Solar-type Stars Observed by LAMOST and Kepler

The stars analyzed in this study have been selected from the sixth Data Release (DR6) of the LAMOST survey.⁶ Here, we define solar-type stars as stars with ${T}_{\mathrm{eff}}$, $\mathrm{log}g$ and [Fe/H] in the ranges 5500–6000 K, 4.14–4.74, and −0.2–0.2, respectively. All these parameters are taken from DR6 of the LAMOST survey, which is based on the LAMOST stellar parameter pipeline (Zhao et al. 2012; Luo et al. 2015). The solar values were taken to be: ${T}_{\mathrm{eff}}$ = 5777 K and $\mathrm{log}g$ = 4.44. To place a lower limit on the quality of the spectroscopic observations, we only considered stars with the signal-to-noise ratios (S/Ns) at the blue end of the spectra higher than 30. With these constraints, we collected 341,557 spectra for 272,854 solar-type stars, which we denote as LAMOST sample. Among them, there are 6626 stars that have been also observed by Kepler mission (Zong et al. 2018), which we denote as the L-K sample.

We cross-matched the selected 6626 stars with the catalog of McQuillan et al. (2014). This is a catalog of Kepler stars containing 34,030 stars with detected rotation periods and 99,000 stars with non-detected rotation periods. Following Reinhold et al. (2020) we concentrated on stars with rotation periods in the range 20–30 days (hereafter, solar-type stars) and stars with non-detected rotation period (hereafter, non-periodic sample). Furthermore, we have also selected stars with periods in the range 10–20 days (hereafter, short-periodic sample). Such a classification results in 254 solar-type stars and 793 short-periodic stars; 1556 stars were deemed as non-periodic. These stars can be considered as pseudo solar-type stars because their rotation periods are unknown. As discussed in Section 1 the Sun would most probably be allocated to the non-periodic sample of pseudo solar-type stars. In the Appendix we also consider stars with rotation periods shorter than 10 days to better illustrate the effect of the rotation period on photometric variability and on Ca ii H and K emission.

Using the LAMOST spectra, we measured the magnetic activity proxy S-index as

$$\begin{eqnarray}&&{S}_{\mathrm{LAMOST}}=\alpha \cdot \displaystyle \frac{H+K}{R+V},\end{eqnarray} \tag{ 1 }$$

where H and K are the integrated fluxes in the cores of Ca ii H and K lines, respectively. The integration is performed using a triangle function with a FWHM of 1.09 Å centered at 3968 Å and 3934 Å, respectively. The parameters R and V are the integrated fluxes in the nearby pseudo-continuum. The integration in pseudo-continuum is performed using a rectangular function with 20 Å width centered at 4001 Å and 3901 Å, respectively. Following Karoff et al. (2016) we put calibration factor α to 14.4. Karoff et al. (2016) argued that such a choice of the calibration factor leads to a distribution of LAMOST S-index values being consistent with that derived from Isaacson & Fischer (2010) around S = 0.2. For stars with multiple observations, the S-indexes were determined by using the weighted mean values of these multiple spectra with the weights being the S/Ns of the spectra. We refer to Zhang et al. (2020) for a detailed discussion of S-index measurements.

One of the main limitations of the current study is that Kepler and LAMOST observations are not performed at the same moment in time. While Kepler data considered in this study were obtained from 2009 June until 2013 May, most of the LAMOST spectra were taken from 2012 to 2017 June. Consequently, 92% of LAMOST spectra used in this study have been taken outside of the period of Kepler observations. Nevertheless, we do not expect S-index to change significantly between the periods of Kepler and LAMOST observations. Indeed, the amplitude of the S-index variability on the rotation and activity cycle timescales is proportional to the time-averaged values of the S-index (see, e.g., Egeland 2017; Radick et al. 2018, and references therein). Consequently, the changes of S-index values of stars in the solar-type and non-periodic samples are expected to be similar to those of the Sun (i.e., about 10% from the mean value). Furthermore, spectra of 577 stars in our samples have been recorded by LAMOST more than once with the intervals among observations often reaching a couple of years. Comparison of these spectra did not reveal any significant changes of the corresponding S-index values with time (e.g., standard deviation among S-index values for the majority of the non-periodic stars was below 0.01–0.015).

