---
author:
- |
    R.R. Ferreira , R. Barbosa , M. Castro , G. Guerrero , L. de Almeida ,\
    P. Boumier
- 'J.-D. do Nascimento Jr.'
bibliography:
- 'bibliography.bib'
date: 'Received , ; accepted , '
title: 'HD43587: a primary CoRoT target under a Maunder minimum phase?'
---

[ One of the most enigmatic phenomena related to solar activity is the so-called Maunder minimum phase. It consists of the lowest sunspot’s counting ever registered for the Sun, and never so far confirmed for other stars. Since the spectroscopic observations of stellar activity at the Mount Wilson Observatory, the solar analog HD43587 has shown a very low and apparently invariant activity level, which makes it a Maunder minimum candidate. ]{} [We analyze the chromospheric activity evolution of HD43587 and its evolutive status, intending to unravel the reasons for this low and flat activity.]{} [We use an activity measurements dataset available in the literature, and compute the activity S-index from HARPS and NARVAL spectra, to infer a cycle period. Besides, we analyze the CoRoT light-curve of HD43587, and apply gyrochronology and activity calibrations, to determine its rotation period. Finally, based on an evolutionary model and the inferred rotation period, we use the EULAG-MHD code to perform global MHD simulations of HD43587 with the aim of getting some insights about its dynamo process.]{} [We confirm the almost flat activity profile, with a cycle period $P_{\mathrm{cyc}} = 10.44\pm3.03$ yrs deduced from the S-index time series, and a long-term trend that might be a period larger than 50 yrs. It was impossible to define a rotation period from the light-curve, however gyrochronology and activity calibrations allow to infer an indirect estimate of $\overline{P}_{\mathrm{rot}} = 22.6 \pm 1.9$ d. Furthermore, the MHD simulations confirm an oscillatory dynamo with a cycle period in good agreement with the observations and a low level of surface magnetic activity. ]{} [ We conclude that this object might be experiencing a “natural” decrease in magnetic activity as a consequence of its age. Nevertheless, the possibility that HD43587 is in a Maunder minimum phase cannot be ruled out. ]{}

Introduction
============

The study of the sunspots number as well as the solar variability have been good indicators of magnetic activity [@eddy; @stuiver; @kivelson]. Continuous records of the sunspots number exist since the first observation by Galileo at the beginning of the 17th-century[^1]. The solar activity from observational data exhibits a period of about 11 years with amplitude variations that remain as an open issue in modern astrophysics [@weissthomas].

One of the most intriguing phenomena observed in the sunspots counting survey is the long period phase, posteriorly called Maunder Minimum (MM hereafter) when almost no sunspot were observed on the solar surface [@eddy]. It lasted for about 70 yrs between the 16th and the 17th centuries. Until now, there is no complete explanation for this period of extremely low activity [@wright; @lubin], however, it is historically known that it coincided with a short glacial era in the Earth [@eddy].

The fundamental question about how singular is the magnetic activity evolution of the Sun has motivated long-term surveys of stellar activity of solar-type stars.

Chromospheric stellar activity consists, basically, in the temporal variations of the stellar magnetic field carried out by mass loss and radiation outside the star [@hall]. These variations have a connection with the flux of Ca II H&K lines which has been widely used as a spectroscopical indicator of stellar activity [@auer; @skumanich; @wilson1968]. In the Sun, for instance, [@wilson63; @wilson64; @wilson1968] established the connection between solar activity and the strength of the Ca II H&K lines.

Strong efforts were performed over the last decades seeking to increase the spectroscopic data [@schwarzschild; @auer]. In his seminal work, [@skumanich72] found that activity, as given by the Ca II H&K lines, rotation, as well as the lithium abundance, decay as the inverse square root of time. The first observational program to continuously measure the activity of main sequence stars, the HK project at Mount Wilson Observatory[^2] (MWO hereafter), is described in detail by [@vaughan1978]. Improved bases of the project and its results of spectroscopical measurements were related by [@wilson1978]. Afterwards, [@duncan] presented a large set of time series spectroscopical measurements from the Mt. Wilson survey containing data for 1296 stars observed between 1966 and 1983. [@baliunas95] presented enhanced and extended results of Mt. Wilson stars and flaged the activity of the stars as cyclic and flat.

