behind after extraction of 151 signi fi cant frequencies, to amplitude limits 0.12 mmag. No peak is statistically signi fi cant in the residuals.Zooms into the region around the main frequencies F0 ( left inset ) and F1 ( right inset ) are marked with vertical dashed lines, showing the multiplet structures. The Astrophysical Journal, ( ))
Residuals left behind after extraction of 151 signi fi cant frequencies, to amplitude limits 0.12 mmag. No peak is statistically signi fi cant in the residuals.Zooms into the region around the main frequencies F0 ( left inset ) and F1 ( right inset ) are marked with vertical dashed lines, showing the multiplet structures. The Astrophysical Journal, ( )) , 2019 July 1 Yang & Esamdin m = − and f m = − . From thisperspective, these two quintuplets may have different origins.Yang et al. ( ) reported that a pair of equidistant sidepeaks around the main frequency were detected in thefrequency spectra in a HADS star KIC 5950759 and theywere caused by the the amplitude modulation to its mainpulsation with a stellar rotation of 0.3193 day − ( v sin i ≈
33 km s − ) . In the case of KIC 10284901, the possibility ofamplitude modulation with rotation was considered to explainthe multiplet structures; however, it is hard to imagine how thestellar rotation, and the low rotation velocity commonly found inHADS stars, could produce two different modulation frequen-cies, i.e., f m = − and f m = − . As aresult, there is not enough evidence to support the hypothesisthat these two modulations are from the stellar rotation. In Blazhko RR Lyrae stars, the equidistant triplets are oftenshown in the frequency spectra and the interval of the triplets isusually equal to the modulation frequency ( Bla ž ko 1907;Kolenberg et al. 2006; Soszy ń ski et al. 2016 ) . The modulationfrequency can also be detected directly in the frequencyspectra. Hurta et al. ( ) and Kolenberg et al. ( ) alsofound equidistant quintuplets in the spectrum of RR Lyrae star.Jurcsik et al. ( ) presented a detailed analysis of fourdouble-mode RR Lyrae stars showing the Blazhko effect basedon new time-series photometry of the globular cluster M3.These four double-mode stars show large-amplitude Blazhkomodulations of both radial modes and rapid phase changeconnected to the amplitude minimum of the respective mode.One of them, V13, shows a anti-correlation in both theamplitude- and phase- modulations of the modes. In KIC10284901, the temporal behaviors of the amplitudes and phasesof the two radial pulsation modes were investigated basedon the observational and synthetic data. We took the 151frequency fi t to the full data and then computed a syntheticlight curve to the full data set using the parameters from this fi t,but leaving out F0, F1, and the T1 and T2 groups. Then thedifference of the original data and this fi t was fi tted by F0 andF1 only in a short pieces ( the time interval of 0.5 day was chosen to obtain a much better time resolution ) . The amplitudeand phase variations of the modes are shown in Figure 4.Contrary to the anti-correlation shown in V13 ( see Figure ) , the amplitudes of the two modes in KIC10284901 seem to show obviously cyclic change andsynchronous behavior. Fourier analysis of the amplitudevariations of these two modes reveals two signi fi cantfrequencies of 0.44071 ( ) day − and 0.81243 ( ) day − , whichare equal to f m and f m . Like its amplitude, the phase of modeF0 also shows obviously cyclic change, but the mode F1 doesnot possess similar variation.Smolec et al. ( ) gave an analysis of Blazhko-typemodulation in double-mode RR Lyrae stars in the OpticalGravitational Lensing Experiment photometry of the Galacticbulge, and found the amplitudes and phase of the radial modesvaried irregularly on a long timescale of a few hundred orthousand days; the same is true for the short-term modulation.In Figure 4, the variations of the amplitude of both the radialmodes, which are commonly in the Blazhko RR Lyrae stars ( e.g., CoRoT 105288363: Guggenberger et al. 2011 and V445Lyr: Guggenberger et al. 2012 ) , are obvious and regular. Fromthis perspective, the modulation detected in KIC 10284901seems to differ from that in Blazhko RR Lyrae stars. However,in Figure 2 the multiplet structures in the main pulsations in theresidual spectrum display a similarity to those in RR Lyrae starCoRoT 105288363 ( Guggenberger et al. 2011 ) , and theamplitudes of the modes of KIC 10284901 also show strongvariations, as shown in CoRoT 105288363.Benkó et al. ( ) reported that three Blazhko RR Lyraestars in the Kepler fi eld ( i.e., V355 Lyr, V 366 Lyr, V450 Lyr ) show two modulation periods in their light curves, and the ratiobetween the primary and secondary modulation periods isnearly 1:2. In KIC 10284901, the SC light curve exhibits anobvious modulation feature in its amplitude, as commonlyshown in the Blazhko RR Lyrae stars ( e.g., Benkó et al. 2014 ) .The multiplet structures found in the SC spectrum, as shown inFigure 3, are similar to that in the Blazhko RR Lyrae stars.Moreover, the modulation frequencies ( f m and f m ) , whichwere detected directly in SC data, have a ratio of nearly 1:2 justas shown in the above Blazhko RR Lyrae stars. These features Figure 3.
