Time-scales of stellar rotational variability and starspot diagnostics

Time-scales of stellar rotational variability and starspot diagnostics Abstract The difference in stability of starspot distribution on the global and hemispherical scales is studied in the rotational spot variability of 1998 main-sequence stars observed by Kepler mission. It is found that the largest patterns are much more stable than smaller ones for cool, slow rotators, whereas the difference is less pronounced for hotter stars and/or faster rotators. This distinction is interpreted in terms of two mechanisms: (1) the diffusive decay of long-living spots in activity complexes of stars with saturated magnetic dynamos, and (2) the spot emergence, which is modulated by gigantic turbulent flows in convection zones of stars with a weaker magnetism. This opens a way for investigation of stellar deep convection, which is yet inaccessible for asteroseismology. Moreover, a subdiffusion in stellar photospheres was revealed from observations for the first time. A diagnostic diagram was proposed that allows differentiation and selection of stars for more detailed studies of these phenomena. diffusion, turbulence, stars: activity, stars: interiors, starspots 1 INTRODUCTION Spots in stellar photospheres are extensively studied as an indicator of physical processes in stars and as a proxy of sunspot phenomenology. Nevertheless, particular features in evolution of starspot patterns and their physical drivers in stars of various types are insufficiently explored. For example, the measurements of spot lifetime lead to unexpectedly high estimates of magnetic diffusivity in some stars, supposing an anomalous magnetic diffusion related to the famous giant convection cells (Bradshaw & Hartigan 2014). Besides magnetic cycles, the shorter time-scales of starspot variability traditionally were interpreted in terms of diffusive decay and spot disruption by the shearing of stellar differential rotation (e.g. Bradshaw & Hartigan 2014 and therein). However, the distinction between these effects rarely was a subject of practical study (Hall & Henry 1994). Mainly both the mechanisms were considered in stellar modelling without distinction and clear diagnostics (e.g. Isik, Schüssler & Solanki 2007 and therein). Additionally, the spot emergence frequency was recognized as the third factor affecting the lifetime of starspot pattern. The importance of the emergence modulation for the evolution of spot pattern was confirmed in our solar and stellar studies (Arkhypov, Antonov & Khodachenko 2013; Arkhypov et al. 2015a,b 2016). In particular, a diagnostic method has been proposed there to identify the manifestations of the aforementioned mechanisms of starspot variability. In this Letter, we describe several important express-results of application of this method to an extended set of main-sequence stars, which opens for research community the new ways to probe the stellar physics. 2 METHOD AND DATA SET Detailed descriptions of the used spectral-autocorrelation method can be found in our previous papers (Arkhypov et al. 2015a,b, 2016). Therefore, we briefly describe here only its key ideas. 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. Like in our previous studies (Arkhypov et al. 2015a,b, 2016), we consider here the squared amplitudes $$A_{1}^{2}$$ and $$A_{2}^{2}$$ of the light-curve rotational harmonics with periods P and P/2, respectively, where P is the period of stellar rotation. Following this approach, we found the time-scales of their variability τ1 = −P/ln [r1(P)] and τ2 = −P/ln [r2(P)]. Here, r1(P) and r2(P) are the autocorrelation functions of the chronological series of the $$A_{1}^{2}$$ or $$A_{2}^{2}$$, respectively, at the time lag of one rotational period P. This method is based on the simplest approximation of the logarithm of autocorrelation function at the shortest lag: ln (rm) ≈ −Δt/τm (for details, see in Arkhypov et al. 2016), where m = 1 or 2 is the harmonic number, and Δt = P is the lag. The squared amplitudes of harmonics are used in the analysis because of their statistical proportionality to the solar spot number (Arkhypov et al. 2016). Analogously, Messina et al. (2003) used as an activity index the maximum amplitude (Amax) of rotational variations of stellar light curve, which is a proxy of our amplitude A1 of the fundamental harmonic. They found that their index of various main-sequence stars (0.1 < P < 20  d; M4V to F5V) is related to normalized X-ray luminosity: $$L_{x}/L \varpropto A_{\text{max}}^{b}$$, where b ≈ 2, and L is the stellar bolometric luminosity. The ratio Lx/L is widely used as a standard activity index (e.g. Wright et al. 2011), related to spots (Wagner 1988; Ramesh & Rohini 2008). Since $$A_{1}^{2}$$ and $$A_{2}^{2}$$ are naturally the major contributors to $$A_{\text{max}}^{2}$$, the approximate relation $$L_{x}/L \varpropto A_{\text{max}}^{2}$$ supports applicability of $$A_{1}^{2}$$ and $$A_{2}^{2}$$ as stellar activity indexes for the non-solar-type stars too. Moreover, we found that our activity index $$A_{1}^{2}$$ is related to the Rossby number similarly to the X-ray ratio Rx ≡ Lx/L (see table 3 in Arkhypov et al. 2016), supporting the general statistical proportionality $$L_{x}/L \varpropto A_{1}^{2}$$. Note that the measurements of time-scales τ1 and τ2 are insensitive to this proportionality. For example, the variation cycle of generalized index $$A_{m}^{\gamma}$$ with harmonic number m = 1, 2, and the constant γ has the same duration (i.e. the same time-scale) at any γ. According to Arkhypov et