Meta-analysis of photometric and asteroseismic measurements
of stellar rotation periods: the Lomb-Scargle periodogram, autocorrelation
function, wavelet and rotational splitting analysis for 92 Kepler asteroseismic targets

We use the sample of stars selected by Kamiaka et al. (2018). The sample includes the Kepler dwarfs LEGACY sample which consists of 66 stars with the best asteroseismic data so far in solar-type category (see e.g. Lund et al., 2017; Aguirre et al., 2017), and additional 28 KOI (Kepler Object of Interest) with detectable oscillation amplitudes. The Kepler targets in the LEGACY sample have magnitudes around 9, which are in general brighter than those for the KOI sample (Kepler magnitude around 11). Thus, the LEGACY sample has spectra with higher height to background ratio and hence their periods can be estimated more precisely. The additional 28 KOI targets are useful to examine whether the stellar inclinations are correlated to the orbital axis of their transiting planets. Kamiaka et al. (2018) performed an asteroseismic analysis of this sample, and estimated their stellar rotation periods and inclinations. Figure 1 plots the location of our targets on their surface gravity and effective temperature plane. Stellar evolutionary tracks are computed with MESA code (Modules for Experiments in Stellar Astrophysics; Paxton et al. (2011)) for stars with initial mass of 0.8, 1.0, 1.2, 1.4, and 1.6 $M_{\odot}$ from their pre-main-sequence phase to white dwarf phase, assuming the solar metallicity.

For the photometric analysis, we use the Kepler long-cadence PDC light curve (PDC-msMAP, Stumpe et al., 2014), Data Release 25, from Mikulski Archive for Space Telescope (MAST). The data consist of quarters of $\sim 90$day duration with a cadence of $29.4$ minutes over four years. We choose the light curves for Q2-Q14 quarters.

KOI-268 (KIC 3425851) has one of the best-quality light curves in our sample and a precisely determined photometric period (see Figures 4 and 44 below). We first compute the duty cycle for KOI-268 in each quarter, and find that it varies between 81 and 96 percents, with an average $\sim$90 percents. Then, we remove those quarters for other stars if the duty cycle is less than $75\%$ of the reference value for KOI-268, in order to secure the statistical significance of each quarter. Furthermore, we require that the targets must have at least 7 quarters of available data, which removes 16 Cyg A (KIC 12069424) and 16 Cyg B (KIC 12069449) from our sample. Thus, the number of our final targets for subsequent analyses becomes 92.

We preprocess the light curves of the 92 stars before performing the photometric analysis, which is summarized in Figure 2.

In order to remove the systematic trend in the light curves, we first fit the original data $L_{\rm init}(t)$ in each quarter to a 4-th order polynomial function $p_{4}(t)$. Then, we obtain the detrended light curves in units of the fitted polynomials separately in each quarter: $L(t)=L_{\rm init}(t)/p_{4}(t)-1$ (Step 1 of Figure 2). In order to remove a long-term trend in a light curves, it is fairly common to fit using polynomials. We make sure that detrending with a third-order or fourth-order polynomials does not change the estimate of the rotation periods unless they are longer than $40$ days. Since the duration of each quarter is 90 days and the Kepler instrument performs pointing corrections every 30 days which are known to generate a recurrent noise around that period, periods longer than $40$ days are not reliable in any case. Therefore, we decide to use the fourth polynomials in detrending.

Next, we concatenate the detrended light curves in available quarters between Q2 and Q14 for each target star into a single array of time series while preserving gaps between the quarters (Step 2 of Figure 2). We set the starting time of this series to be $t=0$.

For several stars, the detrended light curves still exhibit large anomalous variations in some quarters. Figure 3 illustrates such an example for Kepler-1655 (KIC 4141376, KOI-280). The light curves in quarters Q4, Q8, and Q12 (plotted in red) show strange behavior that seems to be uncorrelated with nearby quarters. We suspect that they are due to unknown contamination or instrumental effects. Thus, we decided to remove those quarters if the median absolute flux variation in a particular quarter is three times larger than those of the neighboring quarters. We removed such anomalous quarters from six targets: Kepler-1655 (KIC 4141376, KOI-280), Kepler-100 (KIC 6521045), Kepler-409 (KIC 9955598, KOI-1925), KOI-72 (KIC 11904151), KIC 8694723, and KIC 10730618.

As shown in Figure 3, the detrended light curves also contain the dips due to the planetary transits (for KOI stars) and other possible outliers. We remove them according to the following procedure (Steps 3 and 4 of Figure 2).

In order to remove the periodic dips due to the planetary transit, we first fold the light curves of the KOI sample according to the orbital period of the planet candidates. Then we remove the data corresponding to the transit and occultation phases. The occulations might not be visible for all targets. For those targets we remove the expected occultations by assuming a circular orbit. The duration of transits and occultations is of the order of hours as opposed to the orbital period of the order of days. Thus, the amount of data removed is relatively negligible, and does not change the estimate of the stellar rotation period in practice. In addition, we remove the other outliers by $\sigma$-clipping with a threshold of $5\sigma$ and the center value being the median of concatenated light curve. The data removed in this step is less than 1%, and thus do not affect the period estimation either.

