Magnetic Activities of M-type Stars Based on LAMOST DR5 and Kepler and K2 Missions

LAMOST (also known as Guoshoujing Telescope), located in Xinglong Station of the National Astronomical Observatories of the Chinese Academy of Sciences (NAOC), is a special meridian reflecting Schmidt telescope with about 4 meter effective aperture and a 5° field of view (Wang et al. 1996; Su & Cui 2004; Cui et al. 2012). LAMOST is equipped with 16 spectrographs, each of which is fed by 250 fibers. Each spectrograph uses two 4K × 4K CCDs to record the red and blue spectra of the corresponding target sources. Therefore, LAMOST can obtain about 4000 spectra with a wavelength range of 3650–9000 Å and a resolution R ∼ 1800 in one single exposure.

After two years of commissioning and one year of pilot survey, LAMOST officially began a five-year regular survey in 2012 September. This regular survey includes the LAMOST ExtraGAlactic Survey (LEGAS) and the LAMOST Experiment for Galactic Understanding and Exploration (LEGUE; Zhao et al. 2012), where the main goal of the LEGUE is to observe the spectra of stars covering 9–17.8 mag (r band) in the Milky Way. The raw stellar spectra are first processed by the LAMOST 2D pipeline, and then spectral type classification and radial velocity measurement are performed by the LAMOST 1D pipeline (Luo et al. 2012, 2015). The five-year regular survey ended on 2017 June 16, and all spectra were provided through the LAMOST DR5.⁶ The LAMOST DR5 released 8,171,443 stellar spectra, of which 529,629 are M-type stellar spectra. In addition, LAMOST has performed at least one observation of all 14 subfields of the Kepler field in the LK project (De Cat et al. 2015; Ren et al. 2016; Zong et al. 2018). There are also some overlaps between the observation areas of the LAMOST and K2 missions (Howell et al. 2014). This provides an excellent opportunity for us to study the magnetic activity of M-type stars using the spectra of LAMOST and the light curves of the Kepler and K2 missions.

The Hammer spectral typing facility (West et al. 2004; Covey et al. 2007; West et al. 2011a, 2011b), written in IDL code, has an automatic mode and an interactive mode. In order to make the spectral types of our sample more accurate, we use the interactive mode of the Hammer to visually inspect the 529,629 M-type stellar spectra candidates and manually assign spectral types. The Hammer's manual "eye check" spectral types are accurate to within ±1 subclasses (Covey et al. 2007). In the interactive mode, we remove the low signal-to-noise ratio spectra (S/N < 3 at ∼8300 Å) and other spectra except the M-type stellar spectrum, and finally we obtain 516,688 M-type stellar spectra. The parameters of these 516,688 LAMOST spectra are listed in Table 1. In Table 1, the first column is the spectral name, the second column to the fourth column are the magnitude of the riz bands, the fifth column is the exposure time of the spectrum, the sixth column is the beginning of the spectrum exposure time, the seventh column is the spectral type visually calibrated with the Hammer program, and the last column is the S/N of the spectrum in the R band. After statistics, we plot the distribution of the LAMOST DR5 M-type stellar spectra in Figure 1. From Figure 1 we can see that 98.2% of the spectra are between M0 and M3, while the number of spectra from M4 to M9 is small.