al. (2015a, 2016), the effects of starspot variability can be distinguished using the ‘gradient’ ratio   \begin{eqnarray} \beta _{12}=\frac{\log (\tau _{2})-\log (\tau _{1})}{\log (2)-\log (1)}. \end{eqnarray} (1)In fact, this ratio measures the extent to which the spot distribution evolves faster on smaller scale compared to the largest scales. For example, the Kolmogorov’s theory of turbulence (e.g. Lang 1974) predicts a universal relationship between the characteristic size of turbulent eddies (L) and the time-scale of their variability (τL): τL ∝ L2/3 or τm ∝ m−2/3 (taking into account that L ∝ m−1). Substituting this proportionality into equation (1), one can obtain β12 = −2/3 ≃ −0.67. The horizontal diffusion of magnetic elements in photosphere effectively decreases our activity indexes $$A_{m}^{2}$$ (m = 1, 2), when the average displacement of magnetic elements is about the longitudinal period of harmonic 2π/m. For the normal diffusion, this happens during the characteristic time τm ≈ (2π/m)2η − 1 (where η is the diffusion coefficient), i.e. τm ∝ m−2. In this case, equation (1) gives β12 = −2. In highly magnetized sub-photospheric plasma, like porous media, or in a network flow, the dependence of a squared displacement of magnetic element x2 on time t, can differ from the linear one. In general case, it may be represented as x2 ∝ tα, where α is a constant. Therefore, the squared displacement $$x^{2}\approx (2\pi /m)^{2}=\eta \tau _{m}^{\alpha}$$ corresponds to the noticeable decrease of the activity index $$A_{m}^{2}$$ during the harmonic time-scale τm. In this case, τm ∝ m−2/α and, according equation (1), β12 = −2/α. For example, the subdiffusion of magnetic elements with α = 0.6 ± 0.2 was found in the solar photosphere at the spatial scale of supergranulation (∼104 km), which is related to the material network flow that traps these elements in the conjunction points (Iida 2016). The differential rotation of a star stretches an activity region, or a complex of active regions, over the longitudinal harmonic scale Λm = 2π/m during the time τm = Λm/ΔΩ, where ΔΩ is a typical variation of angular velocity Ω over the latitudinal extension of the activity region or complex. Hence, the time-scale of the considered feature blurring (i.e. $$A_{m}^{2}$$ damping) is τm ∝ m−1 that corresponds to β12 = −1 in equation (1). A stellar activity cycle modulates the total spot number with the identical period or time-scale for all rotational harmonics of a light curve. This means β12 = 0. The aforementioned predictions are tested below by measuring the parameter β12 for the main-sequence stars. For this purpose, we combined our previously analysed data set (Arkhypov et al. 2016) that contains the light curves of 1361 main-sequence stars observed by the Kepler space observatory, with the light curves of additionally selected 637 slow rotators. In summary, the extended sample includes 1998 stars with 0.5 < P < 30  d (according to measurements in Nielsen et al. 2013; McQuillan, Mazeh & Aigrain 2014) and the effective temperatures 3227 ≤ Teff ≤ 7171 K (according to Huber et al. (2014) in the Mikulski Archive for Space Telescopes2). All selected stars belong to the main sequence (surface gravity log (g[cm s−2]) > 4 in the used catalogues). The availability of a high-quality light curve without any interferences (i.e. no detectable short period pulsations or double periodicity from companions) was a special criterion for the compiling of analysed sample of stars. Further details on the star selection and light curve preparing (i.e. removing of gaps, flares, artefacts, trend) and processing are described in Arkhypov et al. (2015b, 2016). 3 DIAGNOSTIC DIAGRAM After processing of the light curves, we have measured the parameter β12 for the selected stars. Fig. 1(a) shows the P − Teff distribution of smoothed values 〈β12〉 as a result of β12 averaging over individual stars with log (Pc) − 0.2 < log (P) < log (Pc) + 0.2 and log (Tc) − 0.05 < log (Teff) < log (Tc) + 0.05 in a sliding window with the central values of stellar rotation period Pc and effective temperature Tc. In average, 117 (up to 400) individual estimates of β12 appeared within this sliding window. The average standard error of 〈β12〉 is 〈σ〉 = 13 per cent. The errors σ = [〈(σ − 〈β12〉)2〉]1/2 in the individual windows of averaging does not exceed 15 per cent in the majority (83 per cent) of cases for 〈β12〉. The total distribution of σ/〈β12〉 is shown in Fig. 1(c). One can see that the found pattern of 〈β12〉 is mainly reliable, excluding the corners of the diagram with a depleted stellar population there (Fig. 1d) and increased relative errors (Fig. 1c). Figure 1. View largeDownload slide Diagnostic diagram and related plots: (a) smoothed distribution of 〈β12〉; (b) scheme of regions with predicted values of the diagnostic parameter 〈β12〉 in (a); (c) relative error of 〈β12〉 estimates in (a); (d) distribution of studied stars in the same frame as in (a)–(c). Figure 1. View largeDownload slide Diagnostic diagram and related plots: (a) smoothed distribution of 〈β12〉; (b) scheme of regions with predicted values of the diagnostic parameter 〈β12〉 in (a); (c) relative error of 〈β12〉 estimates in (a); (d) distribution of studied stars in the same frame as in (a)–(c). Fig. 1(a) reveals an interesting pattern, consisting of the two regions with an increased (dark colour) and decreased (brighter tint) value of 〈β12〉. Fig. 1(b) depicts these regions in terms of predictions for different mechanisms of starspot variability. The used intervals for schematics of 〈β12〉 correspond