Incidentally, we are not able to detect the transit signal for KOI-5.02 and Kepler-37b after folding, and we do not remove any transit dips nor the expected eclipse dips. We suspect that KOI-5.02 is a false-positive, and Barclay et al. (2013) reported that the transit signal of Kepler-37b is less than 20 ppm. Thus possible transiting signals of these two targets would have negligible influence on the determination of stellar rotation period.

We note that the 90 day duration of each Kepler quarter effectively puts an upper limit of detectable periods around 45 days. Indeed, the rotation periods of main-sequence stars similar to our sample (Figure 1) are usually less than 50 days (e.g. Kamiaka et al., 2018). So, we apply our own box-car filter of a 50-day width even though the PDC-SAP light curves has applied smoothing to reduce long-term period systematics. Since the Kepler data is nearly evenly sampled, we map the light curve to a uniformly sampled grid at an interval of $\delta t=29.4$ minutes.

Finally, we pad all removed portions in the light curves (removed quarters, planetary transits, secondary eclipses, outliers, and missing data) by Gaussian noise of zero mean and the variance of $\sigma_{\rm noise}$ that is computed from the detrended light curve for each target (Step 5 of Figure 2). We also tried the zero padding for comparison, and made sure that the two padding schemes hardly change the estimated values of the photometric rotation period, implying that the padding does not affect the result in reality.

A significant fraction of stars are in multiple systems. The multiplicity may lead to a systematic bias on the rotation period estimated from the photometric variation of the target star. Thus, we try to identify possible members in binary and multiple-star systems in our sample.

For that purpose, we primarily adopt Renormalized Unit Weight Error (RUWE) value from Gaia DR3, which measures the goodness of fit with respect to a single-star model derived from the Gaia astrometry (Fabricius et al., 2021). A star with RUWE exceeding 1.4 is commonly suspected to have an unresolved companion(Evans, 2018; Wolniewicz et al., 2021). We list 14 target stars with RUWE $\geq 1.4$ from the Gaia DR3 catalog as possible candidates of binary/multiple-star systems.

In addition, we identify three eclipsing binaries from Kepler Eclipsing Binary Catalogue (Kepler Eclipsing Binary working group, 2019), two binary systems from NASA Exoplanet Archive (Kepler Mission, 2019), four multiple-star systems and two binaries from Table 4 of García et al. (2014), one binary from Eylen et al. (2014), and one binary from Lillo-Box et al. (2014). Among the 13 systems, five stars have RUWE $\geq 1.64$, while the other eight stars have RUWE $\geq 1.05$.

In total, we identify 22 stars with possible companions, and summarize them in Table 2. We compute their photometric rotation periods in the same manner as other 70 stars with no reported stellar companion (Appendix A), but they may be interpreted with caution. We separately discuss seven KOI stars out of them in §7.2 since they may be potentially important targets for planetary studies. In what follows, however, we mainly focus on the other 70 stars; their estimated rotation periods are summarized in Tables 3, 4, 5 and 6.

The rotation period estimated using the LS method for the $q$-th quarter $P_{{\rm LS},q}$ may be different from the period $P_{\rm LS}$ computed for the entire observed duration $T_{0}$. Distribution of starspots at different quarters should change with time due to the creation/annihilation processes over a range of latitudes. Thus, combined with their possible latitudinal differential rotation, each quarter may exhibit the fractional variation of the period on the order of ten percent.

If the typical lifetime of starspots is less than $90$ days, their pattern on the surface at different quarters can be regarded as an uncorrelated realization drawn from the same statistical distribution. Even if the lifetime of starspots has a broad distribution, the variance of $P_{{\rm LS},q}$ among different quarters would reflect the differential rotation for stars. This possibility has been theoretically studied by Suto et al. (2022) using analytic model and mock observations. It is also possible that the large variance is simply due to the low signal-to-noise ratio (e.g., due to a very low number of spots or to the fact that the star is very faint) of the light curve for some quarters. These two possibilities could be distinguished by comparing the LS result with those based on ACF, WA, and asteroseismology on an individual basis (see §3.2 below).

In order to quantify the fractional variance of the rotation periods among different quarters, we introduce the following parameter $\Delta_{\rm LS}$ for each star:

where $N_{Q}$ is the number of quarters with measured $P_{{\rm LS},q}$, and $\langle\cdots\rangle$ denotes the median value operator.

Nielsen et al. (2013) proposed a similar indicator called MAD (Median Absolute Deviation) $\langle\left|P_{{\rm LS},q}-\langle P_{{\rm LS},q}\rangle\right|\rangle$. Our indicator, equation (1), differs from MAD in the following two ways. We use the fractional variance, which would be more appropriate in classifying stars with a broad range of rotation periods. In addition, we average the deviation over all the quarters because the median of deviation sometimes underestimates the real intrinsic variation of the distribution. This will be discussed below individually for each target.

