© P. Schöfer et al. 2022

1 Introduction

Current radial velocity (RV) surveys such as the Calar Alto high-Resolution search for M dwarfs with Exo-earths with Near-infrared and optical Échelle Spectrographs (CARMENES; Quirrenbach et al. 2018) are searching for exoplanets that reveal themselves through periodic RV variations of their host stars down to 1m s⁻¹ or less. The RV is derived from the observed wavelengths of spectral lines and can therefore be affected by line profile changes caused by star spots or plages. In consequence, inhomogeneous distributions of stellar surface features that lead to variations over the course of one rotation, the evolution of active regions, and activity cycles can introduce quasi-periodic RV signals of the same order of magnitude as prospective planetary signals (e.g. Saar & Donahue 1997; Hatzes 2013; Barnes et al. 2014, 2015).

To identify activity-induced RV signals, the variations on the stellar surface need to be monitored using indicators that are complementary to the RV. Both a photometric approach using brightness variations induced by active regions (e.g. Kiraga & Stepien 2007; Irwin et al. 2011; Newton et al. 2016; Suárez Mascareño et al. 2016; Díez Alonso et al. 2019) and a spectroscopic approach based on variations of chromospheric emission lines such as Ca ii H&K and Hα (e.g. Noyes et al. 1984; Suárez Mascareño et al. 2015, 2018; Fuhrmeister et al. 2019) have been used to uncover rotation periods and long-term activity cycles of late-type stars. Over time, evolution and migration of active regions change their distribution on the stellar surface. This fact increases the difficulty of identifying the rotation period, as, for example, photometric light curves often show one dip per rotation for some time, but two dips at other times (Basri & Nguyen 2018). Since active regions may predominantly appear at different latitudes at different times, the detected period may also be subject to variations caused by differential rotation (Donahue et al. 1996). More recent works show that other effects can impact the light curve in similar ways to differential rotation (e.g. Basri & Shah 2020).

An activity study of 331 M dwarfs of the CARMENES sample was presented in Schöfer et al. (2019). That work explored the behaviour of eight chromospheric activity indicators and four photospheric absorption band indices for different spectral subtypes and the correlations among these indicators. In addition, three best-fit periodicities for each indicator were derived from generalised Lomb-Scargle (GLS) periodograms (Zechmeister & Kürster 2009) and compared to photometric rotation periods from the literature. This revealed that the photospheric titanium oxide λ7050 Å absorption band head, the middle line of the Ca ii infrared triplet, and the chromospheric Hα emission line most commonly show rotational variation, and that 11% of the stars with previously known, mostly photometrically derived rotation periods, appear to show a rotational signal in more than two activity indicators. Lafarga et al. (2021) studied rotational signals in a different set of ten spectroscopic indicators in the visible-light range for a subset of 98 CARMENES sample stars and found that any given indicator shows a signal most commonly in a specific mass and activity regime. They also indentified five stars that are representative of the types of rotational signals seen in their sample. A detailed study of periodicities and also correlations between spectroscopic indicators was carried out by Jeffers et al. (2022) for the very active M dwarf EV Lac.

In this work, we analysed the rotational variation of 24 spectroscopic indicators in both the visible-light and the near-infrared range in more detail, including the RV, for four M dwarfs that show particularly clear signals. Ross 318, YZ CMi, and EV Lac are representative examples for the different groups defined by Lafarga et al. (2021), while TYC 3529-1437-1 switched groups over time. We describe the target stars in Sect. 2, and our data and indicators in Sect. 3. The method for our periodicity analysis is described in Sect. 4, the results are presented and discussed in Sects. 5 and 6. Finally, in Sect. 7 we summarise our main conclusions.

2 Target Stars

The CARMENES survey focuses on the brightest M dwarfs of each spectral sub-type from M0.0 V to M9.0 V that are observable from the Calar Alto Observatory. Since the start of the survey in January 2016, a sample of more than 380 stars has been observed (Schweitzer et al. 2019; Sabotta et al. 2021). While known binaries with a separation of less than 5 arcsec were excluded, there was no target pre-selection based on further criteria such as stellar activity and metallicity. Therefore, the CARMENES sample contains wide ranges of stellar parameters such as mass, rotation, and chromospheric activity level, as is shown in Fig. 1.

Schöfer et al. (2019) reported that 15 out of 133 CARMENES sample stars with a rotation period P_rot > 1 d known at that date showed this period as one of the three best-fit periodicities in the variations of three or more chromospheric or photospheric activity indicators. However, the periodograms for 11 of these stars generally showed a forest of peaks, and the frequency corresponding to the rotation period was only the second or third strongest signal in the periodogram. It is possible that the strongest signals were introduced by the evolution of active regions similarly to the RV signals in Nava et al. (2020). Several of these stars were observed fewer than 40 times, and thus did not ensure a sufficient sampling rate for a detailed period analysis. In contrast, the four remaining stars were already observed more than 40 times and the rotation period clearly appeared as the most significant frequency in more than two activity indicators. In this work, we studied the rotational variation in the activity indicators of these four stars in more detail. The four stars are highlighted in Fig. 1 and listed in Table 1.

