KIC 5768203: A δ Sct Pulsator Modulated by Rotation and Spots

Six frequencies with S/N > 5.0 are detected in KSCD and listed in Table 3. Among these frequencies, there are three independent frequencies and two harmonics. The two peaks f_h1 and f_h2 with the highest S/N in the KSCD spectrum are in the δ Sct frequency range, and their period ratio (P_h2/P_h1) is 0.7803. According to Stellingwerf (1979), the period ratio of the fundamental mode and first overtone of δ Scts with radial pulsations can be in a range of 0.756–0.787; thus, we infer the two high frequencies might be the radial fundamental mode and first overtone. However, further confirmation is need to verify this. The two frequencies are also detected in KLCD, TLCD, and TSCD, marked as f_P0 and f_P1 in Tables 4, 5, and 6.

In the top panel of Figure 1, it can be seen that the light curve of KIC 5768203 in KSCD is modulated by a periodic signal. The signal has a period about two days. There are three significant frequencies at the low frequencies of KIC 5768203. One of the significant frequencies is at 0.4576 day⁻¹ with the most highest amplitude in the spectrum window. This frequency (f_l2) may be originated from the periodic modulation. The other two frequencies are harmonics of f_l2, and we mark them as f_l1, f_l3 in Table 3. Retaining f_l1, f_l2, f_l3 and excluding other frequencies with S/N higher than 5, we fold the residual light curve of KIC 5768203 by using a period of 2.1853(5) days (1/f_l2) in KSCD. The phase folded light curve is binned every 0.002 phase and showed in Figure 3.

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One frequency (f_l6) adjacent to f_l2 is detected in KSCD, not in KLCD. The separation between f_l6 and f_l2 is 0.0506 day⁻¹. With f_res = 0.05005 day⁻¹ of KSCD, the frequency f_l6 is identified as an alias.

The observation time span of KLCD is much longer than that of KSCD, so more frequencies can be detected in KLCD than that in KSCD. Fifteen frequencies with S/N > 5.0 are detected in KLCD and listed in Table 4. Among these frequencies, there are three independent frequencies, two combinations, and four harmonics. The two frequencies f_h1 and f_h2 have the highest amplitudes in the KLCD spectrum. They have a period ratio of 0.7803 and are probably radial pulsations. The frequency of f_h5 in Table 4 is twice as much as f_h1, therefore it is a possible harmonic of f_h1. The frequency of f_h10 in Table 4 is equal to the sum of f_h1 and f_h2, and should be a combination. The frequencies f_h1, f_h3, f_h4, and f_h2, f_h6, f_h7 form two triplets. There is an interval of 0.00543(7) day⁻¹ between the peaks of each triplet. The interval is marked as shift1 in Table 4. The frequencies f_h1, f_h8, f_h9 form another triplet. There is an interval of 0.0108(1) day⁻¹ between each peak of the triplet, and the interval is marked as shift2 in Table 4. Shift1 is very close to twice the orbital frequency f_orb = 0.00268 day⁻¹ of the Kepler spacecraft in the error range. Shift2 is very close to the rotation frequency 0.011 day⁻¹ of the Kepler spacecraft in the error range. In frequency analysis of KIC 5768203, the PDC _ SAP flux is used. The PDC _ SAP flux has been corrected for the orbital and rotational effects of the Kepler spacecraft. No frequency or frequency interval of 0.00543(7) day⁻¹ has been found in other literature using the PDC _ SAP flux for frequency analysis. The shift1 of 0.00543(7) day⁻¹ for f_h1 and f_h2 may be attributed to the target itself. The shift2 is twice as much as shift1, and it is possibly caused by the same origin. This point is discussed in Section 4.1.

The frequency f_l2 that may be due to periodic modulation is detected with four harmonics in KLCD. The first harmonic f_l1 is the highest peak at the low frequency of KLCD. The double-humped wave of the phase diagram shown in Figure 3 may result in that the first harmonic hashigher amplitude than the frequency f_l2 has (Shi et al. 2021). The five related frequencies are marked as f_l1, f_l2, f_l3, f_l4, and f_l5 in Table 4.

To observe details of brightness variation caused by the periodic modulation, we filter out the high frequencies (pulsation frequencies) with S/N higher than 5 and adopt the residuals of KLCD to analyze the periodic modulation. A modulation period is calculated as 2.18279(5) days by using the frequency f_l2 in Table 4. We use the modulation period to fold the light curve of the residuals of KLCD, and show a phase evolution diagram over a time span of nearly four years in Figure 4. In the figure, each pixel presents one sampling point of the residuals of KLCD. Color, from blue to red, indicates the surface brightness of the star from bright to dark in each (time, phase) pixel, which is marked as the relative mag on the right Y-axis. For ease of visual clarity, the residuals of KLCD is folded with 2 times in phase. The time sequence is from left to right in the figure. The phase evolution diagram over a time span of nearly four years indicates the surface brightness of the star is uneven, and evolves gradually in time.