From these observational results it is accepted that young stars have strong activity levels [@donascimento]. However, some young stars present extremely low activity along many years. This unexpected phase is a puzzle for theoriticians and observers astronomers. Many authors have proposed to use the solar chromosphere as a pattern to study the activity evolution of other solar-type stars. It could be an approach to investigate the occurrence of MM in stars similar to the Sun [@schrijvera; @schrijver1989; @schrijverc; @schroder].

In this work, we investigate the low spectroscopic activity of the solar analog HD43587 over 50 years of measurements. This object has been widely referred as flat activity star and a MM candidate [@baliunas95; @wright; @lubin; @schroder]. We use the fluxes in the Ca II H&K lines, through the Mt. Wilson S-index ($S_{MW}$), whose photometric measurements were made with the 1.5m telescope at MWO. After the MWO survey, this object has been observed by several others surveys (detailed below in the section \[HD43587\]), as well as described by @morel, @wright, and @radick2018. We focused on the continuous low activity level presented from spectroscopic measurements, showed by the time-averaged value $\braket{S_{MW}}$ = 0.154. In additon, the observations from CoRoT satellite provide a photometric activity proxy $S_{ph}$ = $102 ppm$ as described by @mathur2014, sustaining the flat activity status.

This paper is organized as follows: in section 2 we briefly present the observational data and reduction used along this work, in section 3 we present our results. Section 4 shows an attempt of simulating the star HD43587 by using the EULAG-MHD code. Finally, we present our conclusions in section 5.

Observational data {#data}
==================

In this section, we briefly present the characterization of the solar analog HD43587, and explain the whole spectroscopic data used, as well as the $S_{MW}$ calibration for each instrument. In the last subsection we explain the photometric data from CoRoT satellite.

![*Sun’s chromospheric activity over 50yrs, according to [@egeland]. Upper: complete time series measurements made along 50 yrs depicting the $S_{MW}$ periodic profile. Bottom: both the Lomb-Scargle (left), and the autocorrelation function [ACF, see @brockwell2002] (right) periodograms showing the $P_{\mathrm{cyc}}$$\approx$11 yrs solar activity cycle.*[]{data-label="fig:sun"}](figures/FIGURA1.png){width="9cm" height="9cm"}

The solar analog HD43587 {#HD43587}
------------------------

Our object HD43587 (precisely HD43587Aa) belongs to a quadruple system composed of two distant main sequence visual binaries and localized in the Orion constellation. It is a bright sun-like star, and its fundamental parameters are presented in Table \[table:1\]. This system has been largely observed by many authors [@duncan; @baliunas95; @catala; @morel]. This star was also monitored by using the CFHT [*(Canadian-France Hawai Telescope)*]{}, for four seasons: January 2002 (one night), June 2002 (two nights), September 2004 (one night) and September 2005. In all cases were used $H_{2}$ and \[Fe II\] filters [@catala].

HD43587 was a CoRoT primary target in the seismology program with ID CoRoT 3474, it was observed during the LRa03 for 145 days. Complementary observations were done with the spectrograph HARPS [*(High Accuracy Radial velocity Planet Search)*]{}, mounted on the 3.6m telescope in La Silla Observatory/ESO [@boumier]. In addition, it has been monitored by several spectroscopical long-term observational surveys dedicated to activity, such as the HK project of MWO [@duncan; @baliunas95] between 1966 and 1990, and the California Carnegie and Planet Search, at Keck Observatory between 1995 and 2000 [@wright]. This object was also continuously monitored by Automatic Photometric Telescopes of the Tennessee State University at the Fairborn Observatory, and the Solar-Stellar Spectrograph at the Lowell Observatory between 1995 and 2016 [@radick2018]. It also was observed by the NARVAL spectrograph, mounted at the Bernard Lyot Telescope at the Observatoire du Pic du Midi between 2009 and 2010[^3] [@petit]. Spectroscopic analysis of this star performed by the McDonald Observatory 2.1m Telescope and Sandiford Cassegrain Echelle Spectrograph derived a projected rotation velocity $v~sin~i = 4.9$ km.s$^{-1}$ [@Luck2017].

