Starspot variability as an X-ray radiation proxy

Starspot variability as an X-ray radiation proxy Abstract Stellar X-ray emission plays an important role in the study of exoplanets as a proxy for stellar winds and as a basis for the prediction of extreme ultraviolet (EUV) flux, unavailable for direct measurements, which in their turn are important factors for the mass-loss of planetary atmospheres. Unfortunately, the detection thresholds limit the number of stars with the directly measured X-ray fluxes. At the same time, the known connection between the sunspots and X-ray sources allows using of the starspot variability as an accessible proxy for the stellar X-ray emission. To realize this approach, we analysed the light curves of 1729 main-sequence stars with rotation periods 0.5 < P < 30  d and effective temperatures 3236 < Teff < 7166 K observed by the Kepler mission. It was found that the squared amplitude of the first rotational harmonic of a stellar light curve may be used as a kind of activity index. This averaged index revealed practically the same relation with the Rossby number as that in the case of the X-ray to bolometric luminosity ratio Rx. As a result, the regressions for stellar X-ray luminosity Lx(P, Teff) and its related EUV analogue LEUV were obtained for the main-sequence stars. It was shown that these regressions allow prediction of average (over the considered stars) values of log (Lx) and log (LEUV) with typical errors of 0.26 and 0.22 dex, respectively. This, however, does not include the activity variations in particular stars related to their individual magnetic activity cycles. stars: activity, starspots, ultraviolet: stars, X-rays: stars 1 INTRODUCTION The stellar X-ray luminosity is widely used as an index of stellar magnetic activity (Wright et al. 2011). Moreover, the X-ray surface flux is considered as a proxy of mass-loss rate (i.e. stellar wind) for main-sequence stars (Wood et al. 2014). In addition, the stellar X-ray emission plays an important role in exoplanetary science as a basis for prediction of extreme UV flux, which is a crucial impacting factor for the planetary ionospheres and upper atmosphere mass-loss (Shaikhislamov et al. 2014, 2016; Khodachenko et al. 2015, 2017). Unfortunately, only 32.9 per cent of known stars at distances up to ∼25 pc from the Sun were detected as X-ray sources (Hünsch et al. 1999). This problem is especially obvious for more distant stars, where the Kepler mission found most exoplanets. Such stars with undetected X-ray fluxes need a kind of accessible X-ray proxy. Apparently, the stellar starspot variability at optical wavelengths could give such a proxy. For example, solar X-ray emission is associated with active regions (Wagner 1988), hence, with sunspots. Correspondingly, there are high (>0.95) linear correlations between the sunspot number or the total spot area and the monthly averaged solar X-ray background flux (Ramesh & Rohini 2008). An analogous relation was found in other main-sequence stars using spot-induced brightness variations (Messina et al. 2003). To show this, Messina et al. (2003) used the maximum amplitude (Amax) of rotational variations of the stellar light curve. It has been shown that this simple activity index is non-linearly related with X-ray luminosity $$L_{x}/L \propto A_{\rm max}^{b}$$, where b ≈ 2, and L is the stellar bolometric luminosity. In our previous studies, we proposed and advocated another activity index – the squared amplitude $$A_{1}^{2}$$ of the first (fundamental) rotational harmonics of the stellar light curve (Arkhypov et al. 2015a,b, 2016, 2018). Its major advantage consists in the statistical proportionality to the starspot number Ns (see arguments in Arkhypov et al. 2016), and hence, the presumable proportionality to the X-ray emission: $$L_{x}/L \propto A_{1}^{2}$$. In fact, the analogous proportionality $$L_{x}/L \propto A_{\rm max}^{2}$$ was reported by Messina et al. (2003). However, Winter, Pernak & Balasubramaniam (2016) found that the solar X-ray flux $$F_{x} \propto N_{s}^{\beta }$$, where 1.61 ≲ β ≲ 1.86, i.e. somewhat higher than the expected β ≈ 1. This might mean that besides the spot number Ns, the X-ray radiation is also related to another factor, which is probably the total spot area. Since the integral photometry index $$A_{1}^{2}$$ depends on both the starspot number and the spot area, its relation with Fx could be closer to a linear one. Below we test this expectation. We use the empirical relation between Lx/L and $$A_{1}^{2}$$ to predict the stellar X-ray luminosity Lx(P, Teff) as a function of stellar rotation period P and effective temperature Teff. This could be useful for the modelling of exoplanetary environments’ evolution including the systems of non-solar-like stars. Hitherto the atmosphere-magnetosphere modelling is mainly based on the assumption of a solar-like X-ray emission (e.g. Trammell, Arras & Li 2011; Koskinen et al. 2013, Shaikhislamov et al. 2014, 2016; Khodachenko et al. 2015, 2017). However, many of the Kepler stars are distant objects of non-solar type with an enhanced luminosity, whereas the majority of the stellar population is composed of the faint red dwarfs. This makes the importance of the characterization of X-ray luminosity for a broader, than just solar type, class of stars. In Section 2, we outline our approach using an extended stellar sample and our time-tested processing method (Arkhypov et. al. 2015a,b, 2016, 2018). In Section 3, we justify that the used activity index can play a role of X-ray proxy. As a result, we obtain in Section 4 an approximating regression for Lx(P, Teff), which is tested in comparison to observations in Sections 5. The related extreme ultraviolet (EUV) luminosity of stars is considered in Section 6. Section 7 summarizes the obtained results and their applicability. 2 STELLAR SET AND LIGHT-CURVE PROCESSING As in our previous studies (Arkhypov et al. 2015a,b, 2016, 2018), we consider here the squared amplitude $$A_{1}^{2}$$ of the light curve's first (fundamental) harmonic with period P of stellar rotation. This squared amplitude is taken because of its suspected and confirmed statistical proportionality to the number of starspots, at least for the solar-like stars (Arkhypov et al. 2016). Moreover, the amplitude of the first harmonic A1, in differ to the light-curve variance, is practically insensitive to the photon noise. In contrast with other rotational harmonics, it depends minimally on flares and short-period pulsations. Note that since the light integration over the stellar disc reduces the amplitude of harmonics progressively with the increasing harmonic number, the value of A1 can be measured with the maximal accuracy. To calculate the activity index $$A_{1}^{2}$$ with a maximal time resolution, while focusing on the rotational variability of a star, we divided the stellar light curve on to consecutive fragments that have durations of one stellar rotation period P each and removed contaminating signals/features such as sporadic flares and linear trend, as well as performed interpolation of the light curve in short (<0.2 P) gaps (see details in Arkhypov et al. 2015a,b, 2016, 2018). The standard Fourier analysis applied is prepared in such a way that one-period fragments of the light curve give the varying (from one fragment to another) index $$A_{1}^{2}$$ . The aforementioned expectations regarding the proportionality between the X-ray emission and the measured parameter $$A_{1}^{2}$$ are tested below for the main-sequence stars. For this purpose, we extend the previously analysed data set (Arkhypov et al. 2016), which contains the light curves of the main-sequence stars observed by the Kepler space observatory, to include the light curves of an additional set of 637 slow rotators (Arkhypov et al. 2018). In Fig. 1(a), the analysed extended sample (see the electronic version of Table 1) that includes 1729 stars with 0.5 < P < 30  d (according to measurements in Nielsen et al. 2013; McQuillan, Mazeh & Aigrain 2014) and the effective temperatures 3236 ≤ Teff ≤ 7166 K from the last version of the Kepler stellar properties catalogue (Mathur et al. 2017) is shown. Fig. 1(b) indicates that all stars, selected from Mathur et al. (2017), with surface gravity log [g(cm s−2)] > 4.2, belong to the main sequence. The availability of a high-quality light curve (Fig. 2) without perceptible interferences (i.e. no detectable short-period pulsations or double periodicity from companions) was a special criterion for compiling of the analysed sample of stars. Further details on the selection of stars and preparing of light curves (i.e. removing of gaps, flares, artefacts, trends), and further processing are described in Arkhypov et al. (2015b, 2016). Figure 1. View largeDownload slide Stellar set properties: (a) the distribution of 1729 selected stars over the effective temperature Teff (Mathur et al. 2017) and the rotation periods P (Nielsen et al. 2013; McQuillan et al. 2014) and (b) the HR-diagram with respect to surface gravity log (g) versus Teff analogous to Mathur et al. (2017). Figure 1. View largeDownload slide Stellar set properties: (a) the distribution of 1729 selected stars over the effective temperature Teff (Mathur et al. 2017) and the rotation periods P (Nielsen et al. 2013; McQuillan et al. 2014) and (b) the HR-diagram with respect to surface gravity log (g) versus Teff analogous to Mathur et al. (2017). Figure 2. View largeDownload slide Typical light curve (fragment) with a starspot variability from the Kepler archive (KIC 2283703). The time BJD here is the differential barycentric Julian days, counted from the mission starting time. The plot was prepared using NASA Exoplanet Archives service (http://exoplanetarchive.ipac.caltech.edu/). Figure 2. View largeDownload slide Typical light curve (fragment) with a starspot variability from the Kepler archive (KIC 2283703). The time BJD here is the differential barycentric Julian days, counted from the mission starting time. The plot was prepared using NASA Exoplanet Archives service (http://exoplanetarchive.ipac.caltech.edu/). Table 1. The analysed stellar set, applied parameters, and predictions (sampled from the whole electronic version). KIC  Teffa  Pb  $$\log (\langle A_{1}^{2} \rangle ){}^{\textrm{c}}$$  σd  (B − V)oe  τMLTf  log (Rx)regg  log (Lx)regh  number  (K)  (days)        (days)    (erg s−1)  892834  4940  13.765  − 4.94  0.04  0.96  22.60  − 4.95  28.29  1026146  4419  14.891  − 5.24  0.04  1.21  24.62  − 4.94  27.82  1160947  4583  25.155  − 5.51  0.06  1.12  23.95  − 5.59  27.60  1161620  5977  6.636  − 5.16  0.03  0.57  7.60  − 5.37  28.62  1162635  3758  15.678  − 5.57  0.04  1.62  28.09  − 4.85  27.41  1163579  5721  5.429  − 4.98  0.03  0.65  12.27  − 4.57  29.40  1293993  4996  21.216  − 6.00  0.05  0.93  22.33  − 5.47  27.87  1295069  5418  17.187  − 5.64  0.05  0.76  17.85  − 5.49  28.37  KIC  Teffa  Pb  $$\log (\langle A_{1}^{2} \rangle ){}^{\textrm{c}}$$  σd  (B − V)oe  τMLTf  log (Rx)regg  log (Lx)regh  number  (K)  (days)        (days)    (erg s−1)  892834  4940  13.765  − 4.94  0.04  0.96  22.60  − 4.95  28.29  1026146  4419  14.891  − 5.24  0.04  1.21  24.62  − 4.94  27.82  1160947  4583  25.155  − 5.51  0.06  1.12  23.95  − 5.59  27.60  1161620  5977  6.636  − 5.16  0.03  0.57  7.60  − 5.37  28.62  1162635  3758  15.678  − 5.57  0.04  1.62  28.09  − 4.85  27.41  1163579  5721  5.429  − 4.98  0.03  0.65  12.27  − 4.57  29.40  1293993  4996  21.216  − 6.00  0.05  0.93  22.33  − 5.47  27.87  1295069  5418  17.187  − 5.64  0.05  0.76  17.85  − 5.49  28.37  Notes.aStellar effective temperature from Mathur et al. (2017). bRotation period from McQuillan et al. (2014) or Nielsen et al. (2013) if absent in the first source. cOur active index, averaged over a whole light curve. dEstimated error of $$\log (\langle A_{1}^{2} \rangle )$$. eColour index according to the transformation Teff → (B − V)o in equation (1). f Turnover time according to equation (4) in Noyes et al. (1984). g Logarithm of the ratio