Issue 
A&A
Volume 636, April 2020



Article Number  A69  
Number of page(s)  15  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201936925  
Published online  20 April 2020 
Inflection point in the power spectrum of stellar brightness variations
II. The Sun^{★}
^{1}
MaxPlanckInstitut für Sonnensystemforschung,
JustusvonLiebigWeg 3,
37077
Göttingen,
Germany
email: amazo@mps.mpg.de
^{2}
GeorgAugust Universität Göttingen, Institut für Astrophysik,
FriedrichHundPlatz 1,
37077
Göttingen,
Germany
^{3}
School of Space Research, Kyung Hee University,
Yongin,
Gyeonggi
446701,
Korea
^{4}
Laboratory for Atmospheric and Space Physics,
3665 Discovery Dr.,
Boulder,
CO
80303,
USA
^{5}
Instituto de Astrofísica de Canarias (IAC),
38200
La Laguna,
Tenerife,
Spain
Received:
15
October
2019
Accepted:
6
February
2020
Context. Young and active stars generally have regular, almost sinusoidal, patterns of variability attributed to their rotation, while the majority of older and less active stars, including the Sun, have more complex and nonregular light curves, which do not have clear rotationalmodulation signals. Consequently, the rotation periods have been successfully determined only for a small fraction of the Sunlike stars (mainly the active ones) observed by transitbased planethunting missions, such as CoRoT, Kepler, and TESS. This suggests that only a small fraction of such systems have been properly identified as solarlike analogues.
Aims. We aim to apply a new method of determining rotation periods of lowactivity stars, such as the Sun. The method is based on calculating the gradient of the power spectrum (GPS) of stellar brightness variations and identifying a telltale inflection point in the spectrum. The rotation frequency is then proportional to the frequency of that inflection point. In this paper, we compare this GPS method to alreadyavailable photometric records of the Sun.
Methods. We applied GPS, autocorrelation functions, LombScargle periodograms, and wavelet analyses to the total solar irradiance (TSI) time series obtained from the Total Irradiance Monitor on the Solar Radiation and Climate Experiment and the Variability of solar IRradiance and Gravity Oscillations experiment on the SOlar and Heliospheric Observatory missions. We analysed the performance of all methods at various levels of solar activity.
Results. We show that the GPS method returns accurate values of solar rotation independently of the level of solar activity. In particular, it performs well during periods of high solar activity, when TSI variability displays an irregular pattern, and other methods fail. Furthermore, we show that the GPS and light curve skewness can give constraints on facular and spot contributions to brightness variability.
Conclusions. Our results suggest that the GPS method can successfully determine the rotational periods of stars with both regular and nonregular light curves.
Key words: methods: data analysis / stars: solartype / Sun: activity / Sun: fundamental parameters / stars: activity / stars: rotation
The two movies are available at https://www.aanda.org
© E. M. AmazoGómez et al. 2020
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Open Access funding provided by Max Planck Society.
1 Introduction
A star’s rotation period defines the action of the stellar dynamo, transport of magnetic flux through the convective zone, and its emergence on the stellar surface (see Charbonneau 2010 for a general review of dynamo theory, and Reiners et al. 2014, Işık et al. 2018 for the relation between the rotation period and activity). Furthermore, for the unsaturated regime (i.e. when the Rossby number is larger than 0.13, equivalent to rotation periods longer than 1–10 days for solartype stars), the magnetic activity of stellar chromospheres (as indicated by the Ca II H and K emission core lines) and coronae (as indicated by the Xray emission) monotonically increases with the decrease in the rotation period (Pizzolato et al. 2003; Wright et al. 2011; Reiners 2012; Reiners et al. 2014).
Recent studies indicate that the relationships between rotation period and coronal and chromospheric activity work not only for stars with a tachocline (the transition region between the radiative core and convective envelope, see Spiegel & Zahn 1992), but also for slowly rotating, fully convective stars (see Reiners et al. 2012; Wright & Drake 2016; Newton et al. 2017; Wright et al. 2018). The rotation period thus appears to be a good proxy of the overall magnetic activity of a star.
Accurate measurements of rotation periods are important for understanding stellar evolution and for better calibration of the gyrochronology relationship (Ulrich 1986; Barnes 2003). Knowledge of the stellar rotation period helps distinguish the radial velocity jitter of a star from the planetary signal (see, e.g. Dumusque et al. 2011; Oshagh 2018). This is crucial for detection, as well as for confirming Earthsize planets in ongoing and upcoming surveys, such as the ESPRESSO (see Pepe et al. 2010) and PLATO missions (see Roxburgh et al. 2007; Rauer et al. 2016, 2014).
Rotation periods can be determined from photometric observations thanks to the presence of magnetic features on stellar surfaces. Concentrations of strong localised magnetic fields emerge on the stellar surface and lead to the formation of photospheric magnetic features, such as bright faculae and dark spots (Solanki et al. 2006). The transits of these magnetic features over the visible disk as the star rotates imprint particular patterns onto the observed light curve. These patterns provide a means of tracing stellar rotation periods. We note that in the case of the Sun, such brightness variations are well understood (Solanki et al. 2013; Ermolli et al. 2013), and modern models can explain more than 96% of the variability of total solar irradiance (TSI, which is the spectrallyintegrated solar radiative flux at one astronomical unit from the Sun) (see Yeo et al. 2014).
Available methods for retrieving rotation periods from photometric time series, for example, autocorrelation analysis or LombScargle periodograms, appeared to be very successful for determining periods of active stars with periodic patterns of variability. The transiting planethunting missions such as COROT, Kepler, and TESS, (Bordé et al. 2003; Borucki et al. 2010; Ricker et al. 2015) revealed unprecedented possibilities for acquiring accurate highprecision photometric time series of stars different to the Sun. The new data from these missions enabled studies of stellar magnetic activity on a completely new level. In particular, it has become possible to measure rotation periods for tens of thousands of stars (see, Walkowicz & Basri 2013; Reinhold et al. 2013; Nielsen et al. 2013; García et al. 2014; McQuillan et al. 2014; Buzasi et al. 2016; Angus et al. 2018). At the same time, the pattern of brightness variations in slow rotators such as the Sun is often quasiperiodic, and even irregular. The irregularities are mainly caused by the short (in comparison to the stellarrotation period) lifetimes of magnetic features, such as starspots/sunspots, and a large degree of randomness in the time and position of their emergence on the stellar surface. This renders the determination of rotation periods for low activity stars very difficult. For example, van Saders et al. (2019) showed that rotation periods of about 80% of stars in the Kepler field of view with nearsolar effective temperature remain undetected. Consequently, the stars with known rotation periods represent only the tip of the iceberg of Sunlike stars. This can lead to biases in conclusions drawn based on the available surveys of stellar rotation periods. The relatively low efficiency of standard methods in detecting periods of stars with nonregular light curves might affect studies aiming to compare solar and stellar variability. For example, solar variability appears to be normal when compared to mainsequence Kepler stars with nearsolar effective temperatures (Basri et al. 2013; Reinhold et al. 2020). At the same time, when comparisons are limited to mainsequence stars with nearsolar effective temperatures and with known nearsolar rotation periods, the solar variability appears to be anomalously low (Reinhold et al. 2020). One possible explanation of such a paradox is the inability of standard methods to reliably detect rotation periods of stars with light curves similar to that of the Sun (see also discussion in Witzke et al. 2020). Along the same line, Reinhold et al. (2019) proposed that biases in determining rotation periods might contribute to the explanation of a dearth of intermediate rotation periods observed in Kepler stars (see McQuillan et al. 2013, 2014; Reinhold et al. 2013; Davenport 2017).