  ------------------------ --------------------- ----------------------- ------------------ -----------------
                            Morel et al. (2013)   Boumier et al. (2014)                     
                                                                                TGEC             CESTAM
    Mass \[$M_{\odot}$\]     1.049 $\pm$ 0.016       1.04 $\pm$ 0.01      $1.020\pm0.004$    $1.04 \pm 0.01$
   Radius \[$R_{\odot}$\]     1.15 $\pm$ 0.01             1.19             $1.19\pm0.01$          1.18
          \[dex\]            -0.02 $\pm$ 0.02             0.01            $-0.026\pm0.003$        0.025
        Age \[Gyr\]           4.97 $\pm$ 0.52        5.60 $\pm$ 0.16       $6.76\pm0.12$      $5.7 \pm 0.1$
       $T_{eff}$\[K\]          5947 $\pm$ 17              5951              $5952\pm27$           5979
  ------------------------ --------------------- ----------------------- ------------------ -----------------

This star has been considered by various authors as a strong MM candidate [@baliunas95; @lubin; @schroder; @morel]. Its long-time activity series shows a long flat activity lasting about 50 years. Its time-averaged activity index is $\braket{S_{MW}} = 0.154 \pm 0.004$, while for the Sun $\braket{S_{MW}}_{\odot} = 0.167 \pm 0.006$. The observational spectroscopic data considered here were obtained in part from long-term surveys already published, as well as from HARPS and NARVAL spectra.

-   NSO/MWO: measurements from MWO obtained between 1966 and 1983 [@duncan]. Besides, we used largely dataset updated from National Solar Observatory (NSO) with S-index measurements until 1995[^4];

-   NARVAL: 50 high-resolution spectra obtained in the archive available from Narval/Polarbase [@petit];

-   HARPS: 3 high-resolution spectra from HARPS/ESO [@morel];

-   CCPS: S-index activity measurements from the California and Carnegie Planet Search (CCPS) program at Keck Observatory, available in [@wright].

-   LO: a dataset of chromospheric Ca II H+K emission observations between 1994 and 2016 from Lowell Observatory (LO), available in [@radick2018].

![image](figures/FIGURE2.png){width="\textwidth"}

Recently, [@castro] presented an evolution status analysis of this star, by using an optimization approach and by modeling using two different evolution codes, TGEC [@huibonhoa] and CESTAM [@marques], and by using spectroscopic, photometric and seismic constraints. Results of the optimization are presented in Table \[table:1\]. These results confirm the solar analog status of HD43587, and indicate a star slightly more massive and older than the Sun, which is in agreement with its observed low activity level.

The Mount Wilson S-index
------------------------

Long-term observational programs at the MWO have been relevant to detect anomalous behaviour in stellar activity as the extreme low-activity cases, i.e., young solar-like stars probably in MM phase. In fact, the MM has been investigated by many authors since the 70’s [@eddy; @duncan; @baliunas95; @wright; @lubin; @schroder; @radick2018], initially in MW project and afterward in other surveys. Along the years several indexes to measure stellar activity have been proposed. Here, we adopt the well-known MW S-index (see Eq. \[eq1\]), used to measure the stellar chromospheric activity and to investigate the stellar activity evolution.

Stellar spectra provide valuable information about stellar chromospheres. To measure stellar chromospheric activity, we use the flux in the Ca II H&K lines, to derive the $S_{MW}$ index, that was initially proposed in the MW project [@duncan; @baliunas90; @baliunas95].

