A Combined Chandra and LAMOST Study of Stellar Activity

To estimate the R_X ($={L}_{{\rm{X}}}/{L}_{\mathrm{bol}}$) of different stellar types, we first cross-matched the Chandra point-source catalog (Wang et al. 2016b) and the LAMOST database (DR4), using a radius of 3''. This led to 2638 unique Chandra sources with 3502 LAMOST spectral observations. To calculate the likelihood of mismatch, we shifted the positions of Chandra sources by 1' and cross-matched them with the LAMOST catalog again using the same radius. In this case, we obtained 76 matches, and we conclude the likelihood of mismatch is about 2.88%.

In order to have a reliable estimation of stellar parameters, we only used spectra with signal-to-noise ratio (S/N) higher than 7 in the r band. There are four kinds of classes in the LAMOST database, including "STAR," "GALAXY," "QSO," and "Unknown," and we excluded those objects with the latter three classifications. In addition, some objects classified as "STAR" are actually globular clusters or galaxies. To exclude them, we cross-matched the sample sources with the SIMBAD database⁸ and checked the spectra by eye. Young stellar objects were also excluded following SIMBAD classification (e.g., young stellar object, pre-main-sequence star, Orion type, T Tau star, Herbig–Haro Object, and Herbig Ae/Be star). In summary, up to 2023 LAMOST observations were excluded, including 582 observations with low S/N, 311 observations classified as "Unknown," 41 observations of 31 globular clusters, 757 observations of 648 galaxies and QSOs, 254 observations of 179 young stellar objects, 71 observations of 31 binaries, and seven observations of four other sources (one planetary nebula, two white dwarfs, and one cataclysmic variable). This led to 1086 unique X-ray sources with 1479 spectral observations.

The subarcsecond spatial resolution of Chandra allows X-ray sources to be matched unambiguously to multiwavelength counterparts. Our primary database is the Chandra point-source catalog (Wang et al. 2016b), which includes 363,530 source detections, belonging to 217,828 distinct X-ray sources. To determine the photon counts of the sample sources, first we used wavdetect to construct the 3σ elliptical source region for each detection, by fitting a 2D elliptical Gaussian to the distribution of (observed) counts in the source cell. The broadband count rate (0.5–7 keV) and flux were then extracted with srcflux, which determines the net count rate (NET_RATE_APER) and model-independent flux (NET_FLUX_APER) given source and background regions. We defined the background region as an elliptical annulus around the source region, with the inner/outer radii as 2/4 times the radii of the source ellipse.

Some sources in our sample were observed more than once by Chandra. In some cases, a source was detected by wavdetect in one observation but not detected in another observation. We calculate an upper limit using the 4σ detection threshold, which is a function of the off-axis angle and the background level (see Wang et al. 2016a for details).

The LAMOST (also called the Guo Shou Jing Telescope) is a reflecting Schmidt telescope with an effective aperture of 4 m and a field of view of 5° (Cui et al. 2012; Zhao et al. 2012). Four thousand fibers are configured on the focal surface, which enables the observation of up to 4000 objects simultaneously. The spectral resolution is R ∼ 1800 over the wavelength range of 3690–9100 Å. After a five-year regular survey⁹ started in 2012, more than eight million spectra have been obtained. Stellar parameters, including effective temperature, surface gravity, metallicity, radial velocity, distance, and reddening, have been estimated by several studies (e.g., Wu et al. 2011; Wang et al. 2016; Xiang et al. 2017b). Such parameters can be used in many fields and may eventually revolutionize our knowledge of stellar evolution and the structure of the Milky Way. In this paper, we used the spectra from the DR4 database.

We first collected the V-band magnitude from the UCAC4 catalog (Zacharias et al. 2013). For objects without a UCAC4 V-band magnitude, we calculated it using the g and r magnitudes from Pan-STARRS (Flewelling et al. 2016), following Jester et al. (2005):

$$\begin{eqnarray}&&V=g-0.59\times (g-r)-0.01.\end{eqnarray} \tag{ 1 }$$

There is a small systematic offset (∼0.1 mag) between the V magnitudes of UCAC4 and Pan-STARRS, estimated using common objects in the sample (Figure 1).

