Stellar X-Ray Activity Across the Hertzsprung–Russell Diagram. I. Catalogs

The unprecedented subarcsecond spatial resolution of the Chandra telescope allows X-ray sources to be unambiguously matched to their optical counterparts. Our primary database is the Chandra point-source catalog (Wang et al. 2016), which includes 217,828 distinct X-ray sources with 363,530 detections. The Gaia DR2 released about 1.33 billion celestial objects with estimated distances (Bailer-Jones et al. 2018), which can be used to evaluate X-ray luminosity and bolometric luminosity of the stars. First, we cross-matched the Chandra point-source catalog and the Gaia DR2 catalog, using a radius of 3''. In some cases there are multiple matches, and we only selected the closest neighbor as the counterpart. This led to more than 60 thousand unique sources with Chandra detections.

Second, we used a machine learning method (Bai et al. 2019) to classify possible contamination from non-stellar objects, such as galaxies and quasi-stellar objects (QSO). The classifier uses nine colors (i.e., g − r, r − i, i − J, J − H, $H-{K}_{{\rm{s}}}$, ${K}_{{\rm{s}}}-W1$, W1 − W2, ${\rm{w}}1\mathrm{mag}\_1-{\rm{w}}1\mathrm{mag}\_3$, ${\rm{w}}2\mathrm{mag}\_1\,-{\rm{w}}2\mathrm{mag}\_3$), spanning from optical to far-infrared bands. In order to obtain these colors, we cross-matched the ∼60 thousand sources with the Pan-STARRS DR1 (hereafter PS1; Chambers et al. 2016), UCAC4 (Zacharias et al. 2013), and WISE All-Sky (Cutri et al. 2012) catalogs, by using the TOPCAT⁵ and CasJobs.⁶ The matching radius is 3'', and only the closest counterpart was selected for the multiple coincidences. We first collected the g, r, and i magnitudes from the PS1. For objects without PS1 observations, we derived the magnitudes from the UCAC4 catalog. The WISE All-Sky data release includes the 2MASS magnitudes and WISE magnitudes. The W1 and W2 magnitudes are the w1(2)mpro magnitudes in the WISE catalog; the w1(2)mag_1 and w1(2)mag_3 are magnitudes measured with circular apertures of radii of 55 and 11'', and the differential aperture magnitude is expected to have different distributions for point and extended sources (Bilicki et al. 2014; Krakowski et al. 2016). Furthermore, we set the saturation thresholds for these different surveys: W2 = 6.7 mag for WISE⁷ ; K_S = 8 mag for 2MASS⁸ ; g = 14 mag, r = 14 mag;, and i = 14 mag for PS1⁹ ; V = 7 mag for UCAC4.¹⁰ We also set the detection limits using the g, r, and i magnitudes (∼23.2 mag) for PS1 observations, and the V magnitude (∼17 mag) for UCAC4 observations, respectively. This method led to more than 11,000 stars, ∼800 galaxies, and ∼1100 QSOs (Figure 1). Figure 2 shows the sky distribution of the extragalactic sources in Galactic coordinates. About 490 galaxies lie across the Galactic plane (−10° ≤ l ≤ 10°). Among these sources, about 150/250 sources have a neighbor within 3''/5'', indicating their photometry may suffer from a contamination by nearby objects. Considering that the crowding can result in inaccurate photometry that may move the color index toward the blue band (Gaia Collaboration et al. 2018, 2018), the classification of these galaxies may be unreliable.

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Third, we collected the extinction from the Pan-STARRS 3D dust map, which is constructed with the high-quality stellar photometry of 800 million stars from PS1 and 2MASS (Green et al. 2015). The reddening values E(B − V) were used to estimate the absolute optical magnitude and the unabsorbed X-ray flux.

Fourth, in order to have accurate distance estimations, we excluded the objects with relative parallax uncertainties larger than 0.2, although this may introduce biases toward bright nearby sources in our sample (Luri et al. 2018). Furthermore, we removed those objects with SIMBAD classification as white dwarf, planet nebula, binary, galaxy, and QSO. Finally, there are more than 5900 stars left in our sample.

We extracted the X-ray information (i.e., net counts and count rates in different bands) for each star in each detection from the Chandra point-source catalog (Wang et al. 2016). The hardness ratios (HR1 and HR2) for each detection were then calculated with the net photon counts for the three bands defined by Prestwich et al. (2003), i.e., the soft band (S: 0.3–1.0 keV), the medium band (M: 1.0–2.0 keV), and the hard band (H: 2.0–8.0 keV). The HR1 and HR2 were defined as (M–S)/(M+S) and (H–M)/(H+M), respectively.