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The Balmer lines (Hα, Hβ, Hγ, and Hδ), Ca ii H&K lines, Ca ii IRT lines, and He i D3 line are significant indicators of stellar magnetic activity (Duncan et al. 1991; Huenemoerder 1991; Montes et al. 2004; Hall 2008). In order to accurately determine whether these lines are emitted or absorbed, we calculate the corresponding equivalent widths (EWs) of these lines. The calculation process of EW is as follows. First, we integrate over the specific region of each line (8 Å wide centered on the line) and subtract the mean flux calculated from two adjacent continuum regions (see Table 2 for details). Then we calculate the EWs for each line by dividing the integrated line flux by the mean continuum value (West et al. 2004, 2008, 2011b). Referring to the activity criteria of the EW of the emission lines given by West et al. (2011b), we present the activity threshold for the EW of the Balmer lines, Ca ii H&K lines, Ca ii IRT lines, and He i D3 line, as shown in the last column of Table 2. Active M-type stars are searched by the following criteria: (1) the EW of any one of the special lines is greater than the corresponding threshold, (2) the value of the EW is greater than three times its uncertainty, and (3) the height of the emission line must be larger than three times the noise in the emission line center. It should be pointed out that the above-mentioned spectral lines of some targets have EWs below the corresponding threshold, but they may still be potential active M-type stars (West et al. 2004). The EWs of these spectral lines are listed in Table 3. In Table 3, the first column is the LAMOST spectral name, the second column is the observation time of the corresponding spectrum, and the third column is the spectral type visually calibrated with the Hammer program. The fourth column to the thirteenth column are the EWs of the above 10 special spectral lines, and after each EW value, the letter "Y" indicates that the spectral line conforms to the activity standard, and the letter "N" indicates the corresponding line does not meet the activity criteria. Figure 2 presents two M-type LAMOST spectra (LAMOST J034018.97+264540.4 and LAMOST J055105.96+082234.3) with Balmer lines, Ca ii H&K lines, Ca ii IRT lines, and He i D3 line emission.

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By statistically studying the EWs of Table 3, we list the magnetic activity fraction of Balmer lines, Ca ii H&K lines, Ca ii IRT lines, and He i D3 line in Table 4. Table 4 contains the line names, the number of active spectra and active stars, and the corresponding activity fraction. As can be seen from Table 4, there is the largest number of spectra with Hα activity, and the corresponding activity fraction is also the largest. The activity ratio decreases in the order of Hα, Hβ, Hγ, and Hδ. In addition, we first calculate the activity ratio of the Ca ii H, Ca ii IRT, and He i D3 of the M-type stellar spectrum. The ratio of activities of Ca ii H and Ca ii K is very close, and the proportion of activities of Ca ii IRT lines is very low.

Previous studies (West et al. 2004, 2011b; Yi et al. 2014; Zhang et al. 2016) have shown that the Hα activity ratio is a function of M-type stellar spectral type. In this paper, we also present the Hα activity fraction in M0–M9 spectral types, as shown in panel (a) of Figure 3. From panel (a), we can see that the proportion of Hα activity is gradually increasing from M0 to M3; that is to say, in our sample, the proportion of Hα activity is also a function of spectral type in M0–M3. However, the Hα activity fraction between M4 and M9 varies randomly. It seems that there is a distinction of statistics between M0–M3 and M4–M9 stars, although we cannot confirm that. Because there are only 9296 spectra between M4 and M9 in our sample, accounting for only 1.8% of the total sample, we need more data to confirm that. More importantly, we first present the variation of the activity fraction of Hβ, Hγ, Hδ, Ca ii H&K, and He i D3 between M0 and M9, as shown in panels (b)–(g) of Figure 3. From panel (b) to panel (g), the proportion of activities of Hβ, Hγ, Hδ, Ca ii H&K, and He i D3 gradually increases from M0 to M3. This indicates that the proportion of activities of Hβ, Hγ, Hδ, Ca ii H&K, and He i D3 is in excellent agreement with the Hα activity fraction, and the activity fraction of these lines is a function of spectral type in M0–M3. For Ca ii IRT, we have not found a similar phenomenon because of the small activity fraction of these three lines in our sample.