to ±σ. Since the difference between 〈β12〉 ≈ −2/3 in the ‘turbulent’ region (black colour in Fig. 1b) and the ‘sub-diffusion’ region with 〈β12〉 < −2.3 (grey) is more than 12.5〈σ〉, these diagram details are significant. Fig. 2(a) demonstrates that the discussed regions in the diagnostic diagram are separated approximately along the lines, corresponding to the Rossby number Ro = P/τMLT = 0.13, which was found as a border between the saturated and unsaturated stellar magnetism (Wright et al. 2011). To construct these border lines, we used two versions of the turnover time τMLT in the mixing length theory (MLT): (a) the blue line is based on the classical definition τMLT(B − V) by Noyes et al. (1984), where the standard colour index B − V is related to Teff (Flower 1996); (b) the red line corresponds to the commonly used τMLT(m*) by Wright et al. (2011), where m* is the stellar mass in solar unit related to Teff (Habets & Heintze 1981). Figure 2. View largeDownload slide Backgrounds of the diagnostic diagram: (a) the border lines at Ro = P/τMLT = 0.13 between saturated and unsaturated stellar magnetism (‘red’: τMLT by Wright et al. 2011, ‘blue’: τMLT by Noyes et al. 1984) on the 〈β12〉 pattern; (b) the value 〈log (τ1)〉 averaged in the same sliding windows as in (a); (c) the relative error στ/〈log (τ1)〉 of estimates 〈log (τ1)〉 in (a). Figure 2. View largeDownload slide Backgrounds of the diagnostic diagram: (a) the border lines at Ro = P/τMLT = 0.13 between saturated and unsaturated stellar magnetism (‘red’: τMLT by Wright et al. 2011, ‘blue’: τMLT by Noyes et al. 1984) on the 〈β12〉 pattern; (b) the value 〈log (τ1)〉 averaged in the same sliding windows as in (a); (c) the relative error στ/〈log (τ1)〉 of estimates 〈log (τ1)〉 in (a). Fig. 2(b) illustrates a similar pattern as in Fig. 2(a), but for an analogously smoothed with the sliding-window time-scale 〈log (τ1)〉. Its standard error στ = 〈[log (τ1) − 〈log (τ1)〉]2〉1/2 is shown in Fig. 2(c). While the error, averaged over the whole plot, is 〈στ〉 = 0.03, the value 〈log (τ1)〉 in sliding windows varies from 1.28 to 2.88, corresponding to 19 ≤ τ1 ≤ 765  d in the most reliable region of the diagram at log (P) > 0. Since this variation of 〈log (τ1)〉 is about 53〈στ〉, the found pattern 〈log (τ1)〉 in Fig. 2(b) is significant. The time-scale maximum clearly corresponds to the region with minimal β12 in Fig. 2(a), or the sub-diffusion region in Fig. 1(b). This τ1 ↔ β12 correspondence confirms the reality of stellar sub-diffusion, which prolongs the lifetime of spots due to their suppressed decay. However, the regions of diffusion (β12 ≈ −2) as well as the differential rotation (β12 ≈ −1) in Fig. 1(b) could be artefacts of smoothing in the transition area between the diagram features with values β12 < −2.3 and β12 ≈ −2/3. To test this possibility, we consider the distributions of β12 values, measured for individual stars. Fig. 3(a) shows such distribution for the stars with certainly saturated magnetism at Ro = P/τMLT < 0.13, where τMLT is taken in the conservative version by Noyes et al. (1984). One can see the histogram maximum at the predicted diffusion value β12 = −2, confirming the reality of starspot decay by the normal diffusion process. Fig. 3(b) demonstrates the β12-histograms for the stars with formally unsaturated magnetism at Ro > 0.13. The most probable values β12 ≈ −0.67 correspond to the prediction of the turbulent manifestation (dashed lines). The significance of the main peak is demonstrated using various bins in Figs 3(b) and (c). Considering the binomial distribution of the near-maximum estimates in the limited interval −1.75 < β12 < 0.4 (Fig. 1c), we calculated the probability of the main histogram peak, or a higher one, supposing a random distribution of β12 estimates   \begin{eqnarray} W_{\ge n}=\sum _{n}^{N}\frac{N!}{(N-n)! n!}q^{k}(1-q)^{N-n}=0.0029, \end{eqnarray} (2) where n = 108 is the maximal number of β12 estimates in one bin of the histogram; N = 1062 is the total number of the considered β12 estimates; q = 1/13 is the probability for one estimate to be in a certain bin from the set of 13 bins. Since the obtained probability W≥n is one chance of 344, the considered histogram maximum at β12 ≈ −0.67 is rather not a statistical fluctuation, but an indication of the turbulence manifestation. It is worth to note that the absence in Figs 2(b) and (c) of any hint on the peak at β12 = −1 predicted for the differential rotation. Therefore, the differential rotation effect remains unconfirmed. Figure 3. View largeDownload slide Distributions of non-smoothed values of β12: (a) for the stars with saturated magnetism at the Rossby number Ro < 0.13; (b) for the unsaturated stars at Ro > 0.13 (two histograms with different bin resolution); (c) for the unsaturated stars with more details regarding the histogram features around the distribution maxima in (b) with the labelled probability of exceeding the specified levels (pointed lines), according to the binomial distribution (equation 2). The value n is the number of β12 estimates in one bin of the histogram. The dashed line indicates the prediction of Kolmogorov’s theory of turbulence β12 = −2/3. Figure 3. View largeDownload slide Distributions of non-smoothed values of β12: (a) for the stars with saturated magnetism at the Rossby number Ro < 0.13; (b) for the unsaturated stars at Ro > 0.13 (two histograms with different bin resolution); (c) for the unsaturated stars with more details regarding the histogram features around the distribution maxima in (b) with the labelled probability of exceeding the