Three of our target stars are representative examples for the groups of different rotational signals defined in Lafarga et al. (2021): Ross 318 has a long rotation period above 50 d, for which signals are hard to detect, particularly in chromospheric emission lines (Fuhrmeister et al. 2019), YZ CMi was the example for a star with clear signals at the rotation period, and EV Lac was the example for multiple signals related to the rotation period. TYC 3529-1437-1 showed clear signals at the rotation period in Schöfer et al. (2019), but signals at multiple related frequencies in Lafarga et al. (2021). As we selected the stars with the most significant signals at known rotation periods, this work does not include examples for signals at a previously unknown rotation period or with insignificant signals.

Despite the small sample size, our target stars cover a diverse parameter space. Ross 318 is an inactive mid-type M dwarf with no Hα emission, as suggested by its long rotation period, and has neither a large RV scatter nor a detectable projected rotational velocity. TYC 3529-1437-1 is an early-type M dwarf with a significantly shorter rotation period of 15.8 d and moderate chromospheric activity. In addition, it shows a large, periodically modulated RV scatter with amplitude of A_RV = 11m s⁻¹, while its projected rotational velocity is below the detection limit of 2km s⁻¹ (Reiners et al. 2018). By contrast, the mid-type M dwarfs YZ CMi and EV Lac both show large RV scatters and a detectable projected rotational velocity, and they are therefore part of the CARMENES sample of active RV-loud stars (Tal-Or et al. 2018). Both stars are among the most active stars in the CARMENES sample with normalised Hα luminosities log(L_Hα/L_bol) ≈ –3.6 and rotation periods below 5 d. For YZ CMi, Baroch et al. (2020) used CARMENES spectra and photometric data to constrain star-spot coverage and temperatures and to find a convective redshift.

The rotation periods were derived by Díez Alonso et al. (2019) using photometric time series from the ASAS-SN (Ross 318), MEarth (YZ CMi), and SuperWASP (TYC 3529-1437-1, EV Lac) surveys. Their results are in agreement with previously reported values of 2.78 d for YZ CMi (Chugainov 1974) and 4.376 d for EV Lac (Contadakis 1995), and the more recently measured value of 49.4642 d for Ross 318 (Giacobbe et al. 2020) from independent photometric data. The value of 16.2578 d was also found for TYC 3529-1437-1 by Norton et al. (2007) in an earlier subset of the SuperWASP data. This indicates a high reliability of the rotation periods for these stars.

Fig. 1 Pseudo-equivalent width of Hα line after subtracting an inactive reference star spectrum (pEW′Hα) as a function of the stellar mass for 331 M dwarfs in the CARMENES sample, colour-coded by the rotation period Prot. Black points are stars with unknown Prot. The four stars studied in this work are marked with star symbols and their names. pEW′Hα and Prot are adopted from Schöfer et al. (2019) and references therein, stellar masses were generally computed from luminosities and effective temperatures as in Schweitzer et al. (2019) and Cifuentes et al. (2020), but with the latest Gaia EDR3 parallax and photometry (Gaia Collaboration 2021). More details on these updated masses will be provided in a forthcoming publication.

3 Observations

In this section, we describe the high-resolution spectroscopic CARMENES data and the photometric data obtained with the TESS satellite.

3.1 Carmenes

We analysed a total of 349 CARMENES Guaranteed Time Observations collected between 3 January 2016 and 17 January 2019. The number of observations for each of the four stars is given in Table 1. For all observations, the visible-light (VIS) and near-infrared (NIR) channels of CARMENES were used simultaneously. After spectral reduction with the CARACAL pipeline (Caballero et al. 2016) using flat-relative optimal extraction (Zechmeister et al. 2014), we derived 24 spectroscopic indicators from the spectra. These include pseudo-equivalent widths after subtracting an inactive reference star spectrum of the same spectral type (pEW′, Schöfer et al. 2019) of spectral lines with a chromospheric component, including He i D₃, Na i D doublet, Hα, and the Ca ii infrared-triplet lines (IRT-a, -b, and -c from shortest to longest wavelength) in the VIS channel, and He i λ10833 Å and Paβ in the NIR channel. Next, indices of photospheric absorption bands, including TiO 7050, TiO 8430, VO 7436, and VO 7942 (Schöfer et al. 2019) are present. We also derived a differential line width (dLW, Zechmeister et al. 2018) in both the VIS and the NIR channel, as well as radial velocity (RV VIS and RV NIR) in both spectrograph channels calculated using SERVAL (Zechmeister et al. 2018) with a correction for an instrumental nightly zero-point offset, as described by Trifonov et al. (2018) and Tal-Or et al. (2019), and the corresponding chromatic indices (CRX VIS and CRX NIR, Zechmeister et al. 2018), which measure the wavelength dependence of the RV. Finally, we included a full width at half minimum (FWHM), contrast, and bisector inverse slope (BIS; Queloz et al. 2001) of the cross-correlation function (CCF) with a weighted binary mask from co-added spectra of the star for both spectrograph channels as described in Lafarga et al. (2020).

