Starspot variability as an X-ray radiation proxy

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 < T eff < 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 R x . As a result, the regressions for stellar X-ray luminosity L x ( P , T eff ) and its related EUV analogue L EUV were obtained for the main-sequence stars. It was shown that these regressions allow prediction of average (over the considered stars) values of log ( L x ) and log ( L EUV ) 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.

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 (A max ) 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 ∝ A b max , 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 2 1 of the first (fundamental) rotational harmonics of the stellar light curve (Arkhypov et al. 2015a(Arkhypov et al. ,b, 2016(Arkhypov et al. , 2018. Its major advantage consists in the statistical proportionality to the starspot number N s (see arguments in Arkhypov et al. 2016), and hence, the presumable proportionality to the X-ray emission: L x /L ∝ A 2 1 . In fact, the analogous proportionality C 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.
Downloaded from https://academic.oup.com/mnras/article-abstract/476/1/1224/4839014 by guest on 28 July 2018 L x /L ∝ A 2 max was reported by Messina et al. (2003). However, Winter, Pernak & Balasubramaniam (2016) found that the solar X-ray flux F x ∝ N β s , where 1.61 β 1.86, i.e. somewhat higher than the expected β ≈ 1. This might mean that besides the spot number N s , the X-ray radiation is also related to another factor, which is probably the total spot area. Since the integral photometry index A 2 1 depends on both the starspot number and the spot area, its relation with F x could be closer to a linear one. Below we test this expectation.
We use the empirical relation between L x /L and A 2 1 to predict the stellar X-ray luminosity L x (P, T eff ) as a function of stellar rotation period P and effective temperature T eff . This could be useful for the modelling of exoplanetary environments' evolution including the systems of non-solar-like stars. Hitherto the atmospheremagnetosphere 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, 2016Khodachenko et al. 2015Khodachenko et al. , 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(Arkhypov et. al. ,b, 2016(Arkhypov et. al. , 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 L x (P, T eff ), 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.

S T E L L A R S E T A N D L I G H T-C U RV E P RO C E S S I N G
As in our previous studies (Arkhypov et al. 2015a(Arkhypov et al. ,b, 2016(Arkhypov et al. , 2018, we consider here the squared amplitude A 2 1 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 A 1 , 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 A 1 can be measured with the maximal accuracy. To calculate the activity index A 2 1 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. 2015adetails in Arkhypov et al. ,b, 2016details in Arkhypov et al. , 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 2 1 . The aforementioned expectations regarding the proportionality between the X-ray emission and the measured parameter A 2 1 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 ob-  (Mathur et al. 2017) and the rotation periods P (Nielsen et al. 2013;McQuillan et al. 2014) and (b) the HRdiagram with respect to surface gravity log (g) versus T eff analogous to Mathur et al. (2017). served 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 ≤ T eff ≤ 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. (2015bArkhypov et al. ( , 2016.  Noyes et al. (1984). g Logarithm of the ratio of X-ray to bolometric luminosities which are estimated from equations (6)-(8). h Logarithm of the predicted X-ray luminosity using equations (10)-(14).

Figure 2.
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/). We analyse the rotational modulation of the stellar radiation flux F (PDCSAP_FLUX from the Kepler mission archive 1 ), which reflects the longitudinal distribution of spots. It has been shown in Arkhypov et al. (2016) that the squared amplitude A 2 1 of the first (or fundamental) Fourier harmonic with period P of the stellar oneperiod light curve is proportional to a spot number. Following this, we used the value A 2 1 as an activity index. To exclude the temporal oscillations of stellar activity due to magnetic cycles, we measured a value for A 2 1 that was then averaged over the whole light-curve duration for every star in our set.

