A double-modulation effect detected in a double-mode high-amplitude $\delta$ Scuti star: KIC 10284901

In this paper, we present an analysis of the pulsating behaviour of $Kepler$ target KIC 10284901. The Fourier transform of the high-precision light curve reveals 7 independent frequencies for its light variations. Among them, the first two frequencies are main pulsation modes: F0 = 18.994054(1) $\rm{d^{-1}}$ and F1 = 24.335804(4) $\rm{d^{-1}}$, the ratio of F0/F1 = 0.7805 classify this star as a double-mode high-amplitude $\delta$ Scuti (HADS) star; another two frequencies $f_{m1}$ = 0.4407 d$^{-1}$ and $f_{m2}$ = 0.8125 d$^{-1}$ are detected directly, and the modulations of $f_{m1}$ and $f_{m2}$ to F0 and F1 modes (seen as quintuplet structures centred on these two modes in the frequency spectrum) are also found. This is the first detection of a double-modulation effect in the HADS stars. The features of the frequency pattern and the ratio ($f_{m1}$/$f_{m2}$ $\approx$ 1:2) as well as the cyclic variation of amplitude of the two dominate pulsation modes, which seem to be similar to that in Blazhko RR Lyrae stars, indicate this modulation might be related to the Blazhko effect. The preliminary analysis suggests that KIC 10284901 is in the bottom of the HADS instability strip and situated in the main sequence.


Introduction
The Kepler Space Telescope is designed to search for terrestrial planets orbiting the solartype stars by the transits method Koch et al. 2010). As a supporting program, the Kepler asteroseismolog program possesses an intrinsic important role in the core planet search project (Gilliland et al. 2010). Owning to the ultra-high photometric precision data at the level of µmag, the Kepler mission has significantly improved our understanding of different types of variable star (e.g. Bedding et al. 2011;Giammichele et al. 2018); and the continuous observations -2spanning about 4 yr provide an excellent opportunity to monitor the amplitude modulation 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 of pulsation modes caused by different reasons, e.g. beating, mode coupling and rotation (e.g. Bowman & Kurtz 2014;Bowman et al. 2016;Yang et al. 2018b). These targets are excellent for asteroseismic study as they could increase our knowledge of the stellar structure and evolution for stars.
High amplitude δ Sct (HADS) stars are usually considered as a subclass of δ Sct stars with amplitude of peak-to-peak light variations larger than 0.3 mag (Breger 2000). From the relationship between the amplitude and the measured rotational velocity (vsini) provided by Breger (2000), the HADS stars are typical slow rotator with vsini ≤ 30 km s −1 . They seem to concentrate in the center part of the δ Sct instability region, and occupy a relatively narrow strip with a width of 300 K in temperature (McNamara 2000). With ground-based observation, the HADS stars are usually pulsating with only one or two radial modes (e.g. YZ Boo: Yang et al. 2018a;KIC 5950759: Yang et al. 2018b, etc). In recent decades, some stars also appear to show the non-radial modes with low amplitude owing to the extensive photometric campaigns. With the advent of space missions, especially the Kepler mission, more interesting phenomenon are discovered, including the low-amplitude pulsation modes and the long-term variations in pulsating stars (Balona et al. 2012;Bowman & Kurtz 2014). Several stars observed by Kepler space telescope show triplet or quintuplet structures in their frequency spectra Benkó et al. 2014), for instance, in HADS star KIC 5950759, a pair of triplet structure centred on the main frequency was detected in its frequency spectra, and the cause of the triplet structure is considered as the amplitude modulation of stellar rotation of 0.3193 d −1 (vsini ≈ 33 km s −1 ) (Yang et al. 2018b). These low-amplitude multiplet structures might increase our knowledge of the HADS stars and offer new clues to probe into the stellar interior and the physical processes.
KIC 10284901 (α 2000 =19 h 43 m 46 s .4, δ 2000 =+47 • 20 ′ 32 ′′ .8, 2MASS: J19434637+4720323) was found to be a δ Scuti star in the RApid Temporal Survey of the Kepler field (RATS-Kepler) by Ramsay et al. (2014). In that survey, Ramsay et al. (2014) reported KIC 10284901 was a mid-late A type star and it might be a HADS star due to its high amplitude light variations. Some basic properties of this star from that survey and Kepler Input Catalog (KIC; Brown et al. 2011) are listed in Table 1. This star was also selected as a target in the Kepler Guest Observer program and continuously observed for more than ten months in both long cadence (LC) with 29.4 minute effective integrations and short cadence (SC) with 58.8 s effective integrations (Gilliland et al. 2010). Due to the strong effect of signal averaging in LC data, we only use the SC data in this work. The unique and high precision photometric data make KIC 10284901 unusual to investigate its pulsation behaviour.  (Kjeldsen et al. 2010) provide the short cadence photometric flux data of KIC 10284901 in two types: one is the 'raw' data which in fact has been reduced by the NASA Kepler Science pipeline, and the other is the flux data corrected by KASOC Working Group 4 (WG#4: δ Scuti targets). We use the corrected flux and perform corrections eliminating outliers, as well as the possible linear trends in some quarters. The flux data are converted to magnitude scale, then the mean value of each quarter is subtracted, and the rectified time series is obtained. A portion of the rectified SC light curve is shown in Figure 1. It is clear that the light amplitude of KIC 10284901 is larger than 0.3 mag in SC data. Inspection of the light curve indicate that the amplitude has a repeat with about 2.5 day.