Amplitude spectra after subtracting the main frequencies. The vertical dashed lines indicate the locations of the main frequencies F0 ( left panel ) , F1 ( rightpanel ) , and their multiplets. The two panels clearly show four pairs of side peaks ( f S , f S f S and f S centered on F0; 3: f S , f S ; 4: f S and f S centered on F1 ) around the main frequency F0 and F1 in SC spectrum. The intervals from the side peaks to the center are marked with f m and f m , respectively. The Astrophysical Journal, ( ))
Amplitude spectra after subtracting the main frequencies. The vertical dashed lines indicate the locations of the main frequencies F0 ( left panel ) , F1 ( rightpanel ) , and their multiplets. The two panels clearly show four pairs of side peaks ( f S , f S f S and f S centered on F0; 3: f S , f S ; 4: f S and f S centered on F1 ) around the main frequency F0 and F1 in SC spectrum. The intervals from the side peaks to the center are marked with f m and f m , respectively. The Astrophysical Journal, ( )) , 2019 July 1 Yang & Esamdin hown in KIC 10284901 are similar to the frequency patterns inthe Blazhko RR Lyrae stars.Thus, for the target KIC 10284901, the features, includingthe multiplet structures in the frequency spectrum, the ratio of f m and f m ( nearly 1:2 ) , the obviously cyclic variations of theamplitude of the two dominant pulsation modes, are similar tothe Blazhko effect in RR Lyrae stars. It seems to imply themodulation is related to the Blazhko effect. As the phasevariation of the mode F1 seems to not be obvious, note thatalthough the modulation detected in KIC 10284901 sharessome similarities with the Blazhko effect, it could also belongto a unique effect of HADS stars, which would require furtherinvestigation. More HADS stars with this modulation effectare needed to solve this mystery, especially from spacemissions. – R Diagram
To investigate the evolutionary stage of KIC 10284901, weobtained M V = ( ± ) mag for this star using the period – luminosity relationship M V = − ( ± ) – ( ± ) log P F ( P F is the period of the fundamental mode ) provided byPoretti et al. ( ) and its log P F = − – Rdiagram, KIC 10284901 is located on the bottom of the HADSinstability strip and likely situated in the main sequence, underthe constraints of the above M V = ( ± ) mag and T eff = ( ± ) K derived from GTC spectra in the RATS-
Kepler by Ramsay et al. ( ) . Note that a precise parallax ( ± ) and interstellar extinction value ( A G = ) for this star are available in the Gaia second Data Release ( DR2 ) ( Gaia Collaboration et al. 2018 ) , and the resultingdistance ( d parallax = ( ± ) pc ) is very consistent withthe results ( d = ( ± ) pc ) derived from the period – luminosity relationship by Poretti et al. ( ) .
5. Summary
We analyzed the pulsations of
Kepler target KIC 10284901,and extracted 151 signi fi cant frequencies from the SC data.Among them, seven independent frequencies were found in theSC spectrum. The period ratio ( = ) of the fi rst overtone ( F1 ) and fundamental mode ( F0 ) suggests that this star is adouble-mode HADS star. The derived absolute visual magni-tude M V = ± T eff = ±
180 K obtained from GTC spectra,indicate that KIC 10284901 lies in the bottom of the HADSinstability region and is likely a main-sequence star. With theprecise parallax provided by
Gaia , the distance of KIC10284901 is derived as 3064 ( ± ) pc.KIC 10284901 is the fi rst double-mode HADS star in whichthe quintuplet structures around the main pulsation modes weredetected in the frequency spectra. The quintuplet structures arecaused by two modulation frequencies, f m = − and f m = − . The temporal behavior of the amplitude ofthe two dominant modes reveals that the amplitudes vary withthe same trend, and the phase of F0 also shows obviously cyclicchange. The features of the frequency patterns, the ratio ( f m / f m ≈ ) of the two modulation frequencies, and theobviously cyclic variations of amplitude of the two dominantpulsation modes, suggest that the modulation in this star mightbe related to the Blazhko effect. Nonetheless, the possibility thatthis modulation just belongs to HADS stars cannot be ruled outcompletely, which is worth further study. More HADS stars withthis modulation are needed to verify its nature and investigate therelationship between HADS stars and RR Lyrae stars, andfurther investigations of this modulation could provide a newperspective on the classical instability strip in the H – R diagram.The authors thank the referee for the very helpful commentsand the editor for the careful revision of the manuscript. Thisresearch is supported by the program of the National Natural
Figure 4.
Temporal behavior of the amplitude and phase variations of the two radial pulsation modes in KIC 10284901. The top and middle panels show theamplitude variations of F0 ( in black ) and F1 ( in red ) , respectively. The bottom panels show the phase variations of F0 ( in black ) and F1 ( in red ) , respectively. Forclarity, we only show the data from BJD 2456170-2456205 days ( left panels ) , and BJD 2456355-2456400 days ( right panels ) . Each point contains a non-overlap0.5 day subset derived from the difference of the original light curve and the synthetic data. ESA
Gaia
Archive: http: // gea.esac.esa.int / archive / . The Astrophysical Journal, ( ))
Archive: http: // gea.esac.esa.int / archive / . The Astrophysical Journal, ( )) , 2019 July 1 Yang & Esamdin cience Foundation of China ( grant No.11873081 ) . We thankthe Kepler science team for providing such excellent data.
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