After trial-and-errors, we classify the stars into two groups: Group A with $\Delta_{\rm LS}\leq 0.2$ and Group B with $\Delta_{\rm LS}>0.2$. To select the threshold, we first plot the rotation period against $\Delta_{\rm LS}$, and find that a group of targets with $P_{\rm LS}>$ 30 days and significantly large error-bars start to occur from $\Delta_{\rm LS}\sim 0.2$. Thus, we decide to set a threshold of $\Delta_{\rm LS}$= 0.2 for a reliable group just for convenience. There are 23 stars in Group A and 47 stars in Group B, out of the 70 stars with no reported stellar companion; see §2.3. Figure 4 presents results of our photometric analysis for the reference star KOI-268 (KIC 3425851; see section 2) that is classified in Group A with $\Delta_{\rm LS}=0.014$, as well as for KIC 6116048 (Group B with $\Delta_{\rm LS}=0.372$).

Group A contains targets with a relatively robust periodic component in their light curves, and their $P_{\rm LS}$ values are reliably estimated from the prominent single peak as shown in the left panels of Figure 4.

In contrast, Group B, with $\Delta_{\rm LS}>0.2$, consists of two different types in general. The first type shows either a variation of measured periods among different quarters or multiple co-existing periodic signals (see e.g., the right panels of Figure 4). Possible scenarios for the latter include differential rotation on stellar surface, harmonics (see §A.1.1), unresolved companions, residuals of a systematic trend, and other unknown contaminations. The second type shows very weak periodic signals, whose estimated periods are less reliable and subject to significant uncertainties.

Note that the large variation of period measurements for the first type of targets in Group B may not necessarily indicate that their rotation periods are unreliable, but could be due to the physical variation on stellar surface such as the differential rotation.

We also classify our targets based on asteroseismically estimated periods by Kamiaka et al. (2018), in a complementary manner to the photometric classification. For that purpose, we adopt the ratio $\Delta P_{\rm astero}/P_{\rm astero}$, where $\Delta P_{\rm astero}$ is the sum of upper and lower error of period estimation, i.e., $\Delta P_{\rm astero}=(\Delta P_{\rm astero})_{+}+(\Delta P_{\rm astero})_{-}$; see equations (A7) and (A8) in §A.2.

We set the threshold value of $0.4$ after trial-and-errors, and introduce two groups; Group a with $\Delta P_{\rm astero}/P_{\rm astero}\leq 0.4$ has 33 stars, while Group b with $\Delta P_{\rm astero}/P_{\rm astero}>0.4$ has 37 stars.

Combining the photometric and asteroseismic classification yields four groups labeled as Aa, Ab, Ba, and Bb, which are summarized in Table 1. Figure 5 plots the locations of 70 stars on the $\Delta P_{\rm astero}/P_{\rm astero}$ – $\Delta_{\rm LS}$ plane, and the left and right panels of Figure 6 show the correlations between $\langle P_{{\rm LS},q}\rangle$ and $\Delta_{\rm LS}$, and between $P_{\rm astero}$ and $\Delta P_{\rm astero}/P_{\rm astero}$, respectively.

As Figure 7 clearly indicates, KOI stars in Group Aa show good agreement among $\langle P_{{\rm LS},q}\rangle$, $P_{\rm LS}$, $P_{\rm ACF}$, $P_{\rm WA}$, and $P_{\rm astero}$. Indeed, the major part of the scatter around $P_{\rm LS,q}$ for those stars is dominated by a small number of quarters with low signal-to-noise ratios.

The only exception is Kepler-100 (KIC 6521045, KOI-41) with $\Delta_{\rm LS}=0.18$ that is close to our threshold value of 0.2. Its photometrically estimated periods ($\langle P_{{\rm LS},q}\rangle$, $P_{\rm LS}$, $P_{\rm ACF}$, and $P_{\rm WA}$) consistently point to $\approx 13$ days, while $P_{\rm astero}\approx 25$ days. Intriguingly, McQuillan et al. (2013) and Ceillier et al. (2016) concluded $P_{\rm ACF}\approx 25$ days. Thus, we conduct a further examination of Kepler-100.

The LS periodograms of Kepler-100 for different quarters (Figure 31) shows apparent variations in measured period. While it is not easy to define a robust period, six quarters (Q2, Q4, Q7, Q8, Q9, Q11) out the 12 quarters (except Q6) have the highest peaks around $13$ days. Our analysis excludes Q6 because its median absolute flux variation is more than three times of those of the neighboring quarters (§2.2). Nevertheless, the quarter has a very strong peak at $P_{{\rm LS},q}\approx 25$ days. We found that our own $P_{\rm ACF}$ becomes $\approx 25$ days if we include the Q6 data, which recovers the result of McQuillan et al. (2013) and Ceillier et al. (2016).

