Magnetic activity variability of nearby bright Sun-like stars by 4-year intensive H$\alpha$ line monitoring

We conducted spectroscopic observations of the target stars using the HIgh Dispersion Echelle Spectrograph (HIDES; Izumiura (1999)) of the 1.88-m reflector at Okayama Branch Office, Subaru Telescope. The spectra were acquired by Fiber-Feed mode (HIDES-F; Kambe et al. (2013)), and we utilized high-efficiency mode with an image slicer. In 2018, HIDES was upgraded by removing the slit mode elements from the optical path, and rearranging the optical instruments on a new stabilized platform. The data used in this paper were obtained after the upgrade. The width of the sliced image was $\timeform{1.05^{\prime\prime}}$, corresponding to a spectral resolution of R $\sim$ 55000 achieved by 3.8-pixel sampling. For our analysis, Table 2 lists the observation data of each target star reported in this work. We used one observed data per night. In the case of stars with several observations per night, we averaged the spectra for analyses. The signal-to-noise ratio (S$/$N) for the stars is slightly different depending on weather conditions. For HIDES-F, the three CCDs cover a wavelength region of 3700-7500 Å. However, there were severe aperture overlaps in a wavelength range of 3700-4000 Å owing to the use of the image slicer and hence the scattered light for these apertures could not be subtracted. Consequently, we discard the overlapped apertures and did not analyze the Ca II H&K lines for HIDES-F spectra.

The echelle data reduction was performed in a standard manner using the IRAF¹¹1IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc. under a cooperative agreement with the National Science Foundation, USA packages to extract one-dimensional raw spectra (bias subtraction, flat fielding, scattered-light subtraction and spectrum extraction). The wavelength calibration was determined from Thorium-Argon arcs taken before and after each spectrum. For each night spectrum, we performed the correction for telluric absorption lines using $\alpha$ Leo (B8IV, v sin i = 317 km s^-1), which was taken on the same night as a standard star. We also corrected the dependence on airmass, which varies with the elevation of the object, by using the standard star spectrum (Kawauchi et al. (2018)). The continuum-normalization for the telluric-line-corrected spectra was carried out with a reference spectrum in Table 2 (see Lee et al. (2023)).

We show the result of the data reduction and telluric correction with $\epsilon$ Eri. Figure 1 (a) shows the single raw spectrum and the telluric-line-corrected spectrum for $\epsilon$ Eri obtained after the data reduction. Figure 1 (b) shows the normalized H$\alpha$ line profile and its core for $\epsilon$ Eri. The entire procedure was performed in the same manner for all target stars.

To quantify the variability strength in the H$\alpha$ line core for each star, we averaged all the continuum-normalized spectra and then calculated the relative residuals for each normalized spectrum in relation to the averaged spectrum. Figure 2 shows the relative residuals of the target stars (part of all spectra), which have been smoothed to show the variability in the H$\alpha$ line core more clearly. Each star exhibits nightly variability in the H$\alpha$ line core throughout the observing seasons.

To evaluate the level of the H$\alpha$ variability, we integrated the relative residuals over a $\sim$ 1 Å interval centered around the line core. This is the same method as described in Borsa et al. (2015), which observed one of our target star $\tau$ Boo. In Figure 3, we plot the integrated residual H$\alpha$ flux along the time series of the target stars to present the intensity of the H$\alpha$ line core. Error bars represent the random noise associated with photon fluxes captured by the spectrograph.

Each of the target stars has exhibited distinct variations in the H$\alpha$ intensity. $\epsilon$ Eri shows an extremely clear variation throughout the time series (Fig. 3 (h)). Also, several stars such as $\beta$ Vir, $\alpha$ CMi, and $\pi$¹ UMa, imply periodic variations (Fig. 3 (a),(b),(e)). The remaining stars show a gradual decrease in the integrated residual flux, implying long-term trends. $\kappa$¹ Cet shows some outlier data points (Fig. 3 (d)). Considering that this star is known to exhibit flares (Hamaguchi et al. (2023)), and since other stars did not show similar noise pattern during nearly the same period, the outliers may be derived from sudden increase in intrinsic activity. We obtained the variability of the integrated residual H$\alpha$ flux (hereafter called ‘H$\alpha$ variability’) by calculating the rms of the amplitude for each star, which ranges from 0.9% to 9.4%.