of X-ray to bolometric luminosities which are estimated from equations (6)–(8). hLogarithm of the predicted X-ray luminosity using equations (10)–(14). View Large We analyse the rotational modulation of the stellar radiation flux F (PDCSAP_FLUX from the Kepler mission archive1), which reflects the longitudinal distribution of spots. It has been shown in Arkhypov et al. (2016) that the squared amplitude $$A_{1}^{2}$$ of the first (or fundamental) Fourier harmonic with period P of the stellar one-period light curve is proportional to a spot number. Following this, we used the value $$A_{1}^{2}$$ as an activity index. To exclude the temporal oscillations of stellar activity due to magnetic cycles, we measured a value for $$\langle A_{1}^{2} \rangle$$ that was then averaged over the whole light-curve duration for every star in our set. 3 JUSTIFICATION OF THE OPTICAL PROXY FOR X-RAY EMISSION Because of the absence of common objects in the analysed stellar sample and in the most complete nowadays catalogue of estimates of X-ray ratio Rx ≡ Lx/L by Wright et al. (2011), the direct comparison between Rx and the value $$\langle A_{1}^{2} \rangle$$ is impossible . Since the ratio Rx is commonly considered as a function of the Rossby number Ro ≡ P/τMLT, where τMLT is the turnover time in the mixing length theory (Wright et al. 2011 and therein), instead of direct comparison of the activity indexes Rx and $$\langle A_{1}^{2} \rangle$$ themselves, we compare their dependencies on Ro. Following the arguments by Mamajek & Hillenbrand (2008), we use in this study the classical version of τMLT from Noyes et al. (1984). This approach is valid for Teff ≳ 4000 K corresponding to the colour index (B − V)o < 1.4 of stars used in Noyes et al. (1984). An alternative turnover time approximation by Wright et al. (2011), assuming the linear relation between log (τMLT) and the colour index V − Ks, apparently overestimates τMLT for the stars hotter than the Sun, because theoretically log (τMLT) → −∞ in vanishing convection zones for Teff ≈ 8200 K (Simon et al. 2002), i.e. for V − Ks ∼ 0.5. Equation (11) in Wright et al. (2011), which describes the relation between log (τMLT) and stellar mass, gives an unrealistic independence of the stellar activity on Ro > 1, when the mass estimates for the Kepler stars (Mathur et al. 2017) are used. Another argument for using the mixing time τMLT from Noyes et al. (1984) is provided in Section 5. Since τMLT in Noyes et al. (1984) is defined via the colour index (B − V)o, we use equation (26) from Arkhypov et al. (2016) for the transformation Teff → (B − V)o to make use of the data from the X-ray catalogue by Wright et al. (2011). Analogously, for the analysed set of Kepler Input Catalog (KIC) stars with a slightly different temperature scale, we found the following regression using the bright stars from SIMBAD data base considered in our previous study (Arkhypov et al. 2016):  \begin{equation} (B-V)_{0} = 0.6369 Y^{3} -2.262 Y^{2} -14.38 Y + 52.84, \end{equation} (1)were Y = log (Teff/1K). Since the index $$\langle A_{1}^{2} \rangle$$ (as well as Amax in Messina et al. 2003) depends on a random inclination angle between the stellar rotation axis and the direction to observer, we use the mean activity level $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, defined by averaging (denoted with an upper bar) over many stars with similar parameters Teff and P (for justification see in Section 4.3 by Arkhypov et al. 2016). Fig. 3(a) shows the dependence of stellar activity index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, averaged over star groups in the Rossby number bins log (P/τMLT). The estimates of $$\log (\langle A_{1}^{2} \rangle )$$ for individual stars were used to calculate the converging regression lines for both saturated and non-saturated activity regions with respect to Ro   \begin{equation} \overline{\log (\langle A_{1}^{2} \rangle )} \approx \gamma _{1,2} \log \left(\frac{P}{\tau _{\rm MLT}} \right) + \delta _{1,2} \equiv \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}, \end{equation} (2)where γ1 = −0.00 ± 0.18 and δ1 = −3.99 ± 0.23 are the fitted coefficients for log (P/τMLT) < log (Rosat) = −0.89, whereas γ2 = −1.86 ± 0.04 and δ2 = −5.63 ± 0.02 correspond to the case of log (P/τMLT) > log (Rosat) = −0.89. The saturation Rossby number Rosat = 10−0.89 = 0.13, at which both regressions converge, is the same as one found in Wright et al. (2011). Figure 3. View largeDownload slide Dependence of stellar activity on the Rossby number P/τMLT. (a) Averaged index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ for the star groups in log (P/τMLT) bins, (b) Normalized X-ray luminosity Rx from Wright et al. (2011), and (c) Comparison of the regressions depicted as the lines in (a) and (b). The dashed line corresponds to equality of the predictions. The crosses are the standard error bars ±σreg (see equation 5) of the regressions for equidistantly selected values of the Rossby number. Figure 3. View largeDownload slide Dependence of stellar activity on the Rossby number P/τMLT. (a) Averaged index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ for the star groups in log (P/τMLT) bins, (b) Normalized X-ray luminosity Rx from Wright et al. (2011), and (c) Comparison of the regressions depicted as the lines in (a) and (b). The dashed line corresponds to equality of the predictions. The crosses are the standard error bars ±σreg (see equation 5) of the regressions for equidistantly selected values of the Rossby number. Using Rx ≡ Lx/L as the activity index from the stellar catalogue by Wright et al. (2011), we found the similar pattern in Fig. 3(b) with the following regression:   \begin{equation} \log (R_{x}) \approx \gamma _{1,2}^{x} \log \left(\frac{P}{\tau _{\rm MLT}} \right) + \delta _{1,2}^{x} \equiv \log (R_{x})_{\textrm{reg}}, \end{equation} (3)with close to equation (8) parameters: $$\gamma _{1}^{x}=-0.13 \pm 0.04$$ and $$\delta _{1}^{x}=-3.33 \pm 0.06$$ in the saturation regime and $$\gamma _{2}^{x}=-2.04 \pm 0.08$$ and $$\delta _{2}^{x}=-4.85 \pm 0.04$$ in the non-saturated case. The similarities $$\gamma _{1} \simeq \gamma _{1}^{x}$$, $$\gamma _{2} \approx \gamma _{2}^{x}$$, $$\delta _{1} \sim \delta _{1}^{x}$$, and $$\delta _{2} \sim \delta _{2}^{x}$$ are understandable, because the X-ray emission is associated with the active regions, i.e. with the starspots. In particular, the solar X-ray flux correlates with both sunspot number and their total area (Ramesh & Rohini 2008), which control our index $$A_{1}^{2}$$. Fig. 3(c) shows the comparison between the regressions’ prediction for both activity indexes in the region of measured log (Rx). After the correction $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}-\Delta$$ with an average difference Δ = −0.82 from log (Rx)reg, we arrive at the approximation   \begin{equation} \log (R_{x})_{\textrm{reg}} \approx \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}+0.82. \end{equation} (4)The slight deviations (≲0.15 dex) from the equality $$\log (R_{x})_{\textrm{reg}} = \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}+0.82$$ are consistent with the regression standard errors which are   \begin{equation} \sigma _{\textrm{reg}} = \sqrt{\sigma _{\gamma }^{2}[\log (\text{Ro})]^{2} + \sigma _{\delta }^{2}}, \end{equation} (5)where Ro = P/τMLT is the Rossby number, and the standard errors of the regression coefficients are as follows: σγ = 0.18 and σδ = 0.23 for saturated $$\langle A_{1}^{2} \rangle$$; σγ = 0.04 and σδ = 0.02 for unsaturated $$\langle A_{1}^{2} \rangle$$; σγ = 0.04 and σδ = 0.06 for saturated Rx; and σγ = 0.08 and σδ = 0.04 for unsaturated Rx. The corresponding error bars are shown in Fig. 2(c) for the equidistantly selected vales of log (Ro). The approximate relation $$\log (R_{x})_{\textrm{reg}} \approx \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}+0.82$$ means that the regression $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$ and the related activity index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ can be used as a quasi-proportional proxy for log (Rx). 4 FROM THE OPTICAL PROXY TO THE PREDICTION OF STELLAR X-RAY LUMINOSITY Here, we apply the optical proxy approach to predict stellar average X-ray luminosity, i.e. obtain the regression Lx(P, Teff). Theoretically, using equation (4) and the approximation $$\overline{\log (\langle A_{1}^{2} \rangle )} \approx \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$, following equation (2), the proxy $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$ can be transformed to Rx. However, in practice, an empirical coefficient K should be included in this transformation to take into account a selection effect, i.e. the ignoring of stars with undetected X-ray radiation or low-amplitude light curves, so that   \begin{equation} \log (R_{x})_{\textrm{reg}} \approx \overline{\log (\langle A_{1}^{2} \rangle )}+\log (K), \end{equation} (6)where   \begin{equation} \log (K)=\log \left(R_{x}^{\text{unb}} \right)-\log \left(\left\langle A_{1}^{2} \right\rangle \right)_{\textrm{reg}}. \end{equation} (7)Here, $$R_{x}^{\text{unb}}$$ is the regression for Rx, which was obtained in Wright et al. (2011) using the ‘unbiased sample’ of stars   \begin{equation} \log (R_{x}^{\text{unb}}) \approx \gamma _{1,2}^{\text{unb}} \log \left(\frac{P}{\tau _{\rm MLT}} \right) + \delta _{1,2}^{\text{unb}}, \end{equation} (8)where $$\gamma _{1}^{\text{unb}}=0$$ (assumed) and $$\delta _{1}^{\text{unb}}=-3.13 \pm 0.08$$ are the fitting coefficients for log (Ro) < log (Rosat) = −0.89, whereas $$\gamma _{2}^{\text{unb}}=-2.70 \pm 0.13$$ and $$\delta _{2}^{\text{unb}}=\delta _{1}^{\text{unb}}-\gamma _{2}^{\text{unb}}\log (\text{Ro}_{\text{sat}})=-5.53 \pm 0.14$$ correspond to the case of log (Ro) > log (Rosat) = −0.89. It follows from the definition Rx ≡ Lx/L that in order to estimate Lx, one needs a regression for the bolometric luminosity L(Teff). While the majority of astrophysical studies were focused on the universal mass-luminosity relation, we base our derivation of the needed relation on the apparently most complete compilation of the published values for L and Teff for main-sequence stars (Eker et al. 2015):   \begin{equation} \log (L)=a_{L}Y^{3}+b_{L}Y^{2}+c_{L}Y+d_{L}, \end{equation} (9)where L is the luminosity in solar units, Y = log (Teff/1K), and the fitting coefficients are aL = 3.801, bL = −47.396, cL = 202.329, and dL = −292.539. The standard deviation is εL = 0.20 dex for all used stars (265 objects excluding 31 stars with problematic main-sequence status and >3σ outliers). Higher polynomial power has practically no decreasing effect on εL. Using equations (6)– (9), we calculate Lx = RxL for every star from our set. In fact, we replace in equation (6) the averaged $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ with the values $$\log (\langle A_{1}^{2} \rangle )$$ for individual stars. This substitution is justified, since the further calculation of the regression Lx(P, Teff)reg is equivalent to an averaging over stars. This regression has the following form:   \begin{equation} \log (L_{x}) \approx a_{x0}X^{3}+b_{x0}X^{2}+c_{x0}X+d_{x0} \equiv \log (L_{x})_{\textrm{reg}}, \end{equation} (10)  \begin{equation} a_{x0}=a_{x1}Y^{3}+b_{x1}Y^{2}+c_{x1}Y+d_{x1}, \end{equation} (11)  \begin{equation} b_{x0}=a_{x2}Y^{3}+b_{x2}Y^{2}+c_{x2}Y+d_{x2}, \end{equation} (12)  \begin{equation} c_{x0}=a_{x3}Y^{3}+b_{x3}Y^{2}+c_{x3}Y+d_{x3}, \end{equation} (13)  \begin{equation} d_{x0}=a_{x4}Y^{3}+b_{x4}Y^{2}+c_{x4}Y+d_{x4}, \end{equation} (14)where X = log (P/1 d), Y = log (Teff/1K), and the fitting coefficients axi, bxi, cxi, dxi, found with the least square method, are listed in Table 2. Table 2. Fitting coefficients in equations (11)–(14). Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 17.32  73.48  156.7  − 707.0    2  16.31  − 182.6  675.1  − 827.6    3  13.70  124.5  −1449  2966    4  0.4856  − 108.9  781.9  − 1397  >5500 K  1  − 0.8916  31.44  − 188.9  313.9    2  18.31  − 111.14  72.19  323.6    3  − 5.046  − 116.7  1055  −2047    4  − 36.21  126.4  578.5  −2007  Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 17.32  73.48  156.7  − 707.0    2  16.31  − 182.6  675.1  − 827.6    3  13.70  124.5  −1449  2966    4  0.4856  − 108.9  781.9  − 1397  >5500 K  1  − 0.8916  31.44  − 188.9  313.9    2  18.31  − 111.14  72.19  323.6    3  − 5.046  − 116.7  1055  −2047    4  − 36.21  126.4  578.5  −2007  View Large Fig. 4 shows the deviation $$\varepsilon _{\textrm{reg}} = \overline{\log (L_{x})} - \log (L_{x})_{\textrm{reg}}$$ between the regression and $$\overline{\log (L_{x})}$$, calculated for individual stars (equations 6– 9) and averaged over the considered sample in a sliding window log (Pc) − 0.2 < log (P) < log (Pc) + 0.2 and log (Tc) − 0.05 < log (Teff) < log (Tc) + 0.05 with the central values of stellar rotation period Pc and effective temperature Tc. On average, 104 (up to 403) individual estimates of Lx appeared within this sliding window. Fig. 4(b) demonstrates a histogram of εreg with the standard deviation sreg = 0.14. Hence, the total standard error of the regression log (Lx)reg can be estimated as a combination of the standard errors of the involved regressions, i.e. $$\sigma _{{\rm tot}}=(\sigma _{\textrm{reg}}^{2}+s_{\textrm{reg}}^{2}+\varepsilon _{L}^{2})^{1/2} \approx 0.26$$, where σreg ∼ 0.1 dex for log (Rx)reg (see equation 5 and Fig. 4c for Rx errors), sreg = 0.14 dex for log (Lx)reg, and εL = 0.20 dex for log (L). Therefore, the derived regression (equations 10–14) predicts the average logarithm of stellar X-ray luminosity with typical error 0.26 dex. Figure 4. View largeDownload slide Distributions of the deviation εreg: (a) in relation to stellar parameters P and Teff; (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. Figure 4. View largeDownload slide Distributions of the deviation εreg: (a) in relation to stellar parameters P and Teff; (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. 5 COMPARISON OF THE PREDICTIONS VERSUS OBSERVATIONS For the verification of our predictions, for all 824 objects from the X-ray catalogue by Wright et al. (2011), considered in this paper, we provide in Table 3 the values of log (Lx)reg, calculated using equations (10)– (14), as well as the observed values of log (Lxw). One can compare these values for different stellar clusters using the associated electronic version of Table 3. However, we would like to note that the regressions for luminosity log (L) and activity index $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$ (equations 9 and 2, respectively) were obtained for the main-sequence stars only. Hence, the pre-main-sequence objects in young clusters might appear a wrong example for controlling our estimates. At the same time, the mainly old field stars, which are more numerous than members of any cluster in the used catalogue, are most suitable for the verification of the obtained regressions. Table 3. Parameters and predictions for stars in the catalogue by Wright et al. (2011) (sampled from the complete electronic version). RA 2000a  DE 2000b  Affiliation  Twc  P d  τMLTe  $$\log (R_{x}^{{\rm unb}}){}^{\textrm{f}}$$  log (Lxp)g  log (Lxw)h  (deg.)  (deg.)    (K)  (days)  (days)    (erg s−1)  (erg s−1)  −  −  Sun  5780  26.09  9.97  − 6.66  27.31  27.35  2.02679  47.9507  Field  3201  4.38  32.85  − 3.17  28.84  28.21  2.84350  30.4496  Field  5460  6.05  15.21  − 4.45  29.96  29.14  5.49087  49.2106  Field  3405  6.17  30.69  − 3.65  28.58  28.25  5.71579  − 12.2094  Field  5687  7.78  11.43  − 5.08  29.00  29.15  7.22467  50.3758  Field  3163  1.09  33.30  − 3.13  27.25  28.29  9.14317  55.6267  Field  3654  8.35  28.54  − 4.09  28.17  28.61  9.50008  43.8959  Field  3619  0.55  28.82  − 3.13  28.23  29.02  RA 2000a  DE 2000b  Affiliation  Twc  P d  τMLTe  $$\log (R_{x}^{{\rm unb}}){}^{\textrm{f}}$$  log (Lxp)g  log (Lxw)h  (deg.)  (deg.)    (K)  (days)  (days)    (erg s−1)  (erg s−1)  −  −  Sun  5780  26.09  9.97  − 6.66  27.31  27.35  2.02679  47.9507  Field  3201  4.38  32.85  − 3.17  28.84  28.21  2.84350  30.4496  Field  5460  6.05  15.21  − 4.45  29.96  29.14  5.49087  49.2106  Field  3405  6.17  30.69  − 3.65  28.58  28.25  5.71579  − 12.2094  Field  5687  7.78  11.43  − 5.08  29.00  29.15  7.22467  50.3758  Field  3163  1.09  33.30  − 3.13  27.25  28.29  9.14317  55.6267  Field  3654  8.35  28.54  − 4.09  28.17  28.61  9.50008  43.8959  Field  3619  0.55  28.82  − 3.13  28.23  29.02  Notes.aRight ascension for epoch 2000.0 from Wright et al. (2011). bDeclination for epoch 2000.0 from Wright et al. (2011). cEffective temperature from Wright et al. (2011). dFrom Wright et al. (2011). eTurnover time according to equation (4) in Noyes et al. (1984) and transform Tw → (B − V)o (equation 26 in Arkhypov et al. 2016). fFrom equations (6)– (8); gLogarithm of the predicted X-ray luminosity using equations (10)–(14).  hLogarithm of the measured X-ray luminosity from Wright et al. (2011). View Large Fig. 5 shows the comparison of the predicted X-ray luminosity log (Lxp) = log (Lx)reg (see equation 10) and the observed luminosity log (Lxw) for 443 field stars from the catalogue by Wright et al. (2011). We are focused on the field stars because of the limited and distorted X-ray statistics in more distant stellar clusters, where many of the detections are just above the detection threshold. In Fig. 5(a), one can see the general agreement between the predicted and observed stellar distributions in relation with Teff. However, Fig. 5(b) reveals that some stars show log (Lxw) ≫ log (Lxp) at log (Lxp) ≲ 27 erg s−1. This effect disappears when the faint stars with Teff < 4000 K are omitted in Fig. 5(c). Fig. 5(d) demonstrates that the stars with Teff < 4000 show a clear cut-off of the observed X-ray flux Fxw at the detection threshold of ∼10−13 ergs s−1 cm−2. The stars with Fxw above this threshold are seen in Fig. 5(b) as a specific population of objects with log (Lxw) ≫ log (Lxp). However, at Teff > 4000 K (Fig. 5c), the X-ray luminosity values for the considered stars are clustered along the equality line log (Lxw) = log (Lxp) with a negligible average difference 〈log (Lxw) − log (Lxp)〉 = 0.04 ± 0.04, and the standard deviation of log (Lxw) − log (Lxp) for an individual star is sind = 0.60 dex. Apparently, the Lx variability in time gives the main contribution to sind. Figure 5. View largeDownload slide Comparison of the predicted X-ray luminosity log (Lxp) = log (Lx)reg (diamonds according to equation 10) and the observed luminosity log (Lxw) (filled squares) for the field stars from the catalogue by Wright et al. (2011). (a) The estimates’ distribution in relation with Teff, (b) the cross comparison of the all estimates in (a), (c) the same cross comparison as in (b) but only for stars with Teff > 4000 K, and (d) the comparison of observed Fxw and predicted Fxp X-ray fluxes at the Earth for stars with Teff < 4000 K. The lines depict equalities of the abscissa and ordinate values. Figure 5. View largeDownload slide Comparison of the predicted X-ray luminosity log (Lxp) = log (Lx)reg (diamonds according to equation 10) and the observed luminosity log (Lxw) (filled squares) for the field stars from the catalogue by Wright et al. (2011). (a) The estimates’ distribution in relation with Teff, (b) the cross comparison of the all estimates in (a), (c) the same cross comparison as in (b) but only for stars with Teff > 4000 K, and (d) the comparison of observed Fxw and predicted Fxp X-ray fluxes at the Earth for stars with Teff < 4000 K. The lines depict equalities of the abscissa and ordinate values. In Fig. 6, we test an alternative possible explanation of the aforementioned deviations at log (Lxw) ≫ log (Lxp) in Fig. 5(b) as a result of underestimated τMLT for the red dwarfs outside of the temperature region, for which the used approximation of τMLT was found (Noyes et al. 1984). To do that, we calculated the average values $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ for different Rossby numbers Ro = P/τMLT in two temperature domains Teff > 4000 K and Teff < 4000 K using the different definition versions of τMLT according to Noyes et al. (1984) and equation (10) in Wright et al. (2011), respectively. One can see in Fig. 6(a) that τMLT from Noyes et al. (1984) gives the unified sequence of estimates that are independent on the temperature domain. However, the increased τMLT from Wright et al. (2011) shifts the low-temperature points towards the lower Ro, destroying the similarity between the estimates in Fig. 6(b). Since the temperature independence of the activity-Ro relation is a commonly accepted fact for the estimation of τMLT (e.g. Noyes et al. 1984), Fig. 6 argues for the validity of Noyes’ version of τMLT also at 3300 ≲ Teff < 4000 K. Consequently, the assumption of an increased mixing time τMLT cannot be used for the explanation of the extension of stellar population with log (Lxw) > log (Lxp) in Fig. 5(b). Figure 6. View largeDownload slide Values $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, averaged in the bins of Ro = P/τMLT, for two temperature domains Teff < 4000 K (black squares) and Teff > 4000 K (crosses) with (a) τMLT defined as in Noyes et al. (1984), for all stars and (b) τMLT, defined according to equation (10) in Wright et al. (2011), for the stars with Teff < 4000 K. Figure 6. View largeDownload slide Values $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, averaged in the bins of Ro = P/τMLT, for two temperature domains Teff < 4000 K (black squares) and Teff > 4000 K (crosses) with (a) τMLT defined as in Noyes et al. (1984), for all stars and (b) τMLT, defined according to equation (10) in Wright et al. (2011), for the stars with Teff < 4000 K. For another test of our prediction of the X-ray luminosity, we use the best calibrated and studied case of the solar-type stars. Fig. 7 shows our Lx(P) prediction for the stars with solar Teff = 5770 K (Allen 1973) in comparison with the relations obtained by other authors using independent methods. For example, Mamajek & Hillenbrand (2008) used Ca II H and K emission index as a kind of X-ray proxy. Their regressions (A3), (12)–(14) and Table 10 at the solar colour index B − V = 0.65 (Allen 1973) are shown as a long-dashed curve, which fits sufficiently well with our prediction (solid curve) mainly within its standard error ±σtot dex (see Section 4). Ribas et al. (2005) considered the directly measured X-ray flux from solar analogues with estimated ages. Here, we transformed the stellar ages to the P-scale using two versions of gyrochronological relation: (a) the cited above equations (12)– (14) and Table 10 in Mamajek & Hillenbrand (2008) (dotted line), and (b) equation (1) in García et al. (2014) (dashed line). Both curves fit well with our prediction. Finally, the average X-ray flux of the Sun (Wright et al. 2011), indicated in Fig. 7 with an opened square, coincides with our solid curve. Figure 7. View largeDownload slide Predicted X-ray luminosity versus P is shown as solid curve according to equations (10–14) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 6– 9) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar X-ray luminosity from the catalogue by Wright et al. (2011). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (dotted line). The long-dashed line corresponds to the regressions in Mamajek & Hillenbrand (2008). The standard error σtot of our prediction is depicted as a left-bottom bar. Figure 7. View largeDownload slide Predicted X-ray luminosity versus P is shown as solid curve according to equations (10–14) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 6– 9) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar X-ray luminosity from the catalogue by Wright et al. (2011). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (dotted line). The long-dashed line corresponds to the regressions in Mamajek & Hillenbrand (2008). The standard error σtot of our prediction is depicted as a left-bottom bar. Considering Figs 5 and 7, one can conclude that the obtained regression Lx(P, Teff), defined by equations (10)–(14), generates plausible predictions at least at Teff ≳ 3500 K. Since the reference X-ray catalogue by Wright et al. (2011) is converted to the ROSAT wavelength band, our regression approximates Lx in the same band over the range from 6 to 124 Å. 6 EUV APPLICATION We demonstrate below the application potential of the proposed X-ray proxy for the prediction of related EUV radiation. The EUV radiation at wavelengths from 124 to 912 Å plays an important role in planetary science as a crucial impacting/heating factor for the upper atmospheres. However, in the case of exoplanets it is unobservable because of significant interstellar extinction. That is why there is a common practice to use the observable X-ray radiation as a proxy for the EUV flux. Apparently, the best results were obtained by Chadney et al. (2015) using the empirical relation   \begin{equation} \log (F_{{\rm EUV}}) = 2.63 + 0.58 \log (F_{x}), \end{equation} (15)where $$F_{x}=L_{x}/(4 \pi R_{*}^{2}) ({\rm mW \, m}^{-2})$$ is the stellar surface flux in the ROSAT band 6–124 Å, and FEUV is the EUV surface flux at 124–912 Å. Correspondingly, the stellar EUV luminosity LEUV can be found using the stellar radius R* from the reference catalogue of KIC stellar data by Mathur et al. (2017)   \begin{equation} \log (L_{{\rm EUV}}) = \log (F_{{\rm EUV}}) + \log \left(4 \pi R_{*}^{2} \right) + 4, \end{equation} (16)where term 4 is added for the unit transformation (mW) → (erg s−1). Using equations (15) and (16) in combination with equations (6)–(9), one can calculate LEUV for every KIC star in our data set. These estimates were used to obtain the regression LEUV(P, Teff) as follows:   \begin{equation} \log (L_{{\rm EUV}}) \approx a_{e0}X^{3}+b_{e0}X^{2}+c_{e0}X+d_{e0} \equiv \log (L_{{\rm EUV}})_{\textrm{reg}},\!\!\!\!\!\! \end{equation} (17)  \begin{equation} a_{e0}=a_{e1}Y^{3}+b_{e1}Y^{2}+c_{e1}Y+d_{e1}, \end{equation} (18)  \begin{equation} b_{e0}=a_{e2}Y^{3}+b_{e2}Y^{2}+c_{e2}Y+d_{e2}, \end{equation} (19)  \begin{equation} c_{e0}=a_{e3}Y^{3}+b_{e3}Y^{2}+c_{e3}Y+d_{e3}, \end{equation} (20)  \begin{equation} d_{e0}=a_{e4}Y^{3}+b_{e4}Y^{2}+c_{e4}Y+d_{e4}, \end{equation} (21)where the fitting coefficients aei, bei, cei, dei, obtained with the least square method, are listed in Table 4. Table 4. Fitting coefficients for equations (17)–(21). Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 9.606  42.81  72.39  − 366.5    2  8.348  − 100.3  394.1  − 510.2    3  8.862  64.78  − 822.6  1708    4  − 1.152  − 55.47  455.7  − 838.7  >5500 K  1  − 0.7564  19.33  − 108.2  174.4    2  10.97  − 65.78  36.82  206.3    3  − 2.667  − 71.48  629.6  −1215    4  − 21.71  75.70  349.5  −1201  Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 9.606  42.81  72.39  − 366.5    2  8.348  − 100.3  394.1  − 510.2    3  8.862  64.78  − 822.6  1708    4  − 1.152  − 55.47  455.7  − 838.7  >5500 K  1  − 0.7564  19.33  − 108.2  174.4    2  10.97  − 65.78  36.82  206.3    3  − 2.667  − 71.48  629.6  −1215    4  − 21.71  75.70  349.5  −1201  View Large Fig. 8 shows the deviation $$\varepsilon _{\textrm{reg}}^{{\rm EUV}} = \overline{\log (L_{{\rm EUV}})} - \log (L_{{\rm EUV}})_{\textrm{reg}}$$ between the regression and $$\overline{\log (L_{{\rm EUV}})}$$, calculated for individual stars (equations 15– 16) and averaged over the considered sample in a sliding window log (Pc) − 0.2 < log (P) < log (Pc) + 0.2 and log (Tc) − 0.05 < log (Teff) < log (Tc) + 0.05 with the central values of stellar rotation period Pc and effective temperature Tc. On average, 104 (up to 403) individual estimates of LEUV appeared within this sliding window. Fig. 8(b) shows a histogram of $$\varepsilon _{\textrm{reg}}^{{\rm EUV}}$$ with the standard deviation $$s_{\textrm{reg}}^{{\rm EUV}}=0.08$$. The deviation of the regression (17) from the true value log (LEUV)o is   \begin{equation} d_{{\rm tot}}^{{\rm EUV}} \equiv \log (L_{{\rm EUV}})_{o} - \log (L_{{\rm EUV}})_{\textrm{reg}} = \delta _{{\rm reg}} + \Delta , \end{equation} (22)where δreg = log (LEUV) − log (LEUV)reg and Δ = log (LEUV)o − log (LEUV). Here, log (LEUV) is an estimate obtained using equations (15) and (16). Correspondingly, the standard error of the regression (17) is   \begin{equation} \sigma _{{\rm tot}}^{{\rm EUV}} = \sqrt{\left\langle \left(d_{{\rm tot}}^{{\rm EUV}} \right)^{2} \right\rangle }= \sqrt{\left\langle \delta _{{\rm reg}}^{2} \right\rangle + \langle \Delta ^{2} \rangle }, \end{equation} (23)where $$\langle \delta _{{\rm reg}}^{2} \rangle \approx s_{\textrm{reg}}^{\rm {EUV}}=0.08$$, and 〈Δ2〉 can be obtained from equations (15) and (16) by substitution of variables in the form of a sum of average values with index o plus fluctuation marked with Δ, i.e. log (Lx) = log (Lx)o + Δlog (Lx) and log (R*) = log (R*)o + Δlog (R*). Then the following expression can be obtained:   \begin{equation} \langle \Delta ^{2} \rangle = 0.34 \langle [\Delta \log (L_{x})]^{2} \rangle + 0.71 \langle [\Delta \log (R_{*})]^{2} \rangle + \rho ^{2}, \end{equation} (24)where 〈[Δlog (Lx)]2〉 = σtot, 〈[Δlog (R*)]2〉 ∼ 0.1, which corresponds to the typical error ∼ 27 per cent in Mathur et al. (2017), and ρ ∼ 0.1 is the typical uncertainty of prediction with equation (15) caused by the coefficient errors, which was estimated as a typical deviation of stellar estimates from the regression in Fig. 2 in Chadney et al. (2015). In summary, the total standard error of the regression log (LEUV)reg, i.e. equation (17), is   \begin{equation} \sigma _{\text{tot}}^{\text{EUV}} = \sqrt{\left(s_{\textrm{reg}}^{\text{EUV}} \right)^{2} + 0.34 \sigma _{\text{tot}}^{2} + 0.71 \langle [\Delta \log (R_{*})]^{2} \rangle + \rho ^{2}}. \end{equation} (25)It follows from this equation that the obtained regression (equations 17–21) predicts the average logarithm of stellar EUV luminosity with a typical error $$\sigma _{\text{tot}}^{\text{EUV}} \approx 0.22$$ dex. Figure 8. View largeDownload slide Distributions of the deviation $$\varepsilon _{\textrm{reg}}^{\textrm{EUV}}$$: (a) in relation to stellar parameters P and Teff and (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. Figure 8. View largeDownload slide Distributions of the deviation $$\varepsilon _{\textrm{reg}}^{\textrm{EUV}}$$: (a) in relation to stellar parameters P and Teff and (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. For testing of this prediction we use the best calibrated and studied case of the solar-type stars. Fig. 9 shows our LEUV(P) prediction for the stars with solar Teff = 5770 K (Allen 1973) in comparison with the relations modelled by Ribas et al. (2005) for the solar analogues with estimated ages. Similarly to Section 5, we transformed the stellar ages to the P-scale using two versions of gyrochronological relation: (a) equations (12)–(14) and Table 10 in Mamajek & Hillenbrand (2008) (dot line) and (b) equation (1) in García et al. (2014) (dashed line). Both curves are mainly inside $$\pm \sigma _{\text{tot}}^{\text{EUV}}$$ confidence interval of the prediction. Finally, the average EUV-flux of the Sun (Ribas et al. 2005), indicated in Fig. 9 with an opened square, sufficiently well corresponds to our solid curve. Figure 9. View largeDownload slide Predicted EUV luminosity versus P is shown as solid curve according to equations (17)–(21) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 15–16) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar EUV luminosity from Ribas et al. (2005). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (pointed line). The standard error $$\sigma _{\text{tot}}^{\text{EUV}}$$ of our prediction is depicted as a bar in the lower left of the diagram. Figure 9. View largeDownload slide Predicted EUV luminosity versus P is shown as solid curve according to equations (17)–(21) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 15–16) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar EUV luminosity from Ribas et al. (2005). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (pointed line). The standard error $$\sigma _{\text{tot}}^{\text{EUV}}$$ of our prediction is depicted as a bar in the lower left of the diagram. 7 CONCLUSIONS Since our activity index $$\langle A_{1}^{2} \rangle$$ gives a realistic prediction for Lx and related LEUV, it may be considered as a practical proxy for the stellar X-ray emission. In contrast with the spectral line indexes (e.g. S, RHK, and $$R^{\prime }_{\text{HK}}$$ in Noyes et al. 1984; Mamajek & Hillenbrand 2008), our approach is based on the optical broad-band photometry. Hence, the index $$\langle A_{1}^{2} \rangle$$ is applicable for more faint and numerous stars. Fig. 10 shows Teff, P-patterns of individual predictions for log (Lx) and log (LEUV), averaged in the same sliding window as in Figs 4 and 8 with the dimensions log (Teff) ± 0.05 and log (P) ± 0.2. One can see the similar bright areas in the both plots at 3.6 ≲ log (Teff) ≲ 3.76 and 0 ≲ log (P) ≲ 0.7. The stars in the corresponding intervals 4000 ≲ Teff ≲ 5800 K and 1 ≲ P ≲ 5  d have the enhanced X-ray and EUV luminosities. Therefore, the exoplanets orbiting such stars should experience higher radiative impact that makes of crucial importance the study and an appropriate account of the processes of erosion of upper atmospheres as well as their related magnetospheric features (Khodachenko et al. 2012, 2015; Shaikhislamov et al. 2016). Figure 10. View largeDownload slide Average predictions for (a) log (Lx) using equations (6)–(9) and (b) log (LEUV) using equations (15) and (16). Both plots depict the estimates for individual stars, averaged in the sliding window. Figure 10. View largeDownload slide Average predictions for (a) log (Lx) using equations (6)–(9) and (b) log (LEUV) using equations (15) and (16). Both plots depict the estimates for individual stars, averaged in the sliding window. The obtained regressions (equations 10 and 17) allow characterizing of X/EUV radiation at the distant objects below the sensitivity thresholds of X-ray detectors. This opens the way for statistical studies of exoplanetary environments as well as for exobiological applications. For example, the X/EUV radiation is a crucial factor for (pre)biological evolution and interplanetary panspermia. In summary, the approach we have developed using starspot variability seems to be a useful tool for a broad range of astrophysical studies. Acknowledgements This work was performed as a part of the projects P25587-N27 and S11606-N16 of the Fonds zur Förderung der wissenschaftlichen Forschung, FWF. The authors also acknowledge the FWF projects S11601-N16, S11604-N16, S11607-N16, and I2939-N27. MK was partially supported by Ministry of Education and Science of Russian Federation Grant RFMEFI61617X0084.TL acknowledges also funding via the Austrian Space Application Programme (ASAP) of the Austrian Research Promotion Agency (FFG) within ASAP11. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. Footnotes 1 https://exoplanetarchive.ipac.caltech.edu/ REFERENCES Allen C. W., 1973, Astrophysical Quantities, 3rd ed . Univ. London, Athlone Press, London Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. P., 2015a, A&A , 576, A67 CrossRef Search ADS   Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. P., 2015b, ApJ , 807, 109 CrossRef Search ADS   Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. 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B., Arras P., Li Z.-Y., 2011, ApJ , 728, 152 CrossRef Search ADS   Wagner W. J., 1988, Adv. Space Res. , 8, 67 CrossRef Search ADS   Winter L. M., Pernak R. L., Balasubramaniam K. S., 2016, Sol. Phys. , 291, 3011 CrossRef Search ADS   Wood B. E., Müller H.-R., Redfield S., Edelman E., 2014, ApJ , 781, L33 CrossRef Search ADS   Wright N. J., Drake J. J., Mamajek E. E., Henry G. W., 2011, ApJ , 743, 48 CrossRef Search ADS   SUPPORTING INFORMATION Supplementary data are available at MNRAS online. Table 1. The analysed stellar set, applied parameters, and predictions. Table 3. Parameters and predictions for stars in the catalogue by Wright et al. (2011). Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. © The Author(s) 2018. Published by Oxford University Press on behalf of The Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press