The $S_{MW}$ is computed as follows: $$\begin{aligned}
    \centering
    S_{MW} &=& \alpha \frac{H + K}{R + V}
    \label{eq1}\end{aligned}$$

where $\alpha$ is an observational constant defined by [@duncan], $H$ and $K$ mean the flow strengths in triangular bandpasses with FWHM = 2.18 and centralized at 3933.633 [Å]{} (K) and 3968.469 [Å]{} (H), and $R$ and $V$ are continuous rectangular bands centralized at 3901.07 [Å]{} (V) and 4001.07 [Å]{} (R).\
Since 1960, many authors have investigated the spectroscopic activity using S-index time series [@duncan; @baliunas90; @wright]. Few stars with low-activity or flat-activity, compared to the Sun at the minimum, have been reported. The value of the time average $\braket{S_{MW}}$ for these objects is slightly smaller than $\braket{S_{MW}}_{\odot}$. Subsequently, some studies were performed to understand the nature of flat-activity stars, among them, [@schroder] compared the solar chromospheric activity as a pattern in the context of solar-like stars. In spite of these efforts it has been difficult to precisely determine any star in the MM phase.

Since the first spectroscopic activity measurements, the solar S-index has varied between 0.16 (minimum) and 0.22 (maximum). The S-index time series of the Sun from [@egeland] is shown in Fig. \[fig:sun\]. According to [@schroder] the low-activity levels of the Sun are a good pattern for investigate the activity evolution of other solar-like stars under a long flat-activity profile. These time-lapses of lower activity explicit the chromospheric flat-activity phase in solar-type objects like HD43587. This may be a possible evidence of flat-activity stars at MM phase, as proposed by [@baliunas95].

S-index from NARVAL spectra
---------------------------

We also used spectra obtained with the spectrograph NARVAL, mounted at the Telescope Bernard Lyot (TBL) located at Pic du Midi, France. The spectra are available in the PolarBase[^5] archive [@petit]. We used 50 high-resolution spectra collected on September 2009, January 2010, and February 2010. It consists in 5 spectra per night, totalizing 10 nights. The spectra corresponding to the maximum and minimum of activity obeserved during this campaign are shown in Fig. \[fig:spectra\].

Since different spectrometers have different bandpasses, it is necessary to use a different calibration to derive the $S_{MW}$-index for each instrument. For NARVAL spectra, $S_{MW}$ is derived following the method proposed by [@marsden], where was used a non-renormalized spectrum, similar to that used by [@wright]. The activity index is computed as follows:

$$\begin{aligned}
\centering
S_{MW} &=& \frac{12.873H + 2.502K}{8.877R + 4.271V} + 1.183x10^{-3},
\label{eq.2}\end{aligned}$$

The coefficients in equation (\[eq.2\]) were obtained by using a least-square fit [see fig. 4 in @marsden] to the 113 stars observed by NARVAL [@marsden] and by the California Planet Searcher [@wright].

S-index from HARPS spectra
--------------------------

We used three high-resolution spectra observed between December 2010 and January 2011, obtained with the spectrograph HARPS, in the high-efficiency mode [for detailed information see section 2 of @morel].

We computed the instrumental S-index from HARPS spectra following the same calibration as prescribed by @lovis:

$$\begin{aligned}
\centering
S_{HARPS} &=& \alpha \cdot 8 \cdot \frac{1.09{\AA}}{20{\AA}} \cdot \frac{H + K}{R + V},
\label{eq.3}\end{aligned}$$

The calibration to MW scale was done by using a linear calibration between both indexes obtained with standard stars. According to [@lovis], the activity index transformation is

$$\begin{aligned}
\centering
S_{MW} &=& 1.111 \cdot S_{HARPS} + 0.0153
\label{eq.4}\end{aligned}$$

Photometric observation from CoRoT mission
------------------------------------------

The CoRoT satellite observed the object HD43587 along 145 days [@boumier], during the third long run (LRa03). The LRa03 was in the direction of the galactic anti-center and was performed from October 2009 to March 2010. We made all reductions from the light curve extracted from the level 2 official products [@Samadi2007]. The possibility that the clear absence of a rotational modulation is due to the fact that the star is seen pole on, is ruled out by the determination of the projected rotation velocity by @Luck2017 and mentioned in Sect. \[HD43587\].