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From the statistical analysis of Section 2.3, it can be seen that the activity fractions of Hα, Hβ, Hγ, Hδ, Ca ii H&K, and He i D3 all increase gradually from M0 to M3. This arouses our interest in whether there is some correlation between the EWs of these lines. Previous studies analyzed the relationship between magnetic activity indicators Ca ii H&K, Hα, and Ca ii IRT (Strassmeier et al. 1990; Martínez-Arnáiz et al. 2011), but most of these indicator lines are not observed simultaneously, and the study samples are small. For this research, we use 40,464 M-type LAMOST spectra with Hα activity to study the EW–EW relationships between the spectral lines. Our data set has two noteworthy advantages: the first is that the wavelength range of each M-type LAMOST spectrum covers the Balmer lines, Ca ii H&K lines, Ca ii IRT lines, and He i D3 line simultaneously; the second is that our research sample has greatly improved in number.

In Figure 4, we compare the EWs between Hα and Hβ, Hγ, Hδ, Ca ii H&K, Ca ii IRT, He i D3 and Hγ and Hδ, Ca ii H and Ca ii K, and Ca ii λ8498 Å and Ca ii λ8542 Å. EW–EW relationships between the above lines have been determined by fitting the data to a relation of the type

$$\begin{eqnarray}&&\mathrm{EW}2={\rm{c}}1(\mathrm{err}1)\mathrm{EW}1\ +\ {\rm{c}}2(\mathrm{err}2),\end{eqnarray} \tag{ 1 }$$

where EW1 and EW2 are the equivalent widths of two different lines, c1 and c2 are the fitting parameters, and err1 and err2 are corresponding errors. From panels (a) to (i) in Figure 4, it is obvious that the EWs of Hβ, Hγ, Hδ, Ca ii H&K, Ca ii IRT, and He i D3 are linearly related to the EWs of Hα. Martínez-Arnáiz et al. (2011) attempted to search for a linear correlation between Hα and Ca ii K and Ca ii λ8498 Å through flux–flux relationships using 298 active late-type stars. Panels (e) and (f) in Figure 4 again confirm the linear correlation between Hα and Ca ii K and Ca ii λ8498 Å using the EW–EW relationships. In addition, it can be seen in panels (a)–(c) and (j)–(l) of Figure 4 that the EW–EW relationships between the same line series (such as Balmer lines, Ca ii H&K lines, and Ca ii IRT lines) also have a linear correlation. In the EW–EW relationship diagram, it is important to mention that the slope (c1) between lines of the same series (such as panels (a)–(c) and (j)–(l)) is greater than the slope (c1) between lines from the different series (such as panels (d)–(i)).

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From the 40,464 Hα activity M-type spectra counted in Section 2.3, we find that LAMOST performed at least two observations on 5499 M-type stars (see Table 5 for details) at different times. Among the 5499 M-type stars, their Hα lines have a high S/N. Therefore, we use the spectrum of these 5499 stars to study their Hα variability. First, we search for the maximum and minimum values of their EWs in the LAMOST spectra of these 5499 stars, which are listed in the fifth and seventh columns of Table 5. Subsequently, we calculate the corresponding R_EW value (R_EW = Max(Hα_EW)/Min(Hα_EW)), as shown in the ninth column of Table 5, and plot the relationship between the R_EW value and the spectral type in Figure 5. The distribution of these stars is asymmetrical, and the number of stars from M0 to M3 accounts for 98.6%. In Figure 5, the large R_EW values of M2 and M3 have significant extensions relative to M0 and M1. This is consistent with the previous results (Kruse et al. 2010; Lee et al. 2010; Bell et al. 2012) in which the Hα variability increases at later spectral type. In addition, we also calculate △EW (△EW = Max(Hα_EW) − Min(Hα_EW)), listed in the tenth column of Table 5. Using the same method as Zhang et al. (2016), we find the △EW value is greater than three times its corresponding uncertainty, indicating that the star has Hα variability and is marked with Yes, as shown in the last column of Table 5. We find that 1791 of these 5499 M-type stars have Hα variability. We plot the LAMOST spectra of one of the M-type stars with Hα variability (LAMOST J011921.66−011701.8, panel (a)) and its corresponding EW as a function of observation time (panel (b)) in Figure 6.