Figure 1 shows distributions of S-index values for three stellar samples introduced in Section 2.1. Not surprisingly, the short-periodic sample appears to be on average more active than the sample of solar-type stars. Interestingly, Figure 1 indicates that non-periodic stars are on average less active than solar-type stars. Furthermore, the difference between distributions of solar-type and non-periodic stars is similar to the difference between distributions of short-periodic and solar-type stars.

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To quantify the difference between the distributions we computed the Kolmogorov–Smirnov (K-S) statistic (Hodges 1958). The K-S statistic are measured by the maximum diagonal distance between the empirical cumulative distribution functions of the two samples. The statistic is 0.21 for distributions between short-periodic and solar-type sample, while it is 0.27 for distributions between solar-type and non-periodic sample. The K-S statistic values and the two low p-values (<10⁻⁶) indicate distributions of both sample sets are different, and distributions between the solar-type and non-periodic sample are much more different than those between short-periodic and solar-type sample.

Figure 1 indicates that though the S-index distributions of solar-type and non-periodic stars are different, there is a substantial overlap between them. This implies that stars with detectable and not-detectable rotation periods and, consequently, very different light curves can still have the same levels of the chromospheric activity. The example of two such stars, KIC 10414643 and KIC 5350635, is given in Figure 2 and their main parameters are summarized in Table 1. The photometric variation ${R}_{\mathrm{var}}$ is taken from Reinhold et al. (2020). It is calculated by defining the difference between the 5th and 95th percentile of the sorted differential flux in each Kepler quarter and then taking the median among all quarters value.

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Table 1.  Parameters of KIC 10414643 and KIC 5350635

KIC	${T}_{\mathrm{eff}}$	$\mathrm{log}g$	[Fe/H]	${R}_{\mathrm{var}}$	S-index	${P}_{\mathrm{rot}}$
10414643	5692.39 ± 30.79	4.50 ± 0.05	0.02 ± 0.03	0.2991	0.2018	22.21
5350635	5829.83 ± 23.08	4.46 ± 0.04	−0.11 ± 0.02	0.0638	0.2019

Note. The values of effective temperature, surface gravity, and metallicity are take from DR6 of the LAMOST survey. The values of photometric variations are taken from Reinhold et al. (2020). The value of rotation period is taken from McQuillan et al. (2014).

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Figure 2 shows that while both stars have very similar Ca ii H and K profiles and basically the same values of the S-index, their light curves are pretty different. The light curve of KIC 10414643, on the one hand, is highly regular and its rotation period can be easily determined. On the other hand, the amplitude of variability of KIC 5350635 is roughly five times smaller and no clear periodic signal can be seen behind the noise.

One possible explanation of such behavior is that chromospheric activity is mainly given by the overall coverage of a star by the magnetic features (see, e.g., Shapiro et al. 2014). At the same time, the photometric variability on stellar rotation timescale strongly depends on the surface distribution of magnetic features, their sizes, as well as evolution (see, e.g., Shapiro et al. 2020). In particular, recent studies have been able to reveal the temporal evolution of starspots (see, e.g., Namekata et al. 2020) and also determine their sizes (see, e.g., Morris et al. 2017, who showed that spots of HAT-P-11 and of the Sun have similar sizes).