In Shapiro et al. (2020, hereinafter, Paper I), we proposed a new method for determining stellar rotation periods from the records of their photometric variability. The method is applicable to latetype stars, but particularly beneficial for, and aimed at, stars with low activity and quasiperiodic irregularities in their light curves. The rotation period is determined from the profile of the highfrequency tail (i.e. its part in between about a day and 5–10 days) of the smoothed wavelet power spectrum. For this work, we used a Paul wavelet of order six (see Torrence & Compo 1998). We identified the point where the gradient of the power spectrum (GPS) plotted on a log–log scale (in other words, d (ln P(ν))∕d(lnν), where P is power spectral density and ν is frequency) reaches its maximum value. This point corresponds to the inflection point, which is the concavity of the power spectrum plotted in the log–log scale changes sign there. In Paper I, we showed that the period corresponding to the inflection point is connected to the rotation period of a star by a calibration factor that is a function of stellar effective temperature, metallicity, and activity.
The main goal of the present study is to test and validate the method proposed in Paper I against the Sun, which presents a perfect example of a star with an irregular pattern of variability, but a known rotation period. In particular, by validating the GPS method against the Sun, we show that it has the potential to reduce biases caused by the relative inefficiency of standard methods to determine rotation periods of lowactivity stars. More specifically, we utilised the calibration factor between the position of the inflection point and rotation period corresponding to the solar case, and applied the GPS method to the available photometric records of the Sun. Furthermore, we tested the performance of the GPS method at various levels of solar activity.
In Sect. 2, we give a short overview of the available methods for determining rotation periods as well briefly describing the GPS method. (A more detailed discussion of this method is given in Paper I.) In Sect. 3, we compare the performance of our method with that of other available methods in the exemplary case of the Sun. In Sect. 4, we present the relationship between solar activity and skewness of its photometric light curve. Our main results are summarised in Sect. 5.
2 Methods for determining stellar rotation periods
Highprecision, highcadence photometric timeseries allow the determination of accurate stellarrotation periods, and the scientific community has developed various methods for retrieving periodicities embedded in these data. Current methods include autocorrelation functions, LombScargle periodograms, wavelet power spectra, and very recent techniques based on the Gaussian processes.
The autocorrelation function (ACF) method is based on the estimation of a degree of selfsimilarity in the light curve over time. The time lags at which the degree of selfsimilarity peaks are assumed to correspond to the stellarrotation period and its integer multiplets. This assumption is valid if magnetic features that cause photometric variability are stable over the stellar rotation period.
The ACF method was used by McQuillan et al. (2014) to create the largest available catalogue of rotation periods to date: the rotation periods were found for 34 030 (25.6%) of the 133 030 mainsequence Kepler target stars (observed in the Kepler star field at the time of publication). The ACF calculations presented in our study were performed with the A_ CORRELATE IDL function.
The LombScargle periodogram gives the power of a signal a certain frequency (see Lomb 1976; Scargle 1982). Here, we used the generalised LombScargle (GLS) version v1.03, applying the formalism given by Zechmeister & Kürster (2009). The GLS method is widely used for retrieving periodicities from timeseries, and is applicable to stellar light curves with nonregular sampling. In studies aimed at determining the rotationperiod, the highest peak in the GLS is assumed to correspond to the rotation period (see Reinhold et al. 2013).
A wavelet power spectra (PS) transform analysis is beneficial for timeseries that have a nonstationary signal at many different frequencies. It was employed for determining stellarrotation periods by García et al. (2009). To perform PS, we use the WV_ CWT IDL function based on the sixth order Paul wavelet (see Farge 1992; Torrence & Compo 1998).
Another technique, which is currently actively used, is inferring rotation periods with the help of Gaussian processes (GP, Roberts et al. 2012). A GP can be applied to detect a nonsinusoidal and quasiperiodic behaviour of the signal in light curves. The GP regression has been extensively tested for analysing various time series, and, in particular, for retrieving the periodic modulation from stellar radial velocity (Rajpaul et al. 2015) and photometric (Angus et al. 2018) signals. The GP performancecan be readily compared with other approaches for determining rotation periods in small stellar samples (see, e.g. Faria et al. 2020, who applied various techniques to HD 41284). However, GP calculations demand significant computational resources, for example, analysis of a typical Kepler light curve takes from several to 12 h and longer (Angus et al. 2018). The time efficiency of the GP algorithms is expected to significantly improve with the development of new methods (ForemanMackey et al. 2017a,b). Thus, it might be interesting in future studies to compare rotation periods determined by the GP and GPS algorithms. Such a comparison is, however, out of the scope of the present study.
The methods described above have been very successful for determining rotation periods of active stars with periodic patterns of photometric variability. However, their performance deteriorates for slowerrotating and consequently lessactive stars, and, in particular, for stars with nearsolar levels of magnetic activity. In such stars, the pattern of variability is complex and nonregular, due to the irregular emergence of magnetic features that manifest as spots and faculae. In particular, the dark spots can lead to prominent dips in brightness, and we expect that for stars with solaractivity levels, their decay time is comparable to, or even less than, the stellar rotation period, as it is for the Sun (see Solanki et al. 2006; Shapiro et al. 2017). Furthermore, Shapiro et al. (2017) showed that in the solar case, the superposition of bright facular and dark spot signatures might lead to a disappearance of the rotation peak from the power spectrum of solar brightness variations. This agrees with many studies (see Aigrain et al. 2015; He et al. 2015, 2018) that have found that the rotation period of the Sun can be reliably determined only during the periods of low solar activity. We note that during these periods, the solar variability is caused by longlived facular features, and the spot contribution is small.
Recently, He et al. (2015, 2018) analysed solar and stellar brightness using GLS and introduced two indicators: one describing the degree of periodicity in the light curve, and the other describing the amplitude of the photometric variability. They found that light curve periodicities of theKepler stars are generally stronger during highactivity times. In contrast, solar light curves are more periodic during phases of low activity. By applying GLS to the TSI time series, they could determine the solarrotation period only during periods of low solar activity.
In Paper I, we employed the SATIRE approach (where SATIRE stands for spectral and total irradiance reconstruction, see Fligge et al. 2000; Krivova et al. 2011) to simulate light curves of stars with solar effective temperature and metallicity for various cases of magneticfeature emergence and evolution. We found that the profile of the wavelet power spectrum around the rotation period strongly depends on the decay time of magnetic features and the ratio between coverage of the disc by faculae and spots. At the same time, the highfrequency tail of the power spectrum, particularly the portion with periods between about a day and a quarter of the rotation period, remains much more stable, and mainly depends on the rotation period. To quantitatively characterise the highfrequency tail of the power spectrum, we calculated the ratios R_{k} between the power spectral density, P(ν), at two adjacent frequency grid points: R_{k} ≡ P(ν_{k+1})∕P(ν_{k}). For a frequency grid equidistant in the logarithmic scale (i.e. with a constant value of Δν∕ν), this ratio represents the gradient of the power spectrum plotted on a log–log scale. Consequently, the maxima of the R_{k} values correspond to the positions of the inflection points, where the concavity of the power spectrum plotted on a log–log scale changes sign. Hereinafter, following Paper I, we refer to the R_{k} values as the gradient of the power spectrum, GPS.