While the same VIS channel spectra were also used in the study of rotational signals in a larger sample of stars by Lafarga et al. (2021), we extended the set of indicators by including the NIR channel spectra and the photospheric absorption band indices and used a different measure for the chromospheric lines.

3.2 Tess

All four stars were also observed by the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015): Ross 318 in observing sectors 18, 19, 24, and 25; YZ CMi in sectors 7 and 34; TYC 3529-1437-1 in sectors 14, 25, 26, 40, and 41; and EV Lac in sector 16. While these photometric observations are not contemporaneous with the CARMENES observations, we still use them for comparison.

We retrieved the TESS light curves from the Mikulski Archive for Space Telescopes¹ and used the simple aperture photometry (SAP) flux. For TYC 3529-1437-1 in observing sector 14 and YZ CMi in sector 34, we additionally used the regression corrector from Lightkurve (Lightkurve Collaboration 2018) to remove instrument systematics that dominated the SAP flux.

4 Analysis

We used GLS periodograms (Zechmeister & Kürster 2009) to reveal periodic changes in all our measured indicators. To mitigate the impact of flaring events or bad spectra, we rejected any values that deviate from the respective average values by more than twice the standard deviation (2σ clipping). We also rejected He i λ10833 Å and Paβ measurements of spectra with an observed RV that shifts the neighbouring strong telluric lines into the respective line windows. In the case of YZ CMi, this reduced the number of used measurements of these two indicators by more than 50%.

Because the stars were usually observed at most once per night and all four stars have rotation periods P_rot = 1/f_rot longer than one day, we limited the periodograms to frequencies up to 1 d⁻¹. The nightly sampling introduces alias signals at frequencies f′ = 1 d⁻¹ - f for any signal at frequency f. For the two slower rotating stars, we therefore only show the periodograms (Figs. A.1 and A.2) up to 0.5 d⁻¹ for clarity in the relevant frequency range because the signals at higher frequencies are likely alias signals. We quantified the significance of a signal at frequency f by calculating the probability p(f) that a power higher than the observed GLS power at f was produced by Gaussian noise. We considered a signal significant if its significance exceeds 3σ, that is, log p(f) < −2.87.

In addition to these overall GLS periodograms of the full dataset for each star, we calculated periodograms of subsets of consecutive data points to test the stability of the signals in the overall periodogram and to investigate whether the most significant signals appear at the same frequencies all the time. Our datasets are not evenly sampled in time, but they contain gaps of up to several months, during which the star is not well observable from Calar Alto. Therefore, we split the dataset for each star into two subsets, one before and one after a significant sampling gap. In the case of EV Lac, we further split the subset after the sampling gap into two parts, the second of which contains observations with a higher cadence than the first. Our subsets for EV Lac include, but are not limited to the observations used for the low-resolution Doppler imaging maps 1–4, map 5, and maps 6–8 in the analysis by Jeffers et al. (2022), respectively. A group of six spectra of YZ CMi between October 2017 and January 2018 with a large gap from the preceding subset is too small for further analysis and thus excluded. Similarly, a single isolated spectrum of EV Lac in January 2016 is excluded from further analysis.

5 Spectroscopic periodicities and their stability and evolution

In this section, we present and discuss the results of our periodogram analysis described in the previous section. For each star, we start with the rotational signals in the full datasets and then assess their stability and evolution using the data subsets.

5.1 Ross 318

We show the GLS periodograms of all our indicators for the slow-rotating (P_rot = (51.5 ± 2.6) d) M3.0 V star Ross 318 in Fig. A.1 and tabulate the mean values, standard deviations, and probabilities log p(f_rot) for each indicator in Table A.1. As reported by Schöfer et al. (2019) and Fuhrmeister et al. (2019), the chromospheric lines show only small variations in the least active M dwarfs, and, in consequence, solid detections of P_rot ≳ 50 d using these indicators are elusive. Still, we find that Hα, two of the Ca ii IRT lines, and He i λ10833 Å show the highest periodogram peak with a significance of more than 3σ at f_rot, while Ca ii IRT-a and Paβ show significant power but not the highest peak within the 3σ confidence interval of the rotation frequency.