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Abstract

Abstract Stellar X-ray emission plays an important role in the study of exoplanets as a proxy for stellar winds and as a basis for the prediction of extreme ultraviolet (EUV) flux, unavailable for direct measurements, which in their turn are important factors for the mass-loss of planetary atmospheres. Unfortunately, the detection thresholds limit the number of stars with the directly measured X-ray fluxes. At the same time, the known connection between the sunspots and X-ray sources allows using of the starspot variability as an accessible proxy for the stellar X-ray emission. To realize this approach, we analysed the light curves of 1729 main-sequence stars with rotation periods 0.5 < P < 30  d and effective temperatures 3236 < Teff < 7166 K observed by the Kepler mission. It was found that the squared amplitude of the first rotational harmonic of a stellar light curve may be used as a kind of activity index. This averaged index revealed practically the same relation with the Rossby number as that in the case of the X-ray to bolometric luminosity ratio Rx. As a result, the regressions for stellar X-ray luminosity Lx(P, Teff) and its related EUV analogue LEUV were obtained for the main-sequence stars. It was shown that these regressions allow prediction of average (over the considered stars) values of log (Lx) and log (LEUV) with typical errors of 0.26 and 0.22 dex, respectively. This, however, does not include the activity variations in particular stars related to their individual magnetic activity cycles. stars: activity, starspots, ultraviolet: stars, X-rays: stars 1 INTRODUCTION The stellar X-ray luminosity is widely used as an index of stellar magnetic activity (Wright et al. 2011). Moreover, the X-ray surface flux is considered as a proxy of mass-loss rate (i.e. stellar wind) for main-sequence stars (Wood et al. 2014). In addition, the stellar X-ray emission plays an important role in exoplanetary science as a basis for prediction of extreme UV flux, which is a crucial impacting factor for the planetary ionospheres and upper atmosphere mass-loss (Shaikhislamov et al. 2014, 2016; Khodachenko et al. 2015, 2017). Unfortunately, only 32.9 per cent of known stars at distances up to ∼25 pc from the Sun were detected as X-ray sources (Hünsch et al. 1999). This problem is especially obvious for more distant stars, where the Kepler mission found most exoplanets. Such stars with undetected X-ray fluxes need a kind of accessible X-ray proxy. Apparently, the stellar starspot variability at optical wavelengths could give such a proxy. For example, solar X-ray emission is associated with active regions (Wagner 1988), hence, with sunspots. Correspondingly, there are high (>0.95) linear correlations between the sunspot number or the total spot area and the monthly averaged solar X-ray background flux (Ramesh & Rohini 2008). An analogous relation was found in other main-sequence stars using spot-induced brightness variations (Messina et al. 2003). To show this, Messina et al. (2003) used the maximum amplitude (Amax) of rotational variations of the stellar light curve. It has been shown that this simple activity index is non-linearly related with X-ray luminosity $$L_{x}/L \propto A_{\rm max}^{b}$$, where b ≈ 2, and L is the stellar bolometric luminosity. In our previous studies, we proposed and advocated another activity index – the squared amplitude $$A_{1}^{2}$$ of the first (fundamental) rotational harmonics of the stellar light curve (Arkhypov et al. 2015a,b, 2016, 2018). Its major advantage consists in the statistical proportionality to the starspot number Ns (see arguments in Arkhypov et al. 2016), and hence, the presumable proportionality to the X-ray emission: $$L_{x}/L \propto A_{1}^{2}$$. In fact, the analogous proportionality $$L_{x}/L \propto A_{\rm max}^{2}$$ was reported by Messina et al. (2003). However, Winter, Pernak & Balasubramaniam (2016) found that the solar X-ray flux $$F_{x} \propto N_{s}^{\beta }$$, where 1.61 ≲ β ≲ 1.86, i.e. somewhat higher than the expected β ≈ 1. This might mean that besides the spot number Ns, the X-ray radiation is also related to another factor, which is probably the total spot area. Since the integral photometry index $$A_{1}^{2}$$ depends on both the starspot number and the spot area, its relation with Fx could be closer to a linear one. Below we test this expectation. We use the empirical relation between Lx/L and $$A_{1}^{2}$$ to predict the stellar X-ray luminosity Lx(P, Teff) as a function of stellar rotation period P and effective temperature Teff. This could be useful for the modelling of exoplanetary environments’ evolution including the systems of non-solar-like stars. Hitherto the atmosphere-magnetosphere modelling is mainly based on the assumption of a solar-like X-ray emission (e.g. Trammell, Arras & Li 2011; Koskinen et al. 2013, Shaikhislamov et al. 2014, 2016; Khodachenko et al. 2015, 2017). However, many of the Kepler stars are distant objects of non-solar type with an enhanced luminosity, whereas the majority of the stellar population is composed of the faint red dwarfs. This makes the importance of the characterization of X-ray luminosity for a broader, than just solar type, class of stars. In Section 2, we outline our approach using an extended stellar sample and our time-tested processing method (Arkhypov et. al. 2015a,b, 2016, 2018). In Section 3, we justify that the used activity index can play a role of X-ray proxy. As a result, we obtain in Section 4 an approximating regression for Lx(P, Teff), which is tested in comparison to observations in Sections 5. The related extreme ultraviolet (EUV) luminosity of stars is considered in Section 6. Section 7 summarizes the obtained results and their applicability. 2 STELLAR SET AND LIGHT-CURVE PROCESSING As in our previous studies (Arkhypov et al. 2015a,b, 2016, 2018), we consider here the squared amplitude $$A_{1}^{2}$$ of the light curve's first (fundamental) harmonic with period P of stellar rotation. This squared amplitude is taken because of its suspected and confirmed statistical proportionality to the number of starspots, at least for the solar-like stars (Arkhypov et al. 2016). Moreover, the amplitude of the first harmonic A1, in differ to the light-curve variance, is practically insensitive to the photon noise. In contrast with other rotational harmonics, it depends minimally on flares and short-period pulsations. Note that since the light integration over the stellar disc reduces the amplitude of harmonics progressively with the increasing harmonic number, the value of A1 can be measured with the maximal accuracy. To calculate the activity index $$A_{1}^{2}$$ with a maximal time resolution, while focusing on the rotational variability of a star, we divided the stellar light curve on to consecutive fragments that have durations of one stellar rotation period P each and removed contaminating signals/features such as sporadic flares and linear trend, as well as performed interpolation of the light curve in short (<0.2 P) gaps (see details in Arkhypov et al. 2015a,b, 2016, 2018). The standard Fourier analysis applied is prepared in such a way that one-period fragments of the light curve give the varying (from one fragment to another) index $$A_{1}^{2}$$ . The aforementioned expectations regarding the proportionality between the X-ray emission and the measured parameter $$A_{1}^{2}$$ are tested below for the main-sequence stars. For this purpose, we extend the previously analysed data set (Arkhypov et al. 2016), which contains the light curves of the main-sequence stars observed by the Kepler space observatory, to include the light curves of an additional set of 637 slow rotators (Arkhypov et al. 2018). In Fig. 1(a), the analysed extended sample (see the electronic version of Table 1) that includes 1729 stars with 0.5 < P < 30  d (according to measurements in Nielsen et al. 2013; McQuillan, Mazeh & Aigrain 2014) and the effective temperatures 3236 ≤ Teff ≤ 7166 K from the last version of the Kepler stellar properties catalogue (Mathur et al. 2017) is shown. Fig. 1(b) indicates that all stars, selected from Mathur et al. (2017), with surface gravity log [g(cm s−2)] > 4.2, belong to the main sequence. The availability of a high-quality light curve (Fig. 2) without perceptible interferences (i.e. no detectable short-period pulsations or double periodicity from companions) was a special criterion for compiling of the analysed sample of stars. Further details on the selection of stars and preparing of light curves (i.e. removing of gaps, flares, artefacts, trends), and further processing are described in Arkhypov et al. (2015b, 2016). Figure 1. View largeDownload slide Stellar set properties: (a) the distribution of 1729 selected stars over the effective temperature Teff (Mathur et al. 2017) and the rotation periods P (Nielsen et al. 2013; McQuillan et al. 2014) and (b) the HR-diagram with respect to surface gravity log (g) versus Teff analogous to Mathur et al. (2017). Figure 1. View largeDownload slide Stellar set properties: (a) the distribution of 1729 selected stars over the effective temperature Teff (Mathur et al. 2017) and the rotation periods P (Nielsen et al. 2013; McQuillan et al. 2014) and (b) the HR-diagram with respect to surface gravity log (g) versus Teff analogous to Mathur et al. (2017). Figure 2. View largeDownload slide Typical light curve (fragment) with a starspot variability from the Kepler archive (KIC 2283703). The time BJD here is the differential barycentric Julian days, counted from the mission starting time. The plot was prepared using NASA Exoplanet Archives service (http://exoplanetarchive.ipac.caltech.edu/). Figure 2. View largeDownload slide Typical light curve (fragment) with a starspot variability from the Kepler archive (KIC 2283703). The time BJD here is the differential barycentric Julian days, counted from the mission starting time. The plot was prepared using NASA Exoplanet Archives service (http://exoplanetarchive.ipac.caltech.edu/). Table 1. The analysed stellar set, applied parameters, and predictions (sampled from the whole electronic version). KIC  Teffa  Pb  $$\log (\langle A_{1}^{2} \rangle ){}^{\textrm{c}}$$  σd  (B − V)oe  τMLTf  log (Rx)regg  log (Lx)regh  number  (K)  (days)        (days)    (erg s−1)  892834  4940  13.765  − 4.94  0.04  0.96  22.60  − 4.95  28.29  1026146  4419  14.891  − 5.24  0.04  1.21  24.62  − 4.94  27.82  1160947  4583  25.155  − 5.51  0.06  1.12  23.95  − 5.59  27.60  1161620  5977  6.636  − 5.16  0.03  0.57  7.60  − 5.37  28.62  1162635  3758  15.678  − 5.57  0.04  1.62  28.09  − 4.85  27.41  1163579  5721  5.429  − 4.98  0.03  0.65  12.27  − 4.57  29.40  1293993  4996  21.216  − 6.00  0.05  0.93  22.33  − 5.47  27.87  1295069  5418  17.187  − 5.64  0.05  0.76  17.85  − 5.49  28.37  KIC  Teffa  Pb  $$\log (\langle A_{1}^{2} \rangle ){}^{\textrm{c}}$$  σd  (B − V)oe  τMLTf  log (Rx)regg  log (Lx)regh  number  (K)  (days)        (days)    (erg s−1)  892834  4940  13.765  − 4.94  0.04  0.96  22.60  − 4.95  28.29  1026146  4419  14.891  − 5.24  0.04  1.21  24.62  − 4.94  27.82  1160947  4583  25.155  − 5.51  0.06  1.12  23.95  − 5.59  27.60  1161620  5977  6.636  − 5.16  0.03  0.57  7.60  − 5.37  28.62  1162635  3758  15.678  − 5.57  0.04  1.62  28.09  − 4.85  27.41  1163579  5721  5.429  − 4.98  0.03  0.65  12.27  − 4.57  29.40  1293993  4996  21.216  − 6.00  0.05  0.93  22.33  − 5.47  27.87  1295069  5418  17.187  − 5.64  0.05  0.76  17.85  − 5.49  28.37  Notes.aStellar effective temperature from Mathur et al. (2017). bRotation period from McQuillan et al. (2014) or Nielsen et al. (2013) if absent in the first source. cOur active index, averaged over a whole light curve. dEstimated error of $$\log (\langle A_{1}^{2} \rangle )$$. eColour index