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The Kepler spacecraft was launched in 2009 to search for exoplanets by finding planetary transit events from the high-precision white-light photometric data of Sun-like stars in the Cygnus-Lyra region (Borucki et al. 2010). The Kepler spacecraft was equipped with a telescope with a 95 cm aperture and 105 deg² field of view. The Kepler mission has two observation modes: the mode of sampling once every 30 minutes (long cadence, LC) and the mode of sampling once every 1 minute (short cadence, SC). After four years of observation, the Kepler mission obtained continuous photometric data for nearly 200,000 stars in LC mode (Huber et al. 2014). In 2013 May, the second of four reaction wheels was lost on the Kepler spacecraft, ending Kepler's scientific mission for more than four years. In 2014 June, the K2 mission was fully operational, which represents a new concept for spacecraft operations. The K2 mission is a series of sequential observing "campaigns" of fields distributed around the ecliptic plane and provides a photometric precision close to the original Kepler mission (Howell et al. 2014). The duration of each campaign is approximately 80 days. From Campaign 0 to Campaign 16, the K2 mission has observed more than 470,000 objects. In addition, the Kepler–LAMOST project was proposed by De Cat et al. (2015), which allows many stars in the Kepler observation area to have both photometric and spectroscopic data. The observation area of LAMOST and the observation area of the K2 mission also partially overlap. Therefore, this provides an excellent opportunity to study the magnetic activity of M-type stars using both photometric and spectroscopic data.

Both the Kepler and K2 missions provide both uncorrected simple aperture photometry data and presearch data conditioning (PDC) data, where the instrumental effects of PDC data have been removed. Because of the way K2 is pointed, the center of the stellar point-spread function usually moves about 1 pixel in 6 hr (Van Cleve et al. 2016). Without the use of photometric correction, the error produced on the light curve may be much higher than on Kepler. In our study, considering the effects of Kepler and K2, we finally choose the PDC data. In the second part, we have searched 516,688 M-type stellar spectra from the LAMOST DR5. By cross-matching these LAMOST spectral data with the Kepler catalog (Huber et al. 2014) and the K2 catalog (including Campaigns 0 to 16) with a tolerance of 3'', we obtain the photometric data (LC) of 918 and 8232 M-type stars from Kepler and K2, respectively. After the Kepler spacecraft lost two spacecraft reaction wheels, the Science Data Processing Pipeline of K2 limited the position fitting of the target to 1.5 pixels (about 398) when constructing a model to process K2 data (Jenkins 2017). Therefore, in order to make the flare search more accurate, we removed targets with neighboring stars within 4'' (≤4''), after which we obtain the photometric data (LC) of 834 and 8130 M-type stars from the Kepler and K2 missions, respectively. The Kepler ID or K2 ID of the 8964 M-type stars is listed in the first column of Table 6. The corresponding LAMOST spectral names, spectral observation times, and spectral types are listed in the second, third, and fourth columns of Table 6, respectively. In addition, we list the observation time of the 8964 M-type stars observed by the Kepler or K2 mission in the fifth column of Table 6. The stellar effective temperatures and stellar radii obtained from the literature (Huber et al. 2014, 2016) are listed in the sixth and seventh columns of Table 6, respectively. We also list the EWs of Hα, Ca ii H, and He i D3 in the eighth, ninth, and tenth columns of Table 6, respectively. We will use these cross-matched Kepler and K2 LC photometric data to study the flares of M-type stars.

Flares are sudden explosions on the surface of a star and last from a few minutes to a few hours. Usually, a flare can be divided into three stages: an impulsive rise, an impulsive decay, and a gradual decay. Based on the characteristics of the flare, some methods for detecting flares from the light curve have been developed (Walkowicz et al. 2011; Osten 2012; Shibayama et al. 2013; Wu et al. 2015; Van Doorsselaere et al. 2017; Yang et al. 2017). Wu et al. (2015) used such a method to detect stellar flares by first subtracting the background light curve from the original light curve and then detecting the flare on the detrended data. In this work, we use the same method as Wu et al. (2015) to detect flares.