TiO 7050, CRX VIS, and CCF FWHM in both spectrograph channels all show a significant highest periodogram peak at f_rot, whereas dLW and RV in both channels, VO 7436, CRX NIR, and CCF BIS NIR do not show the highest peak but still significant power at or close to f_rot. In the case of RV VIS, the strongest peak is separated from a weaker peak at f_rot by about 0.003 d⁻¹, and the window function suggests that it is a yearly alias. The signals in dLW, CRX, RV, and CCF FWHM in the VIS channel were also reported by Lafarga et al. (2021). We note that dLW, CRX, CCF FWHM, and CCF BIS show more significant signals in the NIR channel than in the VIS channel. This is unexpected because the NIR wavelength range contains substantially fewer spectroscopic features (Reiners et al. 2018), resulting in larger statistical fluctuations of the NIR parameters. A possible explanation is an asymmetric magnetic topology, which would result in larger variations of the NIR indicators because Zeeman broadening is stronger at larger wavelengths.

As for the full dataset, we calculated the mean values, standard deviations, and log p(f_rot) from the GLS periodogram of each indicator for two data subsets, and we show the results in Table A.1. Several indicators that show significant power at f_rot in the full dataset also show significant power in one or both subsets, with the second subset containing more data points and more significant signals in more indicators. The log p(f_rot) values are generally higher in the full dataset than in either subset, implying that the signals are rather stable and become stronger as more data points are considered.

5.2 TYC 3529-1437-1

For the moderately active M2.0 V star TYC 3529-1437-1, with a rotation period of 15.8 d, we show the GLS periodograms in Fig. A.2. Hα, two Ca ii IRT lines, and RV NIR all show the most significant signal at f_rot or its daily alias frequency. Na i D, Ca ii IRT-a, and RV VIS also show significant power, but not the highest periodogram peak at the rotation frequency. For CRX VIS and RV VIS, the most significant peak instead appears at its second harmonic 2 f_rot or the yearly alias thereof. We therefore tabulate not only the mean values, standard deviations, and log p(f_rot), but also log p(2 f_rot) in Table A.2. Lafarga et al. (2021) reported the same signals and an additional weak signal in CCF BIS VIS that does not fulfil our 3σ significance criterion.

The same quantities as for the full dataset are also tabulated for two data subsets in Table A.2. In the first subset, both He i lines, Hα, all three Ca ii IRT lines, TiO 7050, VO 7436, CRX VIS, RV VIS, and CCF FWHM NIR all show significant power at f_rot, whereas no indicator shows significant power at 2 f_rot. In contrast, in the second subset, TiO 7050, CRX VIS, RV VIS, and RV NIR show significant power at 2 f_rot, but only dLW NIR shows significant power at f_rot. The signals at f_rot in the first subset and 2 f_rot in the second subset are particularly strong in TiO 7050 and RV VIS, but their presence in only one subset leads to less significant or insignificant signals in the full dataset.

We also find that the standard deviations of most indicators are larger in the second subset, and the pEW′ values of chromospheric lines and the indices of photosperic bands are lower, indicating stronger chromospheric emission and stronger magnetic fields or a higher spot coverage in the photosphere. The stronger chromospheric emission can explain why the chromospheric indicators show no significant signals in the second subset, similarly to the more active YZ CMi not showing any significant signals in the chromospheric indicators as described in Sect. 5.4. A higher number of active regions in the photosphere leads more likely to a more symmetric distribution over the hemispheres that can cause the signals at 2 f_rot, similarly to more symmetric star-spot distributions being able to cause two dips per rotation in photometric light curves, while an asymmetric distribution causes one dip (e.g. Basri & Shah 2020). We explore this similarity further in Sect. 6.

5.3 EV Lac

The periodicities in activity indicators of the very active M3.5 V star EV Lac and their relation to the rotation period of 4.38 d were studied in great detail by Jeffers et al. (2022). We show the GLS periodograms of all our indicators in Fig. A.3 and tabulate the mean values, standard deviations, and probabilities log p(f_rot), log p(2 f_rot), and log p(3 f_rot) in Table A.3. We find that dLW and the CCF contrast in both channels, Hα, two of the Ca ii IRT lines, both TiO band indices, and CCF FWHM NIR show a significant peak at f_rot, and in several cases weaker but still significant peaks at the higher harmonics. All three Ca ii IRT lines actually show the most significant peaks at f_rot + 0.03 d⁻¹, which appear to be alias signals caused by the observation sampling. In contrast, CRX, RV, and CCF BIS in both channels, both He i lines, and CCF FWHM VIS show the most significant signal at 2 f_rot and less significant or insignificant power at f_rot or 3 f_rot.