We converted the net count rate (0.3–8 keV) into unabsorbed flux using PIMMS¹¹ with a model called Astrophysical Plasma Emission Code (APEC). APEC is a model of emission spectrum from collisionally ionized diffuse gas in XSPEC and is often used to describe the X-ray emission of stars. First, we constructed spectral tracks of absorbed APEC spectra with the coronal temperature covering from $\mathrm{log}T({\rm{K}})=6$ to 8.5 with ΔlogT = 0.1, and the hydrogen column density ${N}_{{\rm{H}}}\ ({\mathrm{cm}}^{-2})$ covering from 0 to 10²³ with ${\rm{\Delta }}\mathrm{log}{N}_{{\rm{H}}}=0.1$. The HRs of each model in the resulting grid were calculated. Second, we derived the coronal temperature, for each detection of each object in our catalog, by selecting the model with closest HRs (Figure 3). The models (solid lines) can not predict temperatures of the sources with low HR1 values (in the left part of Figure 3). The low HR1 values can be mostly due to statistical fluctuation of the photons, especially for low-count detections. A few sources show a soft excess (i.e., an additional thermal component) in their spectra, which can also reduce HR1 values (see Appendix A). Here we simply extended each APEC model (dashed line) to derive the temperatures of those sources. Third, we determined the observation cycle (or AO) for each detection, which was used in PIMMS to convert the count rate to flux for Chandra archive data. Finally, for each detection, we calculated the flux using the APEC model (Z = 0.5 Z_⊙) with individual coronal temperature and N_H, the latter of which was estimated from individual optical extinction (Zhu et al. 2017):

$$\begin{eqnarray}&&{N}_{{\rm{H}}}({\mathrm{cm}}^{-2})=2.19\times {10}^{21}{A}_{V}(\mathrm{mag}).\end{eqnarray} \tag{ 1 }$$

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Before we estimated an averaged quiescent flux for each star, we removed those detections with X-ray flares. We first used the standard nonparametric Kolmogorov–Smirnov (K-S) test to quantitatively test the source variability during each an observation. The observations with variable light curve were identified with K-S probability P_K−S < 0.1. Then we applied Bayesian block analysis (astropy.stats.bayesian_blocks; Scargle et al. 2013) to these observations to detect flares (Figure 4). By operating on unbinned events, this method can detect and characterize local variance in the count rate, and works well in flare identification (e.g., Neilsen et al. 2013; Ponti et al. 2015). The false alarm probability is set as p0 = 0.01. A detailed study of the X-ray flares in our sample is currently in preparation as an independent work. The vignetting-corrected net count rate and unabsorbed flux for each observation are listed in Table 1. Finally, we calculated an exposure-weighted averaged X-ray flux for each star. The X-ray luminosity (L_X) was determined from the unabsorbed averaged flux (f_X) and distance (Table 2).