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In summary, there are 484 X-ray sources with complete stellar parameters (i.e., V mag, reddening E(B − V), effective temperature T_eff, surface gravity log g, and distance). We listed X-ray properties in each Chandra observation for the sample sources in Table 1. We also listed LAMOST parameters for the sample stars in Table 2, including the subclass classification from Luo et al. (2015); the T_eff, log g, $E(B-V)$, [Fe/H], [α/Fe], and distance from Xiang et al. (2017b); and the age and mass from Xiang et al. (2017a). Figure 2 shows the sky distribution of the sample sources in Galactic coordinates.

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Table 1.  X-Ray Information for the Stars in Our Sample

Object	ObsID	Detector	Exp. Time	MJD	VigF	Count Rate	Absorbed Flux	Flag
			(s)			(ks−1)	(10−15 erg cm−2 s−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
J001208.1+503015	1864	s3	5876.3	51898	0.947	${1.53}_{+1.06}^{-0.75}$	${3.3}_{+2.3}^{-1.6}$	⋯
J001313.2+000250	4829	s3	6658.7	53183	0.975	${0.58}_{+0.66}^{-0.37}$	${4.2}_{+4.9}^{-2.8}$	⋯
J002756.6+261651	14012	i3	21692.2	56194	0.762	${16.0}_{+1.4}^{-1.4}$	${184}_{+16}^{-16}$	⋯
	3249	i3	9977.0	52451	0.975	${34.7}_{+3.1}^{-3.1}$	${259}_{+23}^{-23}$	⋯
J003254.4+393821	12991	s2	34587.1	55738	0.636	${0.18}_{+0.16}^{-0.1}$	${3.2}_{+2.7}^{-1.8}$	⋯
	9525	s3	34732.2	54739	0.841	${0.27}_{+0.19}^{-0.13}$	${1.61}_{+1.11}^{-0.80}$	⋯
J003802.9+400824	2046	s2	14765.9	51853	0.579	${1.36}_{+0.65}^{-0.53}$	${18.8}_{+9.0}^{-7.3}$	⋯
J003823.9+401250	2046	s2	14765.9	51853	0.67	${28.5}_{+2.3}^{-2.3}$	${176}_{+14}^{-14}$	⋯
	2048	s3	13772.2	52093	0.513	${19.6}_{+2.0}^{-2.0}$	${93.5}_{+9.4}^{-9.3}$
J003831.2+401711	2046	s3	14762.8	51853	0.994	${0.8}_{+0.46}^{-0.33}$	${1.59}_{+0.91}^{-0.66}$	⋯
	2047	s3	14585.9	51974	0.974	${1.02}_{+0.5}^{-0.38}$	${2.9}_{+1.5}^{-1.1}$	⋯
	2048	s3	13772.2	52093	0.924	${0.35}_{+0.34}^{-0.21}$	${1.01}_{+1.01}^{-0.61}$	⋯
J004118.6+405159	2049	s2	14568.4	51853	0.564	${8.6}_{+1.3}^{-1.3}$	${50.4}_{+7.7}^{-7.6}$	⋯
	2902	i0	4695.0	52614	0.798	${7.3}_{+2.3}^{-1.9}$	${57}_{+18}^{-15}$	⋯
J004301.5+411052	10551	i2	3964.7	54840	0.805	<2.26	<29.5	u

Note. The columns are: (1) object; (2) observation ID; (3) the detector on Chandra ACIS; (3) exposure time after deadtime correction; (4) Modified Julian Date for the beginning point of the observation; (6) vignetting factor; (7) net count rate in the 0.5–7 keV; (8) absorbed X-ray flux in the 0.5–7 keV; (9) "u" means the source is not detected in this observation, and an upper limit is estimated.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 2.  LAMOST Information for the Stars in Our Sample