We note that the indicators that predominantly show the rotation frequency are equally affected by all features on the hemisphere facing the observer (i.e. they depend on the ‘strength’ of all visible surface features). In contrast, most of the indicators that show mostly the second harmonic are affected in different ways by features on the part of the visible hemisphere that moves towards the observer, and on the part that moves away from the observer (i.e. they depend on the ‘positions’ of the signatures of the surface feature in the spectrum). This may explain the different behaviour of the indicators. However, the He i lines differ from the other chromospheric lines, so it is also possible that different distributions of the features tracked by each indicator cause their different behaviour. For further discussion including correlations between the indicators, we refer the reader to Jeffers et al. (2022). The broader periodicity analysis by Lafarga et al. (2021) found the same signals in the CARMENES VIS data and presented EV Lac as an example of a star that shows signals at multiple frequencies related to the rotation period.

In the analysis of the data subsets, for which the results are also tabulated in Table A.3, EV Lac shows the most complex behaviour of the four stars considered in this work. CRX, RV, and CCF BIS in both spectrograph channels, He i λ10833 Å, dLW VIS, and CCF FWHM VIS show significant signals at 2 f_rot, whereas TiO 7050 shows the most significant signals at f_rot and its daily alias in the first subset. In the second subset, CRX, RV, and CCF BIS in the VIS channel favour f_rot whereas He i λ10833 Å still favours 2 f_rot, and the other indicators show no significant power at the rotation frequency or its second and third harmonics. Finally, in the third subset, CRX and RV in both channels, He i λ10833 Å, and CCF FWHM and CCF BIS in the VIS channel show the most significant signals at 2 f_rot, whereas dLW and the CCF contrast in both channels, TiO 7050, and the chromospheric indicators except for the He i lines favour f_rot. Our results agree with the results reported by Jeffers et al. (2022) for their combined maps 6 and 7, which correspond to the first 26 data points in our third subset. They compared their results with a different subset of 26 randomly selected data points, for which they found similar results to the full dataset. In our larger first and third subsets, some indicators also show significant power at 3 f_rot.

There are no general trends in the standard deviations or mean values between the subsets except for a slight decrease in Ha emission. While this suggests a lower activity level in the third subset that could explain why most chromospheric indicators show significant rotation signals only in this subset, it remains unclear why different indicators favour different periods in the different subsets. We do not find similar behaviour for YZ CMi in Sect. 5.4, although both the normalised Hα luminosity and the average magnetic fields of EV Lac and YZ CMi are similar (Shulyak et al. 2019). However, the Zeeman-Doppler images from Morin et al. (2008) reveal a more complex magnetic topology for EV Lac, which results in a more complex surface feature distribution for EV Lac (e.g. Jeffers et al. 2022) than for YZ CMi (e.g. Baroch et al. 2020). This is likely the reason for the very different behaviour we find for these two stars.

5.4 YZ CMi

YZ CMi is one of the most active M dwarfs in the CARMENES sample, and with spectral type M4.5 V and a rotation period of 2.78 d, it is both the latest type and the fastest rotating star considered in this work. All GLS periodograms for YZ CMi are shown in Fig. A.4, and the mean values, standard deviations, and log p(f_rot) are given in Table A.4. CRX, RV, and the CCF parameters in both channels, the two TiO indices, VO 7436, and dLW VIS all show a significant peak at f_rot that is also the highest peak in the periodogram except for CCF contrast in the NIR. YZ CMi was therefore also the example of a star that shows clear signals at the rotation period in Lafarga et al. (2021). As CRX and RV are strongly anti-correlated (Baroch et al. 2020) for this star, we expect both these indicators to show similar signals.

Another noteworthy point is that none of the chromospheric line indicators show any significant signals at all. This might be caused by flaring events that lead to sporadic, non-periodic variations in the chromospheric emission lines on timescales ranging from minutes up to several hours (e.g. Kowalski et al. 2013), that is, significantly shorter than P_rot, whereas the other indicators are less sensitive to flaring events (e.g. Zechmeister et al. 2009). While the 2σ clipping removes the strongest flaring events from our analysis, minor events or the decay phase of major events might still be included. However, EV Lac and TYC 3529-1437-1 still show rotational signals in the chromospheric indicators despite occasional flaring events. It is possible that flaring events occur more randomly on YZ CMi than on EV Lac, where they occur more often at specific rotation phases (Muheki et al. 2020). Another possible explanation for the missing signals in chromospheric line indicators is that there may be more plage regions in the chromosphere than active regions in the photosphere. A higher number of chromospheric features can lead to a more homogeneous distribution of them and, thus, less rotational variation in the chromospheric lines. In a sample of 13 active mid-to-late-type M dwarfs, Medina et al. (2022) also found no rotational modulation in Hα, except, curiously, for the most active star in their sample.