In Paper I, we found that the position of the highfrequency inflection point (i.e. the inflection point (IP) with highestfrequency, see discussion below) is related to the stellar rotation period by a certain calibration factor defined as: α = HFIP∕P_{rot}, where HFIP is a period corresponding to the highfrequency inflection point and P_{rot} is the rotation period. This allowed us to propose a new method for determining the rotation period: calculate the position of the inflection point and scale it with an appropriate value for thefactor α to determine the rotation period.
The choice of the scaling factor α is one of the most delicate steps in the GPS method and also one of the main sources of its uncertainty. In Paper I, we found that the value of α is fairly insensitive to the parameters describing the evolution of magnetic features, and, in particular, to the decay time of spots. It shows a stronger dependence on the area ratio between the facular and spot components of active regions at the time of maximum area, S_{fac}∕S_{spot}. We estimate that over the 2010–2014 period, the solar S_{fac}∕S_{spot} value was about 3 (see Appendix B of Paper I). For the calculations presented in this paper, we adopt α_{Sun}± 2σ = 0.158 ± 0.014, corresponding to S_{fac}∕S_{spot} = 3 and the spot decay rate of 25 MSH/day. Such a value of spot decay time is intermediate between the 10 MSH day^{−1} estimate by Waldmeier (1955) and 41 MSH/day estimate by Martínez Pillet et al. (1993).
The uncertainty of the α value is computed in Paper I as the standard deviation between positions of inflection points corresponding to different realisations of spot emergences (but the same values of the S_{fac}∕S_{spot} ratio and spot decay rates). We stress that, when applied to other stars, the uncertainty of our method will be significantly larger, since neither the S_{fac}∕S_{spot} value nor spot decay rates are known a priori. For example, in Paper I we estimated that the internal uncertainty of our method is 25%. Its main advantage, however, is that, in contrast to other methods, it is applicable to inactive stars with irregular patterns of brightness variability.
In summary, we calculate the rotation period and its uncertainty as: (1)
where HFIP represents the value of the highfrequency inflection point in the power spectrum of brightness variations.
3 Validation of GPS method for the solar case
In the present work, we evaluate the GPS method for determining the solar rotation period against available records of its brightness variation. We consider the performance of the method at different levels of solar activity using two TSI datasets.
3.1 Records of total solar irradiance, TSI
Driven by the interest from the climate community, the solar irradiance has been measured by various space radiometric instruments and reconstructed with many models, see, for example, reviews by Ermolli et al. (2013) and Solanki et al. (2013). Among all available solarirradiance records, the TSI time series are the most accurate and possess the longest time coverage (see e.g. review by Kopp 2014). In this context, we opted for testing the performance of the GPS method against the TSI time series and focused on what are generally considered to be the two most reliable of the available datasets: one obtained by the Variability of solar IRradiance and Gravity Oscillations (VIRGO) experiment on the ESA/NASA SOlar and HeliosphericObservatory (SoHO) Mission (see Fröhlich et al. 1997), and another obtained by the Total Irradiance Monitor (TIM) on the Solar Radiation and Climate Experiment (SORCE; see Kopp & Lawrence 2005; Kopp et al. 2005a,b).
VIRGO provides more than 21 yr of continuous highprecision, highstability, and highaccuracy TSI measurements. Our analysis is based on the latest update available at the beg inning of this work, version 6.4 : 6_005_1705, level 2.0 VIRGO/ PMO6V observations from January 1996 until June 2017, with a cadence of 1 data point per hour^{1}. The data are available viaftp^{2}.
The TIM data used are version 17, level 3.0, available from February 25, 2003^{3}^{,}^{4}. While TIM data cover a shorter time interval than VIRGO, they have a lower noise level (Kopp 2014), which is particularly important for our analysis of TSI variations during the minimum of solar activity. Here, we use an average version of TSI with a cadence of 1 data point per 1.6 hours, obtained by averaging over a full orbit of the spacecraft^{5}.
3.2 Brightness signature of spot transits
In this section, we analyse the performance of the GPS method during a 90day time interval when TSI variability was generated by three consecutive spot transits. Sunspot transits imprint characteristic signals on the TSI, diminishing the observed solar brightness. Transits of wellresolved sunspots were recorded by TIM/SORCE from December 2006 to February 2007 (see panel a of Fig. 1). One can see a clear signal from spots tran siting the solar disc with a recognisable “Vlike” shape brought about by the combination of foreshortening effects, the Wilson depression (i.e. spot umbrae are slightly depressed from the photospheric opticaldepth unity level due to higher transparency of the spot atmosphere compared to the quiet photosphere (see Wilson 1965), and the centretolimb dependence of spotintensity contrast.
Interestingly, the time separation between consecutive transits is very close to the solar rotation period (compare sine functions and TSI timeseries in Fig. 1a). A fluctuating transitamplitude behaviour (i.e. decrease in the amplitude from first to second transit and increase of the amplitude from second to third transit) suggests that we are observing transits of different spots, even though the emergence of all three spots occurred at the same location on the solar surface. We confirm this by comparing simultaneous records of the MDI intensity maps and TSI^{6}. Such a nesting of spot emergence (Brouwer & Zwaan 1990; Gaizauskas et al. 1994) affects the power spectrum of brightness variations exactly as the increase of the lifetime of the magnetic features. In particular, it makes light curves more regular and helps to determine rotation period.
We perform a comparison between the GLS, ACF, PS, and GPS methods for the considered epoch of spot transits. In Fig. 1b, we plot the results of the GLS analysis. A prominent peak is clearly observed at 27.2 days, which is very close to the synodic Carrington rotation period of 27.27 days. This peak has a power of 0.37, where GLS is normalised to unity. A second peak appears at 13.1 days with a normalised power of 0.23, which is a harmonic of the solar rotation period. A large value of the normalised power in the rotation peak is not surprising: the corresponding light curve has a clear periodicity, which results in a clear peak in the power spectrum. The purple dashed line in Fig. 1a fits a sine wave with period determined from the GLS method.
Figure 1c shows the ACF analysis. Two clear peaks are visible at 26.0 and 52.0 days, with amplitudes of about 0.5 and 0.3, respectively. The PS analysis is shown in Fig. 1d. We use a Paul wavelet basis function with order m = 6 to calculate the global wavelet power spectrum. The PS show a pronounced peak around 26.6 days. One can also see a shoulderlike feature at about 13 days, which is generated due to the ingress and egress of magnetic features over the visible solar disc. The contrast differences between faculae and spots transiting the limbs and the centre will imprint a characteristic pattern in the light curve, detected by the implementation of the GPS.