Object	Subclass	${T}_{\mathrm{eff}}$	log g	$E(B-V)$	[Fe/H]	[α/Fe]	Distance	Age	Mass
		(K)					(pc)	(Gyr)	(M⊙)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
J001208.1+503015	G2	6010 ± 61	4.56 ± 0.11	0.07	−0.07 ± 0.06	0.04 ± 0.1	265 ± 26	...	...
J001313.2+000250	F6	6340 ± 117	4.3 ± 0.22	0.09	0.08 ± 0.17	0.15 ± 0.14	641 ± 218	...	...
J002756.6+261651	G9	5388 ± 146	4.33 ± 0.11	0.04	−0.05 ± 0.08	0.12 ± 0.1	147 ± 13	...	...
J003254.4+393821	K1	5446 ± 211	4.41 ± 0.11	0.08	0.31 ± 0.11	...	723 ± 127	...	...
J003802.9+400824	G2	5871 ± 60	4.61 ± 0.11	0.05	0.02 ± 0.06	0.06 ± 0.1	272 ± 23	...	...
J003823.9+401250	F9	5729 ± 60	3.38 ± 0.09	0.06	−0.04 ± 0.1	0.04 ± 0.1	1286 ± 138	...	...
J003831.2+401711	G8	5259 ± 121	4.03 ± 0.18	0.1	−0.01 ± 0.14	0.03 ± 0.11	1337 ± 364	...	...
J004118.6+405159	G3	5732 ± 68	4.36 ± 0.11	0.11	−0.13 ± 0.09	0.04 ± 0.1	506 ± 87	12.0 ± 2.9	0.91 ± 0.06

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Figure 3 displays the distributions of L_X and R_X for stars from spectral type A through M. Most M-type stars in LAMOST have poorly constrained values of log g and distance. Therefore, only a few M-type stars were included in our sample.

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The L_X distribution of late-type stars (G type or later) extends to significantly lower luminosities than early-type stars. The F, G, and K stars show a fairly wide range of emission levels, with X-ray luminosity ranging from 10²⁷ to 10³³ erg s⁻¹. In these types, giants are clearly X-ray brighter than dwarfs. The F, G, and K giants are the brightest X-ray sources in our sample.

In general, late-type stars have more active ones (higher R_X) than early-type stars. Considering the similarity with solar emission, such as thermal emission-line X-ray spectrum and flares, it was suggested that X-ray emission from late-type stars originates from stellar coronae that are similar to the Sun, although sometimes much more active (Peres et al. 2000). The R_X values exhibit a wide range for stars with similar effective temperatures and surface gravities (also see Figure 4). This may represent an intrinsic range of stellar activity, although it may suffer from some uncertain factors, e.g., solar-like cyclic behavior due to dynamos (Baliunas et al. 1995; Oláh & Strassmeier 2002), long-term variation of stellar activity such as the Maunder minimum phase of solar-type stars (Baliunas & Jastrow 1990; Henry et al. 1996; Saar et al. 1998; Gray et al. 2003), and variable X-ray emission (Soderblom 2010). The R_X ranges for each stellar type found in this paper are consistent with those observed by previous studies (Pizzolato et al. 2003; Mamajek & Hillenbrand 2008; Wright et al. 2011). This suggests that, as expected, the vast majority of these LAMOST spectra are Chandra source counterparts.

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There are about 20 A-type stars showing X-ray activity. Generally, A-type stars are thought to be X-ray dark, as none of the X-ray production mechanisms operating in early- or late-type stars, i.e., strong winds and convection zones, ought to be valid for A-type stars. Previous studies suggested that the X-ray-emitting A types are mostly pre-main-sequence (i.e., Herbig Ae/Be) stars or binaries (Zinnecker & Preibisch 1994; Pease et al. 2006; Schröder & Schmitt 2007). In the binary scenario, the X-ray emission is from an unresolved low-mass companion, often of spectral type dK or dM (Golub et al. 1983; Panzera et al. 1999). We have excluded the identified pre-main sequences and binaries (eclipsing binaries and spectral binaries) from our sample. This does not imply that the X-ray emission from the remaining A-type stars is intrinsic from the stars, because even those sources could still be unrecognized binaries.