specified levels (pointed lines), according to the binomial distribution (equation 2). The value n is the number of β12 estimates in one bin of the histogram. The dashed line indicates the prediction of Kolmogorov’s theory of turbulence β12 = −2/3. Apparently, the difference of selected stars in typical light-curve amplitude and measured rotational periods lead to the dispersion of the autocorrelation accuracy, hence, to the scattering of the estimated β12 in Fig. 3a,b. For example, the more noisy low-amplitude light curves of the hot and slow rotators could give somewhat decreased time-scales τ1 and τ2 caused by decrease of the related autocorrelation coefficients r1(P) and r2(P), respectively. Since generally A2 < A1, this effect could decrease τ2 more than τ1, decreasing therefore the gradient ratio β12. However, one can see in Fig. 1(a) an opposite increase of 〈β12〉 in the hot and slow rotators. Hence, the observed difference in 〈β12〉 for hot-slow and cold-fast rotators is not a result of the selection effect. The quasi-symmetric core of the distribution of β12 estimates in Fig. 3(b) and the correspondence of the histogram peaks to the predicted value β12 = −2/3 demonstrate insignificance of the selection effect for our conclusions. Since the decrease of β12 due to the light-curve noise is statistically insignificant in the considered stars with unsaturated magnetism, this effect must be weaker and negligible in the high-amplitude light curves of saturated stars. Hence, the decreased β12 for cold rotators can be interpreted as real manifestation of the sub-diffusion decay of starspot patterns in stars with saturated magnetism. 4 DISCUSSION AND CONCLUSIONS In general, we found that the global spot distribution evolves slower than the hemispherical one. Particularly, the largest scale is much more stable than the smaller one for cold-slow rotators, whereas the difference is less pronounced for hotter stars and/or faster rotators. The corresponding gradient ratio 〈β12〉 < 0 appears an applicable indicator for a dominating mechanism of starspot pattern evolution. Thus, the constructed diagnostic diagram in Fig. 1(b) opens the way to differentiate stars with respect to the processes which control the stochastic variability of starspot pattern. For example, it is first found that the sub-diffusion (−3.0 < β12 < −2.3 in Fig. 1(a), hence, 0.7 < α < 0.9) is a dominating process in the decay of activity complexes in stars with saturated magnetism and certain values Teff and P. Note that the solar sub-diffusion is a result of deceleration of the normal diffusion process in the converging sub-photospheric plasma flows between supergranules at the scale ∼104 km (Iida 2016). Analogously, the sub-diffusion at much larger scales up to ∼106 km (m = 1 and 2) requires regular mega-flows in the sub-photospheric plasma. Such flows, converging in active regions, are found in the Sun (Hindman, Haber & Toomre 2009). The turbulent convection generates non-regular sub-photospheric flows at the local height scale (m ≫ 1) in the standard MLT approach. For example, in the solar photosphere the turbulent cascade has been described at scales smaller than that of supergranules, i.e. m > 200 (Abramenko et al. 2001; Stenflo 2012). However, Figs 1(b), 3(b) and (c) argue for the turbulence manifestation (β12 ≈ −0.67) at m = 1 and 2 in solar-type stars (4500 ≲ Teff ≲ 6900 K and 6 ≲ P ≲ 30  d). This incommensurability of the photospheric and found global turbulence means that the last is the manifestation of plasma mixing in deep layers of the stellar convection zones. The connection between starspot pattern and the deep convection was predicted in numerical models of magnetic tube emergence which take into account the modulation effect from the convective flows (e.g. Weber, Fan & Miesch 2013). However, the deep-mixing effect is masked when its time-scale becomes shorter than the typical spot lifetime in the stars with saturated magnetism. Figs 1(a) and (b) do not reveal any signs of dominating manifestation of magnetic cycles. Note, the period (>103 d) of a typical activity cycle, which standardly described in terms of MFT (Brandenburg & Subramanian 2005), must be much longer than the time-scales of the aforementioned processes. Our autocorrelation method for the estimation of τm is focused on the shortest time-scale, i.e. on the activity complex decay or turbulence manifestation. Therefore, the described diagnostic approach appears as a useful instrument for future studies of starspot phenomenology, magnetic diffusion and flows at stellar photospheres as well as in deep layers of convection zones. Acknowledgements This work was performed as a part of the project P25587-N27 of the Fonds zur Förderung der wissenschaftlichen Forschung, FWF. We also acknowledge the FWF projects S11606-N16, S11604-N16, S11607-N16 and I2939-N27. MLK acknowledges grant 14.616.21.0084 of the Ministry of Education and Science of the Russian Federation. 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. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. 1 https://exoplanetarchive.ipac.caltech.edu/ 2 http://archive.stsci.edu/kepler/stellar17/search.php REFERENCES Abramenko V., Yurchyshyn V., Wang H., Goode P. R., 2001, Sol. Phys. , 201, 225 CrossRef Search ADS   Arkhypov O. V., Antonov O. V., Khodachenko M. L., 2013, Sol. Phys. , 282, 39 CrossRef Search ADS   Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. 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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: Letters Oxford University Press