We refer to spectroscopic measurement of rotational velocity. According to Table 1 of Marcy et al. (2014), Kepler-100 have $v\sin i_{*}=3.7{\rm kms}^{-1}$ and $R=1.49\pm 0.04R_{\odot}$. Thus equation (3) implies that $P_{\rm spec}\approx 20.4\sin i_{*}$ days. A value of $\sin i_{*}\sim 0.6$ is required to agree with $\langle P_{{\rm LS},q}\rangle\sim 13$days, but it is inconsistent with the asteroseismic result of $\sin i_{*}\approx 1$ and $P_{\rm astero}\approx 26.4^{+1.2}_{-1.7}$ days ($\delta\nu_{*}=0.44^{+0.03}_{-0.02}\,\mu$Hz; see Figure 31).

We note that the orbital periods of its three transiting planets (Marcy et al., 2014) are $6.9$ days (Kepler-100 b), $12.8$ days (Kepler-100 c), and $35.3$ days (Kepler-100 d), one of which are close to our measured rotation period ($\sim$13 days). We visual inspect the light curves to check whether our photometric estimate is somehow contaminated by the removal of the transit signal of Kepler-100 b. As Figure 8 suggests, however, the lightcurves exhibit clear periodic oscillations that are not aligned with the location of the transit dip. Therefore, it is unlikely to be an artifact of the transit signal removal. We also confirm that different padding schemes (with and without noise) give consistent measurements of the rotation period.

A possible explanation is that the 13-day period is the second harmonics of 25 days. The variation of the period in different quarters may suggest that Kepler-100 has multiple large active spots over the surface. Indeed, our visual inspection of light curve (Figure 8) shows a pattern of a deeper trough accompanied by a shallower trough. This pattern may be a signature of multiple spots, which leads to the presence of the harmonics (peak at $P_{\rm rot}$/2) in periodogram. Without visual inspection, the automatic period detection method may the period associated with harmonics as the actual rotation period. In any case, we conclude that Kepler-100 remains as one of the intriguing systems that deserve for further investigation.

Photometric rotation periods $\langle P_{{\rm LS},q}\rangle$, $P_{\rm LS}$, $P_{\rm ACF}$, and $P_{\rm WA}$ agree within error-bars for all the non-KOI targets in Figure 9. In contrast, $P_{\rm astero}\approx 6.5$ days of KIC 5773345 is a factor of two smaller than the photometrically derived values of $12$ days. Thus, we examine KIC 5773345 in detail below.

Figure 32 indicate that KIC 5773345 has persistent periodic signals of P$\sim 12$ days over the entire observed span (except for the removed quarters), while there is no significant signal around $6.5$ days. Asteroseismic constraints on $i_{*}$ and $\delta\nu_{*}$ for KIC 5773345 are shown in Figure 33.

Another complementary constraint on the rotation period $P_{\rm rot}$ is provided from the spectroscopic data. For KIC 5773345, Bruntt et al. (2012) and Molenda-Żakowicz et al. (2013) obtained $v\sin i_{*}=6.6\pm 1.46$ km/s and $v\sin i_{*}=3.4\pm 1.1$ km/s, respectively. The factor of two difference likely comes from the difficulty of modeling the turbulence as pointed out by Kamiaka et al. (2018); see their Figure 8. Adopting $R=2.00^{+0.05}_{-0.07}R_{\odot}$ (Chaplin et al., 2013), equation (3) gives $P_{\rm spec}/\sin i_{*}\approx 15.2^{+4.3}_{-2.8}$ days and $29.5^{+14.1}_{-7.2}$ days, respectively.

Our photometric estimate of $P_{\rm rot}\sim 12$ days becomes consistent with $P_{\rm spec}/\sin i_{*}$ of Bruntt et al. (2012) and Molenda-Żakowicz et al. (2013) if $\sin i_{*}\sim 0.8$ and $\sin i_{*}\sim 0.4$, respectively. Incidentally, it is known that the asteroseismology constrains the parameter $\delta\nu_{*}\sin i_{*}$ primarily, instead of $\delta\nu_{*}$ or $\sin i_{*}$ (Kamiaka et al., 2018). Substituting $\delta\nu_{*}\sin i_{*}=0.93^{+0.1}_{-0.09}\,\mu$Hz (Figure 33) into equation (4) implies $P_{\rm astero}/\sin i_{*}=12.5^{+1.3}_{-1.2}$ days, which agrees with the result of Bruntt et al. (2012) within their 1$\sigma$ uncertainty while inconsistent with that of Molenda-Żakowicz et al. (2013).

Therefore, the discrepancy between our photometric and asteroseismic periods for KIC 5773345 may come from a intrinsic problem of asteroseismic analysis in separating $\delta\nu_{*}\sin i_{*}$ into $\delta\nu_{*}(=1/P_{\rm astero})$ and $\sin i_{*}$ in asteroseismic analysis. In such case, the photometric estimate of $P\sim 12$ is likely the true rotation period. Nevertheless, it may be premature to conclude at this point, and KIC 5773345 should be regarded as another interesting target for follow-up observation.