FREQUENCY ANALYSIS
We performed Fourier transform for the rectified SC light curve using PERIOD04 (Lenz & Breger 2005). The light curve is fitted with the following formula, where m 0 denotes the zero-point, A i is the amplitude, f i is the frequency, and φ i is the corresponding phase.
To detect more significant frequencies, a frequency range of 0 < ν < 50 d −1 , which covers the typical pulsation frequency of the δ Sct stars, was chosen in this work. In the extraction of significant frequencies, the highest peak in the frequency spectrum was considered as a significant frequency, then a multi-frequency least-squares fit using formula 1 was conducted to the light curve with all the significant frequencies detected, resulting to the solutions of all the significant frequencies. Next, all the frequencies of combination signals are fixed to the exact values they are supposed to be, and only leave the independent frequencies, all amplitude and phase as free parameters to be improved. A constructed light curve using the above solutions was subtracted from the data, and the residual was obtained to search for significant term in next step. The above steps were repeated until there was no significant peak in the residual. The criterion (S/N > 4.0) suggested by Breger et al. (1993) was adopted to judge the significant peaks. The uncertainties of frequencies were calculated following Montgomery & Odonoghue (1999).
A total of 151 significant frequencies were extracted in this work, and a full list of the extracted frequencies ( f S1 to f S151 ) with their corresponding amplitude, and identifications, is given in Table  2. After pre-whitening of the 151 frequencies, the amplitude spectrum of the residual is shown in Figure 2. No peak is statistically significant in the residual and the overall distribution of the residual is typical of noise.
Seven high-amplitude independent frequencies were detected, five of which were found in the range of 18-25 d −1 , and the other two (i.e. f S3 and f S4 ) were in 0-1 d −1 . Among these independent frequencies, f S1 and f S2 give a ratio of 0.7805, which is in the typical period ratio range of the first overtone and the fundamental mode for the double-mode HADS star. If the highest frequency f S1 is assumed to be the fundamental mode, then KIC 10284901 is classified as a double mode HADS star. We thus marked the frequencies f S1 and f S2 with 'F0' and 'F1' in the last column of Table 2, respectively. Petersen & Christensen-Dalsgaard (1996) presented a diagram in detail for doublemode HADS stars of different metallicities in their fig.3. Their study showed that higher values of the P(F1)/P(F0) ratio are found for mental-poor stars. For KIC 10284901, the higher period ratio (0.7805) indicate that it may belong to Population II stars.
f S3 and f S4 are interesting as they are out of the typical frequency range of the δ Scuti stars, and they are not combinations of F0 and F1. Considering a repeating with about 2.5 d in the light curve, the frequency f S3 was marked with f m1 , and f S4 with f m2 in Table 2, respectively. Three other frequencies, i.e. f S5 , f S6 and f S7 were also considered as independent frequencies, as they are neither any combinations nor harmonics of other frequencies, we marked these three frequencies with 'independent' in the last column of Table 2. Another eight frequencies (marked with 'T1' and 'T2' in the last column of Table 2) can be divided into two groups: one group consists of the frequencies marked with 'T1', which are combinations of f m1 and the main frequencies (i.e. F0 and F1); the other group is composed of the frequencies marked with 'T2', they are combinations of f m2 and the main frequencies.      Figure 3; and f S12 = 24.7766 d −1 and f S13 = 23.8952 d −1 centred on F1 with interval of f m1 , f S14 = 25.1483 d −1 and f S15 = 23.5233 d −1 centred on F1 with interval of f m2 , see right panel in Figure 3) are the most interesting features in the frequency spectrum of KIC 10284901. Some other peaks visible that are not labeled in Figure 3 are also listed in Table 2, e.g. f S5 and f S67 are the peaks to the left of f S8 , three peaks f S6 , f S37 and f S123 are to the left of f S12 .
The side peaks around F0 and F1 in the frequency spectra of KIC 10284901 form four pairs of uniformly-spaced triplets with interval of f m1 = 0.4407 d −1 and f m2 = 0.8125 d −1 . To investigate these side peaks, we firstly checked the possibility that these triplets were from the instrumental effects of Kepler space telescope, as performed by Yang et al. (2018b), and found that none of the known frequencies of instrumental effects from Kepler space telescope was equal to f m1 or f m2 . Hence, the side peaks detected in the frequency spectra of this star are not caused by the known instrumental effects.