From an inspection of the X-ray properties of our sample stars across the Hertzsprung–Russell diagram (Figure 5, left panel), we find that M dwarfs have the highest R_X but the lowest L_X values; this confirms the trend previously noted by Güdel (2004). There are 51 stars with log g smaller than 3.5, which can be classified as giants. This confirms the conclusion of previous studies (Simon & Drake 1989; Aurière et al. 2015) that late-type giants can have (high) stellar activities. Among these giants with high R_X values, some are very interesting. For example, the object J194707.6+445251 is an F5 star with the highest log R_X value (≈−1.2) among giants. It has a radio counterpart within a 15 radius, called 4C 44.35 (Vollmer et al. 2010). The possible radio emission suggests the presence of magnetic fields and high-energy electrons (Güdel 2002). However, the high X-ray and radio emission are too high for a "normal" F-type giant. The X-ray and radio emission are possibly a chance coincidence with one background AGN, or from the accretion process, which means the F star is in an accreting binary.

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Stellar X-ray emissions are variable (Soderblom 2010), and X-ray fluxes vary for any spectral type (Figure 6). In our sample, 126 stars were observed more than once by Chandra. About half of them have an X-ray flux variation higher than 2. The object J131159.5-011705 has the highest flux variation (∼36), and the variation is not due to a flare event. The flux variation would affect the correlations between R_X and other activity indicators and stellar parameters, which may produce dissimilar results for nonsimultaneous observations.

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The R_X distributions of G and K stars show clear bimodality (Figure 3). For G stars, there are more inactive stars than active ones, while for K stars, more active stars are apparent. We used double-Gaussian functions to fit these R_X distributions: the peaks for the G type are −4.35 and −3.3, with σ being 0.47 and 0.33; the peaks for the K type are −4.49 and −3.18, with σ being 0.64 and 0.27.

The Chandra data are drawn from targeted observations, which means that our sample is inhomogeneous. Therefore, it is necessary to check whether the bimodality is a result of the sampling effect, e.g., two populations of stars in different parts of the Galaxy. We divided the G- and K-type stars into two subsamples, corresponding to stars with log R_X > −3.8 and log R_X < −3.8. To identify whether one source belongs to the thin disk or thick disk, we first defined one probability assuming that the velocities (U_LSR, V_LSR, W_LSR) follow a 3D Gaussian distribution (Guo et al. 2016):

$$\begin{eqnarray}&&\mathrm{Prob}=c\cdot \exp \left\{-\displaystyle \frac{{U}_{\mathrm{LSR}}^{2}}{2{\sigma }_{U}^{2}}-\displaystyle \frac{{({V}_{\mathrm{LSR}}-{V}_{\mathrm{asym}})}^{2}}{2{\sigma }_{V}^{2}}-\displaystyle \frac{{W}_{\mathrm{LSR}}^{2}}{2{\sigma }_{W}^{2}}\right\},\end{eqnarray} \tag{ 5 }$$

where $c={(2\pi )}^{-3/2}{({\sigma }_{U}{\sigma }_{V}{\sigma }_{W})}^{-1}$ normalizes the expression. V_asym is the asymmetric drift, and σ_U, σ_V, and σ_W are the velocity dispersions in three dimensions, all of which vary with different components (Guo et al. 2016). Then, we defined the probability ratio of belonging to the thin disk or thick disk as

$$\begin{eqnarray}&&{f}_{\mathrm{thin}/\mathrm{thick}}=\displaystyle \frac{{\mathrm{Prob}}_{\mathrm{thin}/\mathrm{thick}}}{{\mathrm{Prob}}_{\mathrm{thin}}+{\mathrm{Prob}}_{\mathrm{thick}}+{\mathrm{Prob}}_{\mathrm{halo}}}.\end{eqnarray} \tag{ 6 }$$

We classified a star as being a good candidate star in the thin disk or thick disk when this ratio is larger than 80%. Figure 7 shows the classification of the sample stars in the [α/Fe]–[Fe/H] diagram. The circles and triangles represent stars classified as in the thin disk or thick disk, using the kinematic methods. The green dashed line indicates the approximate border of thin disk and thick disk from metallicity (Masseron & Gilmore 2015). Most stars, both active and inactive, are identified as being in the thin disk. Therefore, both the metallicity and velocity information show that the bimodality is not caused by a different location of the stars, i.e., the sampling effect.