Similarly to Ross 318, the results for the two analysed data subsets given in Table A.4 reveal that the indicators that show significant power at f_rot in the full dataset also show significant power in one or both subsets. As in the full dataset, there are no significant signals in the chromospheric line indicators. Again, the second subset contains more data points and generally more significant signals than the first subset. The second subset also shows larger standard deviations in most indicators and more negative pEW′ values for the chromospheric lines. This stronger emission in these lines indicates a slightly higher activity level, which can result in stronger rotational variations. We note that the second subset for YZ CMi contains the spectra that Baroch et al. (2020) used to find a strong anti-correlation between CRX and RV and to constrain star-spot parameters.

6 Comparison with photometry

In this section, we compare our results from the CARMENES spectra with photometric light curves observed by TESS. As the TiO 7050 index showed strong rotational signals for all four stars, we focus on this photospheric activity indicator. The TiO 7050 index is temperature-sensitive and thus affected by star spots that also cause modulations in the photometric light curve. However, it can also be affected by rotational and magnetic broadening (Schöfer et al. 2019). We did not analyse the light curves of Ross 318 because even two adjacent sectors cover only one rotation period.

Using GLS periodograms for TYC 3529-1437-1, we find the period P_{TiO 7050} = (16.03 ± 0.03) d in the time series of the TiO 7050 index, and the periods P_tess14 = (7.97 ± 0.04) d, P_TESS25/26 = (15.22 ± 0.03) d, and P_tess40/41 = (16.10 ± 0.03) d in the TESS light curves from sector 14, sectors 25 and 26, and sectors 40 and 41, respectively. Although the formal uncertainties derived from the curvature of the GLS periogram peak are very small, only P_TESS25/26 does not agree with either P_rot or P_rot/2 within 3σ. In Fig. 2, we show the TiO 7050 time series and the light curves phase-folded to the respective periods, or 2 P_TESS14 in the case of TESS sector 14, using the barycentric Julian date 2457499.66246 of the first CARMENES observation as the epoch. The TiO 7050 time series shows one dip per rotation in the first subset of the CARMENES data, whereas the middle observations in the second subset show two dips and therefore cause the stronger signal at 2 f_rot than at f_rot. The TESS light curves show two dips in sector 14, but only one dip in the later sectors.

In the top panel of Fig. 3, we show the GLS power at the rotation frequency and its second harmonic for the TiO 7050 index in the two CARMENES data subsets and for the TESS light curves in each sector. As can be seen in the time series in the bottom panel, the TiO 7050 index decreased and the Hα emission increased, indicating a higher activity level in the second subset, which shows more power at 2 f_rot than at f_rot in the GLS periodogram. The power at 2 f_rot is also higher in TESS sector 14 with two dips per rotation in the light curve, but lower again in the later sectors with only one dip. Alternating episodes of one dip and two dips per rotation period are common in light curves of late-type stars and do not necessarily reflect the global spot coverage and activity level. One dip means that one hemisphere appears darker than the other, so there may be spots on both hemispheres, but more on one than on the other. In contrast, two dips require a certain phase distribution of the spots, which can also occur with a lower spot coverage (Basri & Shah 2020). We therefore cannot infer a lower activity level from the single dip per rotation in the later TESS sectors. If the activity decreased again, this would be consistent with an activity cycle of the order of a few years, as is plausible for an early M dwarf (e.g. Suárez Mascareño et al. 2016), with the maximum occurring during or after the second subset of the CARMENES observations.

For YZ CMi and EV Lac, we show the phase-folded TiO 7050 time series and TESS light curves in Fig. 4. The epochs are again the barycentric Julian dates of the first CARMENES observations for each star (2457395.59368 for YZ CMi and 2457398.35702 for EV Lac), the periods are P_{TiO 7050} = (2.775 ± 0.001) d, P_tess7 = (2.773 ± 0.001) d, and P_tess34 = (2.775 ± 0.004) d for YZ CMi, and P_{tio 7050} = (4.359 ± 0.004) d and P_TESS16 = (4.325 ± 0.017) d for EV Lac. All periods are in agreement with the respective P_rot. YZ CMi shows one dip per rotation in the TiO 7050 time series and the TESS sector 7 light curve and two dips in TESS sector 34. However, the second dip is considerably weaker and does not cause more power at the second harmonic than at the rotation frequency in the GLS periodogram. Given the long-term stability of active regions suggested by Baroch et al. (2020) and the stable rotational signals found in the spectroscopic indicators, it is remarkable that we see hints at their evolution in the light curves. While EV Lac also shows two dips per rotation in the TESS light curve, but more GLS power at f_rot, in this case it is the second peak that is considerably weaker than the first peak. We note that EV Lac showed only one dip per rotation period in the photometric data from automated surveys used by Díez Alonso et al. (2019), and the long-term light curves from Alekseev & Kozhevnikova (2017) include both episodes with one dip and episodes with two dips per rotation period for this star.