The modulations of F0 and F1
Kepler mission has found that the multiplet structures can be shown in 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 space telescope, Breger et al. (2011) noticed that in several stars the equally-spaced frequency components were presented in their frequency spectra and these multiplet structures were considered to be from the stellar intrinsic variations. With asteroseismology, these equally-spaced frequencies might provide more information about the global properties. Hence, it is astrophysically interesting to explore the nature of the triplet structures in the frequency spectra of KIC 10284901.

New radial modes or nonradial modes?
For the equidistant or nearly equidistant frequency triplets shown in the spectra of the pulsating stars, Breger & Kolenberg (2006) proposed an explanation called as "The Combination Mode Hypothesis". In this scenario, the frequency f S8 (=19.43478 d −1 ) detected in SC spectrum of KIC 10284901 can be regarded as the third mode (F0 and F1 are the fundamental and the first over-tone mode, respectively), and f S9 (=18.55332 d −1 ) can be considered as the combination of 2F0 and f S8 , i.e. f S9 = 2F0 − f S8 . Some other frequencies are therefore the combinations of F0, F1, and f S8 . Similarly, f S11 (=18.18152 d −1 ) can be regard as the fourth mode and some frequencies are also combinations of F0, F1, and f S11 . Thus, KIC 10284901 might be a multi-mode radial variables. However, the ratios of F0/ f S8 (=0.9773) and F0/ f S11 (=1.0447) are far away from the typical ratio for the P 2O /P F (0.611 ∼ 0.632) and P 3O /P F (0.500 ∼ 0.525)(P F is the period of the fundamental radial mode, P 2O and P 3O are periods of the second and third radial over-tone, respectively) (Stellingwerf 1979). It seems to rule out the possibility that these two frequencies belong to the radial modes.
If f S8 or f S11 is assumed to be a nonradial mode, the equidistant triplets structure formed by the nonradial modes can occur only when the stars rotate with an extremely low velocity. Even the stars rotate only about a few km s −1 , the rotationally split multiplet will not be symmetric and the equidistant structure in the spectrum will also not be seen (Pamyatnykh 2000). Consequently, f S8 and f S11 are unlikely nonradial modes of KIC 10284901. The side peaks might be caused by some modulation effects.