First, we normalize the photometric data (PDC Flux) of the 8964 M-type stars downloaded from the Kepler and K2 missions using the following two formulas:

$$\begin{eqnarray}&&{F}_{\mathrm{average}}=\displaystyle \frac{{F}_{\mathrm{PDC}}(\max )+{F}_{\mathrm{PDC}}(\min )}{2},\end{eqnarray} \tag{ 2 }$$

where F_average is the average value of the PDC Flux of each data set downloaded from the Kepler or K2 missions, F_PDC(max) is the maximum value of the PDC Flux of each data set, and F_PDC(min) is the minimum value.

$$\begin{eqnarray}&&{F}_{\mathrm{norm}}=\displaystyle \frac{{F}_{\mathrm{PDC}}-{F}_{\mathrm{average}}}{{F}_{\mathrm{average}}},\end{eqnarray} \tag{ 3 }$$

where F_norm is the normalized flux, and F_PDC is the PDC Flux of each data set. The top panel of Figure 7 is the normalized light curve. To obtain the stellar background flux (F_background), we use the following formula (4) to calculate the average of the four adjacent points (F_ave) in F_norm:

$$\begin{eqnarray}&&{F}_{\mathrm{ave}}(i)=\displaystyle \sum _{t=0}^{3}\displaystyle \frac{{F}_{\mathrm{norm}}(i+t)}{4},i=1,2,3,\,\ldots ,\,n\end{eqnarray} \tag{ 4 }$$

where i is the serial number of each data point in the data set. Then, we calculate the median absolute deviation (MAD) of the residuals (F_norm − F_ave). Subsequently, we use MAD to scan each point in the data set. If the residuals (F_norm − F_ave) are greater than six times the corresponding MAD, they are treated as outliers and removed from the second four-point average. The schematic of F_background is shown in the middle panel of Figure 7. Finally, we calculate the detrended flux (F_flare) by subtracting F_background from F_norm. The schematic of F_flare is shown in the bottom panel of Figure 7. In the detrended flux (F_flare), the flare can be easily distinguished. Therefore, we can programmatically search for flares.

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After completing the above steps, we use the program to automatically search for the flare candidates from the detrended data. The process of automatically searching for flare candidates is as follows. First, the standard deviation (σ) of each detrended data set is calculated. Then we search for the phase of sudden brightness increase from the detrended data set, and we record the peak of the phase as A, and the point after the peak as B. If the value of A is greater than three times the corresponding σ and the value of B is greater than two times the corresponding σ, our program will list the phase as a flare candidate and display the corresponding sudden brightness increase in F_norm in the form of a picture. At the same time, the program will automatically determine the start and end times of the flare candidate by the following two criteria: (1) at the impulse increase phase of the flare candidate, a point below 5% of the peak flare amplitude is defined as the starting point of the flare; and (2) at the gradual decay phase, a point below 5% of the peak flare amplitude is defined as the flare end moment.

In order to accurately search for flares, we visually inspect the pictures of the flare candidates automatically searched by the program. In the automatic detection of flares, the light curves of some eclipsing binaries and some transit events can become contaminated (Shibayama et al. 2013). These contaminants will be removed during visual inspection of the flare candidates. After visual inspection, the contaminants we removed accounted for about 20% of the total flare candidates.