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Previous studies with Ca ii HK, Hα, and X-ray emission have found the bimodality of stellar activity, with an active and inactive peak (e.g., Stocke et al. 1991; Henry et al. 1996; Wright 2004; Jenkins et al. 2006, 2008, 2011; Agüeros et al. 2009; Martínez-Arnáiz et al. 2011; Pace 2013). The bimodality is explained as one saturated and one nonsaturated subpopulation (Martínez-Arnáiz et al. 2011; Katsova et al. 2016) and can further be explained as one young and one old subpopulation. The latter one is often thought to be inactive in chromospheric and X-ray emission, as the magnetic activity decreases simultaneously as the rotation decelerates with age (e.g., Mamajek & Hillenbrand 2008; Katsova & Livshits 2011).

The coronal temperature is known to be positively correlated with X-ray luminosity and stellar activity (e.g., Vaiana 1983; Güdel 2004; Jeffries et al. 2006). The cause of this relation between coronal temperature and luminosity could be that they are both functions of magnetic activity (Güdel 2004). A more efficient dynamo inside active stars (for example, because of faster rotation) produces stronger magnetic fields in the corona and, consequently, a higher rate of field line reconnections and flares. This results both in a larger density of energetic electrons in the corona and in higher temperatures. Therefore, we suggest that the double-peaked distribution of R_X represents a double-peaked distribution of heating rates, and therefore coronal temperatures. We do not have direct measurements of coronal temperatures for the stars in our sample; however, we take the X-ray hardness ratio as a proxy for the coronal temperature, because the thermal plasma emission shifts progressively to higher photon energies for higher plasma temperatures. There is a clear positive correlation between R_X and HR (Figure 8), which means stronger X-ray emitters (higher R_X) have higher coronal temperatures. Chandra sources are sometimes empirically classified into three groups: super-soft, quasi-soft, and hard, based on their hardness ratios (Di Stefano & Kong 2003). Applying the same classification to our sample stars, we find (Figure 9) that the higher activity peak of the R_X distribution in G- and K-type stars is dominated by hard sources, and the lower activity peak by quasi-soft and super-soft sources.

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Previous studies have shown that X-ray-luminous, single, low-mass stars are the youngest (e.g., Telleschi et al. 2007), while older stars have typically lower X-ray luminosities (Flaccomio et al. 2003; Feigelson et al. 2004; Wright et al. 2010). This means that the X-ray emission decreases over the lifetime of the star. This is explained as a consequence of the stellar dynamo (e.g., Skumanich 1972; Flaccomio et al. 2003; Feigelson et al. 2004; Cranmer & Saar 2011): over time, a magnetized stellar wind driven by the stellar dynamo leads to the loss of angular momentum and the rotational spin-down of the star, which reduces velocity shear at the tachocline between the radiative and convective zones, resulting in a reduced magnetic field (and therefore reduced activity).

Recently, many new results have been obtained about the age–activity relation. An L-shaped relation was found between the chromospheric activity (${R}_{\mathrm{HK}}^{{\prime} }$) and age (Pace 2013): the chromospheric activity decreases with age for stars younger than 2 Gyr, and there is no decay of chromospheric activity after about 1.5–2 Gyr (Lyra & Porto de Mello 2005; Sissa et al. 2016). However, using SDSS observations of open clusters, Zhao et al. (2013) found that the chromospheric activity gradually declines with age, extending to at least ∼8 Gyr. A steepening of the relation between activity and rotation was reported for stars with ages between 1 and 10 Gyr, but was possibly attributed to a nonrepresentative sample of a few target objects. Several improved age–activity relations extending to old late-type stars have also been obtained (Mamajek & Hillenbrand 2008; Žerjal et al. 2017).

Xiang et al. (2017a) estimated ages and masses for 0.93 million main-sequence turnoff and subgiant stars by matching with stellar isochrones employing a Bayesian algorithm, using effective temperatures, absolute magnitudes, metallicities ([Fe/H]), and α-element to iron abundance ratios ([α/Fe]) deduced from the LAMOST spectra. There are 16 A types, 60 F types, and 34 G types with estimated ages and masses in our sample. We do not find a clear trend between R_X and age (Figure 10). A trough with the lowest R_X values around 2 Gyr is consistent with Pace (2013), but the reason is not clear. There are so few sources younger than 1 Gyr that we cannot confirm whether the activity keeps decreasing over the first several hundred Myr. There is also no clear trend between HR and age.