Fig. 2 Phase-folded time series of TiO 7050 index in CARMENES spectra and TESS light curves for TYC 3529-1437-1.

Fig. 3 Evolution of periodogram signals and activity indicators for TYC 3529-1437-1 overtime. Top: GLS periodogram power at frot (filled symbols) and 2 frot (open symbols) of the TiO 7050 index in the two CARMENES data subsets (orange boxes) and of TESS photometry in five TESS observation sectors (green diamonds) as a function of the barycentric Julian date. The bars reflect the time spanned by each set of data. Bottom: time series of TiO 7050 index (orange circles) and pEW′Hα (black circles) derived from CARMENES spectra.

7 Summary and Conclusions

We investigated periodicities in the strength of eight chromospheric emission lines, four photospheric absorption bands, and in dLW, CRX, RV, and CCF parameters from the CARMENES VIS and NIR channels for four early-to-mid type M dwarfs with different activity levels. While the four stars were selected because they all showed clear rotational signals in more than two activity indicators in a previous activity analysis of the CARMENES sample (Schöfer et al. 2019), we find that they show the signals in different indicators and that these signals evolve in different ways.

The histogram in Fig. 5 summarises how many chromospheric, photospheric, and CCF or RV-related indicators show a modulation with either the rotation frequency or its second harmonic. While the photospheric indicators and the CCF and RV-related indicators, particularly the TiO 7050 index and the RV itself, show rotational modulation at all activity levels, rotational signals in the chromospheric indicators are more common at lower activity levels. This is also true for observations of the same star at different activity levels and can be explained with an increasing flaring rate or an increasing number of plage regions leading to a more homogeneous distribution of active regions in the chromosphere with increasing global activity. Overall, our results agree with the findings of Lafarga et al. (2021) for a larger sample of M dwarfs.

Ross 318 and YZ CMi, the least and the most active of the four studied stars, both show stable signals corresponding to P_rot throughout the CARMENES observations, whereas the other two stars exhibit changing signals over time. The moderately active star TYC 3529-1437-1 shows signals at the rotation frequency in several indicators during a less active episode, but signals at the second harmonic 2 f_rot during a more active episode. Our comparison with TESS photometry suggests that this corresponds to the common phenomenon of light curves showing one dip per rotation at certain times, but two dips per rotation at other times (e.g. Basri & Nguyen 2018). For the very active star EV Lac, with its non-axisymmetric magnetic field (Morin et al. 2008), we find a more complex behaviour with different indicators showing the strongest rotational signal at different frequencies during the same episode, and some indicators favouring different frequencies during other episodes in agreement with the results of Jeffers et al. (2022). Particularly for more active stars, it is therefore useful to also search for signals in activity indicators in parts of the dataset, not only in the full dataset.

Fig. 4 Phase-folded time series of TiO 7050 index in CARMENES spectra and TESS light curves for YZ CMi (top) and EV Lac (bottom).

Fig. 5 Fractions of eight spectral lines with a chromospheric component, six photospheric indicators (band indices and dLW), and ten RV and CCF indicators that show a significant signal at frot (solid) or 2 frot (open) in the GLS periodograms. The bars for TYC 3529-1437-1 and EV Lac are split into halves and thirds, respectively, corresponding to the analysed data subsets in chronological order.

Acknowledgements

CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar Alto (CAHA, Almería, Spain). CARMENES is funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Científicas (CSIC), the European Union through FEDER/ERF FICTS-2011-02 funds, and the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai, Institut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-Alemán), with additional contributions by the Spanish Ministry of Science, the German Science Foundation through the Major Research Instrumentation Programme and DFG Research Unit FOR2544 “Blue Planets around Red Stars”, the Klaus Tschira Stiftung, the states of Baden-Württemberg and Niedersachsen, and by the Junta de Andalucía. S.V.J. also acknowledges the support of the DFG priority program SPP 1992 “Exploring the Diversity of Extrasolar Planets (JE 701/5-1)”. L.T.-O. also acknowledges support from the Israel Science Foundation (grant no. 848/16). This paper includes data collected by the TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST). Funding for the TESS mission is provided by NASA’s Science Mission directorate.