Table 2.  Main Properties for the Stars in our Sample

Object	R.A.	Decl.	Class	D	E(B − V)	fX	LX	logRX	HR1	HR2
	(o)	(o)		(pc)	(mag)	(10−15 erg cm−2 s−1)	(erg s−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
J000012.8+622947	0.05374	62.49657	G	${899}_{-20}^{+21}$	0.26	11 ± 2	1.0e+30 ± 2.4e+29	−4.0 ± 0.1	0.59 ± 0.22	−0.84 ± 0.29
J000100.1-245742	0.25066	−24.96192	M	${376}_{-19}^{+21}$	0.02	2.8 ± 1.0	4.7e+28 ± 1.6e+28	−3.52 ± 0.15	0.15 ± 0.4	−0.7 ± 0.52
J000101.9-250431	0.25818	−25.07535	Kd	${242}_{-2}^{+2}$	0.01	1.8 ± 0.8	1.3e+28 ± 5.7e+27	−5.0 ± 0.2	0.0 ± 0.57	−0.38 ± 0.74
J000136.1+130639	0.40042	13.11095	Fd	${381}_{-6}^{+7}$	0.08	149 ± 5	2.6e+30 ± 9.6e+28	−3.74 ± 0.02	0.21 ± 0.04	−0.66 ± 0.04
J000238.8+255219	0.66206	25.87221	K	${703}_{-27}^{+30}$	0.03	40 ± 6	2.4e+30 ± 3.6e+29	−2.66 ± 0.07	0.09 ± 0.17	−0.71 ± 0.18
J000611.4+725929	1.54788	72.99162	G	${1680}_{-119}^{+138}$	0.37	17 ± 2	5.7e+30 ± 7.8e+29	−3.09 ± 0.06	−0.01 ± 0.15	−0.42 ± 0.2
J000645.6+730635	1.69003	73.10997	K	${368}_{-3}^{+3}$	0.33	2.4 ± 1.1	3.9e+28 ± 1.8e+28	−4.4 ± 0.2	−0.1 ± 0.52	−0.5 ± 0.83
J000753.8+512400	1.97436	51.40010	Ky	${597}_{-26}^{+28}$	0.12	2.1 ± 0.9	1.5e+29 ± 3.4e+28	−3.48 ± 0.1	−0.27 ± 0.25	−0.47 ± 0.43
J000759.3+512655	1.99728	51.44876	My	${290}_{-10}^{+10}$	0.05	3.8 ± 1.8	7.4e+28 ± 1.0e+28	−3.15 ± 0.06	−0.19 ± 0.15	−0.59 ± 0.24
J000833.8+512412	2.14095	51.40359	Ad	${434}_{-8}^{+8}$	0.09	11 ± 1	4.7e+28 ± 1.6e+28	−6.53 ± 0.15	−0.14 ± 0.38	−0.76 ± 0.61
J000835.3+512142	2.14724	51.36168	Kd	${50}_{-2}^{+3}$	0.01	3.5 ± 0.8	1.1e+27 ± 2.5e+26	−4.81 ± 0.09	0.07 ± 0.24	−0.83 ± 0.29
J000836.5+512616	2.15234	51.43779	G	${1497}_{-100}^{+115}$	0.11	7.3 ± 1.0	4.2e+29 ± 2.1e+29	−4.14 ± 0.22	−0.12 ± 0.56	−0.91 ± 1.17
J000849.5+512514	2.20651	51.42067	G	${268}_{-15}^{+17}$	0.07	205 ± 4	2.1e+29 ± 1.8e+28	−3.97 ± 0.04	−0.11 ± 0.09	−0.77 ± 0.11
J001051.6-120543	2.71524	−12.09552	—	${522}_{-56}^{+70}$	0.02	2.1 ± 0.7	3.1e+29 ± 1.2e+29	−2.66 ± 0.17	0.33 ± 0.69	0.19 ± 0.48
J001116.8-151526	2.82019	−15.25745	—	${375}_{-48}^{+64}$	0.02	3.8 ± 0.8	2.4e+29 ± 5.6e+28	−2.12 ± 0.1	0.33 ± 0.69	0.19 ± 0.48
J001123.9-151541	2.84977	−15.26147	Kd	${533}_{-14}^{+15}$	0.02	1.6 ± 0.8	1.4e+29 ± 4.6e+28	−4.01 ± 0.14	−0.21 ± 0.39	−0.29 ± 0.6
J001144.7+522859	2.93639	52.48312	K	${393}_{-9}^{+10}$	0.12	24 ± 2	3.1e+29 ± 6.1e+28	−3.04 ± 0.08	0.28 ± 0.21	−0.66 ± 0.23
J001145.8-285501	2.94100	−28.91695	M	${129}_{-2}^{+2}$	0.01	9.6 ± 3.7	1.4e+28 ± 6.3e+27	−3.29 ± 0.2	−0.42 ± 0.73	0.41 ± 0.76
J001147.4-152319	2.94754	−15.38880	K	${374}_{-13}^{+13}$	0.01	14 ± 3	2.3e+29 ± 3.0e+28	−3.08 ± 0.06	−0.33 ± 0.14	−0.53 ± 0.23
J001152.5+523845	2.96884	52.64607	F	${1303}_{-69}^{+77}$	0.15	4.1 ± 1.4	2.5e+30 ± 7.5e+29	−3.31 ± 0.13	0.36 ± 0.33	−0.72 ± 0.42

Note. The columns are: (1) Object. (2) Right ascension in degree. (3) decl. in degree. (4) Stellar classification. (5) Distance from Gaia DR2 data. (6) $E(B-V)$ from PS1 3D map. (7) Averaged unabsorbed X-ray flux in the 0.3–8 keV. (8) X-ray luminosity in the 0.3–8 keV. (9) X-ray-to-bolometric luminosity ratio. (10) Hardness ratio ${HR}1=(M-S)/(M+S)$. S/M represents background-subtracted counts in soft (0.3–1 keV) and medium (1–2 keV) bands. (11) Hardness ratio ${HR}2=(H-M)/(H+M)$. M/H represents background-subtracted counts in medium (1–2 keV) and hard (2–8 keV) bands. (This table is available in its entirety in machine-readable and Virtual Observatory (VO) forms in the online journal. A portion is shown here for guidance regarding its form and content.)