After the above flare search steps, we finally detect 17,432 flares (flare energy > 10³⁰ erg) from 4143 M-type stars, including 9041 flares from 578 M-type stars from the Kepler mission, and another 8391 flares from 3565 M-type stars from the K2 mission. That is to say, among our 8964 M-type stars, 4143 are flare stars, accounting for about 46.2%. We list the number of flares for each source in the eleventh column of Table 6. In addition, we list the parameters of these 17,432 M-type stellar flares in Table 7, where the first column is the Kepler or K2 ID, the second column is the flare start time, the third column is the flare peak moment, the fourth column is the flare end time, the fifth column is the flare duration, and the sixth column is the amplitude of the flare. We also plot eight M-type stellar flares as an example in Figure 8. In comparison, Yang et al. (2017) studied the LC data of 4664 M-type stars in the Kepler field and found that 540 of them are M-type flaring stars. Chang et al. (2017) studied 54 M-type stars by combining LAMOST spectra with Kepler LC data and found that 21 of them are flaring stars. Doyle et al. (2018) studied the SC data of 34 K2 M-type stars and found that 31 of them are flaring stars. Schmidt et al. (2019) found 47 M-type flaring stars from the ASAS-SN data. Günther et al. (2019) found 632 M-type flaring stars from the First Tess Data Release. Therefore, the 4143 M-type flaring stars we found constitute the largest M-type flaring star sample found so far.

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In this work, we use the following equation to calculate the flare energy:

$$\begin{eqnarray}&&{E}_{\mathrm{flare}}=4\pi {R}_{* }^{2}{\sigma }_{\mathrm{sb}}{T}^{4}\int {F}_{\mathrm{flare}}(t){dt}(\mathrm{erg}),\end{eqnarray} \tag{ 5 }$$

where σ_sb is the Stefan–Boltzmann constant, F_flare is the normalized flare flux (the flare amplitude is the peak value of this parameter), R_* is the stellar radius, and T is the stellar effective temperature; these last two parameters come from the stellar parameter catalog of Kepler and K2 (Huber et al. 2014, 2016). The energy of each flare is listed in the last column of Table 7. Figure 9 represents the frequency distribution of flares in M-type stars as a function of the flare energy. The values of the power-law index α are estimated to be 2.02 ± 0.11, which is very close to the results (α ∼ 2.07 ± 0.35) of Yang et al. (2017). In the study of superflares of G-type stars, the values of the power-law index α obtained by Maehara et al. (2012) were from 2.0 to 2.3, the α value obtained by Shibayama et al. (2013) was 2.2, and the α value obtained by Wu et al. (2015) was 2.04 ± 0.17. By comparing the values of the power-law index α of the M-type star with the α values of the G-type star, we found that their power-law index α is close to 2. Therefore, we concluded that the mechanism of flares on G-type stars may be similar to that of M-type stars. In addition, we also use the Lomb–Scargle method (Lomb 1976; Scargle 1982) to analyze the light curve of the 8964 M-type stars and calculate the corresponding stellar rotational period. First, we use the Lomb–Scargle method (Lomb 1976; Scargle 1982) to calculate the possible rotation periods of each M-type star in each data set (these data sets come from different quarters (Kepler mission) or different campaigns (K2 mission)), and we then average these possible rotation periods to obtain the final rotation period of the corresponding source. The results are listed in the twelfth column of Table 6.

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Stellar flares are a sudden explosion of magnetic energy on the stellar surface, which means that the stellar flare is closely related to the stellar magnetic activity. Figure 10 presents the number distribution of 8964 M-type stars observed simultaneously by LAMOST DR5 and the Kepler or K2 mission, 4143 M-type flaring stars, and flare frequency of M-type stars as a function of spectral type. It can be seen from panel (a) of Figure 10 that the M-type stars (black line) and M-type flaring stars (blue line) are mainly distributed in M0–M3, followed by M4–M7, and M8 and M9 have few. There are many sources of M0 type, and the corresponding M0-type flaring stars are also more abundant. But from the distribution of flaring stars from M1 to M3, and from M4 to M7, we can see that both parts have shown an increasing trend. In addition, from panel (b) of Figure 10, it can be seen from the trend of the flare frequency (red line) that the frequency of flare activity from M0 to M3 is gradually increasing, which is similar to the trend of Hα activity from M0 to M3 in Section 2.3. The flare frequency also shows an increasing trend from M5 to M8. In previous studies, West et al. (2008) analyzed the spectra of 38,000 M dwarfs of SDSS DR5 and found that the magnetic activity distribution of M dwarfs is a function of stellar spectral type, and the magnetic activity ratio increases from M0 to M7. On the other hand, Yang et al. (2017) analyzed the flaring fraction of 540 M dwarfs in Kepler and found that the flaring ratio is a function of the spectral type, and the flaring ratio tends to increase from M0 to M4. Therefore, flaring activity and Balmer lines, Ca ii H&K lines, Ca ii IRT lines, and He i D3 are important parameters for measuring the magnetic activity of M-type stars. But we also need to continue to observe more samples in the future, especially to increase the stellar samples from the M4 to M9 spectral type in order to more clearly understand the magnetic activity of the M-type stars.