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In Figure 4, there are three wide vertical white gaps and each gap corresponds to a time span of over 70 days without data in KLCD. We divide the residuals of KLCD into four segments based on the three gaps, and the time span of each segment is 320 days, 263 days, 276 days, and 283 days in chronological order. In order to study variation of frequencies associated with the period modulation, these frequencies (f_l1, f_l2, f_l3, f_l4, f_l5) are reextracted from each segment separately, and are listed in Table 7. The amplitudes, phases, and S/N of these frequencies detected from each segment are different. The values of the modulation frequency (f_l2) of each segment differ slightly, but are nearly identical within the margin of error. Using the modulation frequency (f_l2) of each segment, we fold the light curve of each segment, separately. The phase folded curves and amplitude spectra of each segment display different characteristics, and are shown in Figure 5. This indicates that the surface brightness of KIC 5768203 changes with time.

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The two pulsation frequencies detected in the Kepler data are also found in TSCD and TLCD, and the results of each data set are the same within the error range. The two other frequencies (f_l2, f_l6) at low frequency are detected in TSCD. They are consistent with the modulation frequency and the alias in KSCD within twice the margin of error. The separation of f_l2 and f_l6 in TSCD is 0.045 day⁻¹, which is much larger than the frequency resolution of TSCD. However, the frequency resolution of TSCD is dubious as discussed in Section 2. The identification for f_l6 should be based on the result of KSCD. In TLCD, only the first harmonic of the periodic modulation is detected. The discovery of the modulation frequency in the TESS data infers that KIC 5768203 still has periodic motion and asymmetrical brightness.

By using KLCD to analyze the phase modulation of pulsations, Murphy et al. (2018) proposed that KIC 5768203 was a δ Sct pulsator in a binary system with an orbit period of 183.7(1) days. Subdividing KLCD into multiple 10 day segments as suggested by Murphy et al. (2018), we probe the phase variation of the two pulsation frequencies f_h1 and f_h2 in these segments. The result shows that phases of f_h1 and f_h2 have the same cyclic variation with an period of 184(2) days, as showed in Figure 6. In the amplitude spectrum of KLCD of KIC 5768203, there are two triplets with an interval of 0.00543(7) day⁻¹. If the interval is due to the orbital period of the binary system, then the orbital period should be 184(2) days. The orbit period is consistent with that derived by Murphy et al. (2018) in the error ranges. KIC 5768203 could be a δ Sct pulsator in a binary system.

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The light curve of KIC 5768203 is modulated by a periodic signal. Analyzing the amplitude spectrum of KIC 5768203, the frequency of the periodic signal and its several harmonics are found at the low frequency in KSCD and KLCD. As can be seen from the phase evolution diagram over a time span of nearly four years for the residuals of KLCD in Figure 4, the brightness of the star varies unevenly with phase and time after eliminating the effect of pulsations. The variation of the brightness is most likely caused by rotation and starspots. With a higher S/N in KLCD than KSCD, the frequency of f_l2 in KLCD is adopted as the rotation frequency. Several harmonics should be blamed on stellar spots. The rotation period is derived as 2.18279(5) days with the rotation frequency f_l2 in Table 4. With the rotation period and the radius R = 3.2(2) R_☉, the rotaion velocity of the star is estimated to be about 75(3) km s⁻¹. HADS usually pulsate in either one radial mode or two (or more) radial modes, along with a range of brightness variation larger than 300 mmag. KIC 5768203 has only two independent pulsation modes, which is similar to pulsation characteristics of HADS. However, the range of brightness of the light curve of KIC 5768203 is about 5 mmag, which is much smaller than the minimum brightness variation of 300 mmag of HADS. The relatively fast rotation speed of KIC 5768203 may be one of the reasons for its small pulsation amplitudes.

In addition to its own pulsations, the starspots on the surface are also one of the main causes of brightness variation for KIC 5768203. In four segments divided by three vertical white gaps, features of starspots can be seen from the phase evolution diagram in Figure 4 and the phase folded light curves Figure 5. The yellow bands in Figure 4 correspond to the dips in brightness. One band at Phase ≈ 0.1 (or 1.1) extends throughout the whole time span of the Kepler long-cadence data. It is due to one long-term starspot. There seem to be other evolving starspots between Phase ≈ 0.4 and 0.9 in Figure 4. The evolving spots are close in longitude.