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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To derive individual stellar parameters (i.e., effective temperature, surface gravity, and metallicity), our primary choice is the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (hereafter LAMOST, also called the Guo Shou Jing Telescope) database. LAMOST is a reflecting Schmidt telescope, with an effective aperture of 4 m and a field of view of 5 degrees (Cui et al. 2012; Zhao et al. 2012). The LAMOST DR7 data set released more than 14 million spectra, and presented parameter estimations for more than 6 million A, F, G, and K stars. For stars not included in the LAMOST DR7 catalog, we adopted the parameters (teff50, logg50, met50) from Anders et al. (2019), who derived Bayesian stellar parameters and distances for 265 million stars using the code StarHorse, based on the combination of Gaia DR2 and the photometric catalogs of PS1, 2MASS, and AllWISE. There are 447 and 3550 objects in common between our sample and LAMOST DR7 catalog and Anders et al. (2019), respectively. We defined the stars with parameter estimations as the "parameter" sample, and other stars as the "non-parameter" sample. Using the effective temperatures, we classified the "parameter" sample into different subclasses: O type with T_eff ≥ 30,000 K; B type with 10,000 K ≤ T_eff < 30,000 K; A type with 7500 K ≤ T_eff < 10,000 K; F type with 6000 K ≤ T_eff < 7500 K; G type with 5200 K ≤ T_eff < 6000 K; K type with 3700 K ≤ T_eff < 5200 K; M type with T_eff < 3700 K.

We further classified the sample into giants, young stellar objects (YSOs), and dwarfs. First, a star was considered as a giant if the logg value is smaller than that specified in the following algorithm (Ciardi et al. 2011):

$$\begin{eqnarray}\mathrm{log}g\lt \left\{\begin{array}{ll}3.5 & \mathrm{if}\ {T}_{\mathrm{eff}}\geqslant 6000,\\ 4.0 & \mathrm{if}\ {T}_{\mathrm{eff}}\leqslant 4250,\\ 5.2-2.8\times {10}^{-4}{T}_{\mathrm{eff}} & \mathrm{if}\ 4250\leqslant {T}_{\mathrm{eff}}\leqslant 6000.\end{array}\right.\end{eqnarray} \tag{ 2 }$$

This leads to 685 giants and 3312 dwarfs and YSOs. We then cross-matched our sample with the catalogs in Marton et al. (2016, 2019). By using the 2MASS and WISE photometric data, combined with Planck dust opacity values, Marton et al. (2016) presented a catalog of Class I/II and III YSO candidates with the support vector machine method. By adding the Gaia database, Marton et al. (2019) presented a new catalog, classifying more than 100 million sources into four classes: YSOs, extragalactic objects, main-sequence stars, and evolved stars. There are 1100 objects in common between our sample and these catalogs, including 68 giants, 900 YSOs, and 132 dwarfs. For sources in Marton et al. (2019), only those with reliable classification probability (P > 90%) were selected. Sometimes one star has different classifications from above methods, we preferentially adopted the classification by using the logg value from LAMOST DR7 and Anders et al. (2019)), followed by the classification of Marton et al. (2019). We classified a total of 697 giants.

Second, we defined some criteria to classify more new YSOs. As Marton et al. (2019) reported, 99% of the known YSOs are located in the regions where the dust opacity value is higher than 1.3 × 10⁻⁵. Thus we labeled as YSO candidates those sources that meet the requirements: (1) τ > 1.3 × 10⁻⁵ and (2) $J-H\gt 1-(H-{K}_{{\rm{S}}})$ or W1 − W2 > 0.04. Appendix B shows more details about the YSO classification. In addition, considering that there may be many asymptotic giant branch (AGB) stars with W2 − W3 < 1 (Koenig & Leisawitz 2014), we add one criterion W2 − W3 ≥ 1 for the "non-parameter" sample. With the constraints of colors and dust opacity, we classified additional ≈300 stars to be YSOs. We finally categorized 1196 YSOs.

Third, for the "parameter" sample, if they were not classified as YSOs or giants from above steps and if their dust opacity values are lower than 1.3 × 10⁻⁵, we flagged them as main-sequence dwarfs.

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In summary, the giants were collected from the "parameter" sample and Marton et al. (2019); the YSOs were classified from previous studies (Marton et al. 2016, 2019) and our criteria; for the dwarfs, one part was selected from Marton et al. (2019), and the other part was classified from the "parameter" sample. Figure 5 summarizes the steps in a flowchart for reference.