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Many observations and theoretical models agree that stellar flares have a great influence on the chromospheric activity of stars. During flares, the energy deposition sites in the chromosphere appear as bright ribbons in Hα and other chromospheric diagnostics such as Ca ii H&K, Mg ii H&K, and Ca ii λ8542 Å (Benz 2008; Fletcher et al. 2011; Danilovic et al. 2014; Sindhuja et al. 2019). The results of Karoff et al. (2016) indicated that superflare stars usually have larger chromospheric emission than other stars. In addition, the emission of the He i D3 line is often observed during flares (Zhang & Gu 2008; Zeng et al. 2014). Figure 11 presents the relationship between the EWs of the spectral lines and the flare amplitude in the 4143 M-type flaring stars. Panels (a), (c), and (e) are the relationship between the EW of Hα, Ca ii H, and He i D3 and the maximum flare amplitude of the corresponding star, respectively. Panels (b), (d), and (f) are the relationship between the EW of Hα, Ca ii H, and He i D3 and the average flare amplitude of the corresponding star, respectively. It can be seen from panels (a) and (b) of Figure 11 that the EW of the Hα line has a significant correlation with the maximum flare amplitude and the average flare amplitude of the corresponding M-type star. In these two panels, we attempt to use parabola to fit the maximum flare amplitude and the average flare amplitude with Hα EW, and the fitting parameters are listed in the corresponding panel. Chang et al. (2017) studied 54 M-type stars observed by Kepler and LAMOST and agreed that there is a strong correlation between the EW of Hα and its corresponding maximum flare amplitude. In our work, the research sample of M-type flaring stars has been greatly increased to 4143, and study of the Hα EW and the average flare amplitude has been added, further confirming that there is a significant correlation between the EW of Hα and the flare amplitude. It can be seen from panels (c) and (d) of Figure 11 that there is also a correlation between the EW of Ca ii H and the corresponding flare amplitude, but not as obvious as the relationship between the Hα EW and the flare amplitude, which may be because the S/N near the Ca ii H line is lower than the S/N near the Hα line in the LAMOST spectrum. In panels (c) and (d), we also attempt to use a parabola to fit the maximum flare amplitude and the average flare amplitude with Ca ii H EW, and the fitting parameters are listed in the corresponding panels. However, there is no obvious correlation between the EW of He i D3 and the flare amplitude from panels (e) and (f) of Figure 11. Libbrecht et al. (2019) observed that both the absorption and emission of the He i D3 line can be present in stellar flares. They believe that whether there is a He i D3 absorption or emission signal does not depend on the intensity of the flare, but on the physical conditions that govern the local plasma and line formation mechanisms.