We determined the period of the integrated residual H$\alpha$ flux and peaks significance, using the generalized Lomb-Scargle (GLS) method and the false alarm probability (FAP) by Zechmeister et al. (2009). Given that the minimum data point interval is approximately one day and our focus on periodic variations longer than the rotational periods of the target stars ($>$ 3-5 days), each periodogram was calculated up to its Nyquist frequency of 0.5 day^-1. We also estimated the error range of the determined period by using normal distribution random numbers within the error bars for each data point of the integrated residual H$\alpha$ flux. We generated 10⁴ data sets, choosing random values corresponding to the error bars at each point in one data set. Then, we calculated the GLS periodogram to estimate the error range.

For stars exhibiting a visible long-term trend in the integrated residual H$\alpha$ flux, we assessed the presence of actual long-term trends, including linear or quadratic trends. To address the significance of the visible trend, we used the F-test in a similar way to that described by Cumming et al. (1999) (see also Lee et al. (2023) for details). The results of the F-test are described with the FAP, that here indicates pure noise fluctuations would produce a trend by chance. The found trends for several stars are depicted in Figure 3, alongside the integrated residual H$\alpha$ flux . Long-term trends might influence the power spectrum of the periodogram, so we checked detrended data in the periodogram after removing the linear or quadratic long-term variations. For stars without a long-term trend found by the F-test, we removed possible periodicity.

We also investigated significant peaks in the short-term period range close to stellar rotational or planetary orbital period. We confirmed that the HIDES H$\alpha$ observations show stellar intrinsic activity, not instruments, seasons, or any non-stellar-induced factors, by comparing them to observations of an inactive star, $\tau$ Cet (Lee et al. (2023)).

Additionally, we adopted a so-called bootstrap randomization method to estimate detection limits for possible periodicity. We generated noise data sets by redistributing the actual fit residuals for each star. We then injected a fake data set of the integrated residual H$\alpha$ flux with a fixed period and certain amplitude as a signal against the noise. We calculated the GLS periodogram, examined the power of the peak at the period, and regarded it as detection if the power exceeded the level of FAP=0.1%. We repeated the procedure for 10⁵ fake data sets, and counted the number of the detection, which was regarded as probability of the detection for the signal with the injected period and the amplitude. We performed this simulation for various amplitudes and set the amplitude with the detection probability of 99% to be the detection limit at the period.

With the obtained H$\alpha$ variability, we investigated the possible relationship between the H$\alpha$ variability of individual stars and their stellar parameters. We plot the H$\alpha$ variability as a function of activity level in Figure 4 (a). The activity level is quantified as the value of log R’_HK, which is derived from R’_HK defined as the ratio of chromospheric emission in the cores of the Ca II H&K line to the total bolometric emission of the star (Noyes et al. (1984)).

In Figure 4 (a), we found a significant correlation (correlation coefficient (r) = 0.85) between the H$\alpha$ variability and activity level. More active stars exhibit greater variability in the H$\alpha$ line. It is widely accepted that more active stars show greater activity variability in the Ca II H&K line. Our result suggests a similar dependence in the H$\alpha$ line. The strong correlation between the H$\alpha$ variability and log R’_HK, which indicates the Ca II H&K core emission flux, suggests that the H$\alpha$ line observed with HIDES-F can effectively trace activity variability in Sun-like stars, similar to the Ca II H&K line.

We also plotted the H$\alpha$ variability as a function of several other parameters: stellar age, Rossby number, and rotational velocity (v sin i) in Figure 4. In Figure 4 (b), a very weak correlation was found between stellar age and the H$\alpha$ variability (r = -0.24). Also, a weak correlation was implied between the Rossby number and H$\alpha$ variability (r = -0.45) (Fig. 4 (c)). We could not find any significant correlation between rotational velocity and the H$\alpha$ variability, or between rotational period and the H$\alpha$ variability (Fig. 4 (d)&(e)). It suggests the possibility of a weak relationship between stellar age or Rossby number and the H$\alpha$ variability. The stars with similar activity effects by similar Rossby numbers show varying degrees of activity variability.

For the Sun-like stars, previous studies have explored the relationship between the Ca II H&K emission and stellar age, mass, or color index (Wright et al. (2004b); Lorenzo-Oliveira et al. (2018); Linsky (2019)). As we did not find a similar dependence with the log R’_HK for several parameters of the target stars (Fig B.1), it is evident that more cases would be necessary to establish a clear relationship between activity variability and stellar parameters.