![image](figures/FIGURA3a_3.png){width="\textwidth"}

Analysis and discussion
=======================

This section is divided in three parts. The first one presents the analysis of the photometric data collected by the CoRoT satellite. The second part presents the long term time series analysis of the $S_{MW}$ and the inferred activity cycle. In the third part, we discuss about the indirect determination of the HD43587 rotation period.

The HD43587‘s CoRoT light curve {#HD43587_light_curve}
-------------------------------

Based on the photometric data from CoRoT satellite, we investigated the variability of the light curve of HD43587. Aiming to determine the rotational period, we computed the Generalized Lomb-Scargle (GLS) periodogram [@gls] as well as the Autocorrelation Function [@brockwell2002] (see Fig. \[fig:HD43587LC\]). We used the “DATEJD” as the time array and the “RAWFLUX” in electrons per second as the raw flux to create the light curve. The error relative to the raw flux, “RAWFLUXDEV” in electrons per second, was used to compute $S_{ph}$ as explained below. We also took into account the “RAWSTATUS” flag to make sure we used valid data. The resulting valid data exhibits a significant jump in the flux in the first 4.65 days. This discontinuity creates a false tendency on the overall aspect of the light curve, and, even when corrected, this discontinuity does not have the same dispersion in flux as the rest of the light curve. Thus, we decided to remove completely the first 4.65 days of the data. Then, we followed two approaches. First, we tried to recover the GLS and ACF periodograms (red lines in the lower panels of Figure \[fig:HD43587LC\]) from the raw light curve (after removing the jump). We initially obtained the periodicities $P = 93.98^{101.31}_{-43.49}$ days with the GLS and $P = 86.81$ days with the ACF, as indicated by the vertical red dashed lines in the lower panels of Fig. \[fig:HD43587LC\]. The modulation of $\sim90$ days is due to the accumulation of discontinuities in the flux data as can be seen in the upper-right panel (red dots), which creates a false tendency. The second approach was to detrend all the light curve in each discontinuity, thus eliminating the long term modulation. We then obtained periodicity peaks at $\sim$7, $\sim$10, $\sim$14 and $\sim$21 days with the GLS and a peak at 16.96 days with the ACF. However, these GLS peaks do not have any statistical significance as they lay far below the 5% false alarm probability, as presented in the left-middle panel of Figure \[fig:HD43587LC\].

In accordance with [@DoNascimento2014], such stars should present a rotation period around the solar value, ranging between 14 and 30 days. Even though the rotation period inferred by the ACF method belongs to this interval, the peak corresponding in the ACF curve is not strong enough, compared to the rest, to be considered as a reliable result. Thus, even eliminating the false long term tendency, it was impossible to determine a valid modulation period for HD43587 from the CoRoT light curve.\
We also derived the photometric activity proxy, by computing the standard deviation of the light curve corrected after substracting the photon noise $S_{ph}$ [@mathur2014]. We obtained $S_{ph} = 102$ ppm, which returns a rotational period of $43.38$ d. This value of $P_{\mathrm{rot}}$ is characteristic of subgiant and RGB stars for this stellar mass, according to [@mathur2014]. On the other hand, the weak variability of this light-curve, represented by $S_{ph} = 102$ ppm over a 50 years time span, reinforces the flat activity profile of this star. However, the lack of modulation could also be an indication of a dynamo shut down. Therefore, these preliminary photometric results strengthen the argument in favor of HD43587 being under a long flat-activity phase.

![image](figures/FIGURA4a.png){width="\textwidth"}

Finally, we investigated the possibility to revisit the photometric rotational modulation of this object through a combination of CoRoT and TESS data, in order to enhance the accuracy of the measurements. Unfortunately, after locating this star in the TESS field of view, we realize that HD43587 was situated between two CCDs in one of the TESS field sectors.