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Abstract

Abstract The difference in stability of starspot distribution on the global and hemispherical scales is studied in the rotational spot variability of 1998 main-sequence stars observed by Kepler mission. It is found that the largest patterns are much more stable than smaller ones for cool, slow rotators, whereas the difference is less pronounced for hotter stars and/or faster rotators. This distinction is interpreted in terms of two mechanisms: (1) the diffusive decay of long-living spots in activity complexes of stars with saturated magnetic dynamos, and (2) the spot emergence, which is modulated by gigantic turbulent flows in convection zones of stars with a weaker magnetism. This opens a way for investigation of stellar deep convection, which is yet inaccessible for asteroseismology. Moreover, a subdiffusion in stellar photospheres was revealed from observations for the first time. A diagnostic diagram was proposed that allows differentiation and selection of stars for more detailed studies of these phenomena. diffusion, turbulence, stars: activity, stars: interiors, starspots 1 INTRODUCTION Spots in stellar photospheres are extensively studied as an indicator of physical processes in stars and as a proxy of sunspot phenomenology. Nevertheless, particular features in evolution of starspot patterns and their physical drivers in stars of various types are insufficiently explored. For example, the measurements of spot lifetime lead to unexpectedly high estimates of magnetic diffusivity in some stars, supposing an anomalous magnetic diffusion related to the famous giant convection cells (Bradshaw & Hartigan 2014). Besides magnetic cycles, the shorter time-scales of starspot variability traditionally were interpreted in terms of diffusive decay and spot disruption by the shearing of stellar differential rotation (e.g. Bradshaw & Hartigan 2014 and therein). However, the distinction between these effects rarely was a subject of practical study (Hall & Henry 1994). Mainly both the mechanisms were considered in stellar modelling without distinction and clear diagnostics (e.g. Isik, Schüssler & Solanki 2007 and therein). Additionally, the spot emergence frequency was recognized as the third factor affecting the lifetime of starspot pattern. The importance of the emergence modulation for the evolution of spot pattern was confirmed in our solar and stellar studies (Arkhypov, Antonov & Khodachenko 2013; Arkhypov et al. 2015a,b 2016). In particular, a diagnostic method has been proposed there to identify the manifestations of the aforementioned mechanisms of starspot variability. In this Letter, we describe several important express-results of application of this method to an extended set of main-sequence stars, which opens for research community the new ways to probe the stellar physics. 2 METHOD AND DATA SET Detailed descriptions of the used spectral-autocorrelation method can be found in our previous papers (Arkhypov et al. 2015a,b, 2016). Therefore, we briefly describe here only its key ideas. 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. Like in our previous studies (Arkhypov et al. 2015a,b, 2016), we consider here the squared amplitudes $$A_{1}^{2}$$ and $$A_{2}^{2}$$ of the light-curve rotational harmonics with periods P and P/2, respectively, where P is the period of stellar rotation. Following this approach, we found the time-scales of their variability τ1 = −P/ln [r1(P)] and τ2 = −P/ln [r2(P)]. Here, r1(P) and r2(P) are the autocorrelation functions of the chronological series of the $$A_{1}^{2}$$ or $$A_{2}^{2}$$, respectively, at the time lag of one rotational period P. This method is based on the simplest approximation of the logarithm of autocorrelation function at the shortest lag: ln (rm) ≈ −Δt/τm (for details, see in Arkhypov et al. 2016), where m = 1 or 2 is the harmonic number, and Δt = P is the lag. The squared amplitudes of harmonics are used in the analysis because of their statistical proportionality to the solar spot number (Arkhypov et al. 2016). Analogously, Messina et al. (2003) used as an activity index the maximum amplitude (Amax) of rotational variations of stellar light curve, which is a proxy of our amplitude A1 of the fundamental harmonic. They found that their index of various main-sequence stars (0.1 < P < 20  d; M4V to F5V) is related to normalized X-ray luminosity: $$L_{x}/L \varpropto A_{\text{max}}^{b}$$, where b ≈ 2, and L is the stellar bolometric luminosity. The ratio Lx/L is widely used as a standard activity index (e.g. Wright et al. 2011), related to spots (Wagner 1988; Ramesh & Rohini 2008). Since $$A_{1}^{2}$$ and $$A_{2}^{2}$$ are naturally the major contributors to $$A_{\text{max}}^{2}$$, the approximate relation $$L_{x}/L \varpropto A_{\text{max}}^{2}$$ supports applicability of $$A_{1}^{2}$$ and $$A_{2}^{2}$$ as stellar activity indexes for the non-solar-type stars too. Moreover, we found that our activity index $$A_{1}^{2}$$ is related to the Rossby number similarly to the X-ray ratio Rx ≡ Lx/L (see table 3 in Arkhypov et al. 2016), supporting the general statistical proportionality $$L_{x}/L \varpropto A_{1}^{2}$$. Note that the measurements of time-scales τ1 and τ2 are insensitive to this proportionality. For example, the variation cycle of generalized index $$A_{m}^{\gamma}$$ with harmonic number m = 1, 2, and the constant γ has the same duration (i.e. the same time-scale) at any γ. According to Arkhypov et al. (2015a, 2016), the effects of starspot variability can be distinguished using the ‘gradient’ ratio   \begin{eqnarray} \beta _{12}=\frac{\log (\tau _{2})-\log (\tau _{1})}{\log (2)-\log (1)}. \end{eqnarray} (1)In fact, this ratio measures the extent to which the spot distribution evolves faster on smaller scale compared to the largest scales. For example, the Kolmogorov’s theory of turbulence (e.g. Lang 1974) predicts a universal relationship between the characteristic size of turbulent eddies (L) and the time-scale of their variability (τL): τL ∝ L2/3 or τm ∝ m−2/3 (taking into account that L ∝ m−1). Substituting this proportionality into equation (1), one can obtain β12 = −2/3 ≃ −0.67. The horizontal diffusion of magnetic elements in photosphere effectively decreases our activity indexes $$A_{m}^{2}$$ (m = 1, 2), when the average displacement of magnetic elements is about the longitudinal period of harmonic 2π/m. For the normal diffusion, this happens during the characteristic time τm ≈ (2π/m)2η − 1 (where η is the diffusion coefficient), i.e. τm ∝ m−2. In this case, equation (1) gives β12 = −2. In highly magnetized sub-photospheric plasma, like porous media, or in a network flow, the dependence of a squared displacement of magnetic element x2 on time t, can differ from the linear one. In general case, it may be represented as x2 ∝ tα, where α is a constant. Therefore, the squared displacement $$x^{2}\approx (2\pi /m)^{2}=\eta \tau _{m}^{\alpha}$$ corresponds to the noticeable decrease of the activity index $$A_{m}^{2}$$ during the harmonic time-scale τm. In this case, τm ∝ m−2/α and, according equation (1), β12 = −2/α. For example, the subdiffusion of magnetic elements with α = 0.6 ± 0.2 was found in the solar photosphere at the spatial scale of supergranulation (∼104 km), which is related to the material network flow that traps these elements in the conjunction points (Iida 2016). The differential rotation of a star stretches an activity region, or a complex of active regions, over the longitudinal harmonic scale Λm = 2π/m during the time τm = Λm/ΔΩ, where ΔΩ is a typical variation of angular velocity Ω over the latitudinal extension of the activity region or complex. Hence, the time-scale of the considered feature blurring (i.e. $$A_{m}^{2}$$ damping) is τm ∝ m−1 that corresponds to β12 = −1 in equation (1). A stellar activity cycle modulates the total spot number with the identical period or time-scale for all rotational harmonics of a light curve. This means β12 = 0. The aforementioned predictions are tested below by measuring the parameter β12 for the main-sequence stars. For this purpose, we combined our previously analysed data set (Arkhypov et al. 2016) that contains the light curves of 1361 main-sequence stars observed by the Kepler space observatory, with the light curves of additionally selected 637 slow rotators. In summary, the extended sample includes 1998 stars with 0.5 < P < 30  d (according to measurements in Nielsen et al. 2013; McQuillan, Mazeh & Aigrain 2014) and the effective temperatures 3227 ≤ Teff ≤ 7171 K (according to Huber et al. (2014) in the Mikulski Archive for Space Telescopes2). All selected stars belong to the main sequence (surface gravity log (g[cm s−2]) > 4 in the used catalogues). The availability of a high-quality light curve without any interferences (i.e. no detectable short period pulsations or double periodicity from companions) was a special criterion for the compiling of analysed sample of stars. Further details on the star selection and light curve preparing (i.e. removing of gaps, flares, artefacts, trend) and processing are described in Arkhypov et al. (2015b, 2016). 3 DIAGNOSTIC DIAGRAM After processing of the light curves, we have measured the parameter β12 for the selected stars. Fig. 1(a) shows the P − Teff distribution of smoothed values 〈β12〉 as a result of β12 averaging over individual stars with log (Pc) − 0.2 < log (P) < log (Pc) + 0.2 and log (Tc) − 0.05 < log (Teff) < log (Tc) + 0.05 in a sliding window with the central values of stellar rotation period Pc and effective temperature Tc. In average, 117 (up to 400) individual estimates of β12 appeared within this sliding window. The average standard error of 〈β12〉 is 〈σ〉 = 13 per cent. The errors σ = [〈(σ − 〈β12〉)2〉]1/2 in the individual windows of averaging does not exceed 15 per cent in the majority (83 per cent) of cases for 〈β12〉. The total distribution of σ/〈β12〉 is shown in Fig. 1(c). One can see that the found pattern of 〈β12〉 is mainly reliable, excluding the corners of the diagram with a depleted stellar population there (Fig. 1d) and increased relative errors (Fig. 1c). Figure 1. View largeDownload slide Diagnostic diagram and related plots: (a) smoothed distribution of 〈β12〉; (b) scheme of regions with predicted values of the diagnostic parameter 〈β12〉 in (a); (c) relative error of 〈β12〉 estimates in (a); (d) distribution of studied stars in the same frame as in (a)–(c). Figure 1. View largeDownload slide Diagnostic diagram and related plots: (a) smoothed distribution of 〈β12〉; (b) scheme of regions with predicted values of the diagnostic parameter 〈β12〉 in (a); (c) relative error of 〈β12〉 estimates in (a); (d) distribution of studied stars in the same frame as in (a)–(c). Fig. 1(a) reveals an interesting pattern, consisting of the two regions with an increased (dark colour) and decreased (brighter tint) value of 〈β12〉. Fig. 1(b) depicts these regions in terms of predictions for different mechanisms of starspot variability. The used intervals for schematics of 〈β12〉 correspond to ±σ. Since the difference between 〈β12〉 ≈ −2/3 in the ‘turbulent’ region (black