The relatively short time-scale magnetic activity cycle appears to be a trend in F-type stars. Brandenburg et al. (2017) proposed a spectral type dependence between the magnetic activity cycle and rotational period for the Sun-like stars, with distinct branches for active and inactive stars. The short-term activity cycles suggested in F-type stars fall within the inactive branch. In this respect, we present the activity cycles of F-, G-, and K-type stars as a function of the rotational periods in Figure 10, combining updated data from previous studies (Brandenburg et al. (2017); Mittag et al. (2019)) with the results of HIDES-F H$\alpha$ observations. The samples of F-, G- and K-type stars, colored light green, yellow, and red, respectively, follow the general trend that the faster stars rotate, the shorter the magnetic activity cycle is. The magnetic activity cycles of F-type stars locate in a relatively short-term period range.

We previously reported an independent confirmation of the persistence of the short-term magnetic activity cycle ($\sim$ 123 days, light green diamond) in the H$\alpha$ variability for $\tau$ Boo (Lee et al. (2023)). Furthermore, we suggest the possibility of short-term activity cycles of 530 and 130 days for $\beta$ Vir and $\alpha$ CMi, respectively (light green triangles). These possible activity cycles in $\beta$ Vir and $\alpha$ CMi are within 1-2 years and shorter than those of G- or K-type stars with similar rotational periods. Our result, obtained through intensive monitoring with high cadence, supports the previous theoretical and observational lack of extremely short-term cycles in G- or K-type stars. The short-term activity cycle trend in F-type stars can be possibly related to the stellar intrinsic dynamo.

Meanwhile, the short period found in $\alpha$ CMi does not fall onto the inactive branch. It could be explained by its distinct characteristic, such as the star is located at edge of the main-sequence stars on the HR diagram. Or, the period relying only on season 3 might not real for the entire cycle period. Another possibility is that short activity cycles result from variations in spot distributions or spot lifetimes (Basri et al. (2020)). To clarify this, it is important to investigate the absorption line profile caused by star spots in addition to continuous cycle observations with high cadence for $\alpha$ CMi.

As described already, it is suggested that the magnetic activity cycle of the Sun-like stars tends to follow one of the two branches, and some G- and K-type stars with a shorter and a longer activity cycle of co-existing cycles are on both two branches (Böhm-Vitense (2007); Brandenburg et al. (2017)). Also, there is the possibility that the branches show more scatters because cycle periods depend on stellar parameters like Rossby number. It is suggested that the active branch may actually consist of two sub-sequences (Böhm-Vitense (2007)), but it is not yet established. We turn to Figure 10 to discuss the H$\alpha$ results of our G- and K-type target stars.

$\beta$ Com and $\kappa$¹ Cet are expected to have a shorter cycle from the previous research. From the results of $\beta$ Com in the H$\alpha$ observation, we found the possibility of a long-term variation beyond 8 yr (yellow square), and it is close to one of the detected cycles in $\beta$ Com. $\beta$ Com has been reported to show two cycles of 9.6 and 16.6-yr, however, the 9.6-yr cycle is not consistent with its expected shorter cycle (2.8-yr, yellow cross) and rather close to the active branch. As our H$\alpha$ observation for $\beta$ Com did not detect the expected shorter cycle even with likely sufficient amplitudes, it might support a sub-sequence in the active branch (Böhm-Vitense (2007)), unless the possible shorter cycle had a much smaller amplitude. It is not clear how stellar parameters affect the possibility of the sub-sequence to appear, so further observations of $\beta$ Com to determine co-existing cycles in the H$\alpha$ line could help establish different behaviors in the active branch.

The amplitude of a longer cycle would be greater than that of a shorter cycle (Saar et al. (2002); Brandenburg et al. (2017)), thus there is the possibility that a longer cycle is dominant among co-existing cycles. However, for $\epsilon$ Eri, the shorter cycle is clearly detected (red diamond), but its longer cycle was not implied in the H$\alpha$ line. Whereas for $\beta$ Com and $\kappa$¹ Cet, the shorter-term period was not found while the long-term trend is implied (yellow squares). Considering the longer cycles that were already found for these three stars, especially for $\beta$ Com, the longer cycle of $\epsilon$ Eri is not too long and therefore it can be implied during our observing time span. Thus, we speculate that a distinct shorter cycle could be dominantly detected. In our F-type stars, $\tau$ Boo and $\beta$ Vir, the clear variations of the short-term period appear dominantly as well (Fig. 5 of Lee et al. (2023) & Fig. A.1 (A).