S-index time series analysis {#sec:ts}
----------------------------

We computed the S-index for the different spectra obtained with the spectrographs NARVAL and HARPS. We used our own tools developed according to the prescriptions of @wright and @marsden. These tools extract the activity index from the Ca II H&K lines emission fluxes, performing a discrete counting from the spectral lines $H$, $K$, $R$ and $V$. We used Eq. (\[eq.2\]) to derive the $S_{MW}$ directly from NARVAL spectra. From HARPS spectra, we computed the instrumental $S_{HARPS}$ following Eq. (\[eq.3\]) and calibrated to the Mt. Wilson scale $S_{MW}$ using Eq. (\[eq.4\]).

Afterward, we combined the whole computed $S_{MW}$ time series with the previous published data as described in Sect. \[data\]. The complete dataset contains 1524 measurements obtained over 50 years between 1966 and 2016. This approach allows us to build a long activity time series calibrated to MW scale, which is presented in the upper panel of Fig. \[fig:HD43587Sindex\]. The standing flat-activity profile, $\braket{S_{MW}} = 0.154$ and $\log \braket{R'_{\mathrm{HK}}} = -4.97$ (see Sect. \[sec.sr\]), is reinforced by the very low variability of the S-index, quantified by the standard deviation $\sigma_{S}$ = 0.0043.

From this complete activity time series, we investigate the periodicity of a reliable activity cycle of HD43587 consistent with a solar analog star by using the GLS and the Bayesian generalized Lomb-Scargle periodogram with a trend (BGLST) as proposed by [@Olspert2018]. The results obtained with both methods are similar conforming detailed below.

### Generalised Lomb-Scargle analysis {#sec:gls}

![Comparison between the analysis made with GLS, GP, and BGLST methods. The green line shows the long trend fit, the red line is a sinusoidal representative fit created with GLS, the magenta line is the BGLST fit. The blue line represents the GP mean and the blue shaded region is the confidence region from the GP fit.[]{data-label="fig:BGLST"}](figures/Analysis_HD43587_sindexfit_B.pdf){width="\columnwidth"}

We used the GLS analysis to compute the activity cycle period, as well as to construct a phase curve for the best determination. The GLS periodogram of the whole $S_{MW}$ time series is represented by the solid gray line in the bottom left panel of Fig. \[fig:HD43587Sindex\]. We found two peaks, one around 10 years, and another one, representing a long term tendency larger than 50 years. This latter trend is represented in the upper panel by the solid blue line. We removed this tendency from the time series and generated a new GLS periodogram represented by the solid black line in the bottom left panel. We used a gaussian fit (solid red line) to find the cycle period without the long term trend. We thus inferred a cycle period, $P_{\mathrm{cyc}} = 10.44 \pm 3.03$ yrs. This result is represented by a sinusoidal fitting in the upper panel (red solid line) along with the blue long term trend. The right bottom panel shows the phase of the folded time series with the sinusoidal model plotted over it. This period of the activity cycle is in good agreement with other solar analog stars [see e.g. @Metcalfe2016]. This result revises the activity cycle length proposed by [@egeland_PHD] who, using a shorter dataset, found $P_{\mathrm{cyc}}$ $\approx$ 36 yrs.

We also found a long-term trend that might be a longer cycle with a length exceeding 50 years. This possible longer period could be similar, in nature, to the period of the solar Gleissberg cycle, which is about 88 yrs. Note that HD43587 could be around the minimum of this cycle as indicated by the blue line in Fig. \[fig:HD43587Sindex\]. A considerable observational difficulty lies in the fact that, for HD43587, but also for other analog stars, this secondary cycle may be too long to be determined by using the currently available data. Nevertheless, in the case of HD43587, this result is consistent with the possibility of a primary and a secondary activity cycles [@Brandenburg2017].