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Stellar rotation plays a significant role in the generation of stellar magnetic fields (Skumanich 1972; Parker 1979). The combination of stellar rotation and convective motion generates a strong magnetic field inside the star and produces different magnetic activity phenomena, including stellar flares and the emission of characteristic magnetic activity lines, such as the Balmer lines and Ca ii H&K lines (Suárez Mascareño et al. 2016; Yang et al. 2017; Mittag et al. 2018). In Section 3.3, we have used the Lomb–Scargle method to analyze the light curve of 8964 M-type stars and obtained the rotation period of 8028 M-type stars. Combining the analysis results of the 8028 M-type stellar LAMOST spectra, we present the relationship between the rotation period and the EWs of Hα, Ca ii H, and He i D3 in panels (a)–(c) of Figure 12, respectively. From panel (a) of Figure 12, it can be clearly seen that the EW value of Hα gradually decreases as the rotation period increases. Therefore, we try to linearly fit it, as shown by the red line in panel (a). The correlation between the EW of Ca ii H and He i D3 and the rotation period is also similar. We performed a linear fitting on them, and the results are shown in the red line in panels (b) and (c). The relationship between the EW of Hα, Ca ii H, and He i D3 and the rotation period presented by panels (a)–(c) of Figure 12 can be explained by the influence of the stellar rotation on the stellar magnetic activity. In addition, we also discuss the relationship between the stellar flare amplitude and the rotation period. Panels (d) and (e) of Figure 12 present the flare amplitude of 3811 M-type flaring stars as a function of the rotation period. It can be seen from panels (d) and (e) that the normalized maximum flare amplitude value and average flare amplitude value of the M-type flaring star decrease with the increase in the stellar rotation period. We linearly fit their trends, and the results are shown by the red lines in the panels. The relationship between the flare amplitude and the rotation period shown in panels (d) and (e) can be explained by the influence of the stellar rotation period on the stellar flare. Combining this with Section 4.2, we can conclude that the stellar flare is affected by both the stellar magnetic activity and the stellar rotation period.

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In this section, we define the stellar flare time frequency as the total duration of the stellar flares divided by the observation time of the corresponding star. Panel (a) of Figure 13 presents the relationship between the stellar flare time frequency and the Hα EW in 4143 M-type flaring stars. The red dashed line in panel (a) is the boundary line of the flare time frequency as a function of the Hα EW. To the left of the red boundary (Hα EW < 0.75 Å), the average of the logarithm of the flare time frequency is −2.7504, and the average of the logarithm of the flare time frequency is −2.3609 on the right side of the red boundary (Hα EW > 0.75 Å). West et al. (2011b) defined magnetically active M-type stars as stars with a detectable Hα emission line in their spectra, and their statistical results indicated that the Hα EW activity criterion is 0.75 Å. The results in panel (a) of Figure 13 confirm that Hα EW = 0.75 Å can be used as a criterion for determining the magnetic activity of M-type stars. Although the M-type star may have flares when Hα EW is less than 0.75 Å, the flare time frequency is statistically smaller than the flare time frequency of Hα EW greater than 0.75 Å. Panel (b) of Figure 13 presents the relationship between the flare time frequency and the rotation period in 3811 M-type flaring stars with rotation period. The red dashed line in panel (b) is the boundary line of the flare time frequency as a function of the rotation period. To the left of the red boundary (P_rot < 10 days), the average of the logarithm of the flare time frequency is −2.3724, and the average of the logarithm of the flare time frequency is −22.7632 on the right side of the red boundary (P_rot > 10 days). When Maehara et al. (2012) and Shibayama et al. (2013) studied superflares on solar-type stars, they all took the stellar rotation period equal to 10 days as the threshold of the superflare frequency. The statistical results of panel (b) in Figure 13 confirm that the stellar rotation period equals 10 days, which can be used as a threshold for flare activity in M-type stars. In addition, we also find an interesting result: using the Hα EW equal to 0.75 Å and using the stellar rotation period equal to 10 days as the threshold for the M-type stellar flare time frequency are almost equivalent, because when Hα EW is greater than 0.75 Å, the average of the logarithm of the flare time frequency (−2.3609) is very close to the average of the logarithm of the flare time frequency (−2.3724) with the rotation period less than 10 days, and the result (−2.7504) when the Hα EW is less than 0.75 Å is also very close to the result (−2.7632) of the rotation period greater than 10 days.

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