The expected shorter cycle for $\beta$ Com and $\kappa$¹ Cet may have just too small amplitude to be detected. However, there is another possibility such as the temporal absence of the short-term variation. As described already, it is reported that the shorter cycle in $\epsilon$ Eri was not significant temporarily in the past. Also, one of our F-type stars, $\alpha$ CMi, has implied the possibility of the temporal absence/existence for the short-term variation. For $\alpha$ CMi, although the star showed the distinct short period in the specific season, the possible long-term trend was detected as the highest peak in the periodogram (Fig. 7). This could be because of the temporal absence/existence of the short period. Similarly, the long-term trend in $\beta$ Com and $\kappa$¹ Cet may be dominant just because their shorter cycle is off at present. The dominance or temporal absence/existence of the short cycle suggested by our observations could explain few cases of stars with detected co-existing cycles.

The common condition of stars with co-existing cycles in the two branches is an age range of 0.6 - 2.3 Gyr (Brandenburg et al. (2017)). Also, the activity level represented as log R’_HK is around -4.75 (Baliunas et al. (1995)). For $\pi$¹ UMa, $\beta$ Aql, 61 Vir, $\eta$ Cep, and HD 102195, we could not find evidence of co-existing cycles. Given that either of their ages or activity levels are not satisfied with the conditions of co-existing cycles, the non-detection of any shorter periodic variations in our H$\alpha$ results for these stars may support the suggested conditions. However, there is still the possibility to find evidence of co-existing cycles for some target stars, although we could not detect periodic variations in this observation. For $\pi$¹ UMa, considering its similarity of age and activity level to $\epsilon$ Eri, we can expect its shorter cycle of co-existing cycles. In practice, $\pi$¹ UMa shows a relatively distinct variation but with a slightly weak periodogram power, perhaps due to fewer data. It suggests that conditions of co-existing cycles could be determined by considering more complicated factors related to stellar parameters. Further monitoring would be essential to find robust periodic variations of $\pi$¹ UMa, to establish possible conditions of co-existing cycles as well.

Based on the results in Table 3 and Section 5.2, we discuss the H$\alpha$ variability of the target stars focusing on exoplanet detection. Since the RV data with HIDES-F in almost the same seasons is not available at this stage, we investigated the previous RV results, the most recent ones possible, for the target stars. Owing to the possibility of changes of the magnetic activity variability with different epochs, we note that the H$\alpha$ results should only be considered as a possible suggestion.

In the previous RV measurements for the four F-type stars (Butler et al. (1997); Howard et al. (2016); Wittenmyer et al. (2006); Huber et al. (2011)), the detected or suggested magnetic activity cycles in this observation are not implied. Also, for $\tau$ Boo and $\upsilon$ And, we did not find short-term periods related to their rotational period or planetary orbital periods in the H$\alpha$ observations (Lee et al. (2023)). It is suggested that the activity induced by the stellar rotation does not affect the H$\alpha$ variability for the F-type stars. Also, the impact of the long-term activity variability induced by the activity cycle might not be significant to the RV measurements.

For the five G-type stars, there is the possibility that the H$\alpha$ variability is affected by the stellar rotation and mimic a RV signal. Also, their long-term activity variations suggested by the H$\alpha$ line were not significant in the RV measurements (Howard et al. (2016)). Meanwhile, for 61 Vir, some activity variability may induce stellar noise to the RV for its Earth-mass planet. In this observation, it is speculated that the G-type stars might be able to induce RV signals in various forms.

The possible impact of the H$\alpha$ variability on the RV measurements was different by individual stars for the three K-type stars. For $\epsilon$ Eri, we found its rotational period in the periodogram, however, the 2.9-yr cycle was not found in the RV (Hatzes et al. (2000); Benedict et al. (2006)). For $\eta$ Cep, in addition to the weak periodogram power of the 163 days peak, as the existence of a hot Jupiter is not clear, we can not suggest the significance of the period. Nevertheless, there is the possibility that the magnetic activity variability might have induced the stellar noise to the RV. For HD 102195, the RV measurements reported a 4.11-day orbit of its hot Jupiter (Melo et al. (2007)). Its substantial variability observed in the H$\alpha$ line could hinder searches for short-term periods. The features of individual stars would be critical factors to make effects on the RV for the K-type stars. To understand the reasons for these differences, further investigation considering more various stellar parameters would be required.

Additionally, the results of this observation by the individual stars or spectral type are based on the observation with precision of few m/s. With higher precision observations, the possible impacts can be detected.