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At the end, we presented stellar classifications for 3005 sources in our sample, including 1196 YSOs, 1112 dwarfs, and 697 giants. Among these sources, 432 YSOs, 1111 dwarfs, and 695 giants have parameter estimations. We defined these 2238 sources as the "A-class" sample, which will be used for following analysis.

X-ray emission is prevalent among almost all stellar classes (e.g., Stocke et al. 1991), although with different mechanisms (see Güdel 2004, for a review). For the hot and massive early-type stars, the emitted X-rays arise from either small-scale shocks in their winds or collisions between the wind and circumstellar material (Lucy & White 1980; Parkin et al. 2009). For late-type stars, the X-ray emission is attributed to the presence of a magnetic corona (Vaiana et al. 1981). The X-ray emission of YSOs may be from accretion shock, star-disk interaction, or solar-like corona (e.g., Preibisch et al. 2005). In this paper, we used the ratio of X-ray-to-bolometric luminosities (${R}_{{\rm{X}}}={L}_{{\rm{X}}}/{L}_{\mathrm{bol}}$) as the X-ray activity indicators. We used the PARSEC theoretical models to determine the bolometric luminosity (see Appendix C).

Figure 6 displays the distributions of L_X and R_X for different stellar types in our "A-class" sample. For each stellar type, giants have the highest luminosities up to 10³²–10³³ erg s⁻¹. For G and K stars, we ran a K-S test (scipy.stats.ks_2samp) to check the similarity of L_X distributions. Since the YSOs, dwarfs, and giants have different numbers, we randomly selected ≈100 objects for each class and calculated the P_K−S values. The selection and calculation was repeated 10⁴ times, allowing us to have an averaged ${P}_{{\rm{K}}-{\rm{S}}}$ estimation. For giants and dwarfs, the ${P}_{{\rm{K}}-{\rm{S}}}$ values are smaller than 10⁻¹⁰ for both G and K stars, indicating a different distribution; for YSOs and dwarfs, the ${P}_{{\rm{K}}-{\rm{S}}}$ values are ≈6 × 10⁻⁴ (G stars) and ≈0.06 (K stars), respectively.

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In previous studies, YSOs were found to have X-ray luminosities 10–10⁴ above those typically seen in main-sequence stars (e.g., Feigelson et al. 2002; Preibisch et al. 2005). However, no such notable difference can be seen in Figure 6. This may be explained by the incompleteness of our dwarf sample. Take K stars as an example, previous observations (ROSAT; 0.1–2.4 keV) showed that the field dwarfs have X-ray luminosities ranging from ≈6 × 10²⁶ erg s⁻¹ to ≈2 × 10³⁰ erg s⁻¹, while the dwarfs in open clusters Pleiades and Hyades have X-ray luminosities spanning from ≈3 × 10²⁸ erg s⁻¹ to ≈4 × 10³⁰ erg s⁻¹ (Wright et al. 2011). The X-ray luminosities (Chandra; 0.5–8 keV) of pre-main-sequence K5-7 stars in the Orion nebula are from ≈3 ×10²⁹ erg s⁻¹ to ≈2 × 10³¹ erg s⁻¹ (Wolk et al. 2005). It seems that most of the K dwarfs in our catalog are in the high-luminosity tail of the L_X distribution of Galactic K stars. This may be due to the Chandra observation mode and our sample selection methods. For example, the requirement of WISE photometry may remove a number of dwarfs.

In general, the R_X distributions are similar among different classes. For giants and dwarfs, the ${P}_{{\rm{K}}-{\rm{S}}}$ values are about 0.14 and 0.20, respectively. For YSOs and dwarfs, the ${P}_{{\rm{K}}-{\rm{S}}}$ values are smaller than 0.01 for both G and K stars. More YSOs have higher R_X values than the dwarfs in the same stellar type.

We noted that there is a large difference between the number of M-type YSOs and dwarfs. Among the 161 M-type YSOs with parameter estimations, about half is from Marton et al. (2016, 2019) and the other half is picked out with our criteria. One caveat is that about 50 have W2 − W3 < 1, some of which may be wrongly classified (i.e., contaminated by dwarfs or giants). However, the statistical distributions will not be much affected.

X-ray emission of late-type stars is from solar-like corona, which suggests a relation between stellar X-ray activity and coronal activity. The positive correlation between X-ray luminosity and coronal temperature has been explored (e.g., Schmitt et al. 1995; Güdel 2004), which can be referred on the basis of a "loop" model for stellar coronae (Vaiana 1983). Active stars with more efficient dynamo have stronger magnetic fields in the corona, and thus a higher rate of field line reconnections and flares. This results in higher coronal temperatures and a larger plasma density of energetic electrons. Here we take the X-ray HR to track the coronal temperature for late-type stars. There is a clear positive correlation between L_X and HR1 (Figure 7), which means stronger X-ray emitters have higher coronal temperatures. Most YSOs are in the saturated regime, with a constant R_X value (logR_X ≈ −3) when HR1 varies (Figure 8). We performed a Spearman correlation test (scipy.stats.spearmanr) for all these relations. Then linear regression fits were done for these relations with higher correlation coefficients (Table 3).

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The YSOs and dwarfs share similar HR1 distribution (Figure 9), while for all late stellar types (from F to K), there are giants showing very high HR1 values, which are good candidates possessing high-temperature corona. A K-S test was also run for the HR1 distributions. For YSOs and dwarfs, the P_K−S values are ≈2 × 10⁻⁴ (G stars) and ≈0.05 (K stars), respectively; for giants and dwarfs, the P_K−S values are smaller than 10⁻⁷ for both G and K stars. Again, we remind that the samples of each stellar type are incomplete.

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Stellar X-ray emission is variable (Soderblom 2010; Stelzer 2017). The most prominent signature of the variability is flares, which usually show sudden and intense brightness increase and decay. The short X-ray variability can be also due to rotational modulation of the structural inhomogeneity of magnetic field (Marino et al. 2003). Significant correlations have been found between the positions of active longitudes in the activity diagnostics, including the X-ray light curve, starspot, and magnetic field maps (Hussain et al. 2007). The long-term X-ray variability may reflect magnetic dynamo cycles (Sanz-Forcada et al. 2013; Ayres 2014). However, there are only few sources showing X-ray cycles, since long-term X-ray monitoring is not easily feasible and it is hard to detect dynamo cycle in the X-ray band (Stelzer 2017). For YSOs, the variability can also be related to the shocks and absorption associated with accretion disk and protostellar jets (e.g., Wolk et al. 2005; Flaccomio et al. 2006; Guarcello et al. 2017; Ustamujic et al. 2018).

In our sample, about 1400 stars were observed more than once by Chandra. We calculated the relative flux variation as the standard flux variation during non-flare observations divided by the averaged flux. Most stars have small relative flux variation (σ/f_X < 0.5). The σ/f_X shows no clear relation with R_X and L_X (Figure 10). It also can be seen that the most variable sources are mainly YSOs. Figure 11 shows some examples of long-term light curves

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Many studies have investigated the correlation between several coronal and chromospheric magnetic activity indicators and stellar rotation rate (e.g., Pizzolato et al. 2003; Mamajek & Hillenbrand 2008; Wright et al. 2011; Lehtinen et al. 2020). The Rossby number is used to trace the stellar rotation, which is defined as the ratio of the rotation period to the convective turnover time (Ro = P/τ).

In order to derive stellar rotation periods from Kepler data, we have developed an online platform, Kepler Data Integration Platform,¹² which integrates query, view, and period calculation on the whole Kepler and K2 data set (see Yang & Liu 2019, for details). In our sample, more than 900 stars were observed by the Kepler telescope, among which four are eclipsing binaries and about 100 ones show rotational modulation. By using the baseline-detrended light curve, their periods were computed with the Lomb–Scargle periodogram (see Gao et al. 2016; Yang & Liu 2019, for details).

The classical empirical estimate of τ from colors and effective temperatures works mainly for main-sequence dwarf (e.g., Noyes et al. 1984; Wright et al. 2011). Since our sample also includes YSOs and giants, we decided to derive the τ value using a grid of stellar evolution models from the YalePotsdam Stellar Isochrones (YaPSI) following Lehtinen et al. (2020). We did a fitting, by using the logT_eff and logg from LAMOST DR7 and Anders et al. (2019), to the model evolutionary tracks. The YAPSI models include five initial metallicities, [Fe/H] = +0.3, 0.0, −0.5, −1.0, and −1.5. For each star, we obtained best-fit models for these metallicities, and calculated the final τ value (and stellar radius) by linear interpolation to the metallicity from LAMOST DR7 and Anders et al. (2019).

Figure 12 shows R_X as a function of Ro. The dashed line is from Wright et al. (2011), but shifted to the left with a ratio of Ro/3, since we derived a ratio between the theoretical and empirical τ values around 3 (Lehtinen et al. 2020 obtained a ratio of ≈2.6). For each stellar spectral type, the X-ray activity increases with decreasing rotation period, following a well-known activity–rotation relation (Figure 13). Most M stars are located in the saturated regime, which is usually explained by the magnetic/dynamo saturation (Reiners et al. 2009) or a change in the dynamo configuration of the star (Wright et al. 2011).

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There are some theories for the saturation (and super-saturation; Prosser et al. 1996) of stellar X-ray emission: a saturation of the dynamo itself (e.g., Gilman 1983; Vilhu & Walter 1987), a saturation of the filling factor of active regions on the stellar surface (Vilhu 1984; Solanki et al. 1997; Stȩpień et al. 2001), or centrifugal stripping of the corona (Jardine & Unruh 1999). The lack of observed saturation in chromospheric emission (Cardini & Cassatella 2007; Mamajek & Hillenbrand 2008; Marsden et al. 2009; Jackson & Jeffries 2010; Argiroffi et al. 2016) are in contrast with the idea that the dynamo itself saturates. Analogously, the scenario of a saturation of the filling factor was argued against by some observations of rotational modulation of X-ray emission (Marino et al. 2003) and small coronal filling factors (Testa et al. 2004) in saturated/supersaturated stars. In addition, compact and dense coronal loops are found far below the corotation radius of the supersaturated star VW Cephei (Huenemoerder et al. 2006), indicating that saturation is not necessarily caused by coronal stripping (Wright et al. 2011). These could suggest that saturation is due to a change in regime of the underlying dynamo.

Some recent studies (e.g., Reiners et al. 2014) reported that the rotation period alone suffices to determine the activity–rotation scaling, and the relation L_X/L_bol ∝ ${P}_{\mathrm{rot}}^{-2}{R}^{-4}$ optimally describes the non-saturated fraction of stars. However, Lehtinen et al. (2020) found that the dwarfs and giants are located in a single sequence in the unsaturated regime in relation to Rossby numbers, while they are clearly separated in the relations ${R}_{{HK}}^{{\prime} }$ versus ${P}_{\mathrm{rot}}$ and ${R}_{{HK}}^{{\prime} }$ versus ${P}_{\mathrm{rot}}^{-2}{R}^{-4}$. In our sample, the giants, with longer rotation periods, do not follow the relation R_X versus ${P}_{\mathrm{rot}}^{-2}{R}^{-4}$ for dwarfs well (Figure 14). More stars with accurate period estimations will help examine these relations.

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A number of proxies have been used to study the activity occurring in the photosphere (star spots), chromosphere (Ca ii HK; Mg ii; Hα; optical flares; UV flux), and corona (X-ray emission). Observations have revealed good correlations between some of these proxies (e.g., Mamajek & Hillenbrand 2008; Stelzer et al. 2013), and the empirical scalings of these proxies with rotation period or Rossby number (e.g., Pallavicini et al. 1981; Pizzolato et al. 2003; Wright et al. 2011).

To have a comparison with the chromospheric activity, we calculated the UV activity index as following:

$$\begin{eqnarray}&&{R}_{\mathrm{UV}}^{{\prime} }=\displaystyle \frac{{f}_{\mathrm{UV},\mathrm{exc}}}{{f}_{\mathrm{bol}}}=\displaystyle \frac{{f}_{\mathrm{UV},\mathrm{obs}}-{f}_{\mathrm{UV},\mathrm{ph}}}{{f}_{\mathrm{bol}}},\end{eqnarray} \tag{ 3 }$$

where "UV" stands for the near-UV (NUV) and the far-UV (FUV) bands, respectively. Here ${f}_{\mathrm{UV},\mathrm{exc}}$ is the UV excess flux due to activity. The observed UV flux ${f}_{\mathrm{UV},\mathrm{obs}}$ was estimated from the GALEX magnitude, by using the transformation relations¹³ as

$$\begin{eqnarray}&&{f}_{\mathrm{FUV},\mathrm{obs}}={10}^{0.4\times (18.82-{M}_{\mathrm{FUV}})}\times 1.40\times {10}^{-15}\times \delta {\lambda }_{\mathrm{FUV}}\end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray}&&{f}_{\mathrm{NUV},\mathrm{obs}}={10}^{0.4\times (20.08-{M}_{\mathrm{NUV}})}\times 2.06\times {10}^{-16}\times \delta {\lambda }_{\mathrm{NUV}}.\end{eqnarray} \tag{ 5 }$$

The M_FUV and M_NUV are absolute magnitudes, obtained from the observed magnitudes adopting the distance from Gaia DR2 and the extinction from the PS1 3D extinction map (See Section 2.1). The extinction coefficients were calculated as 8.11 (FUV) and 8.71 (NUV) following Cardelli et al. (1989). The δ λ_FUV and δ λ_NUV are the effective bandwidth¹⁴ of the FUV and NUV filters (268 Å and 732 Å), respectively. The photospheric flux ${f}_{\mathrm{UV},\mathrm{ph}}$, which means the photospheric contribution to the FUV and NUV emission, were estimated with the PARSEC model. Using the best-fit model (see Appendix C), we derived the FUV and NUV magnitudes attributed to photospheric emission. The PARSEC models present the GALEX absolute magnitudes in VEGA system, therefore we first converted them into AB magnitudes (Bianchi 2011) and then converted them into fluxes using Equations (4) and (5). The bolometric flux f_bol was obtained with the bolometric luminosity from the best-fit PARSEC model. We calculated it using a distance of 10 pc. Table 4 lists the UV activity indexes.

Table 4.  Stars with UV Emission in our Sample

(1) (2) (3) (4) (5)

Object	log${f}_{\mathrm{FUV},\mathrm{exc}}$	log${f}_{\mathrm{NUV},\mathrm{exc}}$	log${R}_{\mathrm{FUV}}^{{\prime} }$	log${R}_{\mathrm{NUV}}^{{\prime} }$
	(erg cm−2 s−1)	(erg cm−2 s−1)
J000136.1+130639	6.12 ± 0.73	8.25 ± 0.07	−4.81 ± 0.73	−2.67 ± 0.07
J000238.8+255219	...	7.03 ± 0.96	...	−3.47 ± 0.96
J000833.8+512412	8.81 ± 0.04	...	−2.66 ± 0.04	...
J000835.3+512142	7.62 ± 0.08	9.13 ± 0.01	−2.68 ± 0.08	−1.17 ± 0.01
J001325.4+791537	6.60 ± 0.16	8.14 ± 0.05	−4.35 ± 0.16	−2.80 ± 0.05
J001507.9-302342	...	8.34 ± 0.44	...	−2.44 ± 0.44
J001801.8+300816	...	7.77 ± 0.05	...	−2.86 ± 0.05
J001944.6+591349	...	10.71 ± 0.38	...	−0.49 ± 0.38
J002023.5+591444	...	8.54 ± 0.05	...	−2.43 ± 0.05
J002101.6+591518	...	8.40 ± 0.41	...	−2.03 ± 0.41
J002111.3-084140	6.53 ± 0.06	6.98 ± 0.03	−3.12 ± 0.06	−2.68 ± 0.03
J002344.4+641111	...	10.90 ± 0.28	...	−0.19 ± 0.28
J002410.3-020127	...	6.26 ± 0.49	...	−3.95 ± 0.49
J002438.4+641122	...	12.04 ± 0.02	...	−0.05 ± 0.02
J002519.7-123303	5.83 ± 0.47	6.66 ± 0.14	−4.37 ± 0.47	−3.54 ± 0.14
J002537.0-121450	...	6.02 ± 0.35	...	−4.31 ± 0.35
J002609.9+171559	...	6.19 ± 0.24	...	−3.93 ± 0.24
J002611.2+171234	5.54 ± 0.29	...	−5.09 ± 0.29	...
J002756.6+261651	...	7.56 ± 0.11	...	−3.07 ± 0.11
J003151.5+003233	6.02 ± 0.23	6.41 ± 0.13	−3.69 ± 0.23	−3.30 ± 0.13

Note. (This table is available in its entirety in machine-readable and Virtual Observatory (VO) forms in the online journal. A portion is shown here for guidance regarding its form and content.)

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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We searched for the relations between FUV and NUV activities for different type stars (Figure 15). For F- and G-type stars, there is no clear relation, while for K and M stars, the FUV and NUV fluxes show an obvious positive relation (Stelzer et al. 2013). By using different stellar models (i.e., BT-Cond grid), Bai et al. (2018) calculated the UV excesses emission for millions of stars, and their sample showed similar trends. The lack of clear relation for F and G stars may be explained as the mixed populations with different ages. As can be seen in Figure 16, the F and G stars with lower FUV/NUV ratio may be older ones, while the young population show higher ratio of log(${{R}^{{\prime} }}_{\mathrm{FUV}}/{{R}^{{\prime} }}_{\mathrm{NUV}}$) ≳ −1 (see also Figure 12 in Richey-Yowell et al. 2019). Similar trends can be found in the comparisons between X-ray and UV activities (Figures 17 and 18). For G, K, and M stars, a positive relation can be seen between ${R}_{X}^{{\prime} }$ and ${R}_{\mathrm{FUV}}^{{\prime} }$, while only M stars show an obvious relation between ${R}_{X}^{{\prime} }$ and ${R}_{\mathrm{NUV}}^{{\prime} }$.

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