In general, stars classified as Group Ab have consistent estimates for the photometric rotation period as shown in Figure 10. The asteroseismic measurements for group b targets have large uncertainties so that targets do not show statistically significant discrepancy between photometric and asteroseismic periods, except for Kepler-409 (KIC 9955598, KOI-1925) and KIC 7970740.

For Kepler-409, photometric results consistently point to $P_{\rm rot}\sim 13$ days, while $P_{\rm astero}\sim 29$ days (Figure 40). We also refer to spectroscopic estimations for comparison. Equation (3) and (4) suggest that $P_{\rm spec}\approx 22.3\sin i_{*}$ days if $v\sin i_{*}=2$ km/s (Marcy et al., 2014) while $P_{\rm astero}\approx 37.4^{+4.0}_{-3.3}\sin i_{*}$ days for $\delta\nu_{*}\sin i_{*}=0.31\pm 0.03\,\mu$Hz (Figure 41).

For KIC 7970740, $P_{{\rm LS},q}$ varies from 9 to 18 days (Figure 42), while $P_{\rm astero}\sim 41.1^{+5.0}_{-4.0}\sin i_{*}$ days for $\delta\nu_{*}=0.28\pm 0.03\,\mu$Hz (Figure 43).

Another interesting target is KIC 3425851 (KOI-268) for which the photometric period is precisely measured (Figure 44). If we perform the asteroseismic analysis with the prior for $\delta\nu_{*}$ from photometric estimate, one may be able to constrain the stellar inclination more tightly; see Figure 45.

There are three planet-hosting stars in Group Ba (Figure 12). Kepler-25 (KIC 4349452, KOI-244) is one of the two transiting planetary systems for which the real spin-orbit misalignment angle has been estimated jointly from the Rossiter-McLaughlin effect and asteroseismology (Benomar et al., 2014; Campante et al., 2016).

Figure 34 indicates that the photometric rotation period for Kepler-25 varies significantly from quarter to quarter and not reliable. On the other hand, the asteroseismic rotation period seems to be very precisely estimated; $P_{\rm astero}\approx 7.9^{+0.5}_{-0.4}$ days. Adopting $v\sin i_{*}\approx 9.5$km/s and $R=(1.31\pm 0.02)R_{\odot}$ (Marcy et al., 2014), $P_{\rm spec}\approx 6.9\sin i_{*}$ which is in rough agreement with $P_{\rm astero}$.

Kepler-93 (KIC 3544595, KOI-69) also exhibits large variation in $P_{{\rm LS},q}$ (Figure 36). Referring to spectroscopic measurement from (Marcy et al., 2014), we obtain $P_{\rm spec}\approx 92\sin i_{*}$ days with $v\sin i_{*}\approx 0.5$km/s and $R=(0.92\pm 0.02)R_{\odot}$. However, such a small value of $v\sin i_{*}$ is not so reliable as it is subject to a large uncertainty in turbulence modeling. On the contrary, asteroseismic analysis provides a relatively precise estimation of $P_{\rm astero}\approx 23.3^{+0.5}_{-0.4}$ days (Figure 37).

Finally, we note that Kepler-128 (KOI-274, KIC 8077137) is classified as Group B because of three apparent outliers, which are $P_{{\rm LS},q}\approx 4$ days for Q11 and Q14 and $P_{{\rm LS},q}\approx 32$ days for Q2. Except for the three quarters, $P_{{\rm LS},q}\approx(13-14)$ days persistently, and indeed agrees well with $P_{\rm astero}=12.3\pm 1.3$ days. Thus, Kepler-128 had better be classified as Group A in reality.

Sixteen non-KOI stars in Group Ba exhibit a diverse distribution of periodicities in the LS periodogram at different quarters. Most of them have multiple peaks whose relative heights vary among different quarters so that it is not easy to determine the photometric rotation period robustly. We show two examples (KIC 6225718 and 9965715) in Figures 46 to 48. As examined in §4.4, $P_{\rm astero}$ is likely to provide better period estimate for stars in Group Ba.

Finally, we show the results for KOI and non-KOI stars in Group Bb (Figures 14 and 15), without discussing them individually.

When multiple periodic components exist in the lightcurve, photometric analyses are not enough to discern the true rotation period. In such case, three different analyses assign different relative power to candidate peaks in spectra which occasionally leads to inconsistencies in period detected. For instance, the GWPS is known to increase the power of the fundamental period of a signal (Ceillier et al., 2017), and the ACF method uses a Gaussian smoothing to suppress the short-term modulation including the possible influence of harmonics. Thus both of them may be biased toward a longer period. Here, we provide our interpretation for 5 stars that exhibit multiple periodic components and have different estimates among $P_{\rm LS}$, $P_{\rm ACF}$, and $P_{\rm WA}$.