Figure 1e shows the GPS profile with two welldefined peaks that correspond to the inflection points in the power spectrum. The lowfrequency inflection point is at 19.2 days and is associated with the transition from the rotation peak to highfrequency modulations forming a shoulderlike feature. The highfrequency inflection point is at 4.21 days and corresponds to the transition from the shoulderlike feature to the plateau at higher frequencies. For the considered period of brightness variations due to spots, the highfrequency point also corresponds to the maximum value of the GPS. In other words, the gradient is larger at the highfrequency point than it is at the lowfrequency point.
In Sect. 3.5, we show that, in agreement with Paper I, the highfrequency inflection point is present in the PS even when the rotation peak is absent. Furthermore, the location of the highfrequency inflection point is stable, independently of the presence or absence of the rotation peak.
Based on the location of the highfrequency inflection point at 4.21 days and using Eq. (1) with the calibration factor α_{Sun} = 0.158 and its 2σ uncertainty of 0.014, we computed the solar rotation period: days. We used the 26.6day value for plotting the orange sine curve in Fig. 1a. This rotation period is in reasonably good agreement with the Carrington period, given the relatively short length of the time series. In summary, all methods applied above allow reasonably accurate determinations of the rotation period for the considered period of pseudoisolated transits of sunspots.
Fig. 1
Example of spotdominated total solar irradiance (TSI) variability. Panel a: SORCE/TIM measurements from 28Nov.2006 to 28Feb.2007. Purple and orange curves indicate sinewave functions with periods of 27.3 d (solar rotation period deduced with the generalised Lomb Scargle periodogram method, GLS) and 26.6 d (solar rotation period deduced with the gradient of power spectra method, GPS). Panels b and c: corresponding GLS periodogram and autocorrelation function, ACF, respectively. Panel d: global wavelet power spectrum, PS, calculated with the 6th order Paul wavelet. Panel e: gradient of the power spectrum plotted in panel d. Blue asterisks in panels b and c represent positions of the peaks in theGLS and ACF. Green dotted lines in panel e indicate the high and lowfrequency inflection points. 
3.3 Brightness signature of facular feature transits
Here, we perform a rotation period analysis of an interval of time between December 2007 and March 2008 (see Fig. 2) when the TSI variability was dominated by three consecutive transits of afacular feature. The simultaneous analyses of MDI intensitigram, magnetograms, and TSI records^{7} ^{,}^{8} show that variability is brought about by a single large facular feature, where size is decreasing from transit to transit. We note the difference with the spot case described in Sect. 3.2, where the three consecutive transits were caused by the nesting effect. Such a behaviour is consistent with the fact that lifetimes of facular features are significantly larger than those of the spots.
The transit of a facular feature has a characteristic doublepeak “Mlike” profile, see Fig. 2a. This Mlike profile can be explained by the increase of facular contrast towards the limb. Such an increase partly compensates the foreshortening effect and, consequently, the maximum brightness occurring when the facular feature is observed at an intermediate disc position.
Figure 2b shows a GLS analysis of the considered time interval. A prominent peak is seen at 26.4 days with a normalised power of 0.68. The purple dash line in Fig. 2a fits to the light curve with a sine wave with a corresponding period. Figure 2c shows the corresponding ACF analysis. A clear series of peaks with a time lag of 26.8 days are observed.
The PS analysis shows a peak at 27.2 days (see Fig. 2d). Similarly to the case of spot transits, one can see a shoulderlike feature, but it is shifted towards higher frequencies in comparison to the spot case. Such a shift of the shoulderlike feature is explained by the Mlike profile of a facular transit in the light curve. It leads to the enhanced variability on timescales shorter than the half of the rotation period and consequently shifts the shoulderlike feature.
Figure 2e shows the GPS profile with two clearly visible peaks corresponding to the inflection points of the PS. As in the case of the variability brought about by sunspots, there are two inflection points, one is closer to the rotation period peak, and the other to the shoulderlike feature. We note that for the faculaedominated case, in contrast to the spotdominated case, the maximum inflection point (i.e. the inflection point with highest value of the GPS) corresponds to the lowfrequency inflection point.
The highfrequency inflection point is located at 4.40 days, which is very close to the location of the inflection point during the spotdominated regime of the variability (4.21 days, see Sect. 3.2). Applying the calibration factor α_{Sun,} one can see that the value given by the GPS method for the solar rotation period is days. The orange line in Fig. 2a fits a sinusoidal function with a period of 27.8 days, as revealed through the GPS analysis. The amplitude of the orange curve is given by the maximum amplitude of the wavelet.
The analyses performed in this section and in Sect. 3.2 show that when solar brightness variability is attributed to the periodic transits of spots and faculae, all four methods (GLS, ACF, PS, and GPS) can accurately retrieve a solar rotation period. However, the recurrent transits of one spot group or nested spots (as plotted in Fig. 1a) are rare. Since faculae last significantly longer than spots, the recurrent transits of the same facular features are much more common than those of spots. However, faculae dominate the solar TSI light curve on rotational timescales only at activity minimum (when no large spots are present, although on the solar cycle timescale, faculae play a dominant role).
Most of the time, solar brightness variations are brought about by the combination of magnetic features coming with random phases. As we show below, this strongly affects the performance of the GLS, ACF, and PS methods, but has a much weaker effect on the GPS method.
Fig. 2
Same as Fig. 1 but for the time interval of faculaedominated TSI variability from 21Dec.2007 to 22Mar.2008. The purple dashed curve in panel a represents a sine wave function with a period of 26.4 d corresponding to the solar rotation period deduced with the GLS method. The orange curve in panel a is a sinusoidal function with a period of 27.8 d corresponding to the solar rotation period deduced with the GPS method. 
3.4 Analysis of the entire dataset
In Sects. 3.2 and 3.3, we considered relatively short intervals of time when solar variability is dominated by either spot or facular components. In this section, we study whether the solar rotation period can be reliably retrieved in a more general case when contributions from faculae and spots are entangled. For this, we considered 21 yr of TSI data from SoHO/VIRGO (see Fig. 3) and 15 yr of TSI SORCE/TIM data (see Fig. 4), and we applied GLS, ACF, PS, and GPS methods to the entire time series.
We plot GLS periodograms for VIRGO and TIM data in panels b of Figs. 3 and 4, respectively. One can see that neither of the two periodograms contain a sufficiently strong peak to provide a clear indication of periodicity. Instead, they contain a series of peaks with similar and relatively small power: up to 0.003 for the VIRGO TSI and 0.007 for the TIM TSI. This is about two orders of magnitude lower than the normalised power obtained for pseudoisolated spots and facular cases considered in Sects. 3.2 and 3.3. Consequently, the GLS method does not allow a definitive detection of the rotation period when the entire TIM and VIRGO time series are considered. The ACF of the VIRGO and TIM TSI time series are shown in panels c of Figs. 3 and 4.
Although we can appreciate small peaks at the expected location of the rotation period in both ACF cases, the significance of the maxima with the corresponding time lag needed for the identification of the rotation period yields only a marginal detection. Panels d in Figs. 3 and 4 display the PS analysis of the data. One can see that instead of the peak at the rotation period, there is a plateau region for both datasets.
All in all, the GLS and PS methods cannot detect the solar rotation period, and ACF gives us a marginal detection when long sets of TSI data are considered. This is in line with the result of Shapiro et al. (2017) who showed that the superposition of facular and spot contributions to solar variability can significantly decrease the rotation signal.