according to the transformation Teff → (B − V)o in equation (1). f Turnover time according to equation (4) in Noyes et al. (1984). g Logarithm of the ratio of X-ray to bolometric luminosities which are estimated from equations (6)–(8). hLogarithm of the predicted X-ray luminosity using equations (10)–(14). View Large We analyse the rotational modulation of the stellar radiation flux F (PDCSAP_FLUX from the Kepler mission archive1), which reflects the longitudinal distribution of spots. It has been shown in Arkhypov et al. (2016) that the squared amplitude $$A_{1}^{2}$$ of the first (or fundamental) Fourier harmonic with period P of the stellar one-period light curve is proportional to a spot number. Following this, we used the value $$A_{1}^{2}$$ as an activity index. To exclude the temporal oscillations of stellar activity due to magnetic cycles, we measured a value for $$\langle A_{1}^{2} \rangle$$ that was then averaged over the whole light-curve duration for every star in our set. 3 JUSTIFICATION OF THE OPTICAL PROXY FOR X-RAY EMISSION Because of the absence of common objects in the analysed stellar sample and in the most complete nowadays catalogue of estimates of X-ray ratio Rx ≡ Lx/L by Wright et al. (2011), the direct comparison between Rx and the value $$\langle A_{1}^{2} \rangle$$ is impossible . Since the ratio Rx is commonly considered as a function of the Rossby number Ro ≡ P/τMLT, where τMLT is the turnover time in the mixing length theory (Wright et al. 2011 and therein), instead of direct comparison of the activity indexes Rx and $$\langle A_{1}^{2} \rangle$$ themselves, we compare their dependencies on Ro. Following the arguments by Mamajek & Hillenbrand (2008), we use in this study the classical version of τMLT from Noyes et al. (1984). This approach is valid for Teff ≳ 4000 K corresponding to the colour index (B − V)o < 1.4 of stars used in Noyes et al. (1984). An alternative turnover time approximation by Wright et al. (2011), assuming the linear relation between log (τMLT) and the colour index V − Ks, apparently overestimates τMLT for the stars hotter than the Sun, because theoretically log (τMLT) → −∞ in vanishing convection zones for Teff ≈ 8200 K (Simon et al. 2002), i.e. for V − Ks ∼ 0.5. Equation (11) in Wright et al. (2011), which describes the relation between log (τMLT) and stellar mass, gives an unrealistic independence of the stellar activity on Ro > 1, when the mass estimates for the Kepler stars (Mathur et al. 2017) are used. Another argument for using the mixing time τMLT from Noyes et al. (1984) is provided in Section 5. Since τMLT in Noyes et al. (1984) is defined via the colour index (B − V)o, we use equation (26) from Arkhypov et al. (2016) for the transformation Teff → (B − V)o to make use of the data from the X-ray catalogue by Wright et al. (2011). Analogously, for the analysed set of Kepler Input Catalog (KIC) stars with a slightly different temperature scale, we found the following regression using the bright stars from SIMBAD data base considered in our previous study (Arkhypov et al. 2016):  \begin{equation} (B-V)_{0} = 0.6369 Y^{3} -2.262 Y^{2} -14.38 Y + 52.84, \end{equation} (1)were Y = log (Teff/1K). Since the index $$\langle A_{1}^{2} \rangle$$ (as well as Amax in Messina et al. 2003) depends on a random inclination angle between the stellar rotation axis and the direction to observer, we use the mean activity level $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, defined by averaging (denoted with an upper bar) over many stars with similar parameters Teff and P (for justification see in Section 4.3 by Arkhypov et al. 2016). Fig. 3(a) shows the dependence of stellar activity index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, averaged over star groups in the Rossby number bins log (P/τMLT). The estimates of $$\log (\langle A_{1}^{2} \rangle )$$ for individual stars were used to calculate the converging regression lines for both saturated and non-saturated activity regions with respect to Ro   \begin{equation} \overline{\log (\langle A_{1}^{2} \rangle )} \approx \gamma _{1,2} \log \left(\frac{P}{\tau _{\rm MLT}} \right) + \delta _{1,2} \equiv \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}, \end{equation} (2)where γ1 = −0.00 ± 0.18 and δ1 = −3.99 ± 0.23 are the fitted coefficients for log (P/τMLT) < log (Rosat) = −0.89, whereas γ2 = −1.86 ± 0.04 and δ2 = −5.63 ± 0.02 correspond to the case of log (P/τMLT) > log (Rosat) = −0.89. The saturation Rossby number Rosat = 10−0.89 = 0.13, at which both regressions converge, is the same as one found in Wright et al. (2011). Figure 3. View largeDownload slide Dependence of stellar activity on the Rossby number P/τMLT. (a) Averaged index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ for the star groups in log (P/τMLT) bins, (b) Normalized X-ray luminosity Rx from Wright et al. (2011), and (c) Comparison of the regressions depicted as the lines in (a) and (b). The dashed line corresponds to equality of the predictions. The crosses are the standard error bars ±σreg (see equation 5) of the regressions for equidistantly selected values of the Rossby number. Figure 3. View largeDownload slide Dependence of stellar activity on the Rossby number P/τMLT. (a) Averaged index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ for the star groups in log (P/τMLT) bins, (b) Normalized X-ray luminosity Rx from Wright et al. (2011), and (c) Comparison of the regressions depicted as the lines in (a) and (b). The dashed line corresponds to equality of the predictions. The crosses are the standard error bars ±σreg (see equation 5) of the regressions for equidistantly selected values of the Rossby number. Using Rx ≡ Lx/L as the activity index from the stellar catalogue by Wright et al. (2011), we found the similar pattern in Fig. 3(b) with the following regression:   \begin{equation} \log (R_{x}) \approx \gamma _{1,2}^{x} \log \left(\frac{P}{\tau _{\rm MLT}} \right) + \delta _{1,2}^{x} \equiv \log (R_{x})_{\textrm{reg}}, \end{equation} (3)with close to equation (8) parameters: $$\gamma _{1}^{x}=-0.13 \pm 0.04$$ and $$\delta _{1}^{x}=-3.33 \pm 0.06$$ in the saturation regime and $$\gamma _{2}^{x}=-2.04 \pm 0.08$$ and $$\delta _{2}^{x}=-4.85 \pm 0.04$$ in the non-saturated case. The similarities $$\gamma _{1} \simeq \gamma _{1}^{x}$$, $$\gamma _{2} \approx \gamma _{2}^{x}$$, $$\delta _{1} \sim \delta _{1}^{x}$$, and $$\delta _{2} \sim \delta _{2}^{x}$$ are understandable, because the X-ray emission is associated with the active regions, i.e. with the starspots. In particular, the solar X-ray flux correlates with both sunspot number and their total area (Ramesh & Rohini 2008), which control our index $$A_{1}^{2}$$. Fig. 3(c) shows the comparison between the regressions’ prediction for both activity indexes in the region of measured log (Rx). After the correction $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}-\Delta$$ with an average difference Δ = −0.82 from log (Rx)reg, we arrive at the approximation   \begin{equation} \log (R_{x})_{\textrm{reg}} \approx \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}+0.82. \end{equation} (4)The slight deviations (≲0.15 dex) from the equality $$\log (R_{x})_{\textrm{reg}} = \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}+0.82$$ are consistent with the regression standard errors which are   \begin{equation} \sigma _{\textrm{reg}} = \sqrt{\sigma _{\gamma }^{2}[\log (\text{Ro})]^{2} + \sigma _{\delta }^{2}}, \end{equation} (5)where Ro = P/τMLT is the Rossby number, and the standard errors of the regression coefficients are as follows: σγ = 0.18 and σδ = 0.23 for saturated $$\langle A_{1}^{2} \rangle$$; σγ = 0.04 and σδ = 0.02 for unsaturated $$\langle A_{1}^{2} \rangle$$; σγ = 0.04 and σδ = 0.06 for saturated Rx; and σγ = 0.08 and σδ = 0.04 for unsaturated Rx. The corresponding error bars are shown in Fig. 2(c) for the equidistantly selected vales of log (Ro). The approximate relation $$\log (R_{x})_{\textrm{reg}} \approx \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}+0.82$$ means that the regression $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$ and the related activity index $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ can be used as a quasi-proportional proxy for log (Rx). 4 FROM THE OPTICAL PROXY TO THE PREDICTION OF STELLAR X-RAY LUMINOSITY Here, we apply the optical proxy approach to predict stellar average X-ray luminosity, i.e. obtain the regression Lx(P, Teff). Theoretically, using equation (4) and the approximation $$\overline{\log (\langle A_{1}^{2} \rangle )} \approx \log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$, following equation (2), the proxy $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$ can be transformed to Rx. However, in practice, an empirical coefficient K should be included in this transformation to take into account a selection effect, i.e. the ignoring of stars with undetected X-ray radiation or low-amplitude light curves, so that   \begin{equation} \log (R_{x})_{\textrm{reg}} \approx \overline{\log (\langle A_{1}^{2} \rangle )}+\log (K), \end{equation} (6)where   \begin{equation} \log (K)=\log \left(R_{x}^{\text{unb}} \right)-\log \left(\left\langle A_{1}^{2} \right\rangle \right)_{\textrm{reg}}. \end{equation} (7)Here, $$R_{x}^{\text{unb}}$$ is the regression for Rx, which was obtained in Wright et al. (2011) using the ‘unbiased sample’ of stars   \begin{equation} \log (R_{x}^{\text{unb}}) \approx \gamma _{1,2}^{\text{unb}} \log \left(\frac{P}{\tau _{\rm MLT}} \right) + \delta _{1,2}^{\text{unb}}, \end{equation} (8)where $$\gamma _{1}^{\text{unb}}=0$$ (assumed) and $$\delta _{1}^{\text{unb}}=-3.13 \pm 0.08$$ are the fitting coefficients for log (Ro) < log (Rosat) = −0.89, whereas $$\gamma _{2}^{\text{unb}}=-2.70 \pm 0.13$$ and $$\delta _{2}^{\text{unb}}=\delta _{1}^{\text{unb}}-\gamma _{2}^{\text{unb}}\log (\text{Ro}_{\text{sat}})=-5.53 \pm 0.14$$ correspond to the case of log (Ro) > log (Rosat) = −0.89. It follows from the definition Rx ≡ Lx/L that in order to estimate Lx, one needs a regression for the bolometric luminosity L(Teff). While the majority of astrophysical studies were focused on the universal mass-luminosity relation, we base our derivation of the needed relation on the apparently most complete compilation of the published values for L and Teff for main-sequence stars (Eker et al. 2015):   \begin{equation} \log (L)=a_{L}Y^{3}+b_{L}Y^{2}+c_{L}Y+d_{L}, \end{equation} (9)where L is the luminosity in solar units, Y = log (Teff/1K), and the fitting coefficients are aL = 3.801, bL = −47.396, cL = 202.329, and dL = −292.539. The standard deviation is εL = 0.20 dex for all used stars (265 objects excluding 31 stars with problematic main-sequence status and >3σ outliers). Higher polynomial power has practically no decreasing effect on εL. Using equations (6)– (9), we calculate Lx = RxL for every star from our set. In fact, we replace in equation (6) the averaged $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ with the values $$\log (\langle A_{1}^{2} \rangle )$$ for individual stars. This substitution is justified, since the further calculation of the regression Lx(P, Teff)reg is equivalent to an averaging over stars. This regression has the following form:   \begin{equation} \log (L_{x}) \approx a_{x0}X^{3}+b_{x0}X^{2}+c_{x0}X+d_{x0} \equiv \log (L_{x})_{\textrm{reg}}, \end{equation} (10)  \begin{equation} a_{x0}=a_{x1}Y^{3}+b_{x1}Y^{2}+c_{x1}Y+d_{x1}, \end{equation} (11)  \begin{equation} b_{x0}=a_{x2}Y^{3}+b_{x2}Y^{2}+c_{x2}Y+d_{x2}, \end{equation} (12)  \begin{equation} c_{x0}=a_{x3}Y^{3}+b_{x3}Y^{2}+c_{x3}Y+d_{x3}, \end{equation} (13)  \begin{equation} d_{x0}=a_{x4}Y^{3}+b_{x4}Y^{2}+c_{x4}Y+d_{x4}, \end{equation} (14)where X = log (P/1 d), Y = log (Teff/1K), and the fitting coefficients axi, bxi, cxi, dxi, found with the least square method, are listed in Table 2. Table 2. Fitting coefficients in equations (11)–(14). Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 17.32  73.48  156.7  − 707.0    2  16.31  − 182.6  675.1  − 827.6    3  13.70  124.5  −1449  2966    4  0.4856  − 108.9  781.9  − 1397  >5500 K  1  − 0.8916  31.44  − 188.9  313.9    2  18.31  − 111.14  72.19  323.6    3  − 5.046  − 116.7  1055  −2047    4  − 36.21  126.4  578.5  −2007  Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 17.32  73.48  156.7  − 707.0    2  16.31  − 182.6  675.1  − 827.6    3  13.70  124.5  −1449  2966    4  0.4856  − 108.9  781.9  − 1397  >5500 K  1  − 0.8916  31.44  − 188.9  313.9    2  18.31  − 111.14  72.19  323.6    3  − 5.046  − 116.7  1055  −2047    4  − 36.21  126.4  578.5  −2007  View Large Fig. 4 shows the deviation $$\varepsilon _{\textrm{reg}} = \overline{\log (L_{x})} - \log (L_{x})_{\textrm{reg}}$$ between the regression and $$\overline{\log (L_{x})}$$, calculated for individual stars (equations 6– 9) and averaged over the considered sample in a sliding window log (Pc) − 0.2 < log (P) < log (Pc) + 0.2 and log (Tc) − 0.05 < log (Teff) < log (Tc) + 0.05 with the central values of stellar rotation period Pc and effective temperature Tc. On average, 104 (up to 403) individual estimates of Lx appeared within this sliding window. Fig. 4(b) demonstrates a histogram of εreg with the standard deviation sreg = 0.14. Hence, the total standard error of the regression log (Lx)reg can be estimated as a combination of the standard errors of the involved regressions, i.e. $$\sigma _{{\rm tot}}=(\sigma _{\textrm{reg}}^{2}+s_{\textrm{reg}}^{2}+\varepsilon _{L}^{2})^{1/2} \approx 0.26$$, where σreg ∼ 0.1 dex for log (Rx)reg (see equation 5 and Fig. 4c for Rx errors), sreg = 0.14 dex for log (Lx)reg, and εL = 0.20 dex for log (L). Therefore, the derived regression (equations 10–14) predicts the average logarithm of stellar X-ray luminosity with typical error 0.26 dex. Figure 4. View largeDownload slide Distributions of the deviation εreg: (a) in relation to stellar parameters P and Teff; (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. Figure 4. View largeDownload slide Distributions of the deviation εreg: (a) in relation to stellar parameters P and Teff; (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. 5 COMPARISON OF THE PREDICTIONS VERSUS OBSERVATIONS For the verification of our predictions, for all 824 objects from the X-ray catalogue by Wright et al. (2011), considered in this paper, we provide in Table 3 the values of log (Lx)reg, calculated using equations (10)– (14), as well as the observed values of log (Lxw). One can compare these values for different stellar clusters using the associated electronic version of Table 3. However, we would like to note that the regressions for luminosity log (L) and activity index $$\log (\langle A_{1}^{2} \rangle )_{\textrm{reg}}$$ (equations 9 and 2, respectively) were obtained for the main-sequence stars only. Hence, the pre-main-sequence objects in young clusters might appear a wrong example for controlling our estimates. At the same time, the mainly old field stars, which are more numerous than members of any cluster in the used catalogue, are most suitable for the verification of the obtained regressions. Table 3. Parameters and predictions for stars in the catalogue by Wright et al. (2011) (sampled from the complete electronic version). RA 2000a  DE 2000b  Affiliation  Twc  P d  τMLTe  $$\log (R_{x}^{{\rm unb}}){}^{\textrm{f}}$$  log (Lxp)g  log (Lxw)h  (deg.)  (deg.)    (K)  (days)  (days)    (erg s−1)  (erg s−1)  −  −  Sun  5780  26.09  9.97  − 6.66  27.31  27.35  2.02679  47.9507  Field  3201  4.38  32.85  − 3.17  28.84  28.21  2.84350  30.4496  Field  5460  6.05  15.21  − 4.45  29.96  29.14  5.49087  49.2106  Field  3405  6.17  30.69  − 3.65  28.58  28.25  5.71579  − 12.2094  Field  5687  7.78  11.43  − 5.08  29.00  29.15  7.22467  50.3758  Field  3163  1.09  33.30  − 3.13  27.25  28.29  9.14317  55.6267  Field  3654  8.35  28.54  − 4.09  28.17  28.61  9.50008  43.8959  Field  3619  0.55  28.82  − 3.13  28.23  29.02  RA 2000a  DE 2000b  Affiliation  Twc  P d  τMLTe  $$\log (R_{x}^{{\rm unb}}){}^{\textrm{f}}$$  log (Lxp)g  log (Lxw)h  (deg.)  (deg.)    (K)  (days)  (days)    (erg s−1)  (erg s−1)  −  −  Sun  5780  26.09  9.97  − 6.66  27.31  27.35  2.02679  47.9507  Field  3201  4.38  32.85  − 3.17  28.84  28.21  2.84350  30.4496  Field  5460  6.05  15.21  − 4.45  29.96  29.14  5.49087  49.2106  Field  3405  6.17  30.69  − 3.65  28.58  28.25  5.71579  − 12.2094  Field  5687  7.78  11.43  − 5.08  29.00  29.15  7.22467  50.3758  Field  3163  1.09  33.30  − 3.13  27.25  28.29  9.14317  55.6267  Field  3654  8.35  28.54  − 4.09  28.17  28.61  9.50008  43.8959  Field  3619  0.55  28.82  − 3.13  28.23  29.02  Notes.aRight ascension for epoch 2000.0 from Wright et al. (2011). bDeclination for epoch 2000.0 from Wright et al. (2011). cEffective temperature from Wright et al. (2011). dFrom Wright et al. (2011). eTurnover time according to equation (4) in Noyes et al. (1984) and transform Tw → (B − V)o (equation 26 in Arkhypov et al. 2016). fFrom equations (6)– (8); gLogarithm of the predicted X-ray luminosity using equations (10)–(14).  hLogarithm of the measured X-ray luminosity from Wright et al. (2011). View Large Fig. 5 shows the comparison of the predicted X-ray luminosity log (Lxp) = log (Lx)reg (see equation 10) and the observed luminosity log (Lxw) for 443 field stars from the catalogue by Wright et al. (2011). We are focused on the field stars because of the limited and distorted X-ray statistics in more distant stellar clusters, where many of the detections are just above the detection threshold. In Fig. 5(a), one can see the general agreement between the predicted and observed stellar distributions in relation with Teff. However, Fig. 5(b) reveals that some stars show log (Lxw) ≫ log (Lxp) at log (Lxp) ≲ 27 erg s−1. This effect disappears when the faint stars with Teff < 4000 K are omitted in Fig. 5(c). Fig. 5(d) demonstrates that the stars with Teff < 4000 show a clear cut-off of the observed X-ray flux Fxw at the detection threshold of ∼10−13 ergs s−1 cm−2. The stars with Fxw above this threshold are seen in Fig. 5(b) as a specific population of objects with log (Lxw) ≫ log (Lxp). However, at Teff > 4000 K (Fig. 5c), the X-ray luminosity values for the considered stars are clustered along the equality line log (Lxw) = log (Lxp) with a negligible average difference 〈log (Lxw) − log (Lxp)〉 = 0.04 ± 0.04, and the standard deviation of log (Lxw) − log (Lxp) for an individual star is sind = 0.60 dex. Apparently, the Lx variability in time gives the main contribution to sind. Figure 5. View largeDownload slide Comparison of the predicted X-ray luminosity log (Lxp) = log (Lx)reg (diamonds according to equation 10) and the observed luminosity log (Lxw) (filled squares) for the field stars from the catalogue by Wright et al. (2011). (a) The estimates’ distribution in relation with Teff, (b) the cross comparison of the all estimates in (a), (c) the same cross comparison as in (b) but only for stars with Teff > 4000 K, and (d) the comparison of observed Fxw and predicted Fxp X-ray fluxes at the Earth for stars with Teff < 4000 K. The lines depict equalities of the abscissa and ordinate values. Figure 5. View largeDownload slide Comparison of the predicted X-ray luminosity log (Lxp) = log (Lx)reg (diamonds according to equation 10) and the observed luminosity log (Lxw) (filled squares) for the field stars from the catalogue by Wright et al. (2011). (a) The estimates’ distribution in relation with Teff, (b) the cross comparison of the all estimates in (a), (c) the same cross comparison as in (b) but only for stars with Teff > 4000 K, and (d) the comparison of observed Fxw and predicted Fxp X-ray fluxes at the Earth for stars with Teff < 4000 K. The lines depict equalities of the abscissa and ordinate values. In Fig. 6, we test an alternative possible explanation of the aforementioned deviations at log (Lxw) ≫ log (Lxp) in Fig. 5(b) as a result of underestimated τMLT for the red dwarfs outside of the temperature region, for which the used approximation of τMLT was found (Noyes et al. 1984). To do that, we calculated the average values $$\overline{\log (\langle A_{1}^{2} \rangle )}$$ for different Rossby numbers Ro = P/τMLT in two temperature domains Teff > 4000 K and Teff < 4000 K using the different definition versions of τMLT according to Noyes et al. (1984) and equation (10) in Wright et al. (2011), respectively. One can see in Fig. 6(a) that τMLT from Noyes et al. (1984) gives the unified sequence of estimates that are independent on the temperature domain. However, the increased τMLT from Wright et al. (2011) shifts the low-temperature points towards the lower Ro, destroying the similarity between the estimates in Fig. 6(b). Since the temperature independence of the activity-Ro relation is a commonly accepted fact for the estimation of τMLT (e.g. Noyes et al. 1984), Fig. 6 argues for the validity of Noyes’ version of τMLT also at 3300 ≲ Teff < 4000 K. Consequently, the assumption of an increased mixing time τMLT cannot be used for the explanation of the extension of stellar population with log (Lxw) > log (Lxp) in Fig. 5(b). Figure 6. View largeDownload slide Values $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, averaged in the bins of Ro = P/τMLT, for two temperature domains Teff < 4000 K (black squares) and Teff > 4000 K (crosses) with (a) τMLT defined as in Noyes et al. (1984), for all stars and (b) τMLT, defined according to equation (10) in Wright et al. (2011), for the stars with Teff < 4000 K. Figure 6. View largeDownload slide Values $$\overline{\log (\langle A_{1}^{2} \rangle )}$$, averaged in the bins of Ro = P/τMLT, for two temperature domains Teff < 4000 K (black squares) and Teff > 4000 K (crosses) with (a) τMLT defined as in Noyes et al. (1984), for all stars and (b) τMLT, defined according to equation (10) in Wright et al. (2011), for the stars with Teff < 4000 K. For another test of our prediction of the X-ray luminosity, we use the best calibrated and studied case of the solar-type stars. Fig. 7 shows our Lx(P) prediction for the stars with solar Teff = 5770 K (Allen 1973) in comparison with the relations obtained by other authors using independent methods. For example, Mamajek & Hillenbrand (2008) used Ca II H and K emission index as a kind of X-ray proxy. Their regressions (A3), (12)–(14) and Table 10 at the solar colour index B − V = 0.65 (Allen 1973) are shown as a long-dashed curve, which fits sufficiently well with our prediction (solid curve) mainly within its standard error ±σtot dex (see Section 4). Ribas et al. (2005) considered the directly measured X-ray flux from solar analogues with estimated ages. Here, we transformed the stellar ages to the P-scale using two versions of gyrochronological relation: (a) the cited above equations (12)– (14) and Table 10 in Mamajek & Hillenbrand (2008) (dotted line), and (b) equation (1) in García et al. (2014) (dashed line). Both curves fit well with our prediction. Finally, the average X-ray flux of the Sun (Wright et al. 2011), indicated in Fig. 7 with an opened square, coincides with our solid curve. Figure 7. View largeDownload slide Predicted X-ray luminosity versus P is shown as solid curve according to equations (10–14) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 6– 9) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar X-ray luminosity from the catalogue by Wright et al. (2011). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (dotted line). The long-dashed line corresponds to the regressions in Mamajek & Hillenbrand (2008). The standard error σtot of our prediction is depicted as a left-bottom bar. Figure 7. View largeDownload slide Predicted X-ray luminosity versus P is shown as solid curve according to equations (10–14) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 6– 9) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar X-ray luminosity from the catalogue by Wright et al. (2011). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (dotted line). The long-dashed line corresponds to the regressions in Mamajek & Hillenbrand (2008). The standard error σtot of our prediction is depicted as a left-bottom bar. Considering Figs 5 and 7, one can conclude that the obtained regression Lx(P, Teff), defined by equations (10)–(14), generates plausible predictions at least at Teff ≳ 3500 K. Since the reference X-ray catalogue by Wright et al. (2011) is converted to the ROSAT wavelength band, our regression approximates Lx in the same band over the range from 6 to 124 Å. 6 EUV APPLICATION We demonstrate below the application potential of the proposed X-ray proxy for the prediction of related EUV radiation. The EUV radiation at wavelengths from 124 to 912 Å plays an important role in planetary science as a crucial impacting/heating factor for the upper atmospheres. However, in the case of exoplanets it is unobservable because of significant interstellar extinction. That is why there is a common practice to use the observable X-ray radiation as a proxy for the EUV flux. Apparently, the best results were obtained by Chadney et al. (2015) using the empirical relation   \begin{equation} \log (F_{{\rm EUV}}) = 2.63 + 0.58 \log (F_{x}), \end{equation} (15)where $$F_{x}=L_{x}/(4 \pi R_{*}^{2}) ({\rm mW \, m}^{-2})$$ is the stellar surface flux in the ROSAT band 6–124 Å, and FEUV is the EUV surface flux at 124–912 Å. Correspondingly, the stellar EUV luminosity LEUV can be found using the stellar radius R* from the reference catalogue of KIC stellar data by Mathur et al. (2017)   \begin{equation} \log (L_{{\rm EUV}}) = \log (F_{{\rm EUV}}) + \log \left(4 \pi R_{*}^{2} \right) + 4, \end{equation} (16)where term 4 is added for the unit transformation (mW) → (erg s−1). Using equations (15) and (16) in combination with equations (6)–(9), one can calculate LEUV for every KIC star in our data set. These estimates were used to obtain the regression LEUV(P, Teff) as follows:   \begin{equation} \log (L_{{\rm EUV}}) \approx a_{e0}X^{3}+b_{e0}X^{2}+c_{e0}X+d_{e0} \equiv \log (L_{{\rm EUV}})_{\textrm{reg}},\!\!\!\!\!\! \end{equation} (17)  \begin{equation} a_{e0}=a_{e1}Y^{3}+b_{e1}Y^{2}+c_{e1}Y+d_{e1}, \end{equation} (18)  \begin{equation} b_{e0}=a_{e2}Y^{3}+b_{e2}Y^{2}+c_{e2}Y+d_{e2}, \end{equation} (19)  \begin{equation} c_{e0}=a_{e3}Y^{3}+b_{e3}Y^{2}+c_{e3}Y+d_{e3}, \end{equation} (20)  \begin{equation} d_{e0}=a_{e4}Y^{3}+b_{e4}Y^{2}+c_{e4}Y+d_{e4}, \end{equation} (21)where the fitting coefficients aei, bei, cei, dei, obtained with the least square method, are listed in Table 4. Table 4. Fitting coefficients for equations (17)–(21). Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 9.606  42.81  72.39  − 366.5    2  8.348  − 100.3  394.1  − 510.2    3  8.862  64.78  − 822.6  1708    4  − 1.152  − 55.47  455.7  − 838.7  >5500 K  1  − 0.7564  19.33  − 108.2  174.4    2  10.97  − 65.78  36.82  206.3    3  − 2.667  − 71.48  629.6  −1215    4  − 21.71  75.70  349.5  −1201  Teff  i  axi  bxi  cxi  dxi  <5500 K  1  − 9.606  42.81  72.39  − 366.5    2  8.348  − 100.3  394.1  − 510.2    3  8.862  64.78  − 822.6  1708    4  − 1.152  − 55.47  455.7  − 838.7  >5500 K  1  − 0.7564  19.33  − 108.2  174.4    2  10.97  − 65.78  36.82  206.3    3  − 2.667  − 71.48  629.6  −1215    4  − 21.71  75.70  349.5  −1201  View Large Fig. 8 shows the deviation $$\varepsilon _{\textrm{reg}}^{{\rm EUV}} = \overline{\log (L_{{\rm EUV}})} - \log (L_{{\rm EUV}})_{\textrm{reg}}$$ between the regression and $$\overline{\log (L_{{\rm EUV}})}$$, calculated for individual stars (equations 15– 16) and averaged over the considered sample in a sliding window log (Pc) − 0.2 < log (P) < log (Pc) + 0.2 and log (Tc) − 0.05 < log (Teff) < log (Tc) + 0.05 with the central values of stellar rotation period Pc and effective temperature Tc. On average, 104 (up to 403) individual estimates of LEUV appeared within this sliding window. Fig. 8(b) shows a histogram of $$\varepsilon _{\textrm{reg}}^{{\rm EUV}}$$ with the standard deviation $$s_{\textrm{reg}}^{{\rm EUV}}=0.08$$. The deviation of the regression (17) from the true value log (LEUV)o is   \begin{equation} d_{{\rm tot}}^{{\rm EUV}} \equiv \log (L_{{\rm EUV}})_{o} - \log (L_{{\rm EUV}})_{\textrm{reg}} = \delta _{{\rm reg}} + \Delta , \end{equation} (22)where δreg = log (LEUV) − log (LEUV)reg and Δ = log (LEUV)o − log (LEUV). Here, log (LEUV) is an estimate obtained using equations (15) and (16). Correspondingly, the standard error of the regression (17) is   \begin{equation} \sigma _{{\rm tot}}^{{\rm EUV}} = \sqrt{\left\langle \left(d_{{\rm tot}}^{{\rm EUV}} \right)^{2} \right\rangle }= \sqrt{\left\langle \delta _{{\rm reg}}^{2} \right\rangle + \langle \Delta ^{2} \rangle }, \end{equation} (23)where $$\langle \delta _{{\rm reg}}^{2} \rangle \approx s_{\textrm{reg}}^{\rm {EUV}}=0.08$$, and 〈Δ2〉 can be obtained from equations (15) and (16) by substitution of variables in the form of a sum of average values with index o plus fluctuation marked with Δ, i.e. log (Lx) = log (Lx)o + Δlog (Lx) and log (R*) = log (R*)o + Δlog (R*). Then the following expression can be obtained:   \begin{equation} \langle \Delta ^{2} \rangle = 0.34 \langle [\Delta \log (L_{x})]^{2} \rangle + 0.71 \langle [\Delta \log (R_{*})]^{2} \rangle + \rho ^{2}, \end{equation} (24)where 〈[Δlog (Lx)]2〉 = σtot, 〈[Δlog (R*)]2〉 ∼ 0.1, which corresponds to the typical error ∼ 27 per cent in Mathur et al. (2017), and ρ ∼ 0.1 is the typical uncertainty of prediction with equation (15) caused by the coefficient errors, which was estimated as a typical deviation of stellar estimates from the regression in Fig. 2 in Chadney et al. (2015). In summary, the total standard error of the regression log (LEUV)reg, i.e. equation (17), is   \begin{equation} \sigma _{\text{tot}}^{\text{EUV}} = \sqrt{\left(s_{\textrm{reg}}^{\text{EUV}} \right)^{2} + 0.34 \sigma _{\text{tot}}^{2} + 0.71 \langle [\Delta \log (R_{*})]^{2} \rangle + \rho ^{2}}. \end{equation} (25)It follows from this equation that the obtained regression (equations 17–21) predicts the average logarithm of stellar EUV luminosity with a typical error $$\sigma _{\text{tot}}^{\text{EUV}} \approx 0.22$$ dex. Figure 8. View largeDownload slide Distributions of the deviation $$\varepsilon _{\textrm{reg}}^{\textrm{EUV}}$$: (a) in relation to stellar parameters P and Teff and (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. Figure 8. View largeDownload slide Distributions of the deviation $$\varepsilon _{\textrm{reg}}^{\textrm{EUV}}$$: (a) in relation to stellar parameters P and Teff and (b) in form of a histogram, where the value n is the number of εreg estimates in one bin of the histogram, and N is the total number of such estimates in a sliding window. For testing of this prediction we use the best calibrated and studied case of the solar-type stars. Fig. 9 shows our LEUV(P) prediction for the stars with solar Teff = 5770 K (Allen 1973) in comparison with the relations modelled by Ribas et al. (2005) for the solar analogues with estimated ages. Similarly to Section 5, we transformed the stellar ages to the P-scale using two versions of gyrochronological relation: (a) equations (12)–(14) and Table 10 in Mamajek & Hillenbrand (2008) (dot line) and (b) equation (1) in García et al. (2014) (dashed line). Both curves are mainly inside $$\pm \sigma _{\text{tot}}^{\text{EUV}}$$ confidence interval of the prediction. Finally, the average EUV-flux of the Sun (Ribas et al. 2005), indicated in Fig. 9 with an opened square, sufficiently well corresponds to our solid curve. Figure 9. View largeDownload slide Predicted EUV luminosity versus P is shown as solid curve according to equations (17)–(21) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 15–16) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar EUV luminosity from Ribas et al. (2005). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (pointed line). The standard error $$\sigma _{\text{tot}}^{\text{EUV}}$$ of our prediction is depicted as a bar in the lower left of the diagram. Figure 9. View largeDownload slide Predicted EUV luminosity versus P is shown as solid curve according to equations (17)–(21) for the solar effective temperature (Teff = 5770 K) in comparison with averaged predictions for individual stars (equations 15–16) with 5500 < Teff < 6000 K (solid squares with error bars). Here, the opened square is the average solar EUV luminosity from Ribas et al. (2005). The dependence by Ribas et al. (2005) is transformed in the P-scale using the gyrochronological relation from García et al. (2014) (dashed line) as well as in Mamajek & Hillenbrand (2008) (pointed line). The standard error $$\sigma _{\text{tot}}^{\text{EUV}}$$ of our prediction is depicted as a bar in the lower left of the diagram. 7 CONCLUSIONS Since our activity index $$\langle A_{1}^{2} \rangle$$ gives a realistic prediction for Lx and related LEUV, it may be considered as a practical proxy for the stellar X-ray emission. In contrast with the spectral line indexes (e.g. S, RHK, and $$R^{\prime }_{\text{HK}}$$ in Noyes et al. 1984; Mamajek & Hillenbrand 2008), our approach is based on the optical broad-band photometry. Hence, the index $$\langle A_{1}^{2} \rangle$$ is applicable for more faint and numerous stars. Fig. 10 shows Teff, P-patterns of individual predictions for log (Lx) and log (LEUV), averaged in the same sliding window as in Figs 4 and 8 with the dimensions log (Teff) ± 0.05 and log (P) ± 0.2. One can see the similar bright areas in the both plots at 3.6 ≲ log (Teff) ≲ 3.76 and 0 ≲ log (P) ≲ 0.7. The stars in the corresponding intervals 4000 ≲ Teff ≲ 5800 K and 1 ≲ P ≲ 5  d have the enhanced X-ray and EUV luminosities. Therefore, the exoplanets orbiting such stars should experience higher radiative impact that makes of crucial importance the study and an appropriate account of the processes of erosion of upper atmospheres as well as their related magnetospheric features (Khodachenko et al. 2012, 2015; Shaikhislamov et al. 2016). Figure 10. View largeDownload slide Average predictions for (a) log (Lx) using equations (6)–(9) and (b) log (LEUV) using equations (15) and (16). Both plots depict the estimates for individual stars, averaged in the sliding window. Figure 10. View largeDownload slide Average predictions for (a) log (Lx) using equations (6)–(9) and (b) log (LEUV) using equations (15) and (16). Both plots depict the estimates for individual stars, averaged in the sliding window. The obtained regressions (equations 10 and 17) allow characterizing of X/EUV radiation at the distant objects below the sensitivity thresholds of X-ray detectors. This opens the way for statistical studies of exoplanetary environments as well as for exobiological applications. For example, the X/EUV radiation is a crucial factor for (pre)biological evolution and interplanetary panspermia. In summary, the approach we have developed using starspot variability seems to be a useful tool for a broad range of astrophysical studies. Acknowledgements This work was performed as a part of the projects P25587-N27 and S11606-N16 of the Fonds zur Förderung der wissenschaftlichen Forschung, FWF. The authors also acknowledge the FWF projects S11601-N16, S11604-N16, S11607-N16, and I2939-N27. MK was partially supported by Ministry of Education and Science of Russian Federation Grant RFMEFI61617X0084.TL acknowledges also funding via the Austrian Space Application Programme (ASAP) of the Austrian Research Promotion Agency (FFG) within ASAP11. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. Footnotes 1 https://exoplanetarchive.ipac.caltech.edu/ REFERENCES Allen C. W., 1973, Astrophysical Quantities, 3rd ed . Univ. London, Athlone Press, London Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. P., 2015a, A&A , 576, A67 CrossRef Search ADS   Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. P., 2015b, ApJ , 807, 109 CrossRef Search ADS   Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. 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Published by Oxford University Press on behalf of The Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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