colour in Fig. 1b) and the ‘sub-diffusion’ region with 〈β12〉 < −2.3 (grey) is more than 12.5〈σ〉, these diagram details are significant. Fig. 2(a) demonstrates that the discussed regions in the diagnostic diagram are separated approximately along the lines, corresponding to the Rossby number Ro = P/τMLT = 0.13, which was found as a border between the saturated and unsaturated stellar magnetism (Wright et al. 2011). To construct these border lines, we used two versions of the turnover time τMLT in the mixing length theory (MLT): (a) the blue line is based on the classical definition τMLT(B − V) by Noyes et al. (1984), where the standard colour index B − V is related to Teff (Flower 1996); (b) the red line corresponds to the commonly used τMLT(m*) by Wright et al. (2011), where m* is the stellar mass in solar unit related to Teff (Habets & Heintze 1981). Figure 2. View largeDownload slide Backgrounds of the diagnostic diagram: (a) the border lines at Ro = P/τMLT = 0.13 between saturated and unsaturated stellar magnetism (‘red’: τMLT by Wright et al. 2011, ‘blue’: τMLT by Noyes et al. 1984) on the 〈β12〉 pattern; (b) the value 〈log (τ1)〉 averaged in the same sliding windows as in (a); (c) the relative error στ/〈log (τ1)〉 of estimates 〈log (τ1)〉 in (a). Figure 2. View largeDownload slide Backgrounds of the diagnostic diagram: (a) the border lines at Ro = P/τMLT = 0.13 between saturated and unsaturated stellar magnetism (‘red’: τMLT by Wright et al. 2011, ‘blue’: τMLT by Noyes et al. 1984) on the 〈β12〉 pattern; (b) the value 〈log (τ1)〉 averaged in the same sliding windows as in (a); (c) the relative error στ/〈log (τ1)〉 of estimates 〈log (τ1)〉 in (a). Fig. 2(b) illustrates a similar pattern as in Fig. 2(a), but for an analogously smoothed with the sliding-window time-scale 〈log (τ1)〉. Its standard error στ = 〈[log (τ1) − 〈log (τ1)〉]2〉1/2 is shown in Fig. 2(c). While the error, averaged over the whole plot, is 〈στ〉 = 0.03, the value 〈log (τ1)〉 in sliding windows varies from 1.28 to 2.88, corresponding to 19 ≤ τ1 ≤ 765  d in the most reliable region of the diagram at log (P) > 0. Since this variation of 〈log (τ1)〉 is about 53〈στ〉, the found pattern 〈log (τ1)〉 in Fig. 2(b) is significant. The time-scale maximum clearly corresponds to the region with minimal β12 in Fig. 2(a), or the sub-diffusion region in Fig. 1(b). This τ1 ↔ β12 correspondence confirms the reality of stellar sub-diffusion, which prolongs the lifetime of spots due to their suppressed decay. However, the regions of diffusion (β12 ≈ −2) as well as the differential rotation (β12 ≈ −1) in Fig. 1(b) could be artefacts of smoothing in the transition area between the diagram features with values β12 < −2.3 and β12 ≈ −2/3. To test this possibility, we consider the distributions of β12 values, measured for individual stars. Fig. 3(a) shows such distribution for the stars with certainly saturated magnetism at Ro = P/τMLT < 0.13, where τMLT is taken in the conservative version by Noyes et al. (1984). One can see the histogram maximum at the predicted diffusion value β12 = −2, confirming the reality of starspot decay by the normal diffusion process. Fig. 3(b) demonstrates the β12-histograms for the stars with formally unsaturated magnetism at Ro > 0.13. The most probable values β12 ≈ −0.67 correspond to the prediction of the turbulent manifestation (dashed lines). The significance of the main peak is demonstrated using various bins in Figs 3(b) and (c). Considering the binomial distribution of the near-maximum estimates in the limited interval −1.75 < β12 < 0.4 (Fig. 1c), we calculated the probability of the main histogram peak, or a higher one, supposing a random distribution of β12 estimates   \begin{eqnarray} W_{\ge n}=\sum _{n}^{N}\frac{N!}{(N-n)! n!}q^{k}(1-q)^{N-n}=0.0029, \end{eqnarray} (2) where n = 108 is the maximal number of β12 estimates in one bin of the histogram; N = 1062 is the total number of the considered β12 estimates; q = 1/13 is the probability for one estimate to be in a certain bin from the set of 13 bins. Since the obtained probability W≥n is one chance of 344, the considered histogram maximum at β12 ≈ −0.67 is rather not a statistical fluctuation, but an indication of the turbulence manifestation. It is worth to note that the absence in Figs 2(b) and (c) of any hint on the peak at β12 = −1 predicted for the differential rotation. Therefore, the differential rotation effect remains unconfirmed. Figure 3. View largeDownload slide Distributions of non-smoothed values of β12: (a) for the stars with saturated magnetism at the Rossby number Ro < 0.13; (b) for the unsaturated stars at Ro > 0.13 (two histograms with different bin resolution); (c) for the unsaturated stars with more details regarding the histogram features around the distribution maxima in (b) with the labelled probability of exceeding the specified levels (pointed lines), according to the binomial distribution (equation 2). The value n is the number of β12 estimates in one bin of the histogram. The dashed line indicates the prediction of Kolmogorov’s theory of turbulence β12 = −2/3. Figure 3. View largeDownload slide Distributions of non-smoothed values of β12: (a) for the stars with saturated magnetism at the Rossby number Ro < 0.13; (b) for the unsaturated stars at Ro > 0.13 (two histograms with different bin resolution); (c) for the unsaturated stars with more details regarding the histogram features around the distribution maxima in (b) with the labelled probability of exceeding the specified levels (pointed lines), according to the binomial distribution (equation 2). The