J U S T I F I C AT I O N O F T H E O P T I C A L P ROX Y F O R X -R AY E M I S S I O N
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 R x ≡ L x /L by Wright et al. (2011), the direct comparison between R x and the value A 2 1 is impossible . Since the ratio R x is commonly considered as a function of the Rossby number Ro ≡ P/τ MLT , where τ MLT is the turnover time in the mixing length theory (Wright et al. 2011 and therein), instead of direct comparison of the activity indexes R x and A 2 1 themselves, we compare their 1 https://exoplanetarchive.ipac.caltech.edu/ dependencies on Ro. Following the arguments by Mamajek & Hillenbrand (2008), we use in this study the classical version of τ MLT from Noyes et al. (1984). This approach is valid for T eff 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 (τ MLT ) and the colour index V − K s , apparently overestimates τ MLT for the stars hotter than the Sun, because theoretically log (τ MLT ) → −∞ in vanishing convection zones for T eff ≈ 8200 K (Simon et al. 2002), i.e. for V − K s ∼ 0.5. Equation (11) in Wright et al. (2011), which describes the relation between log (τ MLT ) 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 τ MLT from Noyes et al. (1984) is provided in Section 5. Since τ MLT 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 T eff → (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):  Since the index A 2 1 (as well as A max 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 log( A 2 1 ), defined by averaging (denoted with an upper bar) over many stars with similar parameters T eff and P (for justification see in Section 4.3 by Arkhypov et al. 2016). Fig. 3(a) shows the dependence of stellar activity index log( A 2 1 ), averaged over star groups in the Rossby number bins log (P/τ MLT ).
Using R x ≡ L x /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: with close to equation (8) parameters: γ x 1 = −0.13 ± 0.04 and δ x 1 = −3.33 ± 0.06 in the saturation regime and γ x 2 = −2.04 ± 0.08 and δ x 2 = −4.85 ± 0.04 in the non-saturated case. The similarities γ 1 γ x 1 , γ 2 ≈ γ x 2 , δ 1 ∼ δ x 1 , and δ 2 ∼ δ x 2 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 2 1 . Fig. 3(c) shows the comparison between the regressions' prediction for both activity indexes in the region of measured log (R x ). After the correction log( A 2 1 ) reg − with an average difference = −0.82 from log (R x ) reg , we arrive at the approximation The slight deviations ( 0.15 dex) from the equality log(R x ) reg = log( A 2 1 ) reg + 0.82 are consistent with the regression standard errors which are where Ro = P/τ MLT is the Rossby number, and the standard errors of the regression coefficients are as follows: σ γ = 0.18 and σ δ = 0.23 for saturated A 2 1 ; σ γ = 0.04 and σ δ = 0.02 for unsaturated A 2 1 ; σ γ = 0.04 and σ δ = 0.06 for saturated R x ; and σ γ = 0.08 and σ δ = 0.04 for unsaturated R x . The corresponding error bars are shown in Fig. 2(c) for the equidistantly selected vales of log (Ro).
The approximate relation log(R x ) reg ≈ log( A 2 1 ) reg + 0.82 means that the regression log( A 2 1 ) reg and the related activity index log( A 2 1 ) can be used as a quasi-proportional proxy for log (R x ).

F RO M T H E O P T I C A L P ROX Y TO T H E P R E D I C T I O N O F S T E L L A R X -R AY L U M I N O S I T Y
Here, we apply the optical proxy approach to predict stellar average X-ray luminosity, i.e. obtain the regression L x (P, T eff ). Theoretically, using equation (4) and the approximation log( A 2 1 ) ≈ log( A 2 1 ) reg , following equation (2), the proxy log( A 2 1 ) reg can be transformed to R x . 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  where Here, R unb x is the regression for R x , which was obtained in Wright et al. (2011) using the 'unbiased sample' of stars It follows from the definition R x ≡ L x /L that in order to estimate L x , one needs a regression for the bolometric luminosity L(T eff ). 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 T eff for main-sequence stars (Eker et al. 2015): where L is the luminosity in solar units, Y = log (T eff /1K), and the fitting coefficients are a L = 3.801, b L = −47.396, c L = 202.329, and d L = −292.539. The standard deviation is ε L = 0.20 dex for all used stars (265 objects excluding 31 stars with problematic mainsequence status and >3σ outliers). Higher polynomial power has practically no decreasing effect on ε L . Using equations (6)-(9), we calculate L x = R x L for every star from our set. In fact, we replace in equation (6) the averaged log( A 2 1 ) with the values log( A 2 1 ) for individual stars. This substitution is justified, since the further calculation of the regression L x (P, T eff ) reg is equivalent to an averaging over stars. This regression has the following form: where X = log (P/1 d), Y = log (T eff /1K), and the fitting coefficients a xi , b xi , c xi , d xi , found with the least square method, are listed in Table 2. Fig. 4 shows the deviation ε reg = log(L x ) − log(L x ) reg between the regression and log(L x ), calculated for individual stars (equations 6-9) and averaged over the considered sample in a sliding window log (P c ) − 0.2 < log (P) < log (P c ) + 0.2 and log (T c ) − 0.05 < log (T eff ) < log (T c ) + 0.05 with the central values of stellar rotation period P c and effective temperature T c . On average, 104 (up to 403) individual estimates of L x appeared within this sliding window. Fig. 4(b) demonstrates a histogram of ε reg with the standard deviation s reg = 0.14. Hence, the total standard error of the regression log (L x ) reg can be estimated as a combination of the standard errors of the involved regressions, i.e. σ tot = (σ 2 reg + s 2 reg + ε 2 L ) 1/2 ≈ 0.26, where σ reg ∼ 0.1 dex for log (R x ) reg (see equation 5 and Fig. 4c for R x errors), s reg = 0.14 dex for log (L x ) reg , and ε L = 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.