• •

  KIC 11081729 and 11253226: their $P_{\rm LS}$ is much shorter than $P_{\rm ACF}$. The visual inspection of their wavelet power spectra indicates that the short period component persists more than half of the entire observed time span. Thus, $P_{\rm LS}$ is more likely to represent their true rotation periods.

• •

  KIC 6508366 and 10068307: they both have two components roughly different by a factor of two. For KIC 6508366, the shorter period component is observed in the wavelet spectra for all quarters, while the longer period is visible only for a few quarters. For KIC 10068307, in contrast, the two components exist simultaneously within all of the quarters. Thus, we interpret the longer period corresponding to $P_{\rm LS}$ as the true rotation period, while the shorter period being its second harmonics.

• •

  KIC 5094751 (Kepler-109, KOI-123): It has multiple periodic components in Q4 and Q6 with $P_{\rm LS}$ being longer than both $P_{\rm ACF}$ and $P_{\rm WA}$; see also the right panel of Figure 4. It could be due to a complex spot configuration and/or stellar surface differential rotation, and we can not be certain about which represents the true rotation period.

As described in Section 2.2, using a systematic method, we removed anomalous quarters whose median absolute flux variations are three times larger than those of the neighboring quarters for six targets in total; Kepler-1655 (KIC 4141376, KOI-280), Kepler-100 (KIC 6521045), Kepler-409 (KIC 9955598, KOI-1925), KOI-72 (KIC 11904151), KIC 8694723, and KIC 10730618. The goal is to reduce the risk of having a biased measurement. However, among the stars for which the procedure does not detect anomalous quarters (that is, stars for which no quarter removal is performed), there still remains some irregular and abrupt increases in flux, though less prominent, which may affect the estimation of the rotation period.

These abnormal parts might be partly due to residual systematics, or a contamination from other neighbor sources. Such abnormal variation usually give a dominating peak in the power spectrum, over-weighing other weaker peaks that are more likely associated with the actual rotational signals. Examples of this type of light curves are given in Figure 17, whose shaded region shows a strong but transient flux variation. We visual inspect the wavelet spectra of the two given targets and notice that measured low frequency peaks ($\sim$ 27 days in $P_{\rm WA}$ for KIC 3456181 and $\sim$ 35 days in $P_{\rm WA}$ and $P_{\rm LS}$ for KIC 7106245) are associated with the abnormal regions instead of periodic modulation that we look for. In such circumstance, $\left<P_{\rm LS,q}\right>$ is less biased by those abnormal quarters.

For all targets in our sample with abrupt increase in flux variation, we check the module and channel number of the abnormal quarters to see whether it comes from a specific CCD plate. However, we notice that such abrupt variations are found in various modules and channels. In addition, we check the position of image on the plate and ensure that the location of image is not around the edge. It is natural to think of the increase in variation as the modulation from a large spot. Nevertheless, known that the lifetime of a spot scales with its size, abrupt variations which last for only 30-40 days seems less likely to be the lifetime of a large spot. Alternatively, such abnormal variation could be the residual of systematics like 30-day re-orientation of telescope. Especially, for relatively ’quiet’ stars with few active regions, there is barely any rotational modulation in the light curve. The residual of systematics would become significant and act like a sudden increase in flux variation.

Figure 18 illustrates part of light curves for KIC 6603624 as an example of stars exhibiting a weak flux variation. The WA power spectrum for this type of targets is generally very noisy, and we find that it is simply dominated by the peak at $P\sim 32$ days in Q14 alone for KIC 6603624. As Figure 18 indicated, however, it lasts just for two cycles and it is not clear if it should be interpreted as a rotational feature. Thus, for this type of targets, the measured period is very unreliable, and should be taken with caution. We identify eight such targets with weak or ambiguous periodic components, and label them by “W” in the ’Remark’ column of Table 3, 4, 5, and 6.

Figures 7 to 15 show that a majority of our targets have a range of different values of $P_{{\rm LS},q}$ in different quarters. The origin of those variations is not easy to identify, and may be a combination of stellar surface activity, instrumental effect, and unknown noise/contamination.

Of course, the variation of $P_{{\rm LS},q}$ could be of physical origin, at least for some targets. The most likely possibility includes an effect of the latitudinal differential rotation coupled with the surface distribution and dynamics of starspots. For instance, the presence of $n$ active regions along the same latitude would result in significant peaks with the $n$-th harmonics of the rotation period in the LS periodogram. Such alias would outweigh the fundamental rotation period if such active regions are equally spaced along the same latitude band (e.g. Chowdhury et al., 2018). Furthermore, the co-existence of dark spots (the central darker part, umbra, and surrounding lighter part, penumbra) and brighter spots (faculae) may cancel out part of the photometric variation, which complicates the extraction of the true rotational signal. Our Sun is an example of stars with complex surface configuration (frequent appearance of multiple active regions) which leads to unstable and occasionally low variability in both chromospheric and photometric measurements (see e.g. Donahue et al., 1996; Reinhold et al., 2020). As Donahue et al. (1996) pointed out, it is very likely solar-like star with similar surface configuration to our Sun may not pass the threshold for reliable detection of rotation period.