### Bayesian generalized Lomb-Scargle periodogram with trend (BGLST) analysis

We also performed a periodogram analysis based on the Bayesian generalized Lomb-Scargle method with linear trend (BGLST), as presented in [@Olspert2018]. When applied to our data we found a similar period of $\sim$10 years. The BGLST method found a linear trend from the beginning of the data until the middle and tried to adjust the same tendency until the final fit. This “linear fit” moves the final fit a bit upwards. Thus, the BGLST method returns a result similar to the GLS analysis presented in the previous section. However, instead of removing a long cyclical trend, it results in a periodic signal on top of a linear trend as shown by the magenta line in Fig. \[fig:BGLST\]. In addition, we performed a Gaussian processes (GP) regression with quasi-periodic cycles by using the GPy code [@gpy2014]. We did not find any meaningful discrepancies on the overall period found by GLS alone. Figure \[fig:BGLST\] shows the analysis made with GLS (red line), GP (blue line and shaded blue region), and BGLST (magenta line). The green solid line corresponds to the long trend fit.\

Stellar Rotation {#sec.sr}
----------------

Since the photometric analysis using the CoRoT light curve does not provide a reliable and consistent value for the rotational period of HD43587, in this section we explore three different alternatives to compute this parameter. First, we directly used the long-term spectroscopic measurements of $S_{MW}$ over 50 yrs to look for a rotation period in the GLS periodogram. We analyzed the time series from 46340.99 to 46525.65 days (JD-2400000), containing 195 data points spread over 184 days, which represents the denser region of data in the time series. The period values found in the GLS were below the $10 \%$ false alarm probability, and thus discarded as true rotation period.

$$\begin{aligned}
    \centering
    P_{\mathrm{rot}} &=& f(B-V) \cdot g(t)\;,
    \label{gyro1}\end{aligned}$$

Alternatively, we estimated the rotation period using gyrochronology as proposed by @barnes and @meibom, presented in Eq. (\[gyro1\]). It consists in computing both, the age dependence function, $g(t) = t^{n}$ (with the age $t$ in Myr) and the mass dependence function, $f(B-V) = a(B-V-c)^{b}$. The constant terms are $a = 0.770 \pm 0.014$, $b = 0.553 \pm 0.052$, $c = 0.472 \pm 0.027$ and $n = 0.519 \pm 0.007$. We used the two ages determined by [@castro], inferred from two different evolution codes, as presented in Table \[table:1\] and described in detail in Sect. \[HD43587\]. The color index used is $B-V$ = 0.61 [@VanLeeuwen2007]. Therefore, we found two rotation periods: $P^{T}_{\mathrm{rot}} = 25.04 \pm 4.4$ d with TGEC age, and $P^{C}_{\mathrm{rot}} = 22.9 \pm 3.7$ d with CESTAM age. From these results, we can deduce a theoretical Rossby number, Ro, which is an important parameter of hydromagnetic dynamo theory. It is defined as the ratio of the rotation period, $P_{\mathrm{rot}}$, to the convective turnover time, $\tau_c$, which, in turn, is computed as the ratio of the mixing length to the turbulent velocity one pressure height scale above the base of the convection zone [@gilman80]. From the inferred rotation period with TGEC models age, and the convective turnover time of TGEC models $\tau_c = 12.8\pm1.0$ d, we deduce a Rossby number Ro$^{T}= 1.96\pm0.38$. Unfortunately, we had no access to the convective turnover time in CESTAM models and could not infer a Rossby number from them.

In addition, we also inferred a rotation period by using the semiempirical calibrations from [@noyes84a], using the time-average observational $S_{MW}$ detailed in the Sect. \[data\]. We derived the chromospheric contribution, $R'_{\mathrm{HK}}$, from the $S_{MW}$ measurements. It is computed as $R'_{\mathrm{HK}} = R_{\mathrm{HK}} - R_{phot}$, where $\log R_{phot} = -4.02 - 1.40 (B-V)$ and $R_{\mathrm{HK}} = 1.340C_{cf}\cdot S_{MW}$, where $C_{cf}$ is a color-dependent factor proposed by [@middelkoop]. [@noyes84a] found a cubic fit relation between the Rossby number and the activity index, $R'_{\mathrm{HK}}$ [see Eq. 3 in @noyes84a]. We applied this calculation for each value of $S_{MW}$ in our dataset, and determined Ro for each of them. The mean Rossby number with its standard deviation is Ro$^{N}= 2.05 \pm 0.05$. We then determined the convective turnover time by using the empirical relation presented in Eq. (4) of [@noyes84a], which depends on $(B-V)$. With the resulting value, $\tau_c = 9.67$ d, and the definition of the Rossby number Ro = $P_{\mathrm{rot}}/\tau_c$, we derived a mean $P^{N}_{\mathrm{rot}} = 19.8 \pm 0.7$ d.