Amplitude modulation with rotation?
Rotation plays an important role in stellar evolution and pulsations (Pamyatnykh 2000). Several δ Scuti stars found in the Kepler mission show equidistant multiplets in their frequency spectra which caused by modulation of the amplitude with stellar rotation (Breger et al. 2011). A low-amplitude modulation frequency of 0.16 d −1 to the dominant frequencies was detected in KIC 9700322 and it was confirmed to be the stellar rotation frequency by the high-dispersion spectral 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 curve of this star shows apparent amplitude modulation, similar to that in KIC 10284901. The Fourier transform of its light curve also reveals a high number of combination frequencies of the independent pulsation modes. Moreover, a quintuplet, which is assumed to be stellar rotationally split, is detected in the frequency spectra, and its separation is nearly equal with a mean separation of 0.218 d −1 (Murphy et al. 2013). However, it is different from that in KIC 10284901, as the quintuplet in this work include two exact interval, i.e. f m1 = 0.4407 d −1 and f m2 = 0.8125 d −1 . From this perspective, these two quintuplets may be due to different reasons. Yang et al. (2018b) reported that a pair of equidistant side peaks around the main frequency were detected in the frequency spectra in a HADS star KIC 5950759 and they were caused by the the amplitude modulation to its main pulsation with a stellar rotation of 0.3193 d −1 (vsini ≈ 33 km s −1 ). In the case of KIC 10284901, the possibility of the amplitude modulation with rotation was considered to explain the multiplet structures, however, it is hard to image how the stellar rotation, and such low rotation velocity commonly in HADS stars, could produce two different modulation frequencies, i.e. f m1 = 0.4407 d −1 and f m2 = 0.8125 d −1 . As a result, there is no enough evidence to support the hypothesis that these two modulations are from the stellar rotation.

Blazhko-like effect?
In Blazhko RR Lyrae stars, the equidistant triplets are often shown in the frequency spectra and the interval of the triplets is usually equal to the modulation frequency (Blažko 1907;Kolenberg et al. 2006;Soszyński et al. 2016). The modulation frequency can also be detected directly in the frequency spectra. Hurta et al. (2008) and Kolenberg et al. (2011) also found the equidistant quintuplets in the spectrum of RR Lyrae star. Jurcsik et al. (2014) presented a detailed analysis of four double-mode RR Lyrae stars showing the Blazhko effect based on new time-series photometry of the globular cluster M3. These four double-mode stars show large-amplitude Blazhko modulation of both radial modes and rapid phase change connected to the amplitude minimum of the respective mode. One of them, V13, shows a anti-correlation in both the amplitude-and phase-modulation of the modes. In KIC 10284901, the temporal behavior of the amplitudes and phases of the two radial pulsation modes were investigated based on the observational and synthetic data. We took the 151-frequency fit to the full data and then computed a synthetic light curve to the full data set using the papameters from this fit, but leaving out F0, F1, and the T1 and T2 groups. Then the difference of the original data and this fit was fitted by F0 and F1 only in a short pieces (the time interval of 0.5 day was chosen to obtain a much better time resolution). The amplitude and phase variations of the modes are shown in Figure 4. Contrary to the anti-correlation shown in V13 (see figure 2 in Jurcsik et al. 2014), the amplitudes of the two modes in KIC 10284901 seem to show obviously cyclic change and synchronous behavior. Fourier analysis of the amplitude variations of these two modes reveals two significant frequencies of 0.44071(1) d −1 and 0.81243(4) d −1 , which are equal to f m1 and f m2 . Just like its amplitude, the phase of mode F0 also shows obviously cyclic change, but the mode F1 does not possess the similar variation. Smolec et al. (2015) gave an analysis of Blazhko-type modulation in double-mode RR Lyrae stars in the Optical Gravitational Lensing Experiment photometry of the Galactic bulge, and found the amplitudes and phase of the radial modes varied irregularly on a long time-scale of a few hundred or thousand days, the same is true for the short-term modulation. In Figure 4, the variations of the amplitude of both the radial modes, which are commonly in the Blazhko RR Lyrae stars (e.g. CoRoT 105288363: Guggenberger et al. 2011 andV445 Lyr: Guggenberger et al. 2012), are obvious and regular. From this view, it seems that the modulation detected in KIC 10284901 may differ from that in Blazhko RR Lyrae stars. However, from Figure 2, the multiplet structures in the main pulsations in the resudial spectrum display the similarity to that in RR Lyrae star CoRoT 105288363 (Guggenberger et al. 2011), and the amplitudes of the modes of KIC 10284901 also show strong variations as shown in CoRoT 105288363. Benkó et al. (2014) reported three Blazhko RR Lyrae stars in the Kepler field (i.e. V355 Lyr, V 366 Lyr, V450 Lyr) show two modulation periods in their light curves, and the ratio between the primary and secondary modulation periods is nearly 1:2. In KIC 10284901, the SC light curve exhibits obvious modulation feature in its amplitude as commonly shown in the Blazhko RR Lyrae stars (e.g. Benkó et al. 2014). The multiplet structures found in SC spectrum, as shown in Figure  3, are similar to that in the Blazhko RR Lyrae stars. Moreover, the modulation frequencies ( f m1 and f m2 ), which were detected directly in SC data, have a ratio of nearly 1:2 just as shown in the above Blazhko RR Lyrae stars. These features shown in KIC 10284901 are similar to the frequency patterns in the Blazhko RR Lyrae stars.
Thus, for the target KIC 10284901, the features, including the multiplet structures in the frequency spectrum, the ratio of f m1 and f m2 (nearly 1:2), the obviously cyclic variations of the amplitude of the two dominate pulsation modes, are similar to the Blazhko effect in RR Lyrae stars. It seems to imply the modulation is related to the Blazhko effect. As the phase variation of the mode F1 seems to be not obvious, it is worth to notice that although the modulation detected in KIC 10284901 share some similarities with the Blazhko effect, it could also belong to a unique effect of HADS star and deserve to be investigated further. More HADS stars with this similar modulation effect are needed to solve this mystery, especially from space missions.