C O M PA R I S O N O F T H E P R E D I C T I O N S V E R S U S O B S E RVAT I O N S
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 (L x ) reg , calculated using equations (10)-(14), as well as the observed values of log (L xw ). One can compare these values for different stellar clusters using the associated electronic version of   (8); g Logarithm of the predicted X-ray luminosity using equations (10)- (14). h Logarithm of the measured X-ray luminosity from Wright et al. (2011). to note that the regressions for luminosity log (L) and activity index log( A 2 1 ) 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. Fig. 5 shows the comparison of the predicted X-ray luminosity log (L xp ) = log (L x ) reg (see equation 10) and the observed luminosity log (L xw ) 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 T eff . However, Fig. 5(b) reveals that some stars show log (L xw ) log (L xp ) at log (L xp ) 27 erg s −1 . This effect disappears when the faint stars with T eff < 4000 K are omitted in Fig. 5(c). Fig. 5(d) demonstrates that the stars with T eff < 4000 show a clear cutoff of the observed X-ray flux F xw at the detection threshold of ∼10 −13 ergs s −1 cm −2 . The stars with F xw above this threshold are seen in Fig. 5(b) as a specific population of objects with log (L xw ) log (L xp ). However, at T eff > 4000 K (Fig. 5c), the X-ray luminosity values for the considered stars are clustered along the equality line log (L xw ) = log (L xp ) with a negligible average difference log (L xw ) − log (L xp ) = 0.04 ± 0.04, and the standard deviation of log (L xw ) − log (L xp ) for an individual star is s ind = 0.60 dex. Apparently, the L x variability in time gives the main contribution to s ind .
In Fig. 6, we test an alternative possible explanation of the aforementioned deviations at log (L xw ) log (L xp ) in Fig. 5(b) as a result of underestimated τ MLT for the red dwarfs outside of the temperature region, for which the used approximation of τ MLT was found (Noyes et al. 1984). To do that, we calculated the average values log( A 2 1 ) for different Rossby numbers Ro = P/τ MLT in two temperature domains T eff > 4000 K and T eff < 4000 K using the different definition versions of τ MLT according to Noyes et al. (1984) and equation (10) in Wright et al. (2011), respectively. One can see in Fig. 6(a) that τ MLT from Noyes et al. (1984) gives the unified sequence of estimates that are independent on the temperature domain. However, the increased τ MLT 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 τ MLT (e.g. Noyes et al. 1984), Fig. 6 argues for the validity of Noyes' version of τ MLT also at 3300 T eff < 4000 K. Consequently, the assumption of an increased mixing time τ MLT cannot be used for the explanation of the extension of stellar population with log (L xw ) > log (L xp ) in Fig. 5(b).
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 L x (P) prediction for the stars with solar T eff = 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 longdashed curve, which fits sufficiently well with our prediction (solid curve) mainly within its standard error ±σ tot 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) Fig. 7 with an opened square, coincides with our solid curve.
Considering Figs 5 and 7, one can conclude that the obtained regression L x (P, T eff ), defined by equations (10)-(14), generates plausible predictions at least at T eff 3500 K. Since the reference X-ray catalogue by Wright et al. (2011) is converted to the ROSAT wavelength band, our regression approximates L x in the same band over the range from 6 to 124 Å.