With these results we infer an average rotation period, $\overline{P}_{\mathrm{rot}} = 22.6 \pm 1.9$ d for HD43587, which is consistent with the rotation of a solar analog. The good estimate of $\overline{P}_{\mathrm{rot}}$ found is reinforced when we examine the ratio of the cycle frequency to the rotation frequency given by $\omega_{\mathrm{cyc}} / \Omega = P_{\mathrm{rot}} / P_{\mathrm{cyc}}$ as suggested by @Brandenburg2017. In the case of the primary cycle, $P_{\mathrm{cyc}} = 10.44$ yrs, we have $\log (\overline{P}_{\mathrm{rot}} / P_{\mathrm{cyc}}) = -2.23$, while the solar value is $\log (P_{\mathrm{rot}} / P_{\mathrm{cyc}})_{\odot} = -2.20$. In the case of the possible secondary cycle, $P_{\mathrm{cyc}} \geq  50$ yrs, we found $\log (\overline{P}_{\mathrm{rot}} / P_{\mathrm{cyc}}) \leqslant -2.87$. For the Sun, the long Gleissberg cycle (GC) gives a ratio $\log (P_{\mathrm{rot}} / P_{\mathrm{cyc}})_{\odot,GC} = -3.10$. If we examine the position of HD43587 in the Fig. 5 of @Brandenburg2017, which plots this ratio as a function of $\log \braket{R'_{\mathrm{HK}}}$, we can see that HD43587 lies on the inactive branch region, highlighting that the $\log \braket{R'_{\mathrm{HK}}} = -4.97$ value is below the Vaughan-Preston gap ($\log \braket{R'_{\mathrm{HK}}} = -4.75$).

Finally, another interesting fact concerning HD43587 is that its Rossby number, as determined above, is close to the critical Rossby number, Ro $\sim 2$, beyond which the efficiency of magnetic braking is dramatically reduced [@VanSaders2016; @Metcalfe2016]. This region corresponds to an activity level $\log R'_{\mathrm{HK}} \sim -4.95$. This suggests that HD43587 lies in a zone below the Vaughan-Preston gap where the differential rotation, key mechanism of the dynamo process, becomes inefficient [@Metcalfe2016].

![Radial profiles of the temperature, $T$ (dashed lines), and density, $\rho$ (solid lines). The blue and red lines correspond to the ambient and isentropic states, respectively. The yellow lines show the TGEC stratification.[]{data-label="fig:strat"}](figures/temp_rho.pdf){width="\linewidth"}

Simulating HD43587 with the EULAG-MHD code
==========================================

A relevant tool currently available to study stellar magnetism is global numerical modeling. Several recent works have considered this approach to study the magnetic activity of solar-like stars rotating at different rates [e.g., @BMBBT11; @ABT14; @SBCBdN17; @warnecke18; @Viviani+18; @G+19a]. The results have provided a new understanding on the relation between stellar rotation and magnetic activity. To substantiate our observational analysis we also perform numerical simulations of an atmosphere that mimics the stratification of HD43587. Our goal is not to make a detailed analysis of the simulations but getting some insights about the internal activity of this star providing support to the observational conclusions.

We simulated the internal dynamics of the star HD43587 with the EULAG-MHD code [@SC13]. The fundamental difference between the models presented here and previous solar global dynamo simulations is the adoption of a fiducial fit for its stratification. Thus, structure parameters as the location of the tachocline or the velocity field in the convection zone are properly captured. The code solves the anelastic equations of [@LH] extended to the MHD case and in the inviscid regime. One relevant point distinguishing EULAG-MHD simulations from global dynamo modeling with other codes is the use of the implicit sub-grid scale (SGS) method MPDATA. It allows to keep the numerical viscosity sufficiently low such that it is possible to resolve the transition between convective and radiative zones without much transfer of angular momentum. Within the convection zