Activity and rotation of the X-ray emitting Kepler stars

TL;DR

This study investigates the relationship between magnetic activity and rotation in X-ray emitting late-type stars using combined X-ray and Kepler data, revealing a saturated and correlated activity regime and identifying simultaneous X-ray and white-light flares.

Contribution

It provides a homogeneous analysis of activity and rotation in late-type stars by combining X-ray and Kepler data, including new rotation periods and flare detections.

Findings

Evidence for saturated and correlated activity regimes.

Identification of 74 stars with measured rotation periods.

Detection of 7 X-ray flares with white-light counterparts.

Abstract

The relation between magnetic activity and rotation in late-type stars provides fundamental information on stellar dynamos and angular momentum evolution. Rotation/activity studies found in the literature suffer from inhomogeneity in the measure of activity indexes and rotation periods. We overcome this limitation with a study of the X-ray emitting late-type main-sequence stars observed by XMM-Newton and Kepler. We measure rotation periods from photometric variability in Kepler light curves. As activity indicators, we adopt the X-ray luminosity, the number frequency of white-light flares, the amplitude of the rotational photometric modulation, and the standard deviation in the Kepler light curves. The search for X-ray flares in the light curves provided by the EXTraS (Exploring the X-ray Transient and variable Sky) FP-7 project allows us to identify simultaneous X-ray and white-light flares. A careful selection of the X-ray sources in the Kepler field yields 102 main-sequence stars with spectral types from A to M. We find rotation periods for 74 X-ray emitting main-sequence stars, 22 of which without period reported in the previous literature. In the X-ray activity/rotation relation, we see evidence for the traditional distinction of a saturated and a correlated part, the latter presenting a continuous decrease in activity towards slower rotators. For the optical activity indicators the transition is abrupt and located at a period of ~ 10 d but it can be probed only marginally with this sample which is biased towards fast rotators due to the X-ray selection. We observe 7 bona-fide X-ray flares with evidence for a white-light counterpart in simultaneous Kepler data. We derive an X-ray flare frequency of ~ 0.15 d^{-1} , consistent with the optical flare frequency obtained from the much longer Kepler time-series.

Tables

Table 1. Table 1: XMM-Newton observations from the 3XMM-DR5 Catalogue in the Kepler FoV. Next to observation ID (col. 1), pointing center (cols. 2 and 3), observing date (cols. 4) and exposure time (col. 5), the flux sensitivity limit is given (col. 6) calculated as described in Sect.  2.2 .

OBS_ID	RA(J2000)	DEC(J2000)	Obs. date	Exp. time	$f_{X,\lim }$
				[ks]	[$\mathrm{erg}/{\mathrm{cm}}^2/s$]
0302150101	$\mathrm{19\hspace{0.17em}21\hspace{0.17em}05.99}$	$+\mathrm{43\hspace{0.17em}58\hspace{0.17em}24.0}$	2005-10-10	16.9	$1.46\cdot {10}^{-14}$
0302800101	$\mathrm{19\hspace{0.17em}59\hspace{0.17em}28.28}$	$+\mathrm{40\hspace{0.17em}44\hspace{0.17em}02.0}$	2005-10-14	22.5	$3.35\cdot {10}^{-14}$
0302800201	$\mathrm{19\hspace{0.17em}59\hspace{0.17em}28.28}$	$+\mathrm{40\hspace{0.17em}44\hspace{0.17em}02.0}$	2005-10-16	18.8	$1.43\cdot {10}^{-14}$
0302150201	$\mathrm{19\hspace{0.17em}21\hspace{0.17em}05.99}$	$+\mathrm{43\hspace{0.17em}58\hspace{0.17em}24.0}$	2005-11-14	16.9	$6.27\cdot {10}^{-15}$
0553510201	$\mathrm{19\hspace{0.17em}41\hspace{0.17em}17.98}$	$+\mathrm{40\hspace{0.17em}11\hspace{0.17em}12.0}$	2008-05-18	26.9	$2.8\cdot {10}^{-15}$
0551021701	$\mathrm{19\hspace{0.17em}41\hspace{0.17em}51.86}$	$+\mathrm{50\hspace{0.17em}31\hspace{0.17em}01.8}$	2008-11-08	11.7	$2.27\cdot {10}^{-15}$
0550451901	$\mathrm{19\hspace{0.17em}47\hspace{0.17em}19.42}$	$+\mathrm{44\hspace{0.17em}49\hspace{0.17em}40.8}$	2008-11-20	17.9	$1.16\cdot {10}^{-14}$
0600040101	$\mathrm{19\hspace{0.17em}21\hspace{0.17em}11.41}$	$+\mathrm{43\hspace{0.17em}56\hspace{0.17em}56.2}$	2009-11-29	58.3	$8.35\cdot {10}^{-15}$
0653190201	$\mathrm{20\hspace{0.17em}01\hspace{0.17em}40.00}$	$+\mathrm{43\hspace{0.17em}52\hspace{0.17em}20.0}$	2010-06-23	15.9	$4.02\cdot {10}^{-14}$
0670140501	$\mathrm{19\hspace{0.17em}36\hspace{0.17em}59.90}$	$+\mathrm{46\hspace{0.17em}06\hspace{0.17em}28.0}$	2011-05-10	31.9	$2.44\cdot {10}^{-15}$
0672530301	$\mathrm{19\hspace{0.17em}05\hspace{0.17em}25.90}$	$+\mathrm{42\hspace{0.17em}27\hspace{0.17em}40.0}$	2011-06-05	29.1	$1.93\cdot {10}^{-15}$
0671230401	$\mathrm{19\hspace{0.17em}58\hspace{0.17em}00.01}$	$+\mathrm{44\hspace{0.17em}40\hspace{0.17em}00.0}$	2011-06-09	85.9	$2.53\cdot {10}^{-15}$
0671230301	$\mathrm{19\hspace{0.17em}58\hspace{0.17em}00.01}$	$+\mathrm{45\hspace{0.17em}10\hspace{0.17em}00.0}$	2011-06-23	83.9	$3.37\cdot {10}^{-15}$
0671230101	$\mathrm{19\hspace{0.17em}55\hspace{0.17em}59.98}$	$+\mathrm{44\hspace{0.17em}40\hspace{0.17em}00.0}$	2011-09-25	81.9	$2.53\cdot {10}^{-15}$
0671230201	$\mathrm{19\hspace{0.17em}55\hspace{0.17em}59.98}$	$+\mathrm{45\hspace{0.17em}10\hspace{0.17em}00.0}$	2012-03-26	104.5	$1.84\cdot {10}^{-15}$
0671230601	$\mathrm{19\hspace{0.17em}55\hspace{0.17em}59.98}$	$+\mathrm{45\hspace{0.17em}10\hspace{0.17em}00.0}$	2012-03-26	2.7	$1.10\cdot {10}^{-13}$

Table 2. Table 2: Classification of the main-sequence stars in our sample according to the type of variability observed in the Kepler light curves.

SpT	Main Sequence (MS) Stars	With rotation period	No period	Ecl. binaries	Other binaries	Pulsators	Unclear
All	102	74	2	10	5	3	3
A	3	2	0	0	0	1	0
F	37	24	2	5	2	1	1
G	26	19	0	1	0	1	1
K	28	21	0	4	3	0	0
M	8	8	0	0	0	0	1

Table 3. Table 3: Best fit spectral parameters for the 17 17 17 brightest X-ray sources (more than 200 200 200 counts) identified as rotational variables; parameters are for a phabs*apec or phabs*(apec+apec) model.

KIC_ID	SpT(a)	Counts(b)	Model	$N_H$	kT	EM
				[${\mathrm{cm}}^{-2}$]	[keV]	[${\mathrm{cm}}^{-3}$]
5112508	M	520	phabs*apec	$\sim 0$	$0.83$	$1.31\cdot {10}^{52}$
5113557	F	2088	phabs*apec	$1.46\cdot {10}^{20}$	$0.60$	$6.40\cdot {10}^{50}$
5653243	K	316	phabs*apec	$\sim 0$	$0.80$	$5.86\cdot {10}^{52}$
6761532	G	226	phabs*apec	$3.60\cdot {10}^{20}$	$0.63$	$2.43\cdot {10}^{52}$
7018708	G	2637	phabs*(apec+apec)	$6.80\cdot {10}^{20}$	$0.29,1.04$	$1.93\cdot {10}^{53},2.24\cdot {10}^{53}$
8454353	M	1291	phabs*apec	$\sim 0$	$0.99$	$9.69\cdot {10}^{51}$
8517303	K	1763	phabs*apec	$9.10\cdot {10}^{20}$	$1.06$	$1.10\cdot {10}^{54}$
8518250	K	434	phabs*(apec+apec)	$\sim 0$	$0.94,0.23$	$1.24\cdot {10}^{51},1.88\cdot {10}^{51}$
8520065	F	764	phabs*(apec+apec)	$2.33\cdot {10}^{20}$	$1.07,0.54$	$4.30\cdot {10}^{53},3.42\cdot {10}^{53}$
8584672	K	230	phabs*apec	$\sim 0$	$0.83$	$1.21\cdot {10}^{53}$
8647865	F	433	phabs*apec	$\sim 0$	$0.49$	$1.60\cdot {10}^{52}$
8713822	K	675	phabs*apec	$1.43\cdot {10}^{20}$	$1.05$	$2.58\cdot {10}^{54}$
8842083	K	590	phabs*apec	$1.1\cdot {10}^{21}$	$0.76$	$7.01\cdot {10}^{50}$
9048551	K	1634	phabs*apec	$\sim 0$	$1.07$	$8.91\cdot {10}^{51}$
9048949	K	844	phabs*apec	$\sim 0$	$0.94$	$4.31\cdot {10}^{51}$
9048976	K	285	phabs*apec	$\sim 0$	$0.96$	$4.11\cdot {10}^{51}$
11971335	G	696	phabs*apec	$\sim 0$	$1.30$	$2.10\cdot {10}^{53}$
(a) Spectral type has been evaluated from the $T_{\mathrm{eff}}$ according to Pecaut & Mamajek (2013).
(b) This is the sum of the counts in the three EPIC instruments.

Table 4. Table 4: Parameters of the X-ray flares and Kepler flare characteristic of the X-ray flaring stars. The start time of each X-ray flare is reported, together with the peak and off-flare count rate. The number and the frequency of the white-light flares observed for the star is also given, together with the average, minimum and maximum peak amplitude of all its white-light flares in the Kepler light curve.

KIC_ID	Start time	$Rat{e}_{X,\mathrm{quiesc}}$	$Rat{e}_{X,\mathrm{peak}}$	$N_{\mathrm{opt},\mathrm{flares}}$	$F_{\mathrm{opt},\mathrm{flares}}$		Min-max $\log A_{\mathrm{peak}}$
	(Julian day)	(cts/s)	(cts/s)		($d^{-1}$)
7018131	2455718.31	0.006$\pm$0.011	0.035$\pm$0.019	5	0.005	-2.79	-2.89 / -2.67
8454353	2455829.96	0.036$\pm$0.020	0.263$\pm$0.048	297	0.211	-1.86	-0.36 /-2.69
8520065	2455721.96	0.020$\pm$0.016	0.140$\pm$0.040	0	0	0	0
8909598	2455736.55	0.004$\pm$0.011	0.090$\pm$0.003	7	0.022	-1.83	-1.43/-2.21
9048551	2455736.03	0.033$\pm$0.016	0.125$\pm$0.028	233	0.164	-2.41	-1.47/-2.92
9048976	2455735.94,2455736.80	0.009$\pm$0.013	0.06$\pm$0.02,0.060$\pm$0.024	137	0.097	-2.37	-1.57/-2.91

Table 5. Table 5: Mean values and standard deviations measured for photometric activity diagnostics in the Kepler / XMM-Newton sample.

SpT	$N_f/day$	$\log R_{\mathrm{per}}$ [%]	$S_{\mathrm{ph}}$ [ppm]
A	0.001	$-1.02$	$3.47\cdot {10}^2$
F	0.01	$-0.74$	$1.10\cdot {10}^3$
G	0.02	$0.01$	$5.39\cdot {10}^3$
K	0.04	$0.25$	$8.69\cdot {10}^3$
M	0.12	$0.46$	$1.10\cdot {10}^4$

Table 6. Table 6: KIC (Brown et al. 2011 ) photometry all the 125 125 125 stars in the Kepler / XMM-Newton sample, with the corrections by Pinsonneault et al. ( 2012 ) for the SLOAN filters and updated magnitudes from the UCAC-4 Catalogue (Zacharias et al. 2012 )

KIC ID	$J$	$H$	$K_s$	$U$	$B$	$V$	$K_p$	$g_{\mathrm{SDSS}}$	$r_{\mathrm{SDSS}}$	$i_{\mathrm{SDSS}}$	$z_{\mathrm{SDSS}}$	GALEX_FUV	GALEX_NUV
	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]	[mag]
4935950	9.257	8.679	8.513	—	—	—	11.071	12.18	11.022	10.571	10.319	—	—
4937350	13.109	12.756	12.698	15.291	15.222	14.587	14.273	14.664	14.197	14.034	13.946	—	—
5023534	11.52	11.185	11.072	—	13.758	13.061	13.083	13.688	13.013	12.767	—	—	—
5023985	12.412	12.125	12.082	14.489	14.346	13.715	13.495	13.858	13.42	13.267	13.224	—	—
5024138	13.833	13.579	13.58	15.962	15.851	15.29	14.964	15.344	14.887	14.728	14.641	—	—
5024146	13.573	13.24	13.162	—	—	—	14.845	15.44	14.785	14.533	14.431	—	—
5024450	13.942	13.628	13.56	15.976	15.856	15.284	15.063	15.476	14.981	14.816	14.719	—	—
5025233	11.371	11.083	11.012	13.281	13.268	12.693	12.435	12.765	12.366	12.219	12.179	—	—
5025294	12.119	11.795	11.724	14.255	14.144	13.53	13.266	13.65	13.192	13.031	12.956	—	—
5111932	11.371	11.125	11.09	12.804	12.832	12.397	12.288	12.483	12.21	12.122	12.108	—	—
5112483	9.263	8.583	8.365	—	13.369	12.078	11.244	11.795	11.165	10.947	—	—	—
5112508	12.551	11.912	11.71	18.739	17.736	16.403	15.342	17.196	—	14.586	13.964	—	—
5112553	11.185	10.856	10.771	—	—	—	12.314	12.764	12.221	12.053	11.985	—	—
5112741	10.66	10.073	9.923	14.718	13.963	12.966	12.449	13.365	12.38	12.018	11.791	—	—
5113557	5.799	5.75	5.717	—	—	—	6.266	6.308	—	—	—	—	—
5113797	8.696	8.598	8.59	—	—	—	9.152	8.977	9.131	9.215	9.323	—	—
5200185	13.651	13.05	12.903	18.193	17.141	16.131	15.509	16.574	15.496	15.025	14.747	—	—
5397900	13.689	13.652	13.268	16.332	16.1	15.383	15.005	15.512	14.942	14.726	14.667	—	—
5481017	16.679	16.027	16.05	—	19.285	18.444	17.861	18.727	17.828	—	—	—	—
5482181	12.284	11.752	11.603	16.126	15.465	14.469	14.047	14.958	13.99	13.633	13.397	—	—
5483081	8.115	7.129	6.717	—	16.090	14.849	13.023	16.243	14.132	11.836	10.355	—	—
5653243	9.655	9.257	9.146	12.523	12.123	11.273	11.012	12.18	—	—	—	—	—
5739251	11.465	10.826	10.663	17.17	15.976	14.705	13.789	15.342	13.96	13.148	12.676	—	—
6761498	10.781	10.446	10.342	13.302	12.972	12.274	12.027	12.501	11.945	11.757	11.68	—	—
6761532	9.872	9.544	9.458	12.023	11.869	11.235	11.064	11.499	10.988	10.809	10.74	22.479	16.606
6761559	13.808	13.481	13.416	15.913	15.751	15.132	14.955	15.343	14.879	14.717	14.645	—	—
6846570	12.255	11.618	11.468	—	—	—	14.696	16.122	14.815	14.086	13.634	—	—
6846595	11.186	10.994	10.974	12.495	12.565	12.141	12.066	12.193	11.963	11.902	11.917	20.516	15.813
6847018	12.346	12.124	12.071	13.953	13.972	13.476	13.344	13.596	13.252	13.156	13.141	—	—
6932782	10.465	10.262	10.207	—	11.956	11.5	11.394	11.825	—	—	—	21.233	15.711
7017862	12.083	11.766	11.698	14.134	13.979	13.368	13.128	13.472	13.042	12.906	12.858	—	18.408
7018131	12.439	11.974	11.853	15.822	15.122	14.275	13.93	14.636	13.856	13.576	13.452	—	21.257
7018267	11.141	10.573	10.442	15.743	14.593	13.548	13.002	14.077	12.97	12.514	12.254	—	21.926
7018277	14.314	13.884	13.75	17.472	17.0	16.204	15.841	16.494	15.781	15.507	15.369	—	—
7018323	12.957	12.348	12.14	18.512	17.444	16.16	15.308	16.847	15.464	14.658	14.209	—	—
7018521	11.284	10.893	10.839	13.487	13.35	12.707	12.476	12.915	12.409	12.22	12.133	—	18.001
7018708	10.464	10.105	9.942	13.118	12.82	12.115	12.088	12.477	11.658	11.838	11.385	22.556	17.609
7955782	9.561	9.319	9.268	—	11.215	10.698	10.589	11.124	—	—	—	21.022	15.138
8024987	11.579	11.182	11.128	14.552	14.033	13.243	12.938	13.524	12.852	12.627	12.507	22.838	19.819
8025009	14.742	14.476	14.529	17.152	16.886	16.207	15.956	16.383	15.884	15.704	15.62	—	22.214
8058211	12.999	12.314	12.198	18.295	17.127	15.839	15.124	16.378	15.147	14.573	14.206	—	—
8093473	12.006	11.346	11.16	18.267	17.273	15.883	14.768	16.604	15.173	14.017	13.4	22.373	21.908
8094140	13.134	12.534	12.388	—	—	—	15.159	16.314	15.195	14.645	14.296	—	—
8094292	15.826	15.186	15.255	—	—	—	16.966	17.522	—	—	—	—	—
8195877	12.294	11.973	11.88	14.333	14.282	13.69	13.441	13.84	13.387	13.2	13.129	—	—
8229034	12.954	12.548	12.438	16.332	15.682	14.812	14.375	15.023	14.274	14.042	13.9	—	22.015
8454353	12.423	11.795	11.551	18.172	17.297	16.005	14.894	16.575	15.151	14.195	13.685	—	—
8517293	10.649	10.387	10.304	—	12.477	11.882	12.233	12.418	12.128	12.069	—	—	—
8517303	9.541	8.887	8.785	14.053	13.13	11.993	11.43	12.371	11.405	10.992	10.699	—	19.49
8518250	9.558	9.001	8.856	—	12.63	11.617	10.996	12.779	—	—	—	—	19.677
8518323	12.337	12.054	11.992	14.384	14.347	13.73	13.477	13.88	13.415	13.234	13.171	—	—
8520065	12.737	12.385	12.333	15.149	14.977	14.279	13.989	14.525	13.934	13.698	13.583	—	19.48
8520464	12.352	12.032	11.955	14.732	14.576	13.948	13.631	14.117	13.579	13.358	13.254	—	—
8581658	12.253	11.997	11.946	14.239	14.166	13.579	13.396	13.736	13.326	13.176	13.104	—	—
8582224	12.125	11.849	11.763	14.518	14.335	13.663	13.42	13.899	13.347	13.15	13.051	—	19.367
8582406	11.291	10.715	10.526	16.441	15.212	14.022	13.339	14.592	13.338	12.788	12.472	—	21.83
8583756	13.262	12.596	12.403	18.988	17.732	16.498	15.579	17.075	—	14.942	14.5	—	—
8583911	12.939	12.516	12.462	15.9	15.557	14.763	14.395	15.026	14.313	14.068	13.901	—	—
8584439	12.528	12.102	12.099	14.802	14.661	14.033	13.707	14.203	13.673	13.43	13.331	—	—
8584672	8.939	8.217	8.074	13.672	12.615	11.43	10.889	11.966	10.86	10.402	10.121	—	—
8584920	11.699	11.486	11.408	—	13.59	13.046	13.198	13.688	13.124	12.923	—	—	17.854
8584977	10.626	10.431	10.378	—	12.189	11.714	11.565	12.043	—	—	—	20.536	15.934
8584986	13.007	12.381	12.226	18.15	17.014	15.795	15.18	16.358	15.178	14.657	14.298	—	—
8585000	12.026	11.497	11.435	15.63	14.849	13.941	13.561	14.346	13.486	13.179	13.024	—	—
8647149	11.08	10.848	10.755	—	—	—	12.271	12.657	12.195	12.034	11.957	23.179	17.262
8647865	9.962	9.81	9.786	—	11.214	—	10.733	11.114	—	—	—	18.761	14.419
8648195	11.965	11.647	11.567	14.335	14.229	13.536	13.248	13.773	13.194	12.961	12.853	—	18.907
8648671	9.901	9.355	9.231	13.932	13.103	12.043	11.616	12.503	11.543	11.196	10.969	—	20.281
8649876	11.396	11.166	11.092	13.469	13.331	12.754	12.556	12.965	12.49	12.311	12.246	—	—
8650057	10.696	10.574	10.503	12.138	12.179	11.752	11.642	11.84	11.555	11.473	11.476	20.718	15.566
8702424	12.294	11.867	11.801	14.888	14.604	13.897	13.6	14.129	13.535	13.312	13.191	—	—
8702606	8.089	7.743	7.684	—	—	—	9.292	9.729	9.236	9.037	8.946	—	—
8703129	10.375	9.964	9.839	13.904	13.152	12.227	11.893	12.619	11.81	11.532	11.373	—	—
8703413	8.295	8.275	8.227	—	—	—	8.712	8.62	8.665	8.762	8.856	—	—
8703528	11.619	11.272	11.182	14.706	14.393	13.646	13.034	13.663	12.925	12.708	12.647	—	—
8712626	10.433	10.01	9.951	12.792	12.634	11.937	11.634	12.065	11.587	11.382	11.313	—	17.381
8713594	12.451	12.282	12.141	14.516	14.37	13.793	13.613	13.977	13.554	13.384	13.282	—	—
8713822	12.487	12.048	11.913	16.398	15.754	14.783	14.258	15.112	14.232	13.852	13.608	—	22.126
8713828	11.306	11.111	11.019	—	13.208	12.666	12.525	13.122	12.457	12.212	—	21.63	17.139
8767669	11.829	11.239	11.067	16.726	15.577	14.507	13.875	15.062	13.886	13.347	13.011	—	—
8832676	13.122	12.425	12.262	18.833	17.747	16.395	15.466	16.967	15.562	14.827	14.396	—	—
8840075	9.453	9.223	9.202	—	—	—	10.353	10.851	—	—	—	20.471	14.813
8841307	12.108	11.596	11.544	15.545	14.973	14.051	13.703	14.469	13.641	13.327	13.141	—	21.01
8841616	10.231	9.653	9.456	—	13.806	12.697	12.833	13.769	12.795	12.394	—	22.512	19.069
8842083	7.426	6.889	6.796	11.301	—	—	9.091	9.948	9.029	8.683	8.49	—	—
8842170	13.339	13.048	12.978	16.044	15.863	15.115	14.876	15.417	14.813	14.583	14.417	—	—
8909045	9.692	8.804	8.506	—	—	—	12.604	14.47	12.681	11.83	11.248	—	—
8909598	13.361	12.633	12.437	—	18.446	16.955	16.242	17.534	—	—	—	—	—
8909758	11.819	11.547	11.496	14.05	13.84	13.238	13.023	13.502	12.954	12.753	12.666	—	—
8909833	13.436	12.723	12.57	18.853	17.533	16.263	15.608	16.998	15.591	15.006	14.628	—	—
8977176	12.438	12.172	12.099	14.373	14.322	13.779	13.545	13.871	13.495	13.331	13.263	22.528	—
8977179	13.207	12.63	12.55	17.107	16.172	15.186	14.769	15.526	14.65	14.395	14.213	—	—
8977559	12.506	12.17	12.086	14.958	14.812	14.093	13.777	14.288	13.753	13.496	13.379	—	19.486
8977910	12.352	11.737	11.522	18.774	17.632	16.251	15.103	17.011	15.569	14.328	13.701	—	—
8979190	9.44	9.221	9.165	—	10.985	—	10.302	10.891	—	—	—	19.161	14.318
8980010	9.933	9.749	9.724	11.415	11.383	—	10.866	11.049	10.765	10.703	10.719	19.945	14.772
9046085	7.91	7.427	7.304	11.908	10.995	—	9.64	10.489	9.565	9.234	9.027	23.046	18.219
9046855	12.525	11.836	11.704	17.976	16.609	15.322	14.583	15.908	14.574	14.004	13.687	—	—
9046892	12.682	12.365	12.285	15.152	14.953	14.267	13.921	14.424	13.887	13.642	13.516	—	19.708
9048032	13.19	12.651	12.36	—	19.189	17.756	16.57	18.036	—	—	—	—	—
9048114	10.032	9.933	9.907	11.404	11.256	—	10.84	10.88	10.76	10.749	10.769	17.136	14.393
9048551	11.233	10.608	10.465	16.563	15.345	14.085	13.383	14.701	13.407	12.807	12.477	—	21.9
9048949	10.439	10.01	9.849	13.173	12.854	12.119	11.844	12.435	11.738	11.532	11.477	23.15	17.842
9048976	11.82	11.185	11.047	16.628	15.392	14.251	13.713	14.834	13.632	13.206	12.958	—	—
9410279	12.327	11.82	11.708	16.387	15.467	14.5	14.039	14.929	13.995	13.619	13.409	—	—
9470175	14.295	13.839	13.726	17.117	16.789	16.027	15.347	15.928	15.269	15.039	14.928	—	21.727
9471538	12.021	11.731	11.679	13.643	13.705	13.206	13.095	13.316	13.015	12.918	12.894	—	—
9532030	6.621	6.011	5.868	—	—	—	8.441	9.398	8.383	7.997	7.796	—	—
9532379	11.122	10.72	10.64	13.891	13.42	12.678	12.405	12.945	12.318	12.112	11.998	—	—
9532421	14.944	14.564	14.397	18.995	18.119	17.212	16.77	17.687	—	—	16.119	—	—
9532591	14.14	13.698	13.637	16.732	16.501	15.791	15.426	15.986	15.365	15.127	14.976	—	—
9532637	15.602	15.047	15.17	—	19.076	17.974	17.453	18.566	17.442	—	16.66	—	—
9532903	9.464	8.996	8.9	—	—	—	10.936	11.632	10.848	10.585	10.437	—	—
9533014	12.651	12.236	12.125	15.882	15.25	14.435	14.073	14.732	13.999	13.737	13.626	—	—
9594273	13.093	12.683	12.617	15.607	15.354	14.679	14.337	14.809	14.274	14.069	13.938	—	—
9594647	12.823	12.537	12.509	14.541	14.548	14.049	13.846	14.14	13.787	13.643	13.598	—	—
9654881	12.76	12.598	12.537	14.265	14.245	13.797	13.686	13.83	13.578	13.506	13.544	—	—
9655134	12.359	11.944	11.864	14.97	14.64	13.972	13.632	14.154	13.562	13.345	13.232	—	—
11970692	11.859	11.298	11.182	15.573	14.81	13.84	13.516	14.369	13.453	13.109	12.861	—	—
11971335	10.855	10.573	10.518	12.835	12.803	12.216	12.027	12.377	11.939	11.803	11.744	—	—
12020532	9.509	9.147	9.076	12.298	11.83	—	10.887	11.439	10.788	10.587	10.484	—	—
12020628	7.974	7.57	7.464	11.322	—	—	9.505	10.198	9.388	9.154	9.013	—	—
12069262	12.041	11.826	11.766	13.637	13.699	13.194	13.011	13.243	12.939	12.83	12.83	—	—
12118682	10.154	9.608	9.478	14.02	13.216	12.259	11.891	12.734	11.827	11.487	11.289	—	—
12118993	12.458	12.007	11.904	15.044	14.764	14.055	13.854	14.431	13.805	13.548	13.419	—	—

Table 7. Table 7: Fundamental stellar parameters for all 125 125 125 stars in the Kepler / XMM-Newton sample.

KIC ID	$A_V$	$\log g_{\odot }$	${\left[Fe/H\right]}_{\odot }$	$T_{\mathrm{eff}}$	$d$	Flag $d$	$M_{\ast }$	$\log L_{\mathrm{bol}}$	SpT	Variability(a)	Flag
	[mag]			[K]	[pc]		[$M_{\odot }$]	[$L_{\odot }$]
4935950	$0.04$	$4.37$	$0.040$	$4381\pm 124$	$64.7\pm 6.1$	Jmag	$0.71\pm 0.04$	$-0.79\pm 0.11$	K	ROT
4937350	$0.44$	$4.37$	$-0.24$	$6141\pm 186$	$1435.5\pm 38.3$	Gaia	$1.13\pm 0.13$	$0.22\pm 0.45$	F	CONT?
5023534	$=0.38$	$4.15$	$-0.39$	$5842\pm 101$	$658.1\pm 22.7$	Jmag	$0.93\pm 0.00$	$0.50\pm 0.03$	G	ROT
5023985	$0.39$	$4.27$	$0.079$	$6251\pm 185$	$625.3\pm 5.0$	Gaia	$1.18\pm 0.22$	$0.37\pm 0.47$	F	ROT
5024138	$0.54$	$4.48$	$-1.90$	$6156\pm 204$	$774.4\pm 66.4$	Jmag	$0.75\pm 0.03$	$-0.21\pm 0.11$	F	ROT
5024146	$0.57$	$4.24$	$-1.53$	$6121\pm 214$	$772.6\pm 98.8$	Jmag	$0.74\pm 0.03$	$-0.12\pm 0.14$	F	ECL
5024450	$0.54$	$4.42$	$-0.06$	$6199\pm 216$	$2551.9\pm 145.3$	Gaia	$1.24\pm 0.15$	$0.36\pm 0.29$	F	ECL
5025233	$0.25$	$4.30$	$-0.40$	$6103\pm 168$	$401.7\pm 183.5$	Jmag	$0.89\pm 0.13$	$0.18\pm 0.46$	F	ROT
5025294	$0.26$	$4.49$	$-0.45$	$5933\pm 168$	$437.0\pm 39.2$	Jmag	$0.86\pm 0.05$	$-0.07\pm 0.10$	G	ECL
5111932	$0.27$	$4.36$	$-0.41$	$6570\pm 182$	$440.1\pm 158.0$	Jmag	$1.05\pm 0.09$	$0.32\pm 0.44$	F	ROT
5112483	$=0.38$	$4.68$	$-0.16$	$4086\pm 143$	$50.6\pm 5.1$	Jmag	$0.64\pm 0.04$	$-1.04\pm 0.11$	K	ROT
5112508	$0.21$	$3.98$	$0.570$	$3656\pm 127$	$192.8\pm 22.9$	Jmag	$0.57\pm 0.05$	$-1.27\pm 0.15$	M	ROT	DSEP out.
5112553	$0.33$	$3.90$	$0.221$	$5941\pm 177$	$502.7\pm 110.8$	Jmag	$1.21\pm 0.09$	$0.40\pm 0.29$	G	ROT
5112741	$0.61$	$2.67$	$-0.15$	$4752\pm 87$	$2932.0\pm 178.0$	Gaia	$1.91\pm 0.40$	$1.71\pm 0.28$	K	ROT
5113557	$=0.38$	$4.11$	$-0.50$	$7082\pm 247$	$52.2\pm 21.4$	Jmag	$1.21\pm 0.28$	$0.76\pm 0.49$	F	DSCU
5113797	$0.27$	$3.82$	$-0.27$	$8362\pm 292$	$384.4\pm 22.7$	Jmag	$1.76\pm 0.09$	$1.50\pm 0.08$	A	GDOR?
5200185	$0.24$	$4.52$	$0.276$	$4525\pm 158$	$567.3\pm 29.9$	Jmag	$0.80\pm 0.03$	$-0.65\pm 0.07$	K	ROT
5397900	$0.52$	$4.44$	$-0.26$	$5879\pm 190$	$1109.4\pm 30.6$	Gaia	$0.83\pm 0.08$	$-0.05\pm 0.12$	G	ROT
5481017	$=0.38$	$=4.0$	$=0.0$	$5300\pm 200$	$6887.5\pm 210.0$	Jmag	$1.05\pm 0.00$	$0.40\pm 0.01$	K	NOPER	VOSA, *
5482181	$0.72$	$3.32$	$-0.06$	$5025\pm 175$	$2079.0\pm 1135.8$	Jmag	$1.63\pm 0.49$	$1.08\pm 0.55$	K	ROT
5483081	$1.30$	$0.03$	$-0.06$	$5100\pm 200$	$446.8\pm 102.2$	Jmag	$2.11\pm 0.00$	$1.42\pm 0.23$	K	NOPER	VOSA, DSEP out.
5653243	$=0.38$	$3.56$	$-0.54$	$5190\pm 181$	$548.7\pm 8.8$	Gaia	$0.97\pm 0.46$	$0.67\pm 0.57$	K	ROT
5739251	$0.06$	$4.64$	$0.221$	$3963\pm 138$	$137.4\pm 13.0$	Jmag	$0.63\pm 0.04$	$-1.07\pm 0.13$	K	MULT
6761498	$0.21$	$4.30$	$0.020$	$5623\pm 123$	$223.3\pm 1.3$	Gaia	$0.94\pm 0.05$	$-0.01\pm 0.19$	G	ROT
6761532	$0.25$	$4.25$	$0.030$	$5737\pm 91$	$215.6\pm 40.4$	Jmag	$0.99\pm 0.05$	$0.17\pm 0.17$	G	ROT
6761559	$0.37$	$4.56$	$-0.28$	$6071\pm 178$	$1634.1\pm 58.7$	Gaia	$0.90\pm 0.04$	$0.01\pm 0.13$	F	NOPER
6846570	$0.12$	$4.70$	$0.000$	$3834\pm 134$	$250.8\pm 1.4$	Gaia	$0.58\pm 0.06$	$-1.21\pm 0.17$	M	ROT
6846595	$0.24$	$4.33$	$-0.22$	$6718\pm 184$	$468.1\pm 5.4$	Gaia	$1.40\pm 0.05$	$0.62\pm 0.07$	F	ROT
6847018	$0.31$	$4.41$	$-0.03$	$6447\pm 184$	$677.2\pm 6.5$	Gaia	$1.40\pm 0.16$	$0.62\pm 0.26$	F	ECL
6932782	$=0.38$	$4.07$	$-0.11$	$6329\pm 91$	$446.2\pm 72.4$	Jmag	$1.31\pm 0.09$	$0.65\pm 0.14$	F	ROT
7017862	$0.22$	$4.16$	$0.140$	$5831\pm 73$	$652.9\pm 56.4$	Jmag	$1.08\pm 0.04$	$0.26\pm 0.14$	G	ROT
7018131	$0.25$	$4.26$	$-0.12$	$5064\pm 145$	$356.2\pm 1.9$	Gaia	$0.84\pm 0.03$	$-0.42\pm 0.07$	K	ROT
7018267	$0.10$	$4.32$	$0.407$	$4504\pm 129$	$170.6\pm 11.1$	Jmag	$0.78\pm 0.03$	$-0.70\pm 0.08$	K	ROT
7018277	$0.37$	$4.51$	$-0.37$	$5248\pm 183$	$933.8\pm 28.4$	Gaia	$0.74\pm 0.03$	$-0.48\pm 0.10$	K	ROT
7018323	$0.11$	$4.79$	$-0.10$	$3688\pm 129$	$199.9\pm 33.9$	Jmag	$0.52\pm 0.06$	$-1.38\pm 0.17$	M	ROT
7018521	$0.13$	$4.17$	$0.080$	$5775\pm 98$	$410.2\pm 3.5$	Gaia	$1.02\pm 0.07$	$0.28\pm 0.18$	G	ROT
7018708	$0.45$	$2.65$	$-0.12$	$5885\pm 583$	$1331.1\pm 83.6$	Jmag	$2.11\pm 0.00$	$1.53\pm 0.19$	G	GDOR
7955782	$=0.38$	$4.20$	$0.000$	$6124\pm 214$	$226.2\pm 109.5$	Jmag	$1.15\pm 0.24$	$0.39\pm 0.48$	F	ROT
8024987	$0.20$	$4.18$	$0.030$	$5469\pm 187$	$289.4\pm 1.9$	Gaia	$1.04\pm 0.01$	$0.40\pm 0.02$	G	ROT
8025009	$0.47$	$4.68$	$-0.17$	$6086\pm 189$	$1801.4\pm 115.7$	Gaia	$1.12\pm 0.12$	$0.13\pm 0.22$	F	ROT
8058211	$0.18$	$4.45$	$0.217$	$4257\pm 138$	$360.4\pm 18.9$	Jmag	$0.72\pm 0.04$	$-0.80\pm 0.12$	K	ROT
8093473	$0.17$	$4.94$	$0.000$	$3354\pm 117$	$67.2\pm 25.0$	Jmag	$0.29\pm 0.14$	$-1.99\pm 0.34$	M	ROT
8094140	$0.17$	$4.55$	$0.426$	$4432\pm 151$	$417.9\pm 21.3$	Jmag	$0.77\pm 0.03$	$-0.73\pm 0.08$	K	ECL
8094292	$=0.38$	$=4.0$	$=0.0$	$5800\pm 200$	$4791.3\pm 3090.6$	Jmag	$1.11\pm 0.47$	$0.49\pm 0.55$	G	NOPER	VOSA, *
8195877	$0.40$	$4.16$	$-0.24$	$6256\pm 110$	$907.6\pm 11.8$	Gaia	$1.26\pm 0.09$	$0.52\pm 0.18$	F	ECL
8229034	$0.45$	$3.91$	$0.282$	$5453\pm 190$	$1401.8\pm 34.5$	Gaia	$1.19\pm 0.01$	$0.48\pm 0.03$	G	UNCL
8454353	$0.10$	$4.88$	$-0.10$	$3514\pm 122$	$132.1\pm 26.6$	Jmag	$0.47\pm 0.13$	$-1.54\pm 0.33$	M	ROT
8517293	$=0.38$	$4.16$	$-0.05$	$5939\pm 107$	$277.4\pm 1.9$	Gaia	$1.09\pm 0.06$	$0.37\pm 0.16$	G	ROT
8517303	$0.51$	$2.83$	$-0.08$	$4798\pm 149$	$1009.0\pm 354.9$	Jmag	$1.73\pm 0.67$	$1.53\pm 0.36$	K	ROT
8518250	$=0.38$	$4.35$	$0.090$	$4948\pm 249$	$84.4\pm 0.2$	Gaia	$0.79\pm 0.05$	$-0.49\pm 0.15$	K	ROT
8518323	$0.49$	$4.06$	$0.000$	$6261\pm 114$	$761.8\pm 8.2$	Gaia	$1.24\pm 0.16$	$0.61\pm 0.18$	F	ROT
8520065	$0.60$	$4.13$	$-0.03$	$6049\pm 132$	$1048.9\pm 224.5$	Jmag	$1.14\pm 0.11$	$0.45\pm 0.19$	F	ROT UN
8520464	$0.51$	$4.17$	$0.050$	$5866\pm 89$	$660.3\pm 7.1$	Gaia	$1.05\pm 0.06$	$0.32\pm 0.16$	G	ROT
8581658	$0.40$	$4.19$	$-0.48$	$6230\pm 187$	$959.5\pm 13.9$	Gaia	$0.94\pm 0.22$	$0.34\pm 0.49$	F	ECL
8582224	$0.21$	$4.23$	$0.030$	$5798\pm 86$	$467.1\pm 3.3$	Gaia	$1.01\pm 0.05$	$0.22\pm 0.15$	G	ROT
8582406	$0.06$	$4.06$	$-0.11$	$4533\pm 267$	$166.7\pm 0.4$	Gaia	$0.72\pm 0.05$	$-0.73\pm 0.16$	K	ROT
8583756	$0.20$	$4.77$	$-0.20$	$3737\pm 130$	$290.0\pm 2.6$	Gaia	$0.54\pm 0.06$	$-1.31\pm 0.18$	M	ROT
8583911	$0.33$	$4.36$	$-0.34$	$5240\pm 151$	$948.5\pm 17.1$	Gaia	$0.74\pm 0.03$	$-0.48\pm 0.10$	K	ROT
8584439	$0.47$	$4.02$	$-0.49$	$5813\pm 187$	$692.3\pm 8.3$	Gaia	$0.93\pm 0.01$	$0.50\pm 0.04$	G	ROT
8584672	$0.57$	$1.97$	$-0.69$	$4448\pm 135$	$1940.8\pm 420.5$	Jmag	$1.19\pm 0.00$	$2.30\pm 0.18$	K	NOPER
8584920	$=0.38$	$4.04$	$-0.04$	$6285\pm 103$	$544.1\pm 8.1$	Gaia	$1.28\pm 0.10$	$0.65\pm 0.16$	F	ROT
8584977	$=0.38$	$3.92$	$-0.32$	$6421\pm 224$	$522.5\pm 329.4$	Jmag	$1.11\pm 0.41$	$0.75\pm 0.54$	F	ROT
8584986	$1.06$	$2.20$	$-0.39$	$4650\pm 162$	$7377.9\pm 2712.7$	Jmag	$0.99\pm 0.92$	$1.86\pm 0.49$	K	NOPER
8585000	$0.32$	$3.70$	$0.080$	$5002\pm 200$	$1000.9\pm 591.6$	Jmag	$1.15\pm 0.54$	$0.55\pm 0.63$	K	ROT
8647149	$0.19$	$4.11$	$-0.08$	$6285\pm 117$	$555.7\pm 101.7$	Jmag	$1.28\pm 0.09$	$0.59\pm 0.16$	F	MULT
8647865	$=0.38$	$4.06$	$0.020$	$6904\pm 241$	$422.0\pm 200.7$	Jmag	$1.54\pm 0.33$	$0.88\pm 0.48$	F	ROT
8648195	$0.58$	$4.15$	$-0.14$	$6043\pm 132$	$1034.7\pm 14.0$	Gaia	$1.13\pm 0.09$	$0.42\pm 0.17$	F	ROT
8648671	$0.63$	$2.81$	$0.006$	$4997\pm 158$	$1323.3\pm 36.7$	Gaia	$2.11\pm 0.00$	$1.53\pm 0.21$	K	NOPER
8649876	$0.48$	$3.70$	$-0.03$	$6181\pm 191$	$1085.9\pm 700.8$	Jmag	$1.52\pm 0.57$	$1.03\pm 0.54$	F	ROT
8650057	$0.28$	$4.12$	$-0.19$	$6413\pm 92$	$393.9\pm 4.0$	Gaia	$1.33\pm 0.09$	$0.63\pm 0.14$	F	ROT
8702424	$0.23$	$4.29$	$0.130$	$5741\pm 93$	$631.4\pm 85.9$	Jmag	$1.03\pm 0.04$	$0.13\pm 0.14$	G	ROT
8702606	$0.03$	$3.99$	$-0.04$	$5599\pm 183$	$128.5\pm 92.4$	Jmag	$1.05\pm 0.58$	$0.42\pm 0.61$	G	ROT
8703129	$0.41$	$2.99$	$-0.05$	$4964\pm 155$	$698.2\pm 11.5$	Gaia	$2.04\pm 0.56$	$1.49\pm 0.45$	K	ROT
8703413	$0.21$	$3.77$	$0.254$	$7936\pm 277$	$161.8\pm 0.0$	Jmag	$1.50\pm 0.00$	$0.68\pm 0.00$	A	ROT
8703528	$0.42$	$3.70$	$0.245$	$5498\pm 211$	$559.0\pm 6.3$	Gaia	$1.47\pm 0.00$	$0.81\pm 0.06$	G	CONT
8712626	$0.28$	$4.24$	$-0.03$	$5770\pm 89$	$286.2\pm 46.6$	Jmag	$1.00\pm 0.05$	$0.20\pm 0.15$	G	ROT
8713594	$0.69$	$4.11$	$-0.11$	$6563\pm 125$	$1491.1\pm 40.9$	Gaia	$1.39\pm 0.09$	$0.69\pm 0.14$	F	ROT
8713822	$0.67$	$3.38$	$-0.44$	$5028\pm 175$	$800.0\pm 14.9$	Gaia	$0.95\pm 0.01$	$1.09\pm 0.48$	K	ROT
8713828	$=0.38$	$4.13$	$-0.19$	$6591\pm 122$	$1067.2\pm 33.4$	Gaia	$1.38\pm 0.09$	$0.68\pm 0.14$	F	ROT
8767669	$0.12$	$4.39$	$0.312$	$4331\pm 137$	$229.7\pm 0.0$	Jmag	$0.77\pm 0.00$	$-0.72\pm 0.00$	K	MULT
8832676	$0.24$	$4.77$	$-0.20$	$3737\pm 130$	$230.4\pm 31.6$	Jmag	$0.54\pm 0.06$	$-1.31\pm 0.18$	M	ROT
8840075	$=0.38$	$4.28$	$-0.28$	$6421\pm 224$	$190.4\pm 84.0$	Jmag	$0.99\pm 0.15$	$0.34\pm 0.46$	F	ROT
8841307	$0.52$	$3.76$	$0.040$	$5120\pm 227$	$492.0\pm 5.0$	Gaia	$1.20\pm 0.67$	$0.55\pm 0.65$	K	ROT
8841616	$=0.38$	$4.69$	$-0.76$	$4462\pm 156$	$76.4\pm 6.3$	Jmag	$0.54\pm 0.03$	$-1.02\pm 0.11$	K	ECL
8842083	$0.29$	$2.24$	$-0.19$	$4769\pm 135$	$42.2\pm 0.0$	Jmag	$0.91\pm 0.00$	$-0.32\pm 0.00$	K	ROT
8842170	$0.68$	$3.90$	$-0.21$	$5831\pm 204$	$1369.2\pm 63.9$	Gaia	$1.21\pm 0.53$	$0.64\pm 0.56$	G	CONT?
8909045	$0.90$	$1.09$	$-0.08$	$3800\pm 200$	$5142.4\pm 1062.5$	Jmag	$1.44\pm 0.42$	$2.77\pm 0.19$	M	NODATA	VOSA
8909598	$=0.38$	$=4.0$	$=0.0$	$3600\pm 200$	$219.9\pm 66.6$	Jmag	$0.49\pm 0.14$	$-1.47\pm 0.39$	M	ROT	VOSA, *
8909758	$0.51$	$4.01$	$-0.03$	$6299\pm 108$	$910.4\pm 24.9$	Gaia	$1.38\pm 0.13$	$0.72\pm 0.17$	F	ROT
8909833	$0.21$	$4.73$	$-0.20$	$3903\pm 136$	$313.6\pm 43.0$	Jmag	$0.60\pm 0.05$	$-1.15\pm 0.17$	K	MULT
8977176	$0.49$	$4.14$	$-0.22$	$6499\pm 114$	$975.6\pm 18.1$	Gaia	$1.35\pm 0.08$	$0.63\pm 0.14$	F	MULT
8977179	$0.30$	$4.41$	$-0.03$	$4992\pm 152$	$586.0\pm 7.3$	Gaia	$0.82\pm 0.04$	$-0.49\pm 0.10$	K	ROT
8977559	$0.63$	$3.68$	$-0.74$	$5941\pm 209$	$1573.8\pm 1015.6$	Jmag	$1.21\pm 0.57$	$0.89\pm 0.57$	G	ROT
8977910	$0.18$	$5.01$	$0.000$	$3283\pm 114$	$61.2\pm 30.7$	Jmag	$0.23\pm 0.12$	$-2.22\pm 0.62$	M	ROT
8979190	$=0.38$	$4.14$	$-0.25$	$6459\pm 67$	$227.7\pm 32.4$	Jmag	$1.03\pm 0.04$	$0.50\pm 0.13$	F	ROT
8980010	$0.16$	$4.11$	$-0.07$	$6446\pm 71$	$344.8\pm 58.7$	Jmag	$1.35\pm 0.10$	$0.65\pm 0.15$	F	ROT
9046085	$0.38$	$2.67$	$-0.16$	$4857\pm 140$	$595.7\pm 138.3$	Jmag	$2.12\pm 0.00$	$1.72\pm 0.19$	K	NOPER
9046855	$0.13$	$4.47$	$-0.10$	$4133\pm 124$	$254.4\pm 1.3$	Gaia	$0.66\pm 0.04$	$-0.96\pm 0.13$	K	ROT
9046892	$0.64$	$4.09$	$-0.12$	$6304\pm 110$	$838.9\pm 11.5$	Gaia	$1.27\pm 0.13$	$0.61\pm 0.17$	F	ROT
9048032	$=0.38$	$=4.0$	$=0.0$	$3400\pm 200$	$134.3\pm 66.5$	Jmag	$0.34\pm 0.17$	$-1.86\pm 0.59$	M	ROT	VOSA, *
9048114	$0.41$	$3.84$	$-0.03$	$7759\pm 434$	$388.9\pm 0.0$	Jmag	$1.53\pm 0.00$	$0.82\pm 0.00$	A	ROT
9048551	$0.08$	$4.00$	$-0.08$	$4351\pm 273$	$125.9\pm 0.3$	Gaia	$0.68\pm 0.06$	$-0.83\pm 0.19$	K	ROT
9048949	$0.11$	$4.42$	$-0.13$	$5185\pm 161$	$164.1\pm 12.4$	Jmag	$0.86\pm 0.03$	$-0.36\pm 0.09$	K	ROT
9048976	$0.12$	$4.37$	$-0.49$	$4258\pm 125$	$197.2\pm 0.7$	Gaia	$0.58\pm 0.02$	$-1.02\pm 0.07$	K	ROT
9410279	$0.15$	$4.48$	$-0.65$	$4575\pm 132$	$247.5\pm 15.1$	Jmag	$0.63\pm 0.02$	$-0.82\pm 0.07$	K	ROT
9470175	$0.42$	$4.50$	$-0.02$	$5552\pm 170$	$1232.7\pm 47.2$	Gaia	$0.94\pm 0.06$	$-0.16\pm 0.09$	G	CONT
9471538	$0.38$	$4.18$	$-0.29$	$6627\pm 196$	$749.1\pm 11.7$	Gaia	$1.07\pm 0.23$	$0.52\pm 0.48$	F	ROT
9532030	$0.01$	$2.52$	$-0.02$	$4454\pm 77$	$343.7\pm 62.7$	Jmag	$1.07\pm 0.00$	$1.72\pm 0.20$	K	NOPER
9532379	$0.22$	$4.07$	$-0.20$	$5408\pm 151$	$582.3\pm 8.6$	Gaia	$1.04\pm 0.00$	$0.40\pm 0.00$	G	ROT
9532421	$0.53$	$4.25$	$0.051$	$4884\pm 170$	$1112.3\pm 46.2$	Jmag	$0.80\pm 0.03$	$-0.54\pm 0.09$	K	ECL
9532591	$0.39$	$4.58$	$-0.41$	$5502\pm 164$	$1574.4\pm 103.2$	Gaia	$0.77\pm 0.14$	$-0.35\pm 0.25$	G	CONT?
9532637	$0.40$	$4.57$	$-0.16$	$4507\pm 157$	$1327.2\pm 152.4$	Jmag	$0.75\pm 0.05$	$-0.68\pm 0.14$	K	ECL
9532903	$0.49$	$2.70$	$-0.09$	$4984\pm 94$	$977.2\pm 145.7$	Jmag	$2.11\pm 0.00$	$1.55\pm 0.12$	K	ROT
9533014	$0.34$	$4.15$	$-0.02$	$5276\pm 158$	$523.1\pm 5.3$	Gaia	$1.05\pm 0.00$	$0.40\pm 0.01$	K	ROT
9594273	$0.31$	$4.52$	$-0.61$	$5647\pm 168$	$854.9\pm 16.5$	Gaia	$0.78\pm 0.13$	$-0.23\pm 0.14$	G	GDOR?
9594647	$0.34$	$4.50$	$-0.76$	$6296\pm 184$	$956.2\pm 17.3$	Gaia	$0.84\pm 0.04$	$0.02\pm 0.13$	F	NOPER
9654881	$0.51$	$4.12$	$-0.16$	$7050\pm 227$	$1164.8\pm 29.9$	Gaia	$1.53\pm 0.32$	$0.84\pm 0.48$	F	UNCL
9655134	$0.23$	$4.46$	$-0.66$	$5389\pm 153$	$463.9\pm 4.4$	Gaia	$0.77\pm 0.03$	$-0.38\pm 0.10$	G	ROT
11970692	$0.49$	$3.19$	$-0.24$	$4887\pm 145$	$1315.6\pm 26.7$	Gaia	$1.44\pm 0.67$	$1.12\pm 0.56$	K	ROT
11971335	$0.16$	$4.48$	$-0.33$	$5912\pm 161$	$244.2\pm 17.7$	Jmag	$0.85\pm 0.06$	$-0.08\pm 0.07$	G	ROT
12020532	$0.25$	$2.66$	$-0.18$	$5036\pm 211$	$920.3\pm 265.1$	Jmag	$2.11\pm 0.00$	$1.49\pm 0.27$	K	ROT UN
12020628	$0.34$	$2.87$	$0.268$	$5239\pm 166$	$366.7\pm 0.7$	Jmag	$2.18\pm 0.00$	$1.32\pm 0.05$	K	ROT
12069262	$0.37$	$4.16$	$-0.19$	$6589\pm 194$	$871.3\pm 342.4$	Jmag	$1.35\pm 0.27$	$0.63\pm 0.48$	F	ROT
12118682	$0.46$	$2.66$	$-0.20$	$4918\pm 146$	$1754.4\pm 73.3$	Gaia	$2.11\pm 0.00$	$1.64\pm 0.19$	K	ROT
12118993	$0.21$	$4.55$	$-0.92$	$5180\pm 151$	$842.4\pm 12.0$	Gaia	$0.65\pm 0.02$	$-0.63\pm 0.07$	K	ROT
(a)The variability classification code (column $11$) is the following: ROT: rotational variability; ROT UNCL: probably rotational variability but it is not possible
to establish a reliable rotation period; UNCL: unclear periodic variability; NOPER: no apparent periodicity; ECL: eclipsing binary; CONT: contact binary;
MULT: multiperiodic; DSCU: $\delta$ Scuti-like; GDOR: $\gamma$ Dor-like.

Table 8. Table 8: Rotation and flaring parameters derived from the Kepler light curves for all 74 74 74 rotationally-variable main-sequence stars in the Kepler / XMM-Newton sample

KIC ID	$P_{\mathrm{rot}}$	$R_0$	$\log R_{\mathrm{per}}$	$S_{\mathrm{ph}}$	$S_{\mathrm{ph},5}$	$S_{\mathrm{flat}}$		$\log A_{\mathrm{peak},\min }$	$\log A_{\mathrm{peak},\max }$	Flare rate
	[d]		$[\%]$	[ppm]	[ppm]	[ppm]				$\left[d^{-1}\right]$
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
4935950	$12.66$	$0.38$	$-0.15$	$2621$	$2467$	$41.9$	$-3.28$	$-3.56$	$-2.34$	$0.012$
5023534	$17.76$	$1.52$	$-0.16$	$2361$	$2393$	$109.4$	$-2.94$	$-3.17$	$-2.48$	$0.015$
5023985	$6.58$	$1.40$	$0.16$	$5004$	$4911$	$142.0$	$-2.82$	$-2.83$	$-2.46$	$0.003$
5024138	$7.52$	$1.20$	$-0.83$	$471$	$467$	$326.4$	$-$	$-$	$-$	$-$
5025233	$4.04$	$0.56$	$-0.11$	$2866$	$2799$	$93.4$	$-2.83$	$-3.33$	$-2.01$	$0.060$
5111932	$1.45$	$1.56$	$-0.78$	$565$	$542$	$79.0$	$-3.07$	$-3.34$	$-2.79$	$0.015$
5112483	$63.51$	$1.64$	$0.54$	$9582$	$0$	$55.2$	$-3.25$	$-3.58$	$-2.93$	$0.006$
5112508	$0.76$	$0.02$	$0.41$	$8858$	$8309$	$1021.9$	$-1.49$	$-2.17$	$-1.09$	$0.098$
5112553	$5.33$	$0.53$	$-0.07$	$3285$	$3293$	$77.2$	$-3.10$	$-3.48$	$-2.46$	$0.010$
5200185	$3.74$	$0.12$	$0.49$	$11797$	$11809$	$519.4$	$-2.05$	$-2.50$	$-1.23$	$0.052$
5397900	$2.21$	$0.20$	$0.59$	$15368$	$13912$	$515.9$	$-2.11$	$-2.53$	$-1.58$	$0.053$
5653243	$19.66$	$0.92$	$0.27$	$6196$	$6196$	$80.5$	$-3.27$	$-3.31$	$-3.11$	$0.003$
6761498	$7.17$	$0.47$	$0.15$	$5160$	$4612$	$75.4$	$-3.11$	$-3.53$	$-2.36$	$0.017$
6761532	$4.82$	$0.36$	$0.45$	$9436$	$9127$	$69.5$	$-2.82$	$-3.22$	$-2.58$	$0.021$
6846570	$4.13$	$0.09$	$0.73$	$18594$	$18157$	$496.2$	$-2.05$	$-2.55$	$-0.72$	$0.183$
6846595	$1.00$	$3.90$	$-0.56$	$927$	$893$	$81.2$	$-2.74$	$-3.36$	$-2.49$	$0.025$
6932782	$12.87$	$3.67$	$-1.25$	$187$	$171$	$59.0$	$-$	$-$	$-$	$-$
7017862	$2.02$	$0.17$	$-0.08$	$3734$	$3147$	$128.6$	$-2.68$	$-3.27$	$-1.80$	$0.066$
7018131	$10.59$	$0.46$	$0.16$	$4910$	$4454$	$164.5$	$-2.81$	$-2.89$	$-2.67$	$0.005$
7018267	$17.59$	$0.56$	$-0.19$	$2111$	$1986$	$103.7$	$-$	$-$	$-$	$-$
7018277	$1.57$	$0.08$	$0.62$	$16197$	$15241$	$648.3$	$-1.81$	$-2.62$	$-0.55$	$0.078$
7018323	$6.27$	$0.13$	$0.36$	$8191$	$7625$	$533.2$	$-2.12$	$-2.49$	$-0.90$	$0.052$
7018521	$10.79$	$0.84$	$-0.09$	$2832$	$2650$	$73.6$	$-$	$-$	$-$	$-$
7955782	$2.81$	$0.41$	$-0.26$	$2105$	$1973$	$43.2$	$-3.35$	$-3.61$	$-3.01$	$0.009$
8024987	$8.03$	$0.46$	$0.11$	$4804$	$4320$	$102.9$	$-2.93$	$-3.10$	$-2.81$	$0.001$
8025009	$13.97$	$1.87$	$-0.64$	$708$	$709$	$582.2$	$-1.51$	$-1.51$	$-1.51$	$0.001$
8058211	$1.70$	$0.05$	$0.69$	$16504$	$16518$	$623.2$	$-2.03$	$-2.66$	$-0.78$	$0.096$
8093473	$6.03$	$0.11$	$0.69$	$16562$	$16498$	$776.9$	$-1.79$	$-2.29$	$-0.56$	$0.228$
8454353	$1.49$	$0.03$	$0.58$	$14157$	$13433$	$640.0$	$-1.95$	$-2.69$	$-0.36$	$0.211$
8517293	$2.66$	$0.26$	$0.22$	$6293$	$5865$	$114.2$	$-2.68$	$-3.19$	$-2.18$	$0.011$
8518250	$19.62$	$0.80$	$0.21$	$6132$	$5911$	$60.7$	$-3.26$	$-3.43$	$-2.76$	$0.021$
8518323	$2.78$	$0.61$	$-0.23$	$2049$	$1906$	$155.1$	$-2.90$	$-2.91$	$-2.86$	$0.002$
8520464	$7.70$	$0.68$	$0.15$	$4680$	$4634$	$177.0$	$-2.80$	$-2.87$	$-2.68$	$0.002$
8582224	$8.99$	$0.72$	$0.14$	$4944$	$4815$	$157.4$	$-2.72$	$-2.82$	$-2.71$	$0.002$
8582406	$16.04$	$0.52$	$-0.22$	$2187$	$2069$	$155.5$	$-$	$-$	$-$	$-$
8583756	$9.24$	$0.20$	$0.46$	$10156$	$9959$	$614.8$	$-2.25$	$-2.43$	$-2.02$	$0.009$
8583911	$4.21$	$0.20$	$0.16$	$5341$	$5385$	$264.9$	$-2.63$	$-2.72$	$-2.04$	$0.004$
8584439	$2.86$	$0.23$	$-0.06$	$3033$	$2914$	$186.8$	$-2.61$	$-2.86$	$-2.15$	$0.013$
8584920	$4.45$	$1.07$	$-0.54$	$1046$	$1005$	$109.8$	$-$	$-$	$-$	$-$
8584977	$2.15$	$0.93$	$-1.10$	$260$	$258$	$87.1$	$-$	$-$	$-$	$-$
8585000	$11.63$	$0.49$	$0.17$	$5397$	$4924$	$153.6$	$-2.79$	$-2.93$	$-2.56$	$0.013$
8647865	$1.25$	$-$	$-1.38$	$151$	$142$	$44.8$	$-3.34$	$-3.34$	$-3.34$	$0.001$
8648195	$7.36$	$0.89$	$-0.11$	$2832$	$2810$	$137.6$	$-2.71$	$-2.71$	$-2.71$	$0.001$
8649876	$10.81$	$1.85$	$-1.05$	$278$	$275$	$91.3$	$-$	$-$	$-$	$-$
8650057	$2.66$	$1.11$	$-1.13$	$247$	$240$	$60.8$	$-3.35$	$-3.35$	$-3.35$	$0.001$
8702424	$0.90$	$0.07$	$-0.31$	$1784$	$1631$	$147.6$	$-2.54$	$-3.14$	$-1.44$	$0.056$
8702606	$19.86$	$1.28$	$-1.48$	$98$	$98$	$63.1$	$-$	$-$	$-$	$-$
8703413	$6.62$	$-$	$-1.13$	$247$	$248$	$12.7$	$-3.97$	$-4.37$	$-3.73$	$0.002$
8712626	$9.99$	$0.77$	$-0.11$	$3168$	$3100$	$72.5$	$-3.25$	$-3.27$	$-2.72$	$0.002$
8713594	$2.49$	$2.55$	$-0.15$	$2280$	$2261$	$166.8$	$-2.85$	$-3.27$	$-2.58$	$0.008$
8713828	$2.57$	$3.23$	$-1.19$	$222$	$208$	$89.2$	$-3.06$	$-3.06$	$-3.06$	$0.001$
8832676	$10.93$	$0.24$	$0.10$	$4447$	$4310$	$541.9$	$-1.90$	$-2.51$	$-0.61$	$0.116$
8840075	$2.64$	$1.14$	$-1.34$	$198$	$181$	$35.9$	$-3.28$	$-3.28$	$-3.28$	$0.001$
8841307	$6.15$	$0.28$	$0.52$	$12138$	$11358$	$191.2$	$-2.65$	$-3.08$	$-2.10$	$0.018$
8909758	$3.09$	$0.78$	$-0.77$	$591$	$566$	$124.9$	$-2.91$	$-3.12$	$-2.46$	$0.008$
8977179	$1.59$	$0.07$	$-0.08$	$4048$	$3251$	$387.7$	$-2.29$	$-2.81$	$-1.97$	$0.011$
8977559	$2.47$	$0.25$	$-0.22$	$2244$	$2081$	$192.4$	$-2.64$	$-2.83$	$-1.92$	$0.004$
8977910	$10.73$	$0.18$	$0.33$	$7428$	$7109$	$528.5$	$-2.00$	$-2.47$	$-0.46$	$0.090$
8979190	$3.47$	$1.84$	$-1.24$	$214$	$211$	$40.7$	$-$	$-$	$-$	$-$
8980010	$2.27$	$1.12$	$-1.15$	$253$	$256$	$53.4$	$-$	$-$	$-$	$-$
9046855	$26.47$	$0.70$	$-0.07$	$2724$	$0$	$305.6$	$-2.66$	$-2.66$	$-2.66$	$0.001$
9046892	$6.54$	$1.69$	$-1.10$	$246$	$246$	$190.6$	$-$	$-$	$-$	$-$
9048114	$2.80$	$-$	$-0.92$	$446$	$422$	$36.2$	$-3.44$	$-3.44$	$-3.44$	$0.001$
9048551	$8.58$	$0.25$	$0.59$	$14329$	$12329$	$226.0$	$-2.49$	$-2.92$	$-1.47$	$0.164$
9048949	$3.47$	$0.16$	$0.57$	$14386$	$13203$	$192.0$	$-2.31$	$-2.95$	$-1.27$	$0.141$
9048976	$10.95$	$0.31$	$0.37$	$8089$	$7778$	$198.4$	$-2.43$	$-2.91$	$-1.57$	$0.097$
9410279	$1.57$	$0.05$	$-0.75$	$585$	$574$	$175.4$	$-2.61$	$-2.96$	$-1.84$	$0.032$
9471538	$0.90$	$1.50$	$-0.85$	$505$	$502$	$109.6$	$-2.75$	$-3.25$	$-1.41$	$0.047$
9532379	$9.26$	$0.51$	$0.38$	$9012$	$8354$	$94.9$	$-2.94$	$-2.94$	$-2.94$	$0.001$
9533014	$6.77$	$0.34$	$0.44$	$9615$	$9047$	$190.9$	$-2.59$	$-2.99$	$-2.34$	$0.006$
9655134	$15.09$	$0.81$	$-0.02$	$3539$	$2945$	$146.9$	$-2.59$	$-3.26$	$-1.27$	$0.044$
11971335	$0.78$	$0.07$	$0.66$	$16547$	$15162$	$985.7$	$-1.90$	$-2.02$	$-1.34$	$0.003$
12069262	$2.78$	$3.44$	$-0.23$	$2109$	$2010$	$126.3$	$-2.83$	$-3.13$	$-2.40$	$0.006$
12118993	$0.55$	$0.03$	$0.91$	$27536$	$26132$	$4247.5$	$-0.99$	$-1.23$	$-0.77$	$0.009$
(1) - rotation period from ACF, (2) - Rossby number from Eq.36 in Cranmer & Saar (2011), (3) - percentile amplitude of the rotation
cycle after McQuillan et al. (2013), (4) - standard deviation of the whole lightcurve, (5) - average of the standard deviation for $5$ rotation
cycles after Mathur et al. (2014b), (6) - minimum of the standard deviations from all quarters for the “flattened” lightcurves, (7) - median
of relative flare peak amplitudes, (8) - minimum relative flare peak amplitude, (9) - maximum relative flare amplitude, (10) - flare rate.

Table 9. Table 9: X-ray luminosity (in the 0.2 − 2 0.2 2 0.2-2  keV energy band), observed NUV and FUV luminosity, and NUV and FUV chromospheric excess luminosity.

KIC_ID	$L_X$	$L_{\mathrm{NUV}}$	$L_{\mathrm{FUV}}$	$L_{\mathrm{NUV},\mathrm{exc}}$	$L_{\mathrm{FUV},\mathrm{exc}}$
	[${10}^{29}$ erg/s]	[${10}^{29}$ erg/s]	[${10}^{29}$ erg/s]	[${10}^{29}$ erg/s]	[${10}^{29}$ erg/s]
4935950	$0.04\pm 0.02$	$-$	$-$	$-$	$-$
4937350	$21.40\pm 9.08$	$-$	$-$	$-$	$-$
5023534	$2.58\pm 1.34$	$896.0\pm 63.3$	$-$	$406.0$	$-$
5023985	$1.96\pm 0.99$	$490.0\pm 13.4$	$-$	$-$	$-$
5024138	$2.52\pm 1.29$	$-$	$-$	$-$	$-$
5024146	$4.51\pm 1.91$	$175.0\pm 47.6$	$-$	$-$	$-$
5024450	$40.05\pm 15.43$	$-$	$-$	$-$	$-$
5025233	$4.07\pm 3.91$	$674.0\pm 615.0$	$-$	$-$	$-$
5025294	$0.76\pm 0.36$	$-$	$-$	$-$	$-$
5111932	$1.23\pm 0.98$	$2480.0\pm 1780.0$	$-$	$-$	$-$
5112483	$0.04\pm 0.01$	$1.6\pm 0.3$	$-$	$1.5$	$-$
5112508	$1.71\pm 0.44$	$-$	$-$	$-$	$-$
5112553	$2.69\pm 1.32$	$692.0\pm 305.0$	$-$	$-$	$-$
5112741	$97.25\pm 28.24$	$-$	$-$	$-$	$-$
5113557	$0.89\pm 0.73$	$-$	$-$	$-$	$-$
5113797	$15.36\pm 2.94$	$-$	$-$	$-$	$-$
5200185	$4.12\pm 2.08$	$-$	$-$	$-$	$-$
5397900	$9.85\pm 5.35$	$2520.0\pm 155.0$	$-$	$-$	$-$
5481017	$840.55\pm 467.82$	$-$	$-$	$-$	$-$
5482181	$132.70\pm 147.01$	$1060.0\pm 1180.0$	$-$	$714.0$	$-$
5483081	$0.03\pm 0.48$	$-$	$-$	$-$	$-$
5653243	$18.36\pm 2.54$	$1100.0\pm 38.2$	$-$	$221.0$	$-$
5739251	$0.72\pm 0.17$	$-$	$-$	$-$	$-$
6761498	$0.72\pm 0.26$	$-$	$-$	$-$	$-$
6761532	$2.13\pm 0.86$	$393.0\pm 147.0$	$1.3\pm 0.6$	$-$	$1.0$
6761559	$36.58\pm 13.15$	$-$	$-$	$-$	$-$
6846570	$3.42\pm 0.58$	$-$	$-$	$-$	$-$
6846595	$2.28\pm 0.97$	$3700.0\pm 86.3$	$36.1\pm 4.2$	$-$	$-$
6847018	$2.67\pm 1.45$	$1490.0\pm 32.5$	$-$	$475.0$	$-$
6932782	$0.86\pm 0.52$	$5330.0\pm 1730.0$	$23.5\pm 8.0$	$-$	$-$
7017862	$10.88\pm 3.62$	$620.0\pm 107.0$	$-$	$-$	$-$
7018131	$0.54\pm 0.22$	$14.6\pm 0.2$	$-$	$7.8$	$-$
7018267	$0.08\pm 0.04$	$1.2\pm 0.2$	$-$	$1.0$	$-$
7018277	$10.62\pm 2.59$	$65.0\pm 11.4$	$-$	$45.3$	$-$
7018323	$0.39\pm 0.16$	$-$	$-$	$-$	$-$
7018521	$0.54\pm 0.26$	$284.0\pm 5.2$	$-$	$-$	$-$
7018708	$850.95\pm 112.24$	$10200.0\pm 1280.0$	$74.6\pm 18.3$	$7190.0$	$73.8$
7955782	$3.66\pm 3.59$	$2320.0\pm 2250.0$	$7.3\pm 7.1$	$-$	$-$
8024987	$0.44\pm 0.20$	$31.9\pm 0.8$	$1.5\pm 0.5$	$14.1$	$1.5$
8025009	$20.11\pm 9.52$	$280.0\pm 47.6$	$-$	$-$	$-$
8058211	$3.23\pm 0.89$	$-$	$-$	$-$	$-$
8093473	$0.31\pm 0.27$	$0.2\pm 0.2$	$0.1\pm 0.1$	$0.2$	$0.1$
8094140	$2.72\pm 0.56$	$-$	$-$	$-$	$-$
8094292	$478.63\pm 648.79$	$-$	$-$	$-$	$-$
8195877	$32.17\pm 8.51$	$2040.0\pm 59.1$	$-$	$1140.0$	$-$
8229034	$30.87\pm 13.22$	$192.0\pm 21.1$	$-$	$85.1$	$-$
8454353	$1.04\pm 0.43$	$-$	$-$	$-$	$-$
8517293	$1.75\pm 0.39$	$647.0\pm 9.5$	$-$	$305.0$	$-$
8517303	$107.35\pm 75.72$	$1210.0\pm 852.0$	$-$	$1110.0$	$-$
8518250	$0.21\pm 0.02$	$4.9\pm 0.1$	$-$	$-$	$-$
8518323	$1.83\pm 0.54$	$1460.0\pm 41.4$	$-$	$554.0$	$-$
8520065	$54.89\pm 23.92$	$1670.0\pm 774.0$	$-$	$995.0$	$-$
8520464	$1.21\pm 0.46$	$-$	$-$	$-$	$-$
8581658	$4.59\pm 2.58$	$3310.0\pm 104.0$	$-$	$1310.0$	$-$
8582224	$0.37\pm 0.16$	$129.0\pm 2.7$	$-$	$-$	$-$
8582406	$0.06\pm 0.02$	$1.1\pm 0.0$	$-$	$1.0$	$-$
8583756	$0.59\pm 0.12$	$-$	$-$	$-$	$-$
8583911	$3.37\pm 1.42$	$-$	$-$	$-$	$-$
8584439	$1.90\pm 0.50$	$862.0\pm 24.6$	$-$	$454.0$	$-$
8584672	$19.76\pm 9.77$	$-$	$-$	$-$	$-$
8584920	$0.60\pm 0.24$	$1100.0\pm 33.3$	$-$	$472.0$	$-$
8584977	$1.00\pm 1.28$	$5950.0\pm 7500.0$	$61.2\pm 77.2$	$-$	$20.0$
8584986	$117.23\pm 139.06$	$-$	$-$	$-$	$-$
8585000	$8.44\pm 10.25$	$134.0\pm 159.0$	$-$	$67.8$	$-$
8647149	$0.97\pm 0.46$	$1210.0\pm 443.0$	$3.9\pm 1.9$	$-$	$2.3$
8647865	$1.85\pm 1.77$	$15700.0\pm 14900.0$	$205.0\pm 195.0$	$4880.0$	$72.6$
8648195	$4.87\pm 1.20$	$2650.0\pm 77.3$	$-$	$1060.0$	$-$
8648671	$13.10\pm 6.69$	$1370.0\pm 469.0$	$-$	$813.0$	$-$
8649876	$4.45\pm 6.11$	$5220.0\pm 6740.0$	$-$	$-$	$-$
8650057	$0.36\pm 0.12$	$3680.0\pm 95.3$	$23.4\pm 0.5$	$-$	$-$
8702424	$8.31\pm 5.12$	$-$	$-$	$-$	$-$
8702606	$0.09\pm 0.14$	$-$	$-$	$-$	$-$
8703129	$3.61\pm 1.29$	$-$	$-$	$-$	$-$
8703413	$0.38\pm 0.10$	$-$	$-$	$-$	$-$
8703528	$17.32\pm 1.85$	$377.0\pm 8.6$	$-$	$233.0$	$-$
8712626	$0.78\pm 0.31$	$364.0\pm 119.0$	$-$	$-$	$-$
8713594	$6.58\pm 3.10$	$10500.0\pm 587.0$	$-$	$3440.0$	$-$
8713822	$31.57\pm 3.04$	$102.0\pm 10.3$	$-$	$68.1$	$-$
8713828	$6.12\pm 2.38$	$8180.0\pm 588.0$	$93.2\pm 5.8$	$5280.0$	$90.3$
8767669	$0.79\pm 0.52$	$-$	$-$	$-$	$-$
8832676	$0.43\pm 0.24$	$-$	$-$	$-$	$-$
8840075	$0.32\pm 0.29$	$2220.0\pm 1960.0$	$8.6\pm 7.6$	$-$	$-$
8841307	$3.82\pm 0.46$	$72.0\pm 3.7$	$-$	$41.9$	$-$
8841616	$0.03\pm 0.01$	$7.1\pm 1.2$	$0.2\pm 0.1$	$6.9$	$0.2$
8842083	$0.09\pm 0.01$	$-$	$-$	$-$	$-$
8842170	$19.55\pm 5.02$	$1510.0\pm 150.0$	$-$	$1040.0$	$-$
8909045	$29.60\pm 26.21$	$-$	$-$	$-$	$-$
8909598	$0.41\pm 0.27$	$-$	$-$	$-$	$-$
8909758	$7.79\pm 2.72$	$2210.0\pm 123.0$	$-$	$543.0$	$-$
8909833	$0.69\pm 0.25$	$-$	$-$	$-$	$-$
8977176	$4.65\pm 1.06$	$-$	$44.6\pm 15.8$	$-$	$32.5$
8977179	$2.05\pm 0.47$	$-$	$-$	$-$	$-$
8977559	$12.63\pm 16.56$	$4050.0\pm 5220.0$	$-$	$-$	$-$
8977910	$0.06\pm 0.06$	$-$	$-$	$-$	$-$
8979190	$0.28\pm 0.12$	$5010.0\pm 1420.0$	$41.2\pm 11.8$	$2500.0$	$29.3$
8980010	$0.93\pm 0.37$	$4220.0\pm 1440.0$	$27.3\pm 9.4$	$-$	$-$
9046085	$1.62\pm 1.00$	$960.0\pm 446.0$	$8.0\pm 4.6$	$466.0$	$8.0$
9046855	$0.14\pm 0.07$	$-$	$-$	$-$	$-$
9046892	$3.55\pm 1.15$	$973.0\pm 31.6$	$-$	$113.0$	$-$
9048032	$1.19\pm 1.19$	$-$	$-$	$-$	$-$
9048114	$1.30\pm 0.37$	$15100.0\pm 16.4$	$849.0\pm 5.6$	$2490.0$	$564.0$
9048551	$1.13\pm 0.06$	$0.6\pm 0.1$	$-$	$0.6$	$-$
9048949	$4.66\pm 0.79$	$49.0\pm 7.4$	$0.3\pm 0.1$	$28.4$	$0.3$
9048976	$0.70\pm 0.09$	$-$	$-$	$-$	$-$
9410279	$0.78\pm 0.43$	$-$	$-$	$-$	$-$
9470175	$17.29\pm 8.44$	$180.0\pm 21.4$	$-$	$101.0$	$-$
9471538	$10.19\pm 2.28$	$3040.0\pm 104.0$	$-$	$-$	$-$
9532030	$0.51\pm 0.25$	$92200.0\pm 33800.0$	$-$	$-$	$-$
9532379	$3.55\pm 0.87$	$326.0\pm 13.6$	$-$	$157.0$	$-$
9532421	$5.30\pm 2.80$	$-$	$-$	$-$	$-$
9532591	$10.62\pm 5.39$	$207.0\pm 54.1$	$-$	$-$	$-$
9532637	$6.51\pm 3.92$	$-$	$-$	$-$	$-$
9532903	$5.06\pm 2.85$	$-$	$-$	$-$	$-$
9533014	$1.97\pm 0.95$	$39.1\pm 9.3$	$-$	$20.7$	$-$
9594273	$4.98\pm 1.69$	$208.0\pm 18.7$	$-$	$51.5$	$-$
9594647	$5.56\pm 4.52$	$-$	$-$	$-$	$-$
9654881	$32.92\pm 12.78$	$9880.0\pm 526.0$	$-$	$3620.0$	$-$
9655134	$3.88\pm 1.79$	$-$	$-$	$-$	$-$
11970692	$113.75\pm 27.30$	$-$	$-$	$-$	$-$
11971335	$23.96\pm 3.91$	$-$	$-$	$-$	$-$
12020532	$10.54\pm 7.22$	$-$	$-$	$-$	$-$
12020628	$5.09\pm 1.01$	$-$	$-$	$-$	$-$
12069262	$7.16\pm 6.39$	$-$	$-$	$-$	$-$
12118682	$69.36\pm 24.01$	$-$	$-$	$-$	$-$
12118993	$45.21\pm 9.50$	$318.0\pm 12.4$	$-$	$158.0$	$-$

Equations

$$\mathbf{f}=\frac{SC\_FLU{\mathbf{X}}_{0.2-2.0}}{DET\_FLU{\mathbf{X}}_{0.2-2.0}}$$

$$N_{\mathrm{H}}=A_{\mathrm{V}}[\mathrm{mag}]\cdot 1.79\cdot 10^{21}\mathrm{c}{\mathrm{m}}^{-2}.$$

$$A{I}_X=\frac{L_{\mathrm{X}}}{L_{\mathrm{bol}}}.$$

$$A{I}_{\mathrm{UV}}^{\prime }=\frac{f_{\mathrm{UV},\mathrm{exc}}}{f_{\mathrm{bol}}}=\frac{f_{\mathrm{UV},\mathrm{obs}}-f_{\mathrm{UV},\mathrm{ph}}}{f_{\mathrm{bol}}}$$

Full text

1 Introduction

Main sequence stars are characterized by radiative emission processes, such as high-energy (UV and X-ray) emission, flares and enhanced optical line emission (Ca ii, H$\alpha$), collectively referred to as ‘magnetic activity’, as they are ascribed to processes involving magnetic fields in the stellar corona, chromosphere and photosphere. Magnetic activity is allegedly the result of internal magnetic dynamos, arising from the combination of stellar differential rotation and convection in the sub-photospheric layers.

The understanding of stellar magnetic activity is of capital importance, since it is a fundamental diagnostics for the structure and dynamics of stellar magnetic fields, and gives crucial information on the dynamo mechanism responsible for their existence. Beside this, the high-energy emission associated with magnetic activity has a fundamental role in the evolution of the circumstellar environment and on the composition and habitability of planets.

As stated above, rotation is one of the key ingredients of stellar dynamos. In a feedback mechanism, the coupling of the rotation itself with the magnetic field determines the spin evolution of stars. This is true both in the pre-main sequence phase, in which the angular momentum of the accretion disk is transferred to the central forming star through the magnetic field, and in the main-sequence phase, because of the momentum loss due to magnetized winds ejected by the star. Exploring the relation between the stellar magnetic activity and the star’s rotation rate is thus an efficient way to gather information on stellar dynamos.

The connection between stellar rotation and magnetic activity has been studied in many works, since the seminal papers by Wilson (1966) and Kraft (1967). A fundamental contribution was given in the work by Skumanich (1972), the first to interpret the activity-rotation relation as a consequence of a dynamo mechanism. Since then, many authors (e.g., Walter & Bowyer 1981, Dobson & Radick 1989, Micela et al. 1985, Pizzolato et al. 2003) have focused on the relation between the stellar rotation and specific chromospheric and coronal activity indicators. From these works, a bimodal distribution emerged for the rotation-activity relation: for rotation periods longer than a few days (depending on the spectral type of the star, or the equivalent parameters of stellar mass or colour), the activity decreases with the rotation period; for shorter periods, a saturation regime is reached, in which the activity level is independent of the rotation period. Since rotation is an ingredient of the dynamo mechanism, a correlation between rotation and activity is intuitively reasonable. The origin of the saturation, instead, has not been ascertained. It may be due to a change in the behaviour of the dynamo, or due to limits to the coronal emission because the stars run out of the available surface to accomodate more active regions (O’dell et al. 1995) or because the high rotation rate causes centrifugal stripping of the stellar corona (Jardine & Unruh 1999).

X-ray emission is a very good proxy for stellar activity: The X-ray activity of a star is the result of magnetic reconnection in the stellar corona, an abrupt change in the configuration of the magnetic field determining the release of non-potential energy stored in the magnetic field lines. Rotation-activity studies found in the literature typically refer to collections of X-ray data obtained from various instruments, and rotation periods measured with different techniques, combining ground-based photometric measurments with $v\,\sin i$ spectroscopic techniques. The limitations due to the use of inhomogeneous data sets can now be overcome by combining rotation periods from the Kepler optical light curves to X-ray data obtained with XMM-Newton which has observed $\sim 1500$ objects in the *Kepler *field of view.

In the present work we study the relation between the rotational properties and the magnetic activity of the X-ray emitting main-sequence field stars observed with the Kepler mission comparing various indicators for activity (X-ray and UV activity, white-light flaring rate, flare amplitude, Kepler light curve amplitude and standard deviation) and rotation (rotation period and Rossby number; Noyes et al. 1984).

The sample selection is described in Sect. 2. The procedure used to evaluate the physical parameters of all stars in the sample is described in Section 3. In Section 4 we describe the techniques used to determine the rotation period and photometric activity diagnostics. In Section 5, several indicators of stellar activity are analysed: the X-ray luminosity, the ultraviolet excess in the Spectral Energy Distribution (SED), the white-light and X-ray flaring activity. The results are discussed in Sect. 6 and conclusions are presented in Sect. 7.

2 Sample selection

A proper sample selection is crucial in order to obtain a reliable picture of the activity-rotation relation. Many rotation-activity studies have been performed on inhomogeneous samples of stars for which X-ray data had been collected from a set of various databases and using a combination of spectroscopic and photometric techniques for rotation measurements.

We aim to the highest homogeneity in the determination of the rotation period and in the characterization of the activity indicators, first of all the X-ray activity. To this end, we focus in this work on X-ray emitting stars detected by *XMM-Newton *with light curves from Kepler. We perform a positional match between the 3XMM-DR5 catalogue (Rosen et al. 2015) and the *Kepler *Input Catalogue (KIC, Brown et al. 2011), and then remove non-stellar objects and objects with uncertain photometry.

The 3XMM-DR5 catalogue includes data from $16$ XMM-Newton pointings within the field of view (FoV) of the Kepler mission (see Table 1). Their sky position is shown in Fig. 1. The Kepler FoV covers $\sim 105$ square degrees; the $16$ XMM-Newton observations which fall inside this FoV cover only $\sim 2\%$ of that area.

The KIC positional error ($\sim 0.1^{\prime\prime}$) is negligible with respect to the error in the 3XMM-DR5 position ($\lesssim 4\,^{\prime\prime}$ at $3\,\sigma$). We select all the 3XMM-DR5 unique111The 3XMM-DR5 Catalogue contains a row for each X-ray detection. For a certain “unique” source (the astrophysical object), many detections (referred to different observations) of that object may be available. sources that have a positional match (columns SC_RA, SC_DEC in 3XMM-DR5) with one or more KIC objects within a radius given by three times their 3XMM-DR5 individual ($1\,\sigma$) positional error (column SC_POSERR). We calculate the average probability of a chance association between a 3XMM-DR5 source and a KIC source in each XMM-Newton field as $P=1-e^{\pi\mu r^{2}}$, where $\mu$ is the numerical density of KIC sources in the field and has a typical value of $10^{-4}\,{\rm sources/arcsec^{2}}$, and $r$ is three times the average position error of the 3XMM-DR5 source. We obtain an average probability of chance association of $\sim 0.8\,\%$. This translates to $\sim 0.14$ chance associations per XMM-Newton field, i.e. $\sim 2$ spurious matches in the whole sample.

We select only the objects with a detection significance in 3XMM-DR5 (column DET_ML) greater than $6$ (probability of a spurious detection $<0.025$) in at least one EPIC instrument in the energy band $8$ ($0.2-12.0\,$keV), removing the others from the sample ($4$ objects removed). We reject the $21$ sources that are classified as “extended” in 3XMM-DR5 (flag EP_EXTENT$>6$). The resulting sample consists of $145$ matches. There are no multiple associations, that is associations between a KIC object and two or more 3XMM-DR5 objects, or viceversa.

Within this sample, we aim to identify genuine stars, removing (1) galaxies and (2) stars that possibly suffer confusion (stars which are not resolved by Kepler) and contamination from nearby bright sources. To this end, for each KIC object with a 3XMM-DR5 match, we search for a classification in the SIMBAD database222http://simbad.u-strasbg.fr/simbad/ (Wenger et al. 2000) and we inspect the optical and infrared images, when available, using the Aladin service (Bonnarel et al. 2000; Boch & Fernique 2014). The fit of the Spectral Energy Distribution (SED) with a model for the stellar photospheric emission is also useful to identify galaxies in the sample (objects whose SED cannot be fit by a stellar model). Moreover, it allows us to determine the value of the stellar fundamental parameters. The methods to remove non-stellar counterparts to the X-ray sources are explained in more detail in the following.

2.1 Spectral energy distribution (SED)

2.1.1 Multi-wavelength photometry

We extract the IR (2MASS $J$, $H$, $K$), optical (SLOAN $g$, $r$, $i$, $z$, Johnson $U$, $B$, $V$, Kepler $K_{\rm p}$) and UV (GALEX $FUV$, $NUV$) photometry for all stars of our sample from the KIC (Brown et al. 2011). Many sources lack photometry in one or more of these bands. We exclude from the sample the KIC objects for which no 2MASS IR photometry is available ($2$ objects), since the wavelength range covered by 2MASS is crucial in order to perform a reliable spectral classification.

The visual inspection of the optical images via the Aladin server indicates that some stars suffer contamination from brighter objects located nearby. In such cases, even if they are recognised as distinct objects in the KIC,ß the photometry of the fainter one is expected to be significantly contaminated by the brighter one. Other stars suffer confusion between unresolved objects, i.e. they are not resolved in the KIC survey, but can be recognised as two objects through visual inspection of the optical image. When the object is resolved in the UCAC-4 Catalogue (Zacharias et al. 2012), we replace the KIC photometry for these confused objects with the photometry provided by UCAC-4 for the brightest one. If no UCAC-4 photometry is available, we remove the object from our sample ($2$ objects).

According to Pinsonneault et al. (2012), the SLOAN bands photometry reported in the KIC presents a significant systematic error, as observed comparing the KIC photometry in the bands $g$, $r$, $i$, $z$ with the photometry reported in the Sloan Digital Sky Survey (SDSS-DR8) in the same bands, available for $\sim 10\%$ of the full KIC. Pinsonneault et al. (2012) give semi-empirical formulas to correct these systematics (their Eqs. 1, 2, 3, 4). We apply these corrections to the KIC SLOAN bands photometry of all the stars of our sample. The photometry for all the stars in our sample is reported in Table LABEL:tab:photometry.

2.1.2 SED fitting

After the SEDs have been compiled, we fit them with photospheric BT-Settl models (Allard et al. 2012) within the Virtual Observatory SED Analyser (VOSA, Bayo et al. 2008). The parameters of the BT-Settl models are: $A_{\rm V}$, $\log{g}$, $[Fe/H]$ and $T_{\rm eff}$. Due to parameter degeneracy it is not possible to determine all of them from the SED fit. Therefore we adopt for each star for $A_{\rm V}$, $\log{g}$ and $[Fe/H]$ values taken from the literature (see Sect. 3), leaving $T_{\rm eff}$ as the only free parameter. We thus obtain, for each object, a best fit $T_{\rm eff}$ under the hypothesis of a stellar model. As the uncertainty on the effective temperature we assume the typical dispersion of the five best fit values obtained in VOSA for each object ($\pm 200\,$K).

From the shape of their SED, combined with the visual inspection of the Aladin images, sixteen objects are recognised as galaxies, and we remove them from our sample.

The whole sample selection procedure eventually results in a sample of $125$ 3XMM-DR5 unique sources with a reliable stellar counterpart in Kepler, with a SED that is well-fitted by a stellar photosphere model.

2.2 X-ray flux limit

The sample of stars used in the present work has been selected based on the X-ray emission observed by XMM-Newton and as such is biased towards X-ray active stars. A fundamental step in order to understand the results presented in this work is, therefore, to provide an evaluation of the X-ray flux limit of the sample. To this end, we first produce the EPIC PN333The EPIC (European Photon Imaging Camera) consists of three CCD detectors, two MOS and one PN, located at the focus of the three grazing incidence multi-mirror X-ray telescopes which constitute the main instrument onboard *XMM-Newton *. sensitivity map for each *XMM-Newton *observation of Table 1, for the energy range $0.2-2.0\,{\rm keV}$ using the task esensmap of the *XMM-Newton *Science Analysis Software (SAS). The sensitivity map provides in each point of the FoV the limiting count rate needed to have a $3\,\sigma$ detection of a point source. We measure the limiting count rate in the centre of the FoV. We convert these numbers into flux sensitivity limits ($f_{\rm X,lim}$) using WebPIMMS444http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl assuming an APEC model with plasma temperature $0.86\,{\rm keV}$ (see Sect. 5.1.2), abundance $0.2$ in solar units and the galactic hydrogen column density provided by Kalberla et al. (2005). The resulting numbers for $f_{\rm X,lim}$ are reported in the last column of Table 1. These values give an idea of the lower limit for the X-ray flux in these observations (median is $\sim 6\cdot 10^{-15}\,{\rm erg/cm^{2}/s}$), although we caution that the sensitivity varies strongly (up to about one order of magnitude) from the centre to the outer region of the FoV.

3 Fundamental stellar parameters

For $119$ stars out of $125$ we found literature values for effective temperature ($T_{\rm eff}$), visual absorption ($A_{\rm V}$) surface gravity ($\log g$), and metallicity (${\rm[Fe/H]}$) (in Huber et al. 2014 and   Frasca et al. 2016). Huber et al. (2014) present a compilation of literature values for atmospheric properties ($T_{\rm eff}$, $\log g$ and ${\rm[Fe/H]}$) derived from different observational techniques (photometry, spectroscopy, asteroseismology, and exoplanet transits), which were then homogeneously fitted to a grid of isochrones from the Dartmouth Stellar Evolution Program (DSEP; Dotter et al. 2008), for a set of $\sim 200,000$ stars observed by the Kepler mission in Quarters $1-16$. Frasca et al. (2016) present a systematic spectroscopic study of $\sim 50,000$ Kepler stars performed using the LAMOST telescope (Zhao et al. 2012). We compared the parameters obtained by Huber et al. (2014) and Frasca et al. (2016) for our sample, in particular the effective temperatures, and we found that they agree very well within the error bars. When available, however, we prefer the values obtained from Frasca et al. (2016) through the spectroscopic analysis, since in that work the stellar parameters have been evaluated using the same method for all the stars in their sample, while Huber et al. (2014) present a collection of values obtained in several works using different techniques.

Six stars in the sample do not have $T_{\rm eff}$ reported in Huber et al. (2014) or Frasca et al. (2016). For these $6$ stars we adopt as $T_{\rm eff}$ the effective temperature of the best fit of the individual SED with a model of the stellar photosphere, as described in Sect. 2.1. These stars are flagged with the ‘VOSA’ flag in column $12$ of Table $7$. Two out of these $6$ stars have values for $A_{\rm V}$, ${\rm[Fe/H]}$ and $\log g$ in Huber et al. (2014) or Frasca et al. (2016), and we use them in the SED fit. For the others, we adopt the median values of the distribution of $A_{\rm V}$ in our sample ($0.38$), ${\rm[Fe/H]}_{\odot}$ ($0.0$) and $\log{g}$ ($4.0$). Similarly, for the additional $15$ stars that have no literature value for $A_{\rm V}$ we adopt the median of $0.38$ mag. The adopted values for the spectral parameters of all $125$ stars are listed in Table LABEL:table:params.

In order to validate our SED fitting procedure we compare the obtained $T_{\rm eff}$ with the ones from the above-mentioned literature sources. The distribution of $T_{\rm eff}$ and the comparison between the $T_{\rm eff}$ obtained from the SED fitting and the values in the literature are reported in Fig. 2. Within the error bars, the agreement is for most stars very good, but the error bars of the values obtained by SED fitting are typically larger than the error bars in Huber et al. (2014) and especially in Frasca et al. (2016). This justifies our decision to adopt, when available, the literature values for $T_{\rm eff}$. Given the effective temperature, we assign a spectral type to each star according to Table 5 in Pecaut & Mamajek (2013). These values are given in Table LABEL:table:params.

3.1 Distance, mass and bolometric luminosity

Fig. 3 shows our stellar sample together with the DSEP isochrones on the $\log g$ vs $\log{T_{\rm eff}}$ plane. For each star, we estimate absolute $J$ band magnitude, mass ($M$) and bolometric luminosity ($L_{\rm bol}$) by projecting its position and the related uncertainties onto the isochrones (along the vertical axis) in Fig. 3. For this task we make use of DSEP isochrones in the range of $1.0$ to $9.5\,$Gyr and for each star, we select the set of isochrones corresponding to the ${\rm[Fe/H]}$ value closest to the observed value. There are a few stars the position of which on the $\log g$ vs $\log{T_{\rm eff}}$ plane is not consistent, within the error bars, with any of the DSEP isochrones. We mark them with a flag (‘DSEP outlier’) in Table LABEL:table:params.

Inverting the parallaxes provided in the Gaia-DR2 (Gaia Collaboration et al. 2018) we obtain the distances to our targets. While the overall quality of the Gaia-DR2 data is excellent, the mission is too complex to achieve optimal calibrations with less than two years of observations. As a result the Gaia-DR2 still contains many spurious astrometric solutions (Arenou et al. 2018). To remove putative problematic solutions we cleaned our dataset using the filters defined by Lindegren et al. (2018, appendix C, equations C-1 and C-2). Furthermore we used additional quality indicators of the solutions (astrometric_excess_noise $>$ 0, astrometric_gof_al $>$ 5) to discard other potential outliers. Two stars of our sample have no Gaia parallax while the solutions for further $60$ stars might not be reliable and were filtered out. For these $62$ targets we derive the photometric distance from the comparison between the absolute magnitude and the observed $J$ band magnitude. Comparison of the photometric and astrometric distances for the whole sample shows overall good agreement. For $5$ targets with reliable Gaia distances we found the $J$ band distances to have lower uncertainties. In these cases we decided to adopt the $J$ band distances. To summarize, throughout this work we use the Gaia distances for $58$ stars and the photometric $J$ band distance for the remaining stars. Distance, mass and bolometric luminosity are reported for all $125$ stars in Table LABEL:table:params.

We classify the stars in our sample as main-sequence stars or giant stars according to their $\log g$. In the $\log g$ vs $\log{T_{\rm eff}}$ space, the stars in our sample can be separated into two groups, corresponding to the dwarf branch and to the giant branch, respectively (see Fig. 3). On this basis, we consider as main-sequence stars all the stars with $\log g\geq 3.5$, and as giants all the stars with $\log g<3.5$. With this criterion we find $19$ giant stars. The four stars without $\log g$ value in the literature can not be assigned to these groups, and we eliminate them from the sample. These stars have a flag (*) in column $12$ of Table $7$. This work is focused on main-sequence stars. Therefore, in the following, if not differently declared, we consider only the sample of the $102$ main-sequence stars. The distribution of their adopted distances is plotted in Fig. 4.

4 Analysis of Kepler light curves

The brightness modulation in the optical light curve due to the presence of inequally distributed spots on the rotating stellar surface enables the measurement of the stellar rotation period. Rotation periods, together with several photometric activity diagnostics, are extracted from the Kepler light curves following the procedure described by Stelzer et al. (2016). The methods developed therein for the analysis of M dwarf K2 mission lightcurves can readily be applied to the data from the main *Kepler *mission, and we briefly resume the steps in Sect. 4.1.

To obtain a clean sample of stars for the rotation-activity study, periods originating from mechanisms different from the rotational brightness modulation due to starspots must be identified and removed. In particular, our sample covers a broad range of $T_{\rm eff}$ in the Hertzsprung-Russell diagram, including the intersection of the main-sequence with the classical instability strip, where stellar pulsations are expected. Therefore, the light curves of some stars deserved a more detailed study in order to ascertain if pulsation (but also binarity) could be the real cause of the observed variability. We perform this study by means of the Fourier decomposition (Poretti 1994) and the iterative analysis of the whole frequency spectrum (Vaníček 1971), described in Sect. 4.2.

4.1 Search for rotational periodicity

Briefly, the analysis consists of the following steps, implemented in an iterative procedure, which is described in detail by Stelzer et al. (2016): (1) period search with standard time-series analysis techniques (autocorrelation function [ACF] and Lomb-Scargle [LS] periodogram), (2) boxcar smoothing of the lightcurve and subsequent subtraction of the smoothed from the observed lightcurve, effectively removing the rotational modulation, (3) identification of ‘outlier’ data points in the ‘flattened’ lightcurve obtained from step (2) through $\sigma$-clipping. The ‘outliers’ comprise both instrumental artefacts and flares. The latter ones are identified by imposing three criteria: (i) a threshold ($3\sigma$) for the significance of the flare data points above the ‘flattened’ curve, (ii) criterion (i) applies to at least two consecutive such data points, and (iii) the maximum bin of the flare has at least a factor two higher flux than the last bin defining the flare. For a detailed description of the procedure used in each step of the analysis chain see Stelzer et al. (2016).

After the removal of the ‘outliers’ we repeat the period search, but in practice the applied methods are so robust that the cleaning of the lightcurves does not alter the result. What does change after the cleaning is the amplitude of the rotation cycle, when measured between maximum and minimum flux bin. However, as we describe below in Sect. 4.4, the preferred characterization of the spot cycle amplitude involves diagnostics that are little affected by the small fraction of ‘outlier’ data points. We perform this analysis for each Kepler observing Quarter independently. This allows us to cross-check the results by comparing the periods obtained from different Quarters, as explained in Sect. 4.3.

The determination of the rotation period for a given star proceeds in the following way. We use the routines A_CORRELATE and SCARGLE in the IDL environment555IDL is a product of the Exelis Visual Information Solutions, Inc to generate the ACF and LS periodogram. We work independently on the ACF and LS periodogram series. For each Kepler observing Quarter, we inspect visually the ACF and LS periodogram generated by our procedure, and search for a signal of periodic variability (a sharp peak at a certain frequency, possibly followed by the related harmonics). If absent, we reject the Quarter from the analysis. In the ACF periodogram, we visually choose the highest peak of the series, which corresponds to the period of the modulation; this is generally the first one, or the second one if the light curve shows a double-humped pattern (as described below). In the LS periodogram, we likewise choose the highest peak.

For the majority of stars, the dominating periods derived with both techniques (ACF and LS) are consistent with each other. Deviations regard lightcurves with a double-humped shape. In this case, the light curve shows a double peak in at least some of the *Kepler *observing Quarters, while in others a single peak pattern may be observed. We interpret these features as the signature of two groups of spots at a roughly antipodal position on the photosphere of the star, one of which may occasionally disappear. In this case, the ACF periodogram shows generally a first peak (corresponding to half the period) that has a lower amplitude with respect to the second one, and this pattern repeats for the peaks corresponding to integer multiples of the first-peak period and of the second-peak period, respectively. In the Lomb-Scargle periodogram, the peak corresponding to the shorter period often has a larger amplitude than the longer-period peak. The interpretation of these patterns as the effect of two groups of spots on the photosphere allows us to interpret the longer period of the two, corresponding to the double of the other, as the true rotation period of the star.

We obtain for each star $N_{Q}$ periods, where $N_{Q}$ is the number of Quarters in which the star shows a periodic signal. This number varies individually for the stars of our sample, independently for the ACF and the LS method. We adopt as the star’s period the median value of the periods obtained with the ACF method from the individual quarters.

4.2 Non-rotational periodicity

For some of the stars in our sample periodic variability is identified that can not be explained with a simple spot pattern. In such cases the Fourier decomposition and the evaluation of the whole frequency spectra were very helpful. The frequency spectra of the genuine spotted stars are characterized by the harmonics of the rotational period and by low-frequency peaks due to the activity and rotational cycle-to-cycle period variations due to the shift in the spots’ latitude and to stellar differential rotation, and by some rotational cycle-to-cycle variation in the shape and amplitude of the lightcurve due to the variation of the area and shape of starspots. Other kinds of periodicity present different patterns.

We perform a dedicated anlysis of the light curves of the stars which do not show a clear variability pattern in order to classify their non-rotational variability. We find one multiperiodic star showing the high-frequency pulsational regime of $\delta$ Sct stars and three multiperiodic stars showing the low-frequency regime of $\gamma$ Dor stars. In the light curves of $2$ stars the amplitudes of the even Fourier harmonics are much larger than those of the odd ones, as necessary to fit both the sharp minima and the large maxima shown by contact binaries. For $3$ stars it remains ambiguous if the variability is due to star spots or due to orbital motion in a contact binary; another $2$ stars display non-periodic variability that we could not classify. Finally, $5$ stars are likely rotational variables but displaying more than one period possibly indicating a binary composed of two spotted stars or a complex pattern with uncertain period. These $16$ stars are removed from the sample considered for the rotation-activity relation. A summary of the number of main-sequence stars in each variability class for spectral type is given in Table 2.

4.3 Final sample of spotted stars

From the analysis described above, we can derive a rotation period for $74$ stars in our sample, i.e. these stars are inhomogeneously spotted. For the subsequent analysis, we also compute the Rossby number, defined as $R_{0}=P_{rot}/\tau_{conv}$, i.e. the ratio between the rotation period and the convective turnover time. The convective turnover time is the characteristic time of circulation within a convective cell in the stellar sub-photosphere, and not directly observable. We calculate it from $T_{\rm eff}$ using Eq. $36$ in Cranmer & Saar (2011), which is valid in the range $3300\,$K$\,\lesssim\,T_{\rm eff}\,\lesssim\,7000\,$K. All but two of the stars in our sample are within this range of $T_{\rm eff}$. The rotation periods and Rossby numbers for all “spotted” stars are listed in Table LABEL:tab:rot. Our subsequent photometric variability analysis is limited to this sample.

4.4 Photometric activity diagnostics

From the analysis of the Kepler light curves we also obtain various diagnostics for the stellar photospheric activity: the light curve amplitude and the standard deviation of the light curve.

The light curve amplitude is the photometric difference between the maximum and the minimum of the rotationally-modulated light curve, determined (in a light curve cleaned from flares) by the contrast between spotted and unspotted photosphere, combined with the inhomogeneous distribution of spots which causes the spot coverage of the observed stellar hemisphere to vary during the rotation. Analogous to Stelzer et al. (2016) and for consistency with the previous literature we decided to define the spot cycle amplitude as the range between the $5th$ and $95th$ percentile of the observed flux values in a single rotation cycle ($R_{\rm var}$, see Basri et al. 2013), and we adopt the modified definition of $R_{\rm var}$ introduced by McQuillan et al. (2013), $R_{\rm per}$, which is the mean of the $R_{\rm var}$ values measured individually on all observed rotation cycles, expressed in percent. Cutting the upper- and lower-most $5$ % of the data points is another way of removing the ‘outliers’, such that no difference between the $R_{\rm var}$ values obtained from the original light curve with the $5th$ and $95th$ percentile and from the cleaned lightcurve is expected. We have verified this by comparing the $R_{\rm var}$ values extracted from the light curves ‘cleaned’ from flares and outliers and the $R_{\rm var}$ values obtained from the original light curves.

The second activity diagnostic extracted from *Kepler *light curves that we use to characterize the variability during the spot cycle is the standard deviation of the whole light curve ($S_{\rm ph}$), and the average of the standard deviations computed for time intervals $k\cdot P_{\rm rot}$, with $k$ integer, first defined by Mathur et al. (2014b). Mathur et al. (2014b) have shown for a sample of $22$ stars that roughly after five rotation cycles the full range of flux variation is reached. Therefore, we compute $S_{\rm ph}$ and $\langle S_{\rm ph,k=5}\rangle$ for the stars in our sample. The standard deviation of the light curve was used as a proxy of magnetic activity also by He et al. (2015), in a study on the activity of two solar-like *Kepler *stars.

Finally, we have computed the standard deviation of the ‘flattened’ light curves ($S_{\rm flat}$), measured on the light curves cleaned from flares and outliers, and from which the rotational cycle has been removed (see description at the beginning of this Section). Stelzer et al. (2016) have established this parameter as an indicator for low-level unresolved astrophysical variability such as small unresolved flares and/or small and fast-changing spots for their sample of nearby M dwarfs observed in the K2 mission.

All these photometric activity diagnostics are listed in Table LABEL:tab:rot. It is worth noting here that the noise level in the ‘flattened’ lightcurve is expected to depend on the brightness of the star. In fact, Fig. 5 shows a correlation between the Kepler magnitude and $S_{\rm flat}$. As can be seen in Fig. 5, the apparent brightness of our sample stars is on average larger for earlier spectral types. As a consequence, the noise tends to be smaller in those stars. This must be taken into account in the analysis of other kinds of variability; see e.g. Sect. 6.3.

5 X-ray and UV activity

As our sample is X-ray selected, every star in the sample has an associated X-ray detection. In the main-sequence sample, $36$ stars have also been detected in one or both of the UV energy bands (Near Ultra-Violet, NUV, and Far Ultra-Violet, FUV) of GALEX. In this section we analyse the X-ray and UV luminosity, and the corresponding ‘activity indexes’, defined as the ratio between the luminosity in the high-energy band and the bolometric luminosity of the star, as indicators of the stellar magnetic activity.

5.1 X-ray data analysis

5.1.1 Source X-ray luminosity

We calculate the X-ray luminosity of each star in the sample in the energy band $0.2-2.0\,$keV (‘soft’ energy band, 3XMM-DR5 catalogue energy band $6$). This range of energies corresponds approximately to the energy bands used in previous studies of the rotation-activity connection that were mainly based upon ROSAT data (e.g. Wright et al. 2011).

In the 3XMM-DR5 catalogue, there are many objects (‘sources’) that have been observed more than once: so, for a certain ‘source’, there can be many ‘detections’. Each row of the catalogue – which represents an individual detection of a certain source – contains a set of parameters referred to the detection itself, and a set of parameters averaged on all the detections available for that source. The catalogue provides the fluxes expected for a power law emission model for both the ‘detections’ and the ‘source’. However, the power law model, to which the fluxes given in 3XMM-DR5 refer, is not appropriate for describing the X-ray emission of late-type stars. Therefore, we re-calculate the X-ray flux in the bands $0.2-2.0\,$keV from the EPIC PN count rate (or the MOS, if PN is not available), using the HEASARC online tool WebPIMMS, in which we assume for the X-ray emission of the stars a thermal APEC model (Smith et al. 2001). 3XMM-DR5 gives the ‘detection’ count rate for each of the three EPIC (European Photon Imaging Camera) instruments (PN, MOS1 and MOS2) on board the XMM-Newton mission, and the associated uncertainty, but it does not give the ‘source’ count rate. As described above, the ‘detection’ count rate is different from the ‘source’ count rate when the source has multiple detections in the catalogue. This happens for $11$ sources in our sample. So, we calculate the X-ray flux for these objects rescaling one of the individual ‘detection’ count rate for the factor

[TABLE]

where $\mathbf{SC\_FLUX_{0.2-2.0}}$ and $\mathbf{DET\_FLUX_{0.2-2.0}}$ are respectively the ‘source’ and ‘detection’ flux provided by 3XMM-DR5.

The APEC model in WebPIMMS requires as input parameters the hydrogen column density ($N_{\rm H}$), the metallicity and the temperature of the emitting plasma. We estimate the hydrogen column density for each source from the visual absorption $A_{\rm V}$ following Cardelli et al. (1989), as

[TABLE]

We adopt an average abundance of $0.2$ in solar units, which is a typical value for the coronae of X-ray emitting late-type stars (see e.g. Pandey & Singh 2012). We calculate the count-to-flux conversion factor with WebPIMMS. To establish an operational $kT$, we fit the spectra of the brightest X-ray sources in XSPEC, and assume for all stars the $kT$ value in the WebPIMMS grid ($kT=0.86\,{\rm keV}$) which is closest to the average $kT$ obtained from the spectral fits ($kT=0.83\,{\rm keV}$). The spectral analysis of the brightest sources is described in the following.

5.1.2 Spectral analysis

We select the subsample of stars for which we have at least $200\,$ events in the source extraction region in the three EPIC instruments together (PN+MOS1+MOS2). Excluding the stars classified as eclipsing or contact binaries, we have a sample of $19$ stars. For each one, we extract the source and background spectrum in the energy band $0.3-10.0\,$keV, and perform a joint spectral analysis with XSPEC 12.8.1 (Arnaud 1996) of the spectra of all EPIC instruments available for that source. We fit the spectra with an absorbed one-temperature APEC model (phabsapec) or an absorbed two-temperature APEC model (phabs(apec+apec)). We set the coronal abundance to the typical value of $0.2$ in solar units (see above) to reduce the number of free parameters and avoid degeneracy.

Two stars show a significant parameter degeneracy in their spectrum, so we remove them from the analysis. For the stars for which the model requires two temperatures, we calculate an average temperature weighted on the flux of the two APEC components. The results of the best fit for each of the $17$ stars, all characterized by a null-hypothesis probability $>0.3\,\%$, are reported in Table 3. We calculate the average $kT$ over the $17$ stars ($<kT>=0.83\,$keV).

With the parameters given in Sect. 5.1.1, we obtain from WebPIMMS the expected flux for each source. We then compare these fluxes with the fluxes obtained from the spectral analysis, for each of the $17$ sources for which the spectral fit in XSPEC was performed. For each of these stars we calculate the ratio between the XSPEC flux and the flux from WebPIMMS. For PN, MOS1 and MOS2, we found an average ratio of respectively $0.91,0.89,1.00$ in the soft energy band $0.2-2.0\,$keV. We correct the fluxes obtained with WebPIMMS for the faint ($<200$ counts) sources by multiplying them with this factor. This correction is meant to obtain from WebPIMMS a flux that is, on average, as close as possible to the actual flux obtained from a spectral fit.

In the sample of $17$ bright stars, $kT$ ranges from $0.49\,{\rm keV}$ to $1.3\,{\rm keV}$. This range is typical for active stars, and the actual plasma temperatures of the bulk of the faint stars is likely in the same range. This span of temperatures introduces an error in the flux calculated with the average $kT$, allowing fluxes which are up to $\sim 2\%$ lower or $\sim 15\%$ higher than the one calculated from the average temperature of the bright stars. If the $17$ X-ray-brightest sources are not representative of our whole sample, these errors may be somewhat larger for the faint stars.

From the corrected fluxes we calculate the X-ray luminosity using the distances derived in Sect. 3.1. For each source we calculate the X-ray activity index as

[TABLE]

The distributions of the X-ray luminosity and of the corresponding activity index in the soft energy band ($0.2-2.0\,$keV) are reported in Fig. 6. The individual X-ray luminosities are listed in Table LABEL:tab:xray.

5.1.3 Considerations on the X-ray luminosity distribution

We compare the distribution of the X-ray luminosity in the range $0.2-2.0\,{\rm keV}$ for the stars in our sample with those in the NEXXUS sample of Schmitt & Liefke (2004), which consists in a compilation of coronal X-ray emission for nearby late-type stars based on the ROSAT observatory. The distribution of the X-ray luminosity for our main-sequence sample and for the one from Schmitt & Liefke (2004) is plotted, for each spectral type and for the whole sample, in Fig. 7.

To first order we consider NEXXUS as a volume-limited sample of nearby stars, representing the full range of X-ray activity of the solar-like stellar population. Note, however, that Stelzer et al. (2013) showed that even in as small a volume as $10$ pc around the Sun about $40$ % of the M dwarfs have no X-ray detection in the RASS. From the comparison with our distribution, it is evident that our sample presents a significant bias towards high X-ray luminosities most marked for the latest spectral types reflecting the mass (or $T_{\rm eff}$) dependence of X-ray luminosity. The stars in our sample, which is flux-limited, are on average at a much greater distance (see Fig. 7, median distance for the whole sample: $\sim\mathbf{500}\,{\rm pc}$). Therefore, we interpret the bias of our sample towards active stars as due to both the X-ray selection and the large distances.

5.1.4 X-ray light curves

In order to study the variability of the X-ray emission and to search for X-ray flares, we analyse the light curves provided by the EXTraS (Exploring the X-ray Transient and variable Sky) project 666The EXTraS project (www.extras-fp7.eu), aimed at the thorough characterization of the variability of X-ray sources in archival XMM-Newton data, was funded within the EU seventh Framework Programme for a data span of 3 years starting in January 2014. The EXTraS consortium is lead by INAF (Italy) and includes other five institutes in Italy, Germany and the United Kingdom. (De Luca et al. 2015). EXTraS was synchronized with 3XMM-DR4, while we are studying X-ray sources from 3XMM-DR5 in the Kepler field. Therefore, light curves for observations 0671230201 and 0671230601 are not present in the public EXTraS database, but were produced for this work by the EXTraS team, using the EXTraS analysis pipeline.

EXTraS provides a set of uniform bin light curves with different bin size, from $10\,$s up to $5000\,$s and an optimum bin size (chosen to have at least $25$ counts in each bin), both for the source and the background extraction regions. In addition, EXTraS also provides light curves produced via an adaptive binning, namely bayesian blocks (Scargle et al. 2013) algorithm for each source and for the related background region. This algorithm starts with an initial set of cells defined on the basis of the number of events in the source and background region, and provides a final set of different-duration bins, each of which has a count rate that is not consistent, within $3\,\sigma$, with the count rate of the adjacent bins.

In the EXTraS analysis pipeline, all the source light curves, both the ones obtained with the uniform bin algorithm and the bayesian blocks algorithm, are automatically fitted with a series of different models that account for simple variability patterns (constant, linear, quadratic, negative exponential, constant plus flare, constant plus eclipse). This is a standard algorithm that is part of the EXTraS pipeline, and it is applied to all the light curves in the same way. Its purpose is to provide a first-step indication of the kinds of variability possibly present in the light curve, and not to perform a detailed modeling of the variability features observed, nor to determine their parameters. All results are in the database, and online searches can be performed with the query form.

5.1.5 X-ray flaring

For each X-ray source in our sample, we search for X-ray flares in the EXTraS light curves. First, we inspect the results of the fit performed on the light curves with different variability models by the EXTraS pipeline. As stated above, these results can be used for a first assessment for the kind of variability present in the light curve. If at least one among the uniform bin or adaptive bin light curves is better fitted by a constant plus flare model (higher null-hypothesis probability) than the other models, this is a good indication of a possible flare. This flare model consists of a constant flux level over which a simple flare profile is superimposed, that is a steep linear flux increase, followed by an exponential decay.

The bayesian blocks light curves are particularly useful to detect flares, which appear as one or more blocks that show a higher flux than the preceeding and following blocks. If the flare occurred at the beginning or at the end of the observation, the bayesian blocks light curve may show only the rise phase or the decay phase. In this case, we rely on the uniform bin light curves to establish if the event is a true flare or not. We also require that the flare was observed in at least two of the EPIC instruments. The visual inspection of the light curves is however crucial in order to recognize genuine flares, so we inspect all the available EXTraS light curves for each star in our sample.

With this procedure, based on the EXTraS pipeline products we detect $6$ X-ray flares on $5$ stars, and by visual inspection we identify an additional likely flare on a sixth star, KIC 7018131 (Figs. 8, and 9). In view of the low count statistics of the X-ray lightcurves some remarks on the individual events are in order. KIC 9048976 shows two possible flares: the bayesian blocks algorithm shows one block with a higher flux at the beginning and at the end of the light curve, impeding the observation of the full flare profile. KIC 8909598 also shows an X-ray flare at the end of the observation. The uniform bin light curves reveal a quite obvious flare profile both in PN and MOS2 cameras. However, this star does not have a reliable main-sequence classification (unknown $\log{g}$), so we do not consider it in the analysis. For KIC 9048551 the bayesian block algorithm identifies one flare event, but the uniform bin lightcurve shows evidence of substructure contemporaneous with two events seen in the Kepler band.

The X-ray flares shown in Fig. 8 have simultanous white-light flares in the Kepler light curves (see Sect. 6.3). Fig. 9 displays the light curves for the X-ray flares without a simultaneous white-light counterpart revealed with the flare detection algorithm described in Sect. 4.1. Evidence for a weak increase of the optical brightness at times of the X-ray flare is, however, seen in all the contemporaneous Kepler lightcurves.

In Table 4 we report the start time of the X-ray flares (col. 2), according to the bayesian blocks light curve except for KIC 7018131, for which we infer the start time from the $500\,{\rm s}$ uniform bin light curve of EPIC/pn, since the bayesian blocks light curve has only one block. We also report the quiescent (col. 3) and peak (col. 4) X-ray count rate, taken from the $500\,{\rm s}$ uniform bin light curve of EPIC/pn, together with the overall number of white-light flares observed in the whole Kepler light curve for the star (col. 5), its white-light flare frequency (col. 6), the average, maximum and minimum peak amplitude of the *Kepler *flares ($A_{\rm peak}$, photometric ratio between the peak bin and the baseline of the flare, cols. 7 and 8). This parameter is discussed in more detail in Sect. 6.3.1.

5.2 UV activity

The KIC provides UV magnitudes obtained with the Galaxy Evolution Explorer (GALEX). The GALEX satellite performed imaging in two UV bands, far-UV (henceforth FUV; $\lambda_{\rm eff}=1516\,$Å, $\Delta\lambda=268\,$Å , and near-UV (henceforth NUV; $\lambda_{\rm eff}=2267\,$Å, $\Delta\lambda=732\,$Å). The KIC gives a NUV detection for $71$ stars from our main-sequence sample (i.e. $70$ %) and a FUV detection for $20$ main-sequence stars ($20$ %). All but one of the stars with a FUV detection also have a NUV detection. The individual values for the observed NUV and FUV luminosities are listed in cols. 3 and 4 of Table LABEL:tab:xray.

The SED fit provides UV fluxes for the stellar photosphere. In order to validate these measurements, we convert the NUV de-absorbed photospheric fluxes obtained from VOSA into absolute magnitudes and compare the relation between these magnitudes and the $B-V$ colour (derived from Pecaut & Mamajek 2013) with the analogous relation observed for a set of photospheric NUV magnitudes provided by Findeisen et al. (2011) in their Table 1. Fig. 10 shows that our values are in good agreement with the ones of Findeisen et al. (2011).

For some stars in our sample the observed GALEX NUV and FUV fluxes are significantly higher than the prediction of the best-fitting photosphere model. This UV excess, i. e. the positive difference between the observed UV flux ($f_{\rm UV,obs}$) and the photospheric flux of the BT-Settl model in the same UV band ($f_{\rm UV,ph}$) represents the UV emission associated with magnetic activity processes in the stellar chromosphere. Following Stelzer et al. (2013), we calculate the corresponding UV activity index as

[TABLE]

where $f_{\rm UV,exc}$ is the UV excess flux attributed to activity, and $f_{\rm bol}$ is the bolometric flux. The bolometric flux is obtained from interpolation on the DSEP isochrones, as described in Sect. 3.

We calculate the UV excess in both the GALEX FUV and NUV band, where available. After trying different values and visually inspecting the SED, we find that a threshold of $13\%$ on the ratio between the excess flux and the photospheric expected flux is the best choice to select the stars with a true UV excess. We found $45$ main-sequence stars with a NUV excess, ten of them display also a FUV excess, and an additional $4$ have a FUV excess but no NUV excess. The NUV and FUV excess luminosities are provided in cols. 5 and 6 of Table LABEL:tab:xray.

6 Results and discussion

6.1 Rotation periods

For $74$ main-sequence stars out of $102$ ($73\%$) the Kepler light curve is dominated by the rotational brightness modulation due to starspots, with a period in the explored range $P_{\rm rot}<90\,$d. For the individual spectral types, the fraction of main-sequence stars with a rotational brightness modulation is: A $66$ % (2/3), F $65$ % (24/37), G $73$ % (19/26), K $75$ % (21/28) and M $100$ % (8/8). Our period detection rate is higher with respect to other studies based on Kepler data (McQuillan et al. 2013, 2014, $37\%$ in the range $3500\,{\rm K}<T_{\rm eff}<6500\,{\rm K}$, F $\sim 27\%$, G $\sim 25\%$, K $\sim 60\%$, M $\sim 80\%$; Stelzer et al. 2016, $73\%$ for M dwarfs). This is probably an effect of the selection bias towards active (and thus strongly spotted) stars as well as the removal of giant stars from our sample.

In Table 2 a summary of the number of stars with rotation period from our analysis is reported for every spectral type, together with the number of stars with other types of brightness modulation patterns. We refer to Table LABEL:table:params for the classification of the photometric variability for each star in the sample.

Only $54$ stars out of $74$ have previously reported periods from McQuillan et al. (2014), obtained from the same Kepler light curves. For the $54$ stars for which a rotation period is reported in McQuillan et al. (2014), those periods are consistent with our values within uncertainties. The remaining $22$ stars do not have any previously determined period in the literature.

6.2 Activity-rotation relation

In Fig. 11, we present the relation between the X-ray luminosity in the soft energy band $0.2-2.0\,$keV and the rotation period, together with the relation between the X-ray activity index (defined in Eq. 3) and the Rossby number. A clear decrease in X-ray activity levels is observed for slower rotators. This effect is more evident in the $AI_{\rm x}$ versus Rossby number plot, and produces a ‘kink’ in the distribution, which can be seen quite clearly in the overall sample of all stars, and also in the subsamples of K stars, while for M, G and F stars it is not obvious as these subsamples comprise only a limited range of rotation rates. The ‘kink’ suggests the presence of a correlated regime for slow rotators, and of a saturated regime for fast rotators, with a separation occurring at $\sim 8\,{\rm d}$ (from visual inspection). The wide range of $L_{X}$ and $AI_{\rm X}$ independent of $P_{\rm rot}$ for the (generally fast rotating) F stars is remarkable: it is possible that the F stars present a decoupling between rotation and activity because of their shallow convective zones. This group may also include active binary stars (RS CVn) with unknown contribution to the X-ray emission from the cool companions.

In Fig. 11 we show for comparison the literature compilation from Wright et al. (2011) and the previous empirical relations obtained by Pizzolato et al. (2003) based on a small sample with mostly spectroscopic rotation measurements (solid lines). The rotation periods in Wright et al. (2011) were collected from several works which use both spectroscopic and photometric techniques, and the X-ray fluxes were obtained from the analysis of data taken from different missions, such as XMM-Newton and ROSAT. We find excellent agreement of our more homogeneous data with that study. Especially, the scatter clearly decreases when X-ray luminosity and rotation period are replaced by $AI_{\rm x}$ and $R_{0}$, respectively. Contrary to previous work where no uncertainties were estimated, we present here conservative error bars for our sample. These are dominated by the uncertainties in the distances of stars without Gaia parallax that are derived from mapping the stars in the $\log{g}-\log{T_{\rm eff}}$ diagram onto the Dartmouth isochrones (see Sect. 3.1), i.e. they ultimately go back to the uncertainties in the spectroscopic parameters.

The rotation-activity relation of M dwarfs in our sample can be compared to that of M dwarfs observed in the K2 mission studied by Stelzer et al. (2016) with an analogous approach. In Fig. 12 we show the $L_{\rm x}-P_{\rm rot}$ relation for the two samples. Our X-ray selection evidently excludes slowly rotating M stars in the non-saturated regime. This can also be seen from the comparison with the bimodal relation suggested by Pizzolato et al. (2003) (solid black line in Fig. 11) which is, however, itself extremely poorly defined. Recent M dwarf studies by Wright & Drake (2016) and Wright et al. (2018), also shown in the figure, used rotation periods measured with ground-based instruments. They cover the long-period ‘correlated’ region and are complementary to our Kepler study.

The scatter observed in the saturated regime of $L_{\rm X}$ and $AI_{\rm x}$ for M stars of given $P_{\rm rot}$ is large but consistent with that observed by Wright et al. (2011), Pizzolato et al. (2003) and Stelzer et al. (2016), and probably due, at least in part, to the spectral type distribution within the M class, with cooler stars having lower X-ray luminosity. The scatter decreases when considering the relation $AI_{\rm x}$ versus $R_{0}$. Because of the relatively low statistics in our sample, in particular the low number of stars in the correlated regime, it would be difficult to establish with good confidence the turnover point between the correlated and saturated regime, nor the slope of the correlated part of the relation.

6.3 Kepler activity diagnostics

6.3.1 Optical flares

We explore here the optical flaring activity measured in the Kepler lightcurves of the spotted stars. This analysis must consider that the sensitivity for detecting flares depends on the (quiescent) brightness of the star and is different for each star. The criteria in our definition of flare include a $\geq 3\,\sigma$ upward deviation from the ‘flattened’ lightcurve (see Sect. 4.1). As a result of the relation between $K_{\rm p}$ and spectral type in our sample (evident in Fig. 5), the minimum measurable flare amplitude ($A_{\rm peak}$) – defined here as the relative brightness difference between the flare peak and the ‘flattened’ lightcurve – shows a trend with spectral type (Fig. 13).

For a meaningful comparison of our sample which covers a large range in brightness we must convert relative quantities, such as $A_{\rm peak}$ and $S_{\rm flat}$, to absolute ones. The Kepler photometry is not flux calibrated. However, an approximate brightness can be associated to the ‘flattened’ lightcurve assuming that this normalized ‘quiescent’ emission corresponds to the Kepler magnitude of the star. We convert $K_{\rm p}$ to flux using the zero-point and effective bandwidth provided at the filter profile service of the Spanish Virtual Observatory (SVO)777http://svo2.cab.inta-csic.es/svo/theory/fps/. Then we apply the distances derived in Sect. 3.1 to obtain the ‘quiescent’ luminosity in the Kepler band, $L_{\rm K_{p},0}$. Similarly, the flare amplitude is converted from its relative value ($A_{\rm peak}$) to a luminosity, $\Delta L_{\rm F,K_{p}}=L_{\rm K_{p},0}\cdot A_{\rm peak}$. The resulting relation between flare amplitude (in erg/s) and ‘quiescent’ luminosity is shown in Fig. 14. The lower envelope is defined by our flare detection threshold which is marked as a horizontal bar for each star and which is rising with increasing stellar luminosity. The A-type stars form an exception to this trend. They show lower flare amplitudes than expected for their $L_{\rm K_{p},0}$. This finding can easily be explained when the observed flares are attributed to unknown and unresolved later-type companion stars. In this scenario the actual $L_{\rm K_{p},0}$ value of the flare-host would be lower making the true amplitude $\Delta L_{\rm F,K_{p}}$ higher than measured. This would shift the data points to the left and upwards in Fig. 14.

For a given star, the range of flare luminosities, $\Delta L_{\rm F,K_{p}}$, extends up to $\sim 2$ orders of magnitude above the amplitude of the minimum observable flare. The range of flare amplitudes is smaller for G and F stars but this is probably related to the lack of sensitivity for the detection of low-luminosity flares. This is evident from consideration of the $S_{\rm flat}$ values which set the detection threshold (see Fig. 13 and 14).

One of the aims of our study is the investigation of a connection between flaring activity and rotation rate. In their analogous study on M dwarfs observed in the K2 mission, Stelzer et al. (2016) have found a sharp transition in the optical flaring behavior at a period of $\sim 10$ d. This transition is difficult to probe with this X-ray selected Kepler sample for two reasons: (1) the small number of slow rotators and (2) the broad range of $K_{\rm p}$ translating into a dishomogeneous flare detection threshold. If we restrict our sample to M dwarfs which span a relatively narrow range in optical brightness (c.f. Fig. 5), the relative flare amplitudes and the flare frequencies are consistent with the results obtained by Stelzer et al. (2016) but we are covering only the fast periods up to the presumed transition (see Fig. 15). The average flare frequency of the M stars in our sample ($0.12\,{\rm flares/d}$) is in excellent agreement with the average flare frequency in Stelzer et al. (2016) calculated over the $P_{\rm rot}$ range ($P_{\rm rot}\lesssim 10\,{\rm d}$) covered by the Kepler M stars ($0.11\,{\rm flares/d}$). This shows that our Kepler sample, albeit highly incomplete at the slow rotation / low activity side, is representative for fast rotating and active M dwarfs.

Analogous to the study of Stelzer et al. (2016) on K2 lightcurves, the sampling time of $29.4\,$min of the Kepler lightcurves used in this work, together with the characteristics of the flare search algorithm (see Sect. 4.3), prevent the detection of flares with duration below $\sim 1\,{\rm h}$. As a consequence, we detect mostly flares with large-amplitude and relatively long duration. It can be suspected that there is a significant population of smaller and shorter flares that cannot be observed, due to the long cadence of the light curves and to the flare-extraction pipeline. Therefore, the flare frequencies of Fig. 15 and Table 5 likely represent a lower limit to the actual values.

6.3.2 X-ray versus white-light flares

As described in Sect. 5.1.5 we find $7$ X-ray flares on $6$ stars. For these events the count rate at the flare peak (as measured from the EXTraS bayesian light curves) is a factor $3-7$ higher than during the non-flaring, quiescent time-intervals. As a result of the large distances of the Kepler stars, the X-ray count rates are small and the low counts statistics prevent a detailed quantitative analysis such as the determination of decay time and total flare energy.

All the stars with an X-ray flare except KIC $8909598$ are classified as main-sequence: one M-type, three K-type and one F-type star. We do not consider KIC $8909598$ in the further analysis. All but the F star (KIC $8520065$), show a clear rotational modulation in their Kepler light curves. This latter one is according to our analysis probably also a rotator, but the variability pattern is unclear, and it is not possible to estimate a reliable rotation period.

We compare the X-ray flaring with the white-light flaring activity. Of the four stars with a detected rotation period, three show a large number of white-light flares in the Kepler lightcurves, and are among the stars with the highest optical flare frequency. This underlines the strong connection between X-ray and optical flaring mechanism: Since the X-ray observations are relatively short in duration, X-ray flares are observed predominantly in the light curves of the stars which show a higher optical flare frequency. There are simultaneous XMM-Newton/Kepler observations for $69$ stars in our sample. All the X-ray flaring stars in our sample (except for KIC $8909598$, which, as mentioned, we exclude from the analysis) have a *Kepler *observation simultaneous with the *XMM-Newton *observation in which the X-ray flare was detected. No white-light flares without X-ray counterpart were detected during the simultaneous X-ray/white-light observations.

Our automated *Kepler *flare-detection algorithm finds $3$ events occurring within $\sim 1\,{\rm h}$ from the observed X-ray flares (see Fig. 8). However, visual inspection shows that the other X-ray flares likely have optical counterparts as well, which have remained below the detection threshold of our automatic algorithm. Finally, a simultaneous event, detected only by visual inspection of the X-ray (and Kepler) lightcurves, occurred on one star. To summarize, of the $6$ X-ray flares observed from rotating dwarf stars, there is evidence for a contemporaneous optical event in all cases, but only for three of them is the associated white-light flare clearly detected (one M-type and two K-type stars).

6.3.3 Other *Kepler *activity diagnostics

Here we examine the distribution of the photospheric activity diagnostics $S_{\rm ph}$ and $R_{\rm per}$ (see Sect. 4.4 for their definition). As mentioned above, as a result of the bias to active stars, our Kepler sample does not allow us to study the transition of the optical activity diagnostics at $P_{\rm rot}\sim 10$ d revealed in the K2 data of nearby M dwarfs. Fig. 16 shows our M dwarf sample compared to the one studied by Stelzer et al. (2016). Similar to the results on optical flares, the rotation cycle amplitude ($R_{\rm per}$) and the overall variability of the lightcurve ($S_{\rm ph}$) of the two samples follow the same distribution in the fast-rotator regime.

We can examine the spectral type dependence of the photometric activity diagnostics, provided that – analogous to the detection of flares – the brightness dependence of the sensitivity for measuring variations is considered. In Fig. 17 we display $S_{\rm ph}$ and $R_{\rm per}$, versus $K_{\rm p}$ separately for each spectral class. The average values of $S_{\rm ph}$ and $R_{\rm per}$ for each spectral type are listed in Table 5. While the exclusively high values for the activity diagnostics in the M dwarfs may be due to their faintness, preventing the detection of small variations, a curious distinction is seen for F stars. Albeit covering roughly the same level of (quiescent) brightness in the Kepler band as the G dwarfs and some of the K dwarfs and the same range of $P_{\rm rot}$ (see e.g. Fig. 11), the variability of the F stars is markedly smaller. This points at intrinsically weaker variability in F stars. A similar effect is seen in Fig.4 of McQuillan et al. (2014). Our finding is aggravated by the fact that the F stars of our sample are more active, i.e. have higher $S_{\rm ph}$ values, than a sample of $22$ F stars selected from the Kepler Asteroseismic Science Consortium (KASC) programme studied by Mathur et al. (2014a).

6.4 UV activity

As this sample is X-ray selected, all stars for which we detected an UV excess have an X-ray detection. We show in Fig. 18 the correlation between the UV excess luminosity (as calculated in Sect. 5.2) and the X-ray luminosity. With the caveat that the GALEX filters exclude some important chromospheric contributions, most notably the Mg ii doublet, these relations represent a comparison of the chromospheric and the coronal radiative energy output. The sample with FUV excess is too small for any conclusion. We perform the Spearman’s and Kendall’s correlation tests for the NUV and X-ray luminosities of each SpT and find that only the K stars show a significant positive correlation between the two luminosities (p-values $<0.05$) with rank correlation coefficients $\rho_{\rm s}=0.85$ and $\rho_{\rm K}=0.64$. The F and G stars show each a considerable spread in the distribution, similar to the behavior of the F stars in the X-ray / rotation relation of Fig. 11.

We also compare the UV emission levels of our sample to the UV emission of the exoplanet hosts studied by Shkolnik (2013). Deviating from the remainder of this paper, which is focused on the high-energy emission due to stellar activity, we consider here the total observed UV emission which includes photospheric plus chromospheric contribution. This enables a direct comparison to the work of Shkolnik (2013). Secondly, for the irradiation of exoplanets the relevant parameter is the total UV output of the star irrespective of its origin. Fig. 19 shows the distribution of $L_{\rm NUV}/L_{\rm bol}$ versus $T_{\rm eff}$ for our stars and for the sample of exoplanet hosts analysed by Shkolnik (2013). Our UV luminosity values are for given $T_{\rm eff}$ about one order of magnitude above the values of the stars with exoplanets. This might be explained by an oppositely directed bias in both samples: While our sample is biased towards active stars, known exoplanet hosts are typically inactive as a result of the selection criteria for planet search programs. A second reason that might influence the differences between the NUV emission of the two samples is the fact that the stars in the sample of Shkolnik (2013) are typically nearby (tens of parsec), such that many stars were saturated in the GALEX observations and they have been removed from the analysis, thus generating a bias towards small NUV luminosity. Based on these arguments, there is good reason to believe that the two samples taken together represent the full range of NUV luminosity for each spectral type.

6.5 A-type stars

Three stars in our sample are classified as A-type on the basis of their effective temperature. The light curve of one of them (KIC 5113797) shows a variability pattern that suggests an association with the class of $\gamma\,$Dor non-radial pulsators. The others (KIC 8703413, KIC 9048114) have a clear rotational modulation pattern suggesting the presence of star spots. Only $3$ flares have been detected by our algorithm in the *Kepler *lightcurves of the A stars in our sample.

Late A-type stars represent the high-mass borderline of the magnetically active stars. A change in the stellar structure occurs with respect to later-type stars: the outer convective layer disappears, leaving a fully radiative stellar interior. It is generally believed that the standard stellar dynamo model can not be applied to fully radiative stars, which lack the transition between convective and radiative zones. Dynamos may operate in the small convective cores of hot stars, but a major problem is the emergence of the ensuing fields to the surface of the stars where magnetic activity is observed. Considering these difficulties, the magnetic activity apparently observed on A-type stars, and especially the X-ray emission, is often ascribed to unresolved late-type companion stars in binary systems. In a systematic study of a large sample of A-type stars associated with a ROSAT X-ray source about $25$ % are bona-fide single stars (Schröder & Schmitt 2007), and the question on whether A-type stars can maintain a corona has not been conclusively resolved.

Similarly, the occurrence of photospheric magnetic activity in A-type stars is disputed. Balona (2013) and Balona (2015) performed a search for activity in the A stars observed by Kepler. Both rotational variability due to starspots and flaring activity was detected for a significant fraction of stars. Two A stars from our sample (KIC 8703413 and KIC 5113797) were classified as flaring stars by Balona (2013) and Balona (2015). The authors considered the flares on Kepler A-type too energetic to be attributed to possible late-type companions. However, this result has been refuted by Pedersen et al. (2016).

The $T_{\rm eff}$ of the three stars classified as A-type present in the *Kepler */ *XMM-Newton *sample is below the limit set by Simon et al. (2002) for the fully radiative regime ($8250\,{K}$) within uncertainties. KIC 9048114 is also consistent, within uncertainties, with an early-F type classification. The detection of activity from these stars may thus not be inconsistent with the standard dynamo model.

7 Conclusions

Using a complex sample selection approach involving multi-band photometric characterisation and visual inspection of multi-band images, we identify $125$ stars with X-ray detection in the 3XMM-DR5 catalog that are observed by the Kepler mission. The subsample of $102$ dwarf stars studied in this work comprises stars with spectral types from A to M. The distance of these stars range between $\sim\mathbf{40-7400}\,{\rm pc}$, with $90\,\%$ of the stars in the range $\sim\mathbf{40}-1500\,{\rm pc}$. By comparison to the volume-limited NEXXUS sample of nearby stars (Schmitt & Liefke 2004) we estimate that – as a result of the X-ray selection and the large distances of the Kepler stars – we probe only the most active $10$ % of the stellar main-sequence population.

A previous survey of the X-ray (and UV) activity of *Kepler *objects was performed by Smith et al. (2015), in which $\sim 1/5$ of the *Kepler *field of view was surveyed with the Swift instruments XRT and UVOT (X-ray sensitivity in the energy range $0.2-10\,{\rm keV}:2\cdot 10^{-14}\,{\rm erg\,cm^{-2}\,s^{-1}}$ in $10^{4}$ s). Ninety-three KIC objects were found to have an X-ray counterpart, of which $33$ were recognised as stars. None of the $20$ X-ray emitting stars from our joint *Kepler */ *XMM-Newton *study that are in the Swift FoV was detected by Swift. This is not surprising, as the average sensitivity limit of the XMM-Newton observations in the Kepler field is about a factor $3$ deeper than the Swift observations (see Table 1 for the range of flux limits for our data set).

In the *Kepler *light curves of the XMM-DR5 selected sample, we found several types of variability: rotational modulation due to spots, eclipsing binary systems, multi-periodicity possibly associated with rotational modulation in binary systems, contact binaries, variability due to stellar pulsations ($\delta\,$Sct and $\gamma\,$Dor stars), uncorrelated and confuse variability patterns. A large fraction of the stars present rotational modulation due to starspots ($74$, $\sim 73\%$ of the dwarf stars in the sample). The number of stars with rotational modulation may represent a lower limit, since other stars may have a weak rotational modulation which cannot be ascertained with sufficient confidence, or the rotational modulation is superimposed on other kinds of variability, as e.g. in the above-mentioned probable binary systems. The maximum period detectable is limited by the length of the *Kepler *Quarters ($\sim 90\,{\rm d}$). We found periods in the range $0.3-70\,{\rm d}$, with a distribution which does not present significant differences from one spectral type to the other.

We explored the relation of several photometric activity indicators and the coronal X-ray emission with the rotation period. Our data confirm previously observed values and ranges for the rotation-dependence of the star-spot brightness amplitude ($R_{\rm per}$), the overall variability (characterized by the standard deviation of the lightcurve $S_{\rm ph}$) and the residual noise after subtraction of rotational modulation and flares ($S_{\rm flat}$). Due to the predominance of fast rotators in our sample, the bimodality observed in previous works on M dwarfs (Stelzer et al. 2016), with a sharp transition between fast and slow rotators, can not be studied at the same detail. We do find, however, no evidence for high flare rates and amplitudes in the slow rotator regime above $P_{\rm rot}\sim 10$ d, in line with the expectation from these previous findings.

The $R_{\rm per}$ and $S_{\rm ph}$ values of the F stars appear underluminous with respect to later-type stars at the same intrinsic brightness, indicating that this result is not a bias. The same effect was seen in larger and less biased samples, e.g. by McQuillan et al. (2014). Unresolved late-type companion stars are a possible explanation for the low amplitude of the rotational modulation of the F stars. The rotational signal might actually be from such an unknown late-type star, diluted by the bright F star that dominates the unmodulated flux. Alternatively, the low $R_{\rm per}$ and $S_{\rm ph}$ values of the F stars may mark decreasing spot filling factor towards the high-mass limit of partially convective, dynamo-driving stars.

To examine the relation between X-ray activity and rotation we have composed $L_{\rm X}$ vs $P_{\rm rot}$ and $L_{\rm X}/L_{\rm bol}$ vs Rossby number ($R_{0}$) plot. A kink in the rotation / X-ray relation is observed, traditionally described as a transition between a saturated and a correlated regime. The resulting distributions confirm earlier studies of this parameter space using here a much more homogeneous data set (X-ray data and rotation periods from a single instrument, each) and analysis (same flare and rotation search mechanism for all stars). Similar to the most extensive sample studied previously in this respect, Wright et al. (2011), the ‘saturated’ and ‘correlated’ regime are hard to define for subsamples divided by spectral type. The reason is that in the representation involving Rossby number (where the two regimes are usually more marked than when using $P_{\rm rot}$) individual spectral types cover either only the saturated part (M stars) or only the correlated part (FGK stars).

We note that the F stars seem not to participate in the downward trend of the X-ray emission in the $P_{\rm rot}$ (or $R_{0}$) range representing the ‘correlated’ zone for the later spectral types. Also, contrary to the other spectral types, the F stars show no clear correlation between X-ray luminosity and UV excess luminosity. Within the group of F stars there is no correlation of X-ray or UV excess with $T_{\rm eff}$. It is therefore not obvious to ascribe this finding to the transition to fully radiative interior with the ensuing break-down of the solar-type dynamo. One possibility for the disordinated behavior of the F stars is that this group may comprise close binary stars with a later type companion star confusing the activity pattern. In such unresolved binaries the measured period might actually represent the rotation of the (unknown) later-type star with implications on all relations involving $P_{\rm rot}$. In fact, we observe in Fig. 11 that both the range of periods and the range of observed $L_{\rm x}$ are similar for M stars and F stars. This scenario would explain the absence of a correlation between $P_{\rm rot}$ and $L_{\rm x}$ as the longest observed periods ($\sim 10$ d) are still within the range of the ‘canonical’ saturated regime for M dwarfs.

A major asset of this study is the fact that $9$ of the *XMM-Newton *observations were carried out during the Kepler monitoring, providing strictly simultaneous optical and X-ray lightcurves. We discovered seven X-ray flares. For all of them there is evidence of a white-light counterpart in the *Kepler *lightcurves.

Acknowledgements.

This work is based on observations obtained with the Kepler mission and with XMM-Newton. Funding for the Kepler mission is provided by the NASA Science Mission directorate. XMM-Newton is an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. We have made use of data produced by the EXTraS project, funded by the European Union’s Seventh Framework Programme, and we acknowledge its financial support under grant agreement no. 607452 (EXTraS project) and that of the Italian Space Agency (ASI) through the ASI-INAF agreement 2017-14-H.0. This work also includes data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the SVO Filter Profile Service (http://svo2.cab.inta-csic.es/theory/fps/) supported from the Spanish MINECO through grant AyA2014-55216 and of VOSA, developed under the Spanish Virtual Observatory project supported from the Spanish MICINN through grants AYA2008-02156 and AYA2011-24052.