Appendix A Periodograms and Tables

Fig. A.1 GLS periodograms of our indicators and window functions for Ross 318. The indicators are the pEW′ of chromospheric emission lines (He i D3, NaI D, Hα, and Ca ii IRT in the visible-light channel of CARMENES, He i λ10833 Å and Paβ lines in the near-infrared channel), indices of photospheric absorption bands (TiO 7050, TiO 8430, VO 7436, VO 7942), and differential line width (dLW), chromatic index (CRX), radial velocity (RV), CCF FWHM, CCF contrast, and CCF bisector inverse slope (BIS) in each channel. Horizontal lines indicate the analytical 10% (dotted), 1% (dashed), and 0.1% (dash-dotted) false-alarm probability levels. N is the number of used data points after a 2σ clipping for each indicator. For most signals at a frequency f, there are aliases at 1 d−1 − f. The red dotted line marks the rotation frequency frot = 0.0194d−1 (Prot = 51.5 d).

Fig. A.2 Same as Fig. A.1, but for TYC 3529-1437-1. The red dotted line marks the rotation frequency frot = 0.063 d-1 (Prot = 15.8 d), while the blue dotted line marks its second harmonic 2 frot.

Fig. A.3 Same as Fig. A.1, but for EV Lac. The red dotted line marks the rotation frequency frot = 0.228 d−1 (Prot = 4.38 d), while the blue and magenta dotted lines mark its second and third harmonics 2 frot and 3 frot, respectively.

Fig. A.4 Same as Fig. A.1, but for YZ CMi. The red dotted line marks the rotation frequency frot = 0.36 d1 (Prot = 2.78 d).

References

All Tables

All Figures

Fig. 1 Pseudo-equivalent width of Hα line after subtracting an inactive reference star spectrum (pEW′Hα) as a function of the stellar mass for 331 M dwarfs in the CARMENES sample, colour-coded by the rotation period Prot. Black points are stars with unknown Prot. The four stars studied in this work are marked with star symbols and their names. pEW′Hα and Prot are adopted from Schöfer et al. (2019) and references therein, stellar masses were generally computed from luminosities and effective temperatures as in Schweitzer et al. (2019) and Cifuentes et al. (2020), but with the latest Gaia EDR3 parallax and photometry (Gaia Collaboration 2021). More details on these updated masses will be provided in a forthcoming publication.

In the text

	Fig. 2 Phase-folded time series of TiO 7050 index in CARMENES spectra and TESS light curves for TYC 3529-1437-1.
In the text

Fig. 3 Evolution of periodogram signals and activity indicators for TYC 3529-1437-1 overtime. Top: GLS periodogram power at frot (filled symbols) and 2 frot (open symbols) of the TiO 7050 index in the two CARMENES data subsets (orange boxes) and of TESS photometry in five TESS observation sectors (green diamonds) as a function of the barycentric Julian date. The bars reflect the time spanned by each set of data. Bottom: time series of TiO 7050 index (orange circles) and pEW′Hα (black circles) derived from CARMENES spectra.

In the text

	Fig. 4 Phase-folded time series of TiO 7050 index in CARMENES spectra and TESS light curves for YZ CMi (top) and EV Lac (bottom).
In the text

	Fig. 5 Fractions of eight spectral lines with a chromospheric component, six photospheric indicators (band indices and dLW), and ten RV and CCF indicators that show a significant signal at frot (solid) or 2 frot (open) in the GLS periodograms. The bars for TYC 3529-1437-1 and EV Lac are split into halves and thirds, respectively, corresponding to the analysed data subsets in chronological order.
In the text

Fig. A.1 GLS periodograms of our indicators and window functions for Ross 318. The indicators are the pEW′ of chromospheric emission lines (He i D3, NaI D, Hα, and Ca ii IRT in the visible-light channel of CARMENES, He i λ10833 Å and Paβ lines in the near-infrared channel), indices of photospheric absorption bands (TiO 7050, TiO 8430, VO 7436, VO 7942), and differential line width (dLW), chromatic index (CRX), radial velocity (RV), CCF FWHM, CCF contrast, and CCF bisector inverse slope (BIS) in each channel. Horizontal lines indicate the analytical 10% (dotted), 1% (dashed), and 0.1% (dash-dotted) false-alarm probability levels. N is the number of used data points after a 2σ clipping for each indicator. For most signals at a frequency f, there are aliases at 1 d−1 − f. The red dotted line marks the rotation frequency frot = 0.0194d−1 (Prot = 51.5 d).

In the text

	Fig. A.2 Same as Fig. A.1, but for TYC 3529-1437-1. The red dotted line marks the rotation frequency frot = 0.063 d-1 (Prot = 15.8 d), while the blue dotted line marks its second harmonic 2 frot.
In the text

	Fig. A.3 Same as Fig. A.1, but for EV Lac. The red dotted line marks the rotation frequency frot = 0.228 d−1 (Prot = 4.38 d), while the blue and magenta dotted lines mark its second and third harmonics 2 frot and 3 frot, respectively.
In the text

	Fig. A.4 Same as Fig. A.1, but for YZ CMi. The red dotted line marks the rotation frequency frot = 0.36 d1 (Prot = 2.78 d).
In the text