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The X-ray emission and other activity indicators are proxies of stellar magnetic activity (Testa et al. 2015). The X-ray activity shows a good correlation with the activity indicators of the chromosphere and transition region, e.g., Ca ii HK emission (Vaiana 1983; Schrijver et al. 1992; Sterzik & Schmitt 1997), Mg ii emission of dMe stars (Mathioudakis & Doyle 1989), and Hα and UV emission of M dwarfs (Martínez-Arnáiz et al. 2011; Stelzer et al. 2012, 2013). LAMOST provides us with a good opportunity to study the relation between X-ray activity and Hα emission.

First, we calculated the equivalent widths (EWs) of the Hα emission line. The EW was calculated using the following formula:

$$\begin{eqnarray}&&\mathrm{EW}=\int \displaystyle \frac{f(\lambda )-f(0)}{f(0)}d\lambda ,\end{eqnarray} \tag{ 7 }$$

where f(0) denotes the nearby pseudo-continuum flux.

In order to trace pure chromospheric emission, we calculated the excess EW (hereafter EW') by subtracting a "basal line" of chromospheric emissions (Fang et al. 2018), which is constructed using those stars with Hα absorption lines (EW < 0). Then, we used the CK04 stellar atmosphere models (Castelli & Kurucz 2004) to transform the EWs to stellar surface fluxes of Hα emission lines, ${f}_{{\rm{H}}\alpha ,\mathrm{sur}}$ (Yang et al. 2017). For each star, the model with the most similar T_eff, log g, and [Fe/H] was used. Finally, we determined the flux ratio ${f}_{{\rm{H}}\alpha ,\mathrm{sur}}/{f}_{\mathrm{bol},\mathrm{sur}}$ using ${f}_{\mathrm{bol},\mathrm{sur}}=\sigma \,{T}^{4}$ and calculated the Hα emission index, ${R}_{{\rm{H}}\alpha }={L}_{{\rm{H}}\alpha }/{L}_{\mathrm{bol}}={f}_{{\rm{H}}\alpha ,\mathrm{sur}}/{f}_{\mathrm{bol},\mathrm{sur}}$, to quantify stellar chromospheric activity (Table 4).

Table 4.  Stars with Hα Emission Lines in Our Sample

Object	EWHα	${\mathrm{EW}}_{{\rm{H}}\alpha }^{{\prime} }$	log RHα
	Å	Å
(1)	(2)	(3)	(4)
J004322.0+405752	1.95 ± 0.19	3.43	−3.5 ± 0.05
J004458.9+330431	0.51 ± 0.13	1.93	−3.72 ± 0.08
J013729.6+331230	1.82 ± 0.1	2.63	−3.64 ± 0.1
J022641.9+002811	0.35 ± 0.02	2.2	−3.59 ± 0.04
J024132.9+002053	2.48 ± 0.05	3.42	−3.48 ± 0.08
J033102.5+434758	0.65 ± 0.05	2.41	−3.62 ± 0.06
J034501.5+321050	0.78 ± 0.06	2.39	−3.58 ± 0.07
J034631.1+240702	1.64 ± 0.05	2.45	−3.69 ± 0.1
J040355.6+261652	1.35 ± 0.06	2.13	−3.93 ± 0.11
J054143.2−020353	1.03 ± 0.07	2.46	−3.62 ± 0.06
J054211.2−015943	0.85 ± 0.05	2.33	−3.66 ± 0.07
J063005.3+054540	2.1 ± 0.07	3.19	−3.46 ± 0.04
J063324.2+222837	8.96 ± 0.33	9.64	−3.29 ± 0.11
J071618.9+374451	1.69 ± 0.12	3.07	−3.49 ± 0.05
J095817.1+362216	1.17 ± 0.1	1.9	−3.89 ± 0.11
J095948.8−051413	0.93 ± 0.07	2.26	−3.63 ± 0.06
J111032.9+570633	1.99 ± 0.02	3.5	−3.39 ± 0.03
J111637.2+013249	1.26 ± 0.05	2.29	−3.72 ± 0.07
J122837.2+015720	2.19 ± 0.03	3.16	−3.54 ± 0.08
J131156.7−011311	0.78 ± 0.03	2.25	−3.66 ± 0.07
J203323.0+411222	1.69 ± 0.05	3.25	−3.44 ± 0.03
J203552.9+411503	1.39 ± 0.05	2.31	−3.79 ± 0.08
J223606.9+012603	1.31 ± 0.05	2.53	−3.65 ± 0.04
J224506.0+394016	0.98 ± 0.05	2.4	−3.62 ± 0.07