In the first segment (segment1), two dips can be seen on the phase light curve in Figure 5, which are caused by one starspot at Phase ≈ 1.1 and another at Phase ≈ 0.5 in Figure 4. The main hump on the phase light curve represents the surface brightness of the star without starspots. The two dips have almost the same depth. The brightness of the two dips is lower about 0.4 mmag than that of the main hump, and lower about 0.25 mmag than that of the secondary hump. The starspot at Phase ≈ 1.1 is larger than the starspot at Phase ≈ 0.5. There are some tiny dark regions between the two starspots in Figure 4. It is inferred that these regions are covered by small diffuse starspots, which cause slight decreases in brightness of the star.

In the second segment (segment2), there are three dips with unequal depths on the phase curve in Figure 5. The deepest dip corresponds to the long-term starspot at Phase ≈ 0.1 (or 1.1) in Figure 4, and has a brightness difference of about 0.43 mmag with the main hump. One significant starspot appears at Phase ≈ 0.8 in Figure 4. The starspot at Phase ≈ 0.5 becomes less prominent in segment2 than in segment 1, and gradually merges with the starspot at Phase ≈ 0.8 over time. The starspots at Phase ≈ 0.5 and 0.8 in Figure 4 shoud be responsible for the two shallow dips on the phase curve in Figure 5, and the two shallow dipshave brightness differences of about 0.25 and 0.33 mmag with the main hump, respectively.

In the third segment (segment3), there are two dips on the phase light curve in Figure 5, which are attributed to the two starspots at Phase ≈ 0.1 and 0.7 in Figure 4. The two dips have almost equal depth, and their brightness is lower about 0.33 mmag than that of the main hump.

In the fourth segment (segment4), there are still two significant dips on the phase light curve in Figure 5. The dip corresponding to the starspot at Phase ≈ 0.1 has brightness lower about 0.25 mmag than the main hump. The other dip corresponding to the starspot at Phase ≈ 0.7 has brightness lower about 0.2 mmag than the main hump. The two starspots seem to be getting closer and closer and have a tendency to merge in Figure 4.

The dips on the phase light curve of each segment appear at different phases. This indicates that the starspot corresponding to the dip has shifted its position on the surface of the star. The brightness difference between the dip at Phase ≈ 0.1 and the main hump is estimated based on the phase light curve of each segment, and becomes smaller and smaller with time. It infers that the starspot at Phase ≈ 0.1 is in the process of gradually becoming diffuse or disappearing. During the observing time span of KLCD, the starspot at Phase ≈ 0.1 has always existed and will probably continue to exist for some time, and the starspot at Phase ≈ 0.5 has existed for about 550 days and then merges with the starspot at Phase ≈ 0.8 to form another starspot at Phase ≈ 0.7. The merged starspot at Phase ≈ 0.7 has existed for about 800 days, and might merger with the long-term starspot at Phase ≈ 0.1 over time.

In addition to the prominent starspots discussed above, there are other small starspots which are obscure in Figure 4 and Figure 5. These small starspots correspond to harmonics of the rotation frequency of low S/N at the amplitude spectrum of each segment. Each prominent 'starspot' in Figure 4 might be a cluster of many small starspots on the stellar surface. The detailed geometries of these starspots (or clusters of starspots) could not be determined with the Kepler data.

In Figure 4, some blue points corresponding to very bright small areas can be seen. These bright points only appear in the residual data after eliminating the effect of pulsation. Ruling out the possibility for sampling outliers, these points should be attributed to the target itself. 162 flares had been detected on the surface of KIC 5768203 with the Kepler long-cadence data in the literature of Davenport (2016). These bright points may be due to the stellar flares.

The discovery of the rotation frequency and first harmonic in the TESS data infers that there are still spots on the surface of KIC 5768203. It is the first time that a δ Sct pulsator is found to have starspots for a long time. The long-term existence of starspots may prove that KIC 5768203 has a relatively active magnetic field. A necessary condition forming the surface magnetic field is convection in the outer layer, which can generate the operation of the dynamic mechanism (Charbonneau 2014). KIC 5768203 with the effective temperature 6934 ± 134 K has a long-lived active magnetic field. It may be an example of possible convection on F stars, which is consistent with the latest studies about envelopes of A-F stars (Balona 2018, 2019b, 2021; Cantiello & Braithwaite 2019).