The location in H-R diagram
To investigate the evolutionary stage of KIC 10284901, we obtained M V = 2.84(±0.17) mag for this star using the period-luminosity relationship M V = −1.83(±0.08) − 3.65(±0.07) log P F (P F is the period of the fundamental mode) provided by Poretti et al. (2008) and its log P F = −1.279. Thus, in H-R diagram, KIC 10284901 is located in the bottom of the HADS instability strip and likely situated in the main sequence, under the constraints of the above M V = 2.84(±0.17) mag and T e f f = 7710(±180) K derived from GTC spectra in the RATS-Kepler by Ramsay et al. (2014). It is noted that a precise parallax (0.328±0.023 mas) and interstellar extinction value (A G = 0.21 mag) for this star is available in Gaia second Data Release (DR2) 2 (Gaia Collaboration et al. 2018a), and the resulting distance (d parallax = 3064 (± 215) pc) is very consistent with the results (d = 3002 (± 235) pc) derived from period-luminosity relationship by Poretti et al. (2008).

SUMMARY
We analysed the pulsations of Kepler target KIC 10284901, and extracted 151 significant frequencies from the SC data. Among them, seven independent frequencies are found in SC spectrum. The period ratio (=0.7805) of the first over-tone (F1) and fundamental mode (F0) suggests that this star is a double-mode HADS star. The derived absolute visual magnitude M V = 2.84 ±0.17 mag, as well as the effective temperature T e f f = 7710±180 K obtained from GTC spectra, indicate that KIC 10284901 lies in the bottom of the HADS instability region and is likely a main sequence star. With the precise parallax provided by Gaia, the distance of KIC 10284901 is derived as 3064 (± 215) pc.
KIC 10284901 is the first double-mode HADS star in which the quintuplet structures around the main pulsation modes were detected in the frequency spectra. The quintuplet structures are caused by two modulation frequencies, f m1 = 0.4407 d −1 and f m2 = 0.8125 d −1 . The temporal behavior of amplitude of the two dominant modes reveals the amplitudes vary in a same trend, the phase of F0 also shows obviously cyclic change. The features of the frequency patterns, the ratio ( f m1 / f m2 ≈ 1:2) of the two modulation frequencies, and the obviously cyclic variations of amplitude of the two dominate pulsation modes, provide the possibility that the modulation in this star might be related to the Blazhko effect. Nonetheless, the possibility that this modulation just belong to HADS stars can not rule out completely and it is still worth to study further. More HADS stars with this modulation are needed to verify its nature and investigate the relationship between HADS stars and RR Lyrae stars, and further investigations for this modulation could provide a new perspective to the classical instability strip in H-R digram. (in black) and F1 (in red) respectively. The bottom panels show the phase variations of F0 (in black) and F1 (in red) respectively. For clarity, we only show the data from BJD 2456170 -2456205 d (left panels), and BJD 2456355 -2456400 d (right panels). Each point contains a non-overlap 0.5-day subset derived from the difference of the original light curve and the synthetic data.