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Figure 11 shows a general power-law dependence of R_X on R_Hα:

$$\begin{eqnarray}&&\mathrm{log}\,{R}_{{\rm{X}}}=(1.16\pm 0.28)\times \mathrm{log}\,{R}_{{\rm{H}}\alpha }+(1.22\pm 1.03).\end{eqnarray} \tag{ 8 }$$

There are 20 K-type and 4 M-type stars in the R_X–${R}_{{\rm{H}}\alpha }$ diagram. We propose the scatter in the relations is caused by the nonsimultaneous X-ray and Hα observations. However, Stelzer et al. (2013) reported that the scatter may represent an intrinsic range of physical conditions between the corona and chromosphere. It was reported by Martínez-Arnáiz et al. (2011) that in a sample of late-type dwarf active stars with spectral types from F to M, ${F}_{{\rm{X}}}\propto {F}_{{\rm{H}}\alpha }^{1.48\pm 0.07}$. For M dwarfs, Stelzer et al. (2013) derived ${R}_{{\rm{X}}}\propto {R}_{{\rm{H}}\alpha }^{1.90\pm 0.31}$. In this paper, we have obtained that ${R}_{{\rm{X}}}\propto {R}_{{\rm{H}}\alpha }^{1.12\pm 0.30}$, a slightly flatter relation than found in the other studies. The discrepancy may be due to our small sample size. In addition, we are aware that the lack of simultaneous observations of those two intrinsically varying properties (coronal and chromospheric activities) introduces another source of uncertainty in all of these studies (Martínez-Arnáiz et al. 2011).

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Flares are in the center of the debate on the origin of coronal heating (Audard et al. 2000), with evidence showing that (X-ray or U-band) flares can release a sufficient amount of energy to produce quiescent coronal emission (Doyle & Butler 1985; Skumanich 1985; Pallavicini et al. 1990; Pandey & Singh 2008; Pye et al. 2015). The light curves of flares can help us understand the characteristics of the coronal structures and, therefore, of the magnetic field (Schmitt & Favata 1999; Favata et al. 2000; Reale et al. 2004; Pandey & Singh 2008)

The X-ray emission from flares is indicative of very energetic, transient phenomena, associated with energy release via magnetic reconnection (Pye et al. 2015). X-ray flares differ considerably among different stars (Benz & Güdel 2010), e.g., in shape (Cully et al. 1993; Graffagnino et al. 1995; Osten & Brown 1999), peak luminosity (≈10²⁶–10³³ erg s⁻¹) and total energy (≈10²⁸–10³⁷ erg; Kuerster & Schmitt 1996; Güdel et al. 2002; Osten et al. 2007; Getman et al. 2008), and rise and decay times (Haisch et al. 1987; Tagliaferri et al. 1991; Cully et al. 1994; Benz & Güdel 2010).

We selected stars with X-ray flares by eye (Table 5). The light curves of the flares are clearly variable (Figure 12). Some flares are like compact solar flares with short duration (e.g., J122837.2+015720); some flares show quite long duration (e.g., J055207.8+322639); some flares have a low rise shape (e.g., J063324.2+222837); some flares have strong absorption (e.g., J193015.7+493209). The morphological differences indicate different processes of energy release in these flares (Pandey & Singh 2008). For example, compact flares suggest a quick energy release, while a prolonged energy release is preferred for long decay flares (Pallavicini et al. 1977; Pallavicini et al. 1988).

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