E U V A P P L I C AT I O N
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 MNRAS 476, 1224-1233 (2018) Downloaded from https://academic.oup.com/mnras/article-abstract/476/1/1224/4839014 by guest on 28 July 2018 Figure 5. Comparison of the predicted X-ray luminosity log (L xp ) = log (L x ) reg (diamonds according to equation 10) and the observed luminosity log (L xw ) (filled squares) for the field stars from the catalogue by Wright et al. (2011). (a) The estimates' distribution in relation with T eff , (b) the cross comparison of the all estimates in (a), (c) the same cross comparison as in (b) but only for stars with T eff > 4000 K, and (d) the comparison of observed F xw and predicted F xp X-ray fluxes at the Earth for stars with T eff < 4000 K. The lines depict equalities of the abscissa and ordinate values. 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 where F x = L x /(4πR 2 * )(mW m −2 ) is the stellar surface flux in the ROSAT band 6-124 Å, and F EUV is the EUV surface flux at 124-912 Å. Correspondingly, the stellar EUV luminosity L EUV can be found using the stellar radius R * from the reference catalogue of KIC stellar data by Mathur et al. (2017) log(L EUV ) = log(F EUV ) + log 4πR 2 * + 4, 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 L EUV for every KIC star in our data set.
These estimates were used to obtain the regression L EUV (P, T eff ) as follows: where the fitting coefficients a ei , b ei , c ei , d ei , obtained with the least square method, are listed in Table 4. , averaged in the bins of Ro = P/τ MLT , for two temperature domains T eff < 4000 K (black squares) and T eff > 4000 K (crosses) with (a) τ MLT defined as in Noyes et al. (1984), for all stars and (b) τ MLT , defined according to equation (10) in Wright et al. (2011), for the stars with T eff < 4000 K.
values of stellar rotation period P c and effective temperature T c . On average, 104 (up to 403) individual estimates of L EUV appeared within this sliding window. Fig. 8(b) shows a histogram of ε EUV reg with the standard deviation s EUV reg = 0.08. The deviation of the regression (17) from the true value log (L EUV ) o is where δ reg = log (L EUV ) − log (L EUV ) reg and = log (L EUV ) o − log (L EUV ). Here, log (L EUV ) is an estimate obtained using equations (15) and (16). Correspondingly, the standard error of the regression (17) is where δ 2 reg ≈ s EUV reg = 0.08, and 2 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 , i.e. log (L x ) = log (L x ) o + log (L x ) and log (R * ) = log (R * ) o + log (R * ). Then the following expression can be obtained: Figure 7. Predicted X-ray luminosity versus P is shown as solid curve according to equations (10-14) for the solar effective temperature (T eff = 5770 K) in comparison with averaged predictions for individual stars (equations 6-9) with 5500 < T eff < 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 σ tot of our prediction is depicted as a left-bottom bar. where [ log (L x )] 2 = σ tot , [ log (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 (L EUV ) reg , i.e. equation (17), is It follows from this equation that the obtained regression (equations 17-21) predicts the average logarithm of stellar EUV luminosity with a typical error σ EUV tot ≈ 0.22 dex. For testing of this prediction we use the best calibrated and studied case of the solar-type stars. Fig. 9 shows our L EUV (P) prediction for the stars with solar T eff = 5770 K (Allen 1973) in comparison with the relations modelled by Ribas et al. (2005)   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 ±σ EUV tot 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.

C O N C L U S I O N S
Since our activity index A 2 1 gives a realistic prediction for L x and related L EUV , it may be considered as a practical proxy for the stellar X-ray emission. In contrast with the spectral line indexes (e.g. S, R HK , and R HK in Noyes et al. 1984;Mamajek & Hillenbrand 2008), our approach is based on the optical broad-band photometry. Hence, the index A 2 1 is applicable for more faint and numerous stars. Fig. 10 shows T eff , P-patterns of individual predictions for log (L x ) and log (L EUV ), averaged in the same sliding window as in Figs 4 and 8 with the dimensions log (T eff ) ± 0.05 and log (P) ± 0.2. One can see the similar bright areas in the both plots at 3.6 log (T eff ) 3.76 and 0 log (P) 0.7. The stars in the corresponding intervals 4000 T eff 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 Figure 9. Predicted EUV luminosity versus P is shown as solid curve according to equations (17)-(21) for the solar effective temperature (T eff = 5770 K) in comparison with averaged predictions for individual stars (equations 15-16) with 5500 < T eff < 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 σ EUV tot of our prediction is depicted as a bar in the lower left of the diagram.  (Khodachenko et al. 2012(Khodachenko et al. , 2015Shaikhislamov et al. 2016).
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.

AC K N OW L E D G E M E N T S
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.