One interesting effect capable of a strong increase of the photometric variability without a direct influence on the S-index values and, thus, explaining the difference between periodic and non-periodic stars is nesting in the distribution of magnetic features (i.e., the tendency of magnetic features to emerge within certain "nests" of activity; see, e.g., Castenmiller et al. 1986). In contrast to the spatially random distribution of emergences, nesting would lead to a non-axisymmetric distribution of spots (see, e.g., Isık et al. 2018) and, consequently, regular light curves with large amplitudes of the rotational brightness variability. The effect of nesting on the photometric variability will be addressed in the forthcoming study.⁷

Another contributing factor might be the stellar inclination, i.e., the angle between the direction to the observer and stellar rotation axis. While photometric variability and period detectability strongly depends on the inclination of a star (Nèmec et al. 2020), chromospheric activity shows a much weaker dependence (Shapiro et al. 2014). In particular, if a star is observed at a relatively low inclination (i.e., pole-on) its rotational variability would be significantly reduced and a star will be classified as non-periodic despite a large chromospheric activity. Finally, we cannot fully exclude a possible change of the S-index between periods of Kepler and LAMOST observations (although, see the discussion in Section 2.2). We emphasize here the need for the future contemporaneous spectroscopic and photometric observations for a large sample of stars.

In Figure 3 we plot the dependences of photometric variability on S-index for the solar-type and non-periodic samples. For the solar-type sample, the ${R}_{\mathrm{var}}$ significantly increases with S-index. We binned the S-index values into seven equidistant segments within the range 0.12–0.30 for the solar-type sample and within the range 0.1–0.3 for the non-periodic sample. The averaged ${R}_{\mathrm{var}}$ values in each bins were then calculated. The binned values show that for both samples photometric variability somehow increases with the S-index. The increase is, however, not particularly strong and is to a large extent hidden by the large spread of photometric variabilities the stars with the same S-index can have.

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In agreement with Reinhold et al. (2020), Figure 3 shows that photometric variability of solar-type stars is significantly larger than that of non-periodic stars. The stars in the non-periodic sample exhibit variabilities similar to that of the Sun (for which median ${R}_{\mathrm{var}}$ over the last 140 yr was 0.07% and maximum ${R}_{\mathrm{var}}$ was about 0.2%; see Reinhold et al. 2020 for detailed discussion). At the same time, Figure 3 demonstrates that although distributions of S-index values for the periodic and non-periodic stars are different (see Section 3.1), the S-index is not the main factor that defines the morphology of the light curves (regular versus non-regular) and their amplitude. We note that the S-index mainly depends on the total coverage of stellar surface by magnetic features, while the photometric variability additionally depends on the degree of axisymmetry of this distribution (see Section 3.1). Consequently, our result hints that the surface distribution of magnetic features and its degree of axisymmetry plays an important role in defining whether a star is solar-type or non-periodic stars.

Kepler data shows that photometric variability increases with the stellar rotation rate until periods of about 12 days and saturates for faster rotators (see, e.g., Figure 15 in Notsu et al. 2019). Interestingly, the saturation appears to be less pronounced for the S-index values (see Figure 6 in Zhang et al. 2020). We further illustrate this point in the Appendix where we repeat Figures 1 and 3 but for samples of stars with rotational periods P_rot < 8 days, 8 < P_rot < 12 days, 12 < P_rot < 20 days, and 20 < P_rot < 30 days (see Figures A1 and A2). One can see that S-index values and photometric variabilities for the last three samples are clearly different, increasing from slower to faster rotators (see red, blue, and maroon in Figures A1 and A2). For the S-index the same trend is also valid for stars with P_rot < 8 days—they appear to be more active than stars in other samples (see yellow in Figure 1). At the same time there seems to be a saturation in the photometric variabilities as their average values for stars with P_rot < 8 days and for stars with 8 < P_rot < 12 days are very similar (see maroon and yellow in Figure A2). This is consistent with the results of Notsu et al. (2019) and Zhang et al. (2020).