Consider a simple model of the latitudinal differential rotation (Reiners & Schmitt, 2003; Hirano et al., 2012):

where $\omega(\ell)$ is the angular velocity of the stellar surface at latitude $\ell$, $\omega_{\mathrm{eq}}$ is its value at the equator ($\ell=0$), and $\alpha$ is a parameter characterizing the strength of the differential rotation. For the Sun, $\alpha\approx 0.2$.

In order to proceed further, we need to assume the size and latitude distribution functions of starspots and their dynamics (migration over the stellar surface and creation/annihilation timescales), which is unknown. Suto et al. (2022) examined the variation of $P_{{\rm LS},q}$ assuming that the spot distribution in each quarter is independently drawn from the empirical distribution to match the observed flux modulation of the Sun.

For simplicity, we may equate our $\Delta_{\rm LS}$ to the variation of $\alpha\sin^{2}\ell$ in equation (5). Furthermore, if the spot latitude $\ell_{\rm spot}$ varies between $0^{\circ}$ and $45^{\circ}$, $\Delta_{\rm LS}\sim\alpha/2$. Thus, in principle, we may estimate $\alpha$ individually for our targets. In reality, however, most of the variation of $P_{{\rm LS},q}$ in Figures 7 to 15 are very diverse, implying that the differential rotation combined with the spot distribution may not be dominant except for stars exhibiting modest variations. While we do not conduct further quatitative examination, Kepler-408, KOI-974, Kepler-50, KOI-269, KIC 7103006, KIC 7206837, and KOI-268 may be potentially interesting targets for tightly constraining their differential rotation.

We have 22 stars with possible stellar companions (§2.3). Since their lightcurves might be contaminated by the companions, we exclude them from our main analysis. Nevertheless, we examine seven KOI stars in this subsection because they may have interesting implications for their planetary architecture. Figure 21 plots their rotation periods in the same manner as Figure 7. Among them, we select the following three stars that are potentially interesting, the LS periodogram and asteroseismic constraints of which are shown in §C.11, §C.12, and §C.13.

Kepler-2 (HAT-P-7, KOI-2, KIC 10666592) is one of the two transiting planetary systems along with Kepler-25 for which the real spin-orbit misalignment angle has been estimated jointly from the Rossiter-McLaughlin effect and asteroseismology (Benomar et al., 2014; Campante et al., 2016). In reality, however, it is not easy to identify its rotation period photometrically from Figure 50. On the other hand, $P_{\rm astero}\approx 12$ days from Figure 51 seems to be a reliable estimation of rotation period. This is a good example of the complementary power of asteroseismology.

Kepler-410 (KOI-41, KIC 8866102) has precise estimates for both photometric and asteroseismic rotation periods, but their values are very different ($\langle P_{{\rm LS},q}\rangle\sim 4P_{\rm astero}$). Intriguingly, Figure 52 indicates that the LS periodograms for all quarters have persistent secondary peaks at $\approx 10$ days, which is very close to the second harmonics of the starspot modulation analytically predicted by Suto et al. (2022). Thus, it is possible that $\langle P_{{\rm LS},q}\rangle\approx 20$ days, instead of $P_{\rm astero}\approx$ 6 days, is the true rotation period of Kepler-410. On the other hand, Kepler-410 is a binary system(Van Eylen et al., 2014), and it is also likely that either asteroseismic or photometric analyses may be contaminated by the companion star to some extent.

Finally, $\langle P_{{\rm LS},q}\rangle$ and $P_{\rm astero}$ of KOI-288 (KIC 9592705) agree within their error-bars. Figure 54 shows that its photometric periods are very robust. Thus, the joint photometric and asteroseismic analysis of KOI-288 may put a tighter constraint on its inclination. Incidentally, this is an example of systems whose $P_{\rm astero}$ is close to the orbital period of the planetary companion (Suto et al., 2019), although it may be just by chance.

Stars lose the angular momentum through magnetic winds, and the loss rate is supposed to be sensitive to the depth of the convective zone, and therefore to their mass and effective temperature. Indeed, it is known observationally that stars with $T_{\rm eff}<6200$K have a thicker convective envelope and much slower rotation rate than those with $T_{\rm eff}>6200$K (Kraft, 1967). Winn et al. (2010) found that hot Jupiters around stars with $T_{\rm eff}<6200$K prefer the orbital axis well-aligned to the stellar spin axis, and proposed that the spin-orbit alignment results from the stronger tidal interaction between close-in giant planets and their host stars with a thicker convective envelope.

Figure 22 plots the photometric and asteroseismic rotation periods against the stellar effective temperature. We adopt the SDSS $T_{\rm eff}$ from Table 7 of Pinsonneault et al. (2012) for 60 stars. Our result is basically consistent with Figure 3 of Hall et al. (2021) and qualitatively reproduce the Kraft break. On the other hand, our targets with $P_{\rm astero}>30$ days often have smaller values of $\langle P_{{\rm LS},q}\rangle$ and may not properly represent the surface rotation period. Furthermore, the uncertainties of the asteroseismic measurements seem to be anti-correlated with $T_{\rm eff}$ (see Figures 19 and 20).

Finally, we study the stellar inclination as it could be an indicator of misalignment for planetary systems observed by transit and also has an imprint of the genesis of the stars. The stars are considered as a single population, from which we seek to identify the underlying distribution of stellar inclination and its dependence with temperature. The adopted criteria is the $\sin i$, which is obtained following the method described below.

For the asteroseismic sample of stars, we directly convert the posterior distribution of $i_{*}$ by Kamiaka et al. (2018), and compute the distribution of $\sin i_{*,\rm{astero}}$. For the spectroscopic determination of $\sin i_{*,\rm{spec}}$, it is required to use equation (3), which depends on $R$, $P_{\rm spec}$ (substituted by the photometric rotation period $P_{\rm LS}$ for the current purpose) and $v\sin i_{*}$. Furthermore, it is necessary to consider the distribution functions of $R$ and $P_{\rm LS}$ and $v\sin i_{*}$. Here, we assume uncorrelated Gaussian distributions9 for $R$ and $v\sin i_{*}$ with their mean being the measured value and standard deviation being its $1\sigma$ error. Finally, in order to evaluate the distribution of the period, we directly use the LS periodogram of $P_{\rm LS}$ as the underlying probability distribution. However, we only use a range of the LS periodogram that spans over a slice that is twice the HWHM around the highest peak. This distribution is also normalised by integration over the considered range mentioned above. All of the probability distributions of the variables being defined, we then compute the distribution of $\sin i_{*,\rm{spec}}$ via Monte Carlo sampling of $R$, $P_{\rm LS}$ and $v\sin i_{*}$.

Left panels of Figure 24 plot $\sin i_{*,\rm{astero}}$ and $\sin i_{*,\rm{spec}}$ for KOI stars against the stellar effective temperature following the same procedure as described in Kamiaka et al. (2018) (in their Section 5.3), while right panels show the corresponding distribution functions.

Although the sample of KOI stars is small, we note that if one excludes stars from Group Bb (the least reliable stars), KOIs tend to have $\sin i_{*}$ clustered around unity, and that spectroscopy and seismology are in good agreement. This indicates that a majority of stars with planets show a preference for a spin-orbit alignment, which is consistent with the previous conclusion by Kamiaka et al. (2018).

Figure 24 focuses on non-KOI stars. Excluding stars from Group Bb, $\sin i_{*,\rm{astero}}$ exhibits a broader distribution compared to KOI stars. We note that the distribution is not uniform in $\sin i_{*}$ as it would be expected if there was no preference in their rotational spin direction on the sky. As shown by Kamiaka et al. (2018), asteroseismology overestimates inclinations below $i\simeq 30^{\circ}$, corresponding to $\sin i_{*}\lessapprox 0.5$. This is likely the cause for a lack of population with reliable measurement when $\sin i_{*}<0.4$. In addition, as pointed out by Kamiaka et al. (2018), available spectroscopic measurements are unreliable for some stars, in particular when the $v\sin i_{*}$ is low (due to the dominance of the stellar turbulence). Therefore, spectroscopic results from Molenda-Żakowicz et al. (2013) and from Bruntt et al. (2012) sometimes lead to a mathematically inconsistent result of $\sin i_{*}>1$. If we exclude these impossible cases, there is no clear preferential stellar inclination. This is again consistent with an isotropic distribution of stars and contrasts with results from the KOI case shown in Figure 24.

This paper has performed comprehensive photometric analyses of the 92 Kepler stars selected by Kamiaka et al. (2018) for the asterosesmic analysis. The photometric intensity variability measures the period of rotational modulation caused by the active regions at a specific latitude on stellar surface. The asteroseismic analysis, on the other hand, studies the pulsation pattern and derives an averaged rotation period over a range of latitudes and depths inside the star. Since stars are not ideal solid-body rotators, their radial and latitudinal differential rotations would affect the estimates of their rotation periods in different methods. Therefore, the comparison between the photometric and asteroseismic results provides unique and complementary insights into the nature of those targets. One of the most important examples is the stellar inclination angle measured from asteroseismic analysis, which is a key ingredient for measuring a three-dimensional spin-orbit angle of exoplanetary systems (see e.g. Benomar et al., 2014; Winn & Fabrycky, 2015; Kamiaka et al., 2018, 2019; Suto et al., 2019). While the asteroseismic analysis provides a unique relation between the stellar inclination and the rotation period, its degeneracy is not easy to be broken by asteroseismology alone (see Kamiaka et al., 2018). When a more precise estimate of the rotational period is available from the intensity variability, it is very useful to break the degeneracy. Hence, a detailed examination of various photometric methods as presented in this work provides an important framework for those complementary approaches.