value n is the number of β12 estimates in one bin of the histogram. The dashed line indicates the prediction of Kolmogorov’s theory of turbulence β12 = −2/3. Apparently, the difference of selected stars in typical light-curve amplitude and measured rotational periods lead to the dispersion of the autocorrelation accuracy, hence, to the scattering of the estimated β12 in Fig. 3a,b. For example, the more noisy low-amplitude light curves of the hot and slow rotators could give somewhat decreased time-scales τ1 and τ2 caused by decrease of the related autocorrelation coefficients r1(P) and r2(P), respectively. Since generally A2 < A1, this effect could decrease τ2 more than τ1, decreasing therefore the gradient ratio β12. However, one can see in Fig. 1(a) an opposite increase of 〈β12〉 in the hot and slow rotators. Hence, the observed difference in 〈β12〉 for hot-slow and cold-fast rotators is not a result of the selection effect. The quasi-symmetric core of the distribution of β12 estimates in Fig. 3(b) and the correspondence of the histogram peaks to the predicted value β12 = −2/3 demonstrate insignificance of the selection effect for our conclusions. Since the decrease of β12 due to the light-curve noise is statistically insignificant in the considered stars with unsaturated magnetism, this effect must be weaker and negligible in the high-amplitude light curves of saturated stars. Hence, the decreased β12 for cold rotators can be interpreted as real manifestation of the sub-diffusion decay of starspot patterns in stars with saturated magnetism. 4 DISCUSSION AND CONCLUSIONS In general, we found that the global spot distribution evolves slower than the hemispherical one. Particularly, the largest scale is much more stable than the smaller one for cold-slow rotators, whereas the difference is less pronounced for hotter stars and/or faster rotators. The corresponding gradient ratio 〈β12〉 < 0 appears an applicable indicator for a dominating mechanism of starspot pattern evolution. Thus, the constructed diagnostic diagram in Fig. 1(b) opens the way to differentiate stars with respect to the processes which control the stochastic variability of starspot pattern. For example, it is first found that the sub-diffusion (−3.0 < β12 < −2.3 in Fig. 1(a), hence, 0.7 < α < 0.9) is a dominating process in the decay of activity complexes in stars with saturated magnetism and certain values Teff and P. Note that the solar sub-diffusion is a result of deceleration of the normal diffusion process in the converging sub-photospheric plasma flows between supergranules at the scale ∼104 km (Iida 2016). Analogously, the sub-diffusion at much larger scales up to ∼106 km (m = 1 and 2) requires regular mega-flows in the sub-photospheric plasma. Such flows, converging in active regions, are found in the Sun (Hindman, Haber & Toomre 2009). The turbulent convection generates non-regular sub-photospheric flows at the local height scale (m ≫ 1) in the standard MLT approach. For example, in the solar photosphere the turbulent cascade has been described at scales smaller than that of supergranules, i.e. m > 200 (Abramenko et al. 2001; Stenflo 2012). However, Figs 1(b), 3(b) and (c) argue for the turbulence manifestation (β12 ≈ −0.67) at m = 1 and 2 in solar-type stars (4500 ≲ Teff ≲ 6900 K and 6 ≲ P ≲ 30  d). This incommensurability of the photospheric and found global turbulence means that the last is the manifestation of plasma mixing in deep layers of the stellar convection zones. The connection between starspot pattern and the deep convection was predicted in numerical models of magnetic tube emergence which take into account the modulation effect from the convective flows (e.g. Weber, Fan & Miesch 2013). However, the deep-mixing effect is masked when its time-scale becomes shorter than the typical spot lifetime in the stars with saturated magnetism. Figs 1(a) and (b) do not reveal any signs of dominating manifestation of magnetic cycles. Note, the period (>103 d) of a typical activity cycle, which standardly described in terms of MFT (Brandenburg & Subramanian 2005), must be much longer than the time-scales of the aforementioned processes. Our autocorrelation method for the estimation of τm is focused on the shortest time-scale, i.e. on the activity complex decay or turbulence manifestation. Therefore, the described diagnostic approach appears as a useful instrument for future studies of starspot phenomenology, magnetic diffusion and flows at stellar photospheres as well as in deep layers of convection zones. Acknowledgements This work was performed as a part of the project P25587-N27 of the Fonds zur Förderung der wissenschaftlichen Forschung, FWF. We also acknowledge the FWF projects S11606-N16, S11604-N16, S11607-N16 and I2939-N27. MLK acknowledges grant 14.616.21.0084 of the Ministry of Education and Science of the Russian Federation. 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. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. 1 https://exoplanetarchive.ipac.caltech.edu/ 2 http://archive.stsci.edu/kepler/stellar17/search.php REFERENCES Abramenko V., Yurchyshyn V., Wang H., Goode P. R., 2001, Sol. Phys. , 201, 225 CrossRef Search ADS   Arkhypov O. V., Antonov O. V., Khodachenko M. L., 2013, Sol. Phys. , 282, 39 CrossRef Search ADS   Arkhypov O. V., Khodachenko M. L., Güdel M., Lüftinger T., Johnstone C. 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Monthly Notices of the Royal Astronomical Society: LettersOxford University Press

Published: Jan 1, 2018

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