The GPS profiles of VIRGO and TIM time series are given in panels e of Figs. 3 and 4. One can see that both profiles display conspicuous highfrequency inflection points. They are located at 4.15 and 4.12 days for VIRGO and TIM data, respectively. Consequently, we retrieve a rotation period for the Sun: days, and days, for the VIRGO and TIM data, respectively. These P_{rot} values agree within the error bars with the solar synodic Carrington rotation period of 27.27 days, as well with the solar equatorial synodic rotation period for a fixed feature of 26.24 days. Consequently, the GPS method allows a proper determination of solar rotation period over timescales, in comparison with other traditional approaches.
Fig. 3
Same as Fig. 1 but for 21 yr [1996.01.28–2017.05.23] of TSI data from SoHO/VIRGO. In total, 7787 days are considered. This corresponds to about 186 889 data points at an hourly cadence. The GLS, ACF, and PS methods do not show a clear signal of the rotation period. GPS shows a prominent peak at 4.15 days, resulting in a solar rotation period value of about 26.3 days (see text for details). Orange vertical lines and marks in panel a represent a splitting of the VIRGO dataset into Keplerlike timespan K_{n}, each one representing the full lifetimes of the Kepler mission, around 4 yr (see Sect. 3.5). 
3.5 The solar variability in 90day quarters
Here, we consider the exemplary case of the Sun, as it would be observed over the same timespan as the Kepler stellar dataset. The Kepler telescope reoriented itself every 90 days, thus introducing discontinuities into the light curves. To mimic the observational routine of the Kepler telescope, we segment the entire VIRGO and TIM datasets into 86 and 60 quarters of 90day durations, with a cadence of 1.0 and 1.6 h, respectively. The cadence of solar observations are close to regular long cadence Kepler observations of 29.4 min. In this section, we analyse the stability of the rotation signal from quarter to quarter, using the four methods described above.
The inflection points described by the GPS method in all quarters are shown in Fig. 5 for the VIRGO (top panel) and TIM (bottom panel) data. We note that quarters 10, 12, 13, and 52 in the VIRGO dataset are affected by the lack of data due to spacecraft and instrument failures. TIM data quarters number 1, 43, 44, and 45 contain long gaps, some of them larger than five days. For the three recent quarters, this is because of the failing SORCE battery.
We note that power spectra of some quarters have more than one inflection point with considerable amplitude (like in the time intervals shown in Figs. 1 and 2). We analyse the two highest inflection points in the GPS and describe these in terms of its frequency location. The inflection point towards the highest frequency, HFIP, is represented by blue dots, in Fig. 5. The inflection point towards lower frequencies, LFIP, is represented by red dots. In Fig. 5, for each quarter, open black diamonds surround the inflection point corresponding to the maximum amplitude of the gradient of the power spectra, MIP. In most of the cases, these maximum peaks are simultaneously the high frequency inflection points (as in the case of the variability brought about by spots, see Fig. 1), marked with open black diamonds overplotted over the blue dots. Similarly, there are several quarters when the maximum value of the gradient corresponds to the lowfrequency inflection point (like in the case of the variability brought about by faculae, as we explained before, see Fig. 2). These quarters are associated with low solar activity and with TSI variability being dominated by faculae. Interestingly, the highfrequency inflection points are still present in such quarters (even though they no longer correspond to the maximum amplitude of the gradient). In Fig. 5, we indicate when the maximum GPS amplitude corresponds to the lowfrequency inflection points with open black diamonds overplotted over red dots.
Figure 5 demonstrates that positions of the inflection points are stable, and the mean value of positions over all considered quarters are basically the same for the VIRGO and TIM data. We construe this as the proof that the GPS method works for the Sun.
Having shown that the position of the inflection point is stable for the Sun, we could then use the Sun to calibrate the GPS method and apply α_{Sun} value to other Sunlike stars, independently of the simulations presented in Paper I. At the same time, it is reassuring to see that the theoretical α_{Sun} value found in Paper I leads to a reasonable value of the solar rotation period. Indeed, after applying the α_{Sun} calibration factor from Paper I to the position of the inflection point at 4.17 ± 0.59 d for VIRGO data and 4.17 ± 0.59 d for TIM data (where 0.59 d and 0.57 d corresponds to 2σ of the observed distribution of values for all quarters), we obtain 26.4 ± 3.7 d and 26.4 ± 3.6 d, respectively.
We note that the uncertainty of the rotation period is calculated differently here than in Sects. 3.2–3.4, where it was defined via the uncertainty of the theoretical α_{Sun} value (σ_{α}). This σ_{α} uncertainty is brought about by the dependence of the inflection point position on the specific realisation of emergences of magnetic features. In contrast, in this section, we calculate the rotation period and its uncertainty as: (2)
where the σ_{HFIP} value is the standard deviation of the observed distribution of inflection point positions (blue points in Fig. 5). As well as σ_{α}, σ_{HFIP} accounts for the uncertainty due to the randomness in emergences of magnetic features (so that it does not make sense to account for σ_{α} in Eq. (2)). In addition, σ_{HFIP} also accounts for the noise in the TSI data. Hence, we utilise Eq. (2) and σ_{HFIP} here to estimate the uncertainty of the rotation period with the GPS method.
Figure 6 compares the performance of the ACF, GLS, PS, and GPS methods. It shows one value of the rotation period per 90day quarter determined with the four methods for 21 yr of TSI by VIRGO (top panels), and 15 yr of TSI by TIM (bottom panels). In addition, the pale orange bar denotes a range between 23 and 34 days, which we take to be the success range for determining the solar rotation period (which we expect would lie in the range 27–30 d within 4 d error bars). One can see that the distribution of retrieved periods is similar for data from both instruments.
The values obtained by ACF (Maroon dots) range between 2.18 days and 57.6 days for VIRGO data (see left panels of Fig. 6). There are quarters where ACF can accurately detect solar rotation period, but there are many other quarters where the method fails. Overall, the rotation values retrieved by the ACF lie between 23 and 34 days for 32 out of the 86 VIRGO quarters, meaning the ACF method has a success rate of 37.2% when applied to the VIRGO data. For the TIM dataset, the ACF obtained values range from 4.8 to 59 days. The success rate is 36.1%, as shown in Table 1.
The secondfromleft panels in Fig. 6 show the performance of the PS method. The retrieved rotation periods are in between 10.7 and 54.7 days for the VIRGO data, and between 9.8 days and 50.8 days for the TIM data. The success rates for period determination are 51.2 and 54.1% for VIRGO and TIM, respectively.
Using GLS, we were able to retrieve closer solar periodicities per quarter, as is shown in the right panels in Fig. 6. GLS successfully retrieves the solar rotation period for 55% of the quarters using VIRGO data and 47% for TIM data. We notice lower scatter for the TIM dataset, with the returned rotation periods lying in a range between 8.6 and 50.0 days. For VIRGO data, we obtain a rotation period ranging from 8.9 to 56.7 days.
In the right panels of Fig. 6 and Table 1, we show values retrieved with the GPS method. One can seethat the GPS method results in less scatter for the retrieved values of the rotation period, finding rotation periods in the range of [17.8–41.4] days for VIRGO data, and [16.6–34.6] days for the TIM dataset. The GPS method achieves success rates of 88.4 and 82.1% for the VIRGO and TIM data, respectively. The regularity of the signal allows us to analyse the distribution of the highfrequency inflection point and its behaviour over time.
Regular stellar photometric observations are normally performed during unknown stellar activity stages. To characterise the detectability of the rotation period for the different activity timespans, we tested the performance of the GPS method in comparison with the ACF, GLS, and PS methods for periods of relatively low and high solar activity. For this, we used only VIRGO data, since it covers both periods of very high and low solar activity.
To mimic Kepler observations, we split the entire period of VIRGO observations into five segments K_{1} –K_{5} (the length of the segments roughly corresponds to the total duration of the four years of Kepler observations) and the remaining 712day segment K_{6} (see verticaldashed orange lines in Figs. 3–5). We then subdivided each K_{n} segment in 17 quarters of 90 days each, and analysed them separately.
The performance of all four methods is compared in Fig. 7 for segment K_{4} (corresponding to a period of low solar activity) and K_{5} (high solar activity). We observe that for the low period of activity, K_{4}, the values obtained per quarter for PS and GLS show less scatter than for the values shown during high levels of solar activity in the K_{5} segment. The ACF values show similar scattered rotation values for both high and low levels of solar activity. The GPS method recovered rotation period values closer to the solar rotation period range for both K_{4} and K_{5} segments.
Figure 8 shows the rotation period detected with the GPS method per quarter for K_{1} –K_{6} segments. While there is some scatter in the values of the rotation periods deduced from the analysis of segments K_{1} –K_{6} (in particular, values obtained in segment K_{2} are four days lower than sidereal Carrington rotation period), Fig. 8 indicates that the solar rotation period can be successfully retrieved by GPS for all K_{n} analysed segments.
Fig. 4
Same as Fig. 1 but for 15 yr [2003.01.24–2018.04.24] of TSI data from SORCE/TIM. In total, 5508 days are considered. This corresponds to about 82 632 data points at a cadence of 1.6 hours. Orange lines represent the same Keplerlike timespan K_{n} as is shown in Fig. 3. The GLS, ACF, and PS methods do not show a definitive detection signal of the rotation period. GPS method shows a prominent peak at 4.12 days corresponding to the solar rotation period value of 26.1 days. 
Fig. 5
Topleft panel: positions of the inflection points for the 86 (90day) quarters of the VIRGO data. Red dots represent the lowfrequency inflection points (LFIP), blue dots represent the highfrequency inflection points (HFIP), black diamonds indicate inflection points with maximum GPS value (see text). Topright panel: distribution of the maximum inflection point positions. Bottom panels: same as top panels but for 60 quarters of 90days using the TIM data. Orange lines in left panels indicate splitting of the VIRGO and TIM datasets into Keplerlike timespan K_{n}, as in Fig. 3. 
Fig. 6
Values of the rotation periods per 90day quarters returned by the ACF, PS, GLS, and GPS methods. The analysis is performedfor the VIRGO (top panels) and TIM (bottom panels) data. Pale orange shaded areas cover the period range of [23–34] days (see Sect. 3.5 for details). Information about the ranges of rotation period values obtained by each method for different instruments is shown near the top of each panel. 
Compilation of rotation period analysis for the four different methods implemented.
3.6 The impact of white noise in the inflection point position
In Sect. 3.5, we processed solar TSI data to represent the Sun as it would be observed with the timespan of Kepler observations. We considered two TSI datasets, one obtained by SoHO∕VIRGO, another by TIM∕SORCE. While the noise level in these two datasets is rather different (Kopp 2016), the positions of the inflection points are basically independent of the dataset (see, Figs. 5 and 6). This implies that our analysis is only weakly affected by the noise in the TIM and VIRGO data. At the same time, the noise level in Kepler data is normally significantly higher than those in the solar data.
Solar and stellar light curves are recorded differently, so that the noise sources are also substantially different. The TSI is measured using radiometers, while stellar photometric measurements are performed using charged coupled devices (CCD). Photon detection by a CCD is a statistical process associated with several sources of noise, which can be generally approximated by Gaussian white noise.
To assess the impact of noise on the position of the inflection point, we artificially added white noise to the VIRGO TSI data. The amplitude of the noise was chosen following the expected dependence (see Van Cleve & Caldwell 2016) of the noise value per specific Kepler magnitude Kp, which is the measured source intensity observed through the Kepler bandpass.
Figure 9 shows the position of the highfrequency inflection point as a function of the white noise level. The colours of the dots represent theKepler magnitude Kp. For each Kp value (and, corresponding amplitude of the white noise), we calculate five realisations of the noise, add it to the entire VIRGO dataset shown in Fig. 3, and calculate the position of the inflection point, as in Sect. 3.4. One can see that the inflection point shifts to lower frequencies when the noise level is increased.
We tested a simple method for mitigating such a shift. Namely, we utilised the fact that the power spectrum flattens at high frequencies. The power in the flattened part represents the superposition of white noise and granulation (Shapiro et al. 2017). We calculated the mean power between periods of one hour and one day and subtracted this single value from the entire powerspectrum. Then, we recalculated the position of the inflection point. These corrected positions of the inflection points are represented by grey filled star symbols in Fig. 9. One can see that our method is reasonably effective until the noise level reaches about 300–400 ppm, which corresponds to about a Kp magnitude of 14 for a onehour cadence light curve.
Beyond a level of introduced noise of 500 ppm, the error in the estimation of the real high frequency inflection point location starts to become considerable, even though the location of the high frequency inflection point still gets more accurate after the correcting noise procedure, see grey star symbols in Fig. 9. For example, for a star with the level of noise expected for a Kp magnitude of 15, the mean scatter in the high frequency inflection point value corresponds to 0.83 days, which yields a 5.25day deviation in the rotation period value.
Fig. 7
Values of the solar rotation period per 90day quarters of the VIRGO data returned by the ACF, PS, GLS, and GPS methods (from left to right panels). Shown are the K_{4} [2007:09:12–2011:07:27] and K_{5} [2011:07:28–2015:06:11] Keplerlike timespan, corresponding to low and high levels of solar activity, respectively. Pale orange coloured areas indicate the period range of [23–34] days. 
Fig. 8
Same as Fig. 7 but for all Keplerlike (K_{1}–K_{6}) timespans of the VIRGO data. Only GPS values are shown. Blue dots represent the estimation of the rotation period per quarter obtained by GPS. 
Fig. 9
Position of the highfrequency inflection point calculated for the entire VIRGO dataset (as in Fig. 3) as a function of Keplerlike white noise added to the original VIRGO data. The expected level of white noise is a function of the stellar Kepler magnitude, Kp (see main text for more information). The colour of the dots (see colour panel in the top of the figure) indicates Kp magnitude corresponding to the expected level of the white noise. Grey star symbols represent the position of the inflection point after the noise correction (see text for details). There are five different realisations of noise per each value of the Kp magnitude. When the level of noise in the LC is lower than 100 ppm, around Kp = 12, the values of IP values are overlapped and appear as a single point in the plot. 
4 GPS and skewness relation
Distinguishing between facular and spotdominated regimes of brightness variability is important for understanding the structure of the stellar magnetic field and for identifying biases in determination of stellar rotation periods.
While solar rotation variability is predominantly spotdominated, there are also periods of facular domination (see Sect. 3.3). In Sects. 3.2 and 3.3, we demonstrated that the GPS spectrum has a different profile depending on whether variability is facular or spotdominated. In this section, we show that in the solar case these two regimes can also be distinguished based on the skewness of the distribution of TSI values. This suggests that skewness can be a good indicator of the variability regime for lowactivity stars like the Sun.
For a dataset of TSI values, the skewness can give us valuable information about the distribution of maximum and minimum values. In other words when we have a decrease in the intensity due to spots, the distribution will be skewed preferentially towards the left side (i.e. to the lower values) of the maximum peak of the distribution. When an increase in intensity is registered in the light curve due to the presence of brighter facular regions, the skewness will shift to the right side (i.e. to the higher values) of the distribution. The skewness of a distribution can tell us about its degree of symmetry.
In order to analyse the relation between skewness and the regime of solar variability we calculate skewness of the TSI values in each of the 90day quarters introduced in Sect. 3.5. In the upper left panel of Fig. 10 we show the 21yr span of VIRGO TSI data. In the middle left panel we illustrate the location of the maximum inflection point (see Sect. 3.4) per quarter. As discussed in Sects. 3.2 and 3.3, the maximum inflection point (MIP)corresponds to the lowfrequency inflection point for faculaedominated regimes, and the highfrequency inflection point for spotdominated regimes. The bottomleft panel shows the skewness values for all 90day quarters. One can see that periods of low solar activity, when TSI variability is mainly brought about by faculae (see, e.g. discussion in Shapiro et al. 2016), simultaneously correspond to positive skewness and maximum GPS value reached at the lowfrequency inflection point. The observed scatter in skewness values is a higher scatter than that in the MIP by GPS. This implies that the inflection point analysis provides a better indication of when the LC is mainly drawn by spot or facular components.
In the right panel of Fig. 10, we show the distribution of TSI values for the entire VIRGO dataset. The skewness for the entire dataset, Sk_{E} = 0.23, is positive. This is because the skewness value of the entire dataset is affected by the TSI variability on the timescale of the 11yr cycle, which is faculaedominated. To remove the contribution from the 11yr variability, we can also calculate skewness by averaging all individual values per 90day quarters. Since rotation TSI variability is mainly spotdominated, we then get a negative value of Sk_{Q} = −0.69.
Figures 11 and 12 show the skewness analysis for minimum and maximum activity segments, K_{4} and K_{5}, respectively. The comparison between the light curve segmented in quarters and its respective skewness values are in agreement with conclusions drawn analysing the entire VIRGO dataset. In particular, one can see that quarters with prominent positive excursions of brightness caused by faculae correspond to positive skewness, while quarters with negative excursions caused by spot correspond to negative values of skewness. Clearly, the skewness of brightness distribution in the entire K_{4} segment (corresponding to the minimum of solar activity) is also positive, while skewness values for the K_{5} segment with higher value of activity is negative. We will extend the combined, skewness and GPS, analysis to stars observed by Kepler and TESS in the forthcoming publications.
Fig. 10
Top panel of left image: 21yr of TSI data gathered by VIRGO. Middle panel of left image: positions of the maximum inflection point (MIP) per quarter. Bottom left: skewness values per quarter. Right: distribution of the TSI values shown in the topleft panel. We list mean, maximum, and minimum TSI values, skewness for the entire dataset (Sk over thewhole time series), the mean of individual skewness values calculated per quarter (Sk quarters, see text for more details), kurtis (Ku), standard deviation (SDEV), and mean deviation (MDEV). 
5 Discussion and summary
The determination of rotation periods of stars with activity levels similar to those of our Sun is a challenging task, even when using highquality data from spaceborne photometric missions. In Shapiro et al. (2020), we proposed the GPS method specifically aimed at the determination of periods in old inactive stars, like our Sun. The main idea of the method is to calculate the gradient of the power spectrum of stellar brightness variations and identify the inflection point, which is the point where the concavity of the power spectrum changes its sign. The stellar rotation period can then be determined by applying a scaling coefficient to the position of the inflection point.
We applied the GPS method to the available measured records of solar brightness (specifically the total solar irradiance) and compared its performance to that of other methods routinely utilised for the determination of stellar rotation periods. There are time intervals when a solar light curve has a regular pattern, where the GPS and other methods return the correct value of solar rotation period. These intervals correspond to low values of solar activity when variability is either brought about by longliving faculae or nested sunspots. However, most of the time, solar brightness variations are attributed to superposition of simultaneous contributions from several bright and dark magnetic features with random phases. We show that this leads to the failure of other methods to identify a clear signal of the rotation period. At the same time, the GPS method still allows an accurate determination of the rotation period of the Sun independently of its activity level and the number of features contributing to brightness variability and of the ratio of facular to sunspot area.
In particular, we show that GPS method returns accurate values of the solar rotation period for most of the timespan of SoHO/VIRGO and SORCE/TIM measurements, with the exception of several intervals affected by the absence of data. We found that when the entire 21yr VIRGO and 15yr TIM datasets are split in Keplerlike 90day quarters, and inflection points are calculated for each of the quarters, the maximum of the distribution of the inflection point positions peaks at 4.17 ± 0.59 days for the VIRGO dataset, and4.17 ± 0.57 days for the TIMdataset (see Fig. 5). This results in a determination of the solar rotation period of 26.4 ± 3.7 days and 26.4 ± 3.6 days for VIRGO and TIM datasets, respectively. In a series of typical Keplerlike observations of the Sun, the GPS method can correctly determine the rotation period in more than 80% of the cases, while this value is about 50% for GLS and below 40% for ACF.
Typically, solar variability on timescales up to a few months is spotdominated. However, there are also time intervals when it is faculaedominated (see, e.g. Fig. 2). We show that these regimes can be distinguished from the GPS profile thanks to substantially different centretolimb variations of facular and spot contrasts. Furthermore, the two regimes can be separated by analysing the comparison between the inflection point location from GPS and the skewness of light curves: the bright faculae lead to positively skewed light curves and a stronger signal at the lowfrequency inflection point, while dark spots lead to negatively skewed light curves and a dominant signal at the lowfrequency inflection point. However, the skewness values in Fig. 10 show higher scatter than the IP by GPS. This implies that the IP by GPS provide a better indication of when the LC is mainly drawn by spot or facular components.
We construe the success of the GPS method in the solar case as an indication that it can be applied to reliably determine rotation periods in lowactivity stars like the Sun, where other methods generally fail. Furthermore, our analysis demonstrates that photometric records alone can be used to identify the regime of stellar variability, that is, whether it is dominated by the effects of spots or of faculae. In subsequent papers we will apply the GPS method to determine rotation periods and regimes of the variability of Kepler and the longterm follow up of TESS stars.
Fig. 11
Skewness analysis for lowactivity Keplerlike timespan K_{4}. Upperleft panel: VIRGO TSI time series during the K_{4} season, which consists of 1415 days emulating 17 Kepler quarters. Lowerleft panel: skewness values per quarter. Right: same as right panel in Fig. 10, but showing only TSI values in K_{4} season. 
Acknowledgements
We would like to thank the referee for the constructive comments which helped to improve the quality of this paper. The analysis presented in this Paper I based on new scale version 6.4 observations collected by the VIRGO Experiment on the cooperative ESA/NASA Mission SoHO, provided by the VIRGO team through PMOD/WRC, Davos, Switzerland. In addition orbitalaveraged version 17, level 3.0 data from the Total Irradiance Monitor (TIM) on the NASA Earth Observing System (EOS) SOlar Radiation and Climate Experiment (SORCE) where analysed. This work was supported by the International MaxPlanck Research School (IMPRS) for Solar System Science at the University of Göttingen and European Research Council under the European Union Horizon 2020 research and innovation program (grant agreement by the No. 715947). M.O. acknowledges the support of the Deutsche Forschungsgemeinschft (DFG) priority program SPP 1992, Exploring the Diversity of Extrasolar Planets (RE 1664/171). E.M.A.G. and M.O. also acknowledge the support of the FCT/DAAD bilateral grant 2019 (DAAD ID: 57453096). Financial support was also provided by the Brain Korea 21 plus program through the National Research Foundation funded by the Ministry of Education of Korea and by the German Federal Ministry of Education and Research under project 01LG1209A. We would like to thank the International Space Science Institute, Bern, for their support of science team 446 and the resulting helpful discussions.
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See 2006_TSI_Movie.mp4 online and at: ftp://laspftp.colorado.edu/3month/kopp/
See 2007_TSI_Movie.mp4 online and at: https://spot.colorado.edu/~koppg/TSI/
Movies
2006_TSI_Movie (Access here)
2007_TSI_Movie (Access here)
All Tables
Compilation of rotation period analysis for the four different methods implemented.
All Figures
Fig. 1
Example of spotdominated total solar irradiance (TSI) variability. Panel a: SORCE/TIM measurements from 28Nov.2006 to 28Feb.2007. Purple and orange curves indicate sinewave functions with periods of 27.3 d (solar rotation period deduced with the generalised Lomb Scargle periodogram method, GLS) and 26.6 d (solar rotation period deduced with the gradient of power spectra method, GPS). Panels b and c: corresponding GLS periodogram and autocorrelation function, ACF, respectively. Panel d: global wavelet power spectrum, PS, calculated with the 6th order Paul wavelet. Panel e: gradient of the power spectrum plotted in panel d. Blue asterisks in panels b and c represent positions of the peaks in theGLS and ACF. Green dotted lines in panel e indicate the high and lowfrequency inflection points. 

In the text 
Fig. 2
Same as Fig. 1 but for the time interval of faculaedominated TSI variability from 21Dec.2007 to 22Mar.2008. The purple dashed curve in panel a represents a sine wave function with a period of 26.4 d corresponding to the solar rotation period deduced with the GLS method. The orange curve in panel a is a sinusoidal function with a period of 27.8 d corresponding to the solar rotation period deduced with the GPS method. 

In the text 
Fig. 3
Same as Fig. 1 but for 21 yr [1996.01.28–2017.05.23] of TSI data from SoHO/VIRGO. In total, 7787 days are considered. This corresponds to about 186 889 data points at an hourly cadence. The GLS, ACF, and PS methods do not show a clear signal of the rotation period. GPS shows a prominent peak at 4.15 days, resulting in a solar rotation period value of about 26.3 days (see text for details). Orange vertical lines and marks in panel a represent a splitting of the VIRGO dataset into Keplerlike timespan K_{n}, each one representing the full lifetimes of the Kepler mission, around 4 yr (see Sect. 3.5). 

In the text 
Fig. 4
Same as Fig. 1 but for 15 yr [2003.01.24–2018.04.24] of TSI data from SORCE/TIM. In total, 5508 days are considered. This corresponds to about 82 632 data points at a cadence of 1.6 hours. Orange lines represent the same Keplerlike timespan K_{n} as is shown in Fig. 3. The GLS, ACF, and PS methods do not show a definitive detection signal of the rotation period. GPS method shows a prominent peak at 4.12 days corresponding to the solar rotation period value of 26.1 days. 

In the text 
Fig. 5
Topleft panel: positions of the inflection points for the 86 (90day) quarters of the VIRGO data. Red dots represent the lowfrequency inflection points (LFIP), blue dots represent the highfrequency inflection points (HFIP), black diamonds indicate inflection points with maximum GPS value (see text). Topright panel: distribution of the maximum inflection point positions. Bottom panels: same as top panels but for 60 quarters of 90days using the TIM data. Orange lines in left panels indicate splitting of the VIRGO and TIM datasets into Keplerlike timespan K_{n}, as in Fig. 3. 

In the text 
Fig. 6
Values of the rotation periods per 90day quarters returned by the ACF, PS, GLS, and GPS methods. The analysis is performedfor the VIRGO (top panels) and TIM (bottom panels) data. Pale orange shaded areas cover the period range of [23–34] days (see Sect. 3.5 for details). Information about the ranges of rotation period values obtained by each method for different instruments is shown near the top of each panel. 

In the text 
Fig. 7
Values of the solar rotation period per 90day quarters of the VIRGO data returned by the ACF, PS, GLS, and GPS methods (from left to right panels). Shown are the K_{4} [2007:09:12–2011:07:27] and K_{5} [2011:07:28–2015:06:11] Keplerlike timespan, corresponding to low and high levels of solar activity, respectively. Pale orange coloured areas indicate the period range of [23–34] days. 

In the text 
Fig. 8
Same as Fig. 7 but for all Keplerlike (K_{1}–K_{6}) timespans of the VIRGO data. Only GPS values are shown. Blue dots represent the estimation of the rotation period per quarter obtained by GPS. 

In the text 
Fig. 9
Position of the highfrequency inflection point calculated for the entire VIRGO dataset (as in Fig. 3) as a function of Keplerlike white noise added to the original VIRGO data. The expected level of white noise is a function of the stellar Kepler magnitude, Kp (see main text for more information). The colour of the dots (see colour panel in the top of the figure) indicates Kp magnitude corresponding to the expected level of the white noise. Grey star symbols represent the position of the inflection point after the noise correction (see text for details). There are five different realisations of noise per each value of the Kp magnitude. When the level of noise in the LC is lower than 100 ppm, around Kp = 12, the values of IP values are overlapped and appear as a single point in the plot. 

In the text 
Fig. 10
Top panel of left image: 21yr of TSI data gathered by VIRGO. Middle panel of left image: positions of the maximum inflection point (MIP) per quarter. Bottom left: skewness values per quarter. Right: distribution of the TSI values shown in the topleft panel. We list mean, maximum, and minimum TSI values, skewness for the entire dataset (Sk over thewhole time series), the mean of individual skewness values calculated per quarter (Sk quarters, see text for more details), kurtis (Ku), standard deviation (SDEV), and mean deviation (MDEV). 

In the text 
Fig. 11
Skewness analysis for lowactivity Keplerlike timespan K_{4}. Upperleft panel: VIRGO TSI time series during the K_{4} season, which consists of 1415 days emulating 17 Kepler quarters. Lowerleft panel: skewness values per quarter. Right: same as right panel in Fig. 10, but showing only TSI values in K_{4} season. 

In the text 
Fig. 12
Same as Fig. 11 but for highactivity Keplerlike timespan K_{5}. 

In the text 
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