The main difficulty in finding the solar S-index as it would be measured by LAMOST is the relatively low spectral resolution of LAMOST. Although the mean resolution power of LAMOST is about 1800, the resolution strongly depends on wavelength and is expected to be about 1000 around Ca ii H and K lines (Xiang et al. 2015). Furthermore, Xiang et al. (2015) reported that the resolution power of each LAMOST fiber varies considerably, with amplitudes amounting to 1 Å. Potentially such a spread of resolutions might affect not only placing the Sun on the LAMOST S-index scale, but also the distributions plotted in Figure 1. Indeed, one might expect that stars with near-solar activity levels observed at lower spectral resolution will have higher S-index values than stars observed at higher resolution (as the lower the spectral resolution is the stronger the line cores are mixed with wings and, consequently, the larger are the H and K fluxes, see Equation (1)).

To clarify the strength of the impact of the differences in the resolution power between different LAMOST fibers on the distribution of S-index values, we collected the observational resolution curves at the blue end (3700–5900 Å) of the individual spectra that were obtained utilizing the LAMOST arc lamp and sky emission lines.⁸ In the left panel of Figure 4 we show the dependence of the S-index on the LAMOST observational resolution power, i.e., FWHM for stars from Figure 1. Note that the observational resolution curves are available for ∼50% of stars in Figure 1. One can see that the dependence (if any) of the S-index on the LAMOST spectral resolution is rather weak compared to the differences of S-index values between the periodic and non-periodic sample. Consequently, we do not expect that any of our results might be affected by the LAMOST stars being observed at slightly different spectral resolutions.

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It is important to place the Sun and solar-type stars on the same S-index scale for comparing their chromospheric activities. This, however, is hindered by the fact that LAMOST survey has no record of spectra of solar light reflected from minor bodies or from inactive satellites. Therefore, we take the following indirect approach to obtain the S-index of the Sun on LAMOST scale. We use the high-resolution (better than 350,000) solar flux spectrum from Hamburg atlas⁹ (see, e.g., Doerr et al. 2016) created using the data from the Fourier Transform Spectrometer (FTS) at the McMath-Pierce solar telescope at the Kitt Peak National Observatory. We degrade this FTS spectrum by convolving it with Gaussian kernels of varying FWHM in the range 0–5 Å, which covers the range of spectral resolutions achieved from LAMOST fibers. Then we compute the S-index for the high-resolution (undegraded) and degraded spectra. In the right panel of Figure 4 we show the ratio of S-index measured for the degraded spectrum to that measured for the original high-resolution FTS spectrum as a function of the resolution. Due to the increased H and K fluxes resulting from the lower spectral resolution, the ratio increases with decreasing resolution. In the left panel of Figure 4, it appears that the majority of LAMOST stars considered for this plot were observed with spectral resolution between 3.0 and 3.2 Å. For this range of resolution, the ratio varies only slightly, i.e., between 1.45 and 1.49 as indicated by the shaded horizontal bar in the right panel of Figure 4, reinforcing that S-index has a weaker dependence on the variations in LAMOST resolution. However, we note that S-index measured with a low-resolution spectra like LAMOST would be on an average larger by 47% than that measured with the high-resolution spectrum.

Egeland et al. (2017) has accurately placed the solar S-index value on the MWO S-index scale. They found that minimum and maximum values of the solar S-index during cycles 15–24 were 0.162 and 0.177, respectively. We note that these values correspond to α = 19.2 instead of α = 14.4 that we adopted here following Karoff et al. (2016). Thus, we first corrected Egeland et al. (2017) values for the difference in calibration factors and then applied factor 1.47 (see the right panel of Figure 4) to correct for LAMOST spectral resolution. As a result we find that solar S-index on the LAMOST scale varied between 0.179 and 0.194 during cycles 15–24. These values are designated by the dashed vertical lines in Figure 1. Compared to the activity level of the Sun, the peak level of activity of the non-periodic stars is near the range of the Sun, while the majority of the periodic stars have activity levels that are higher than that of the Sun. Consequently, the analysis of the Ca ii H and K data reinforces the conclusion of Reinhold et al. (2020; drawn from the analysis of the Kepler light curves) that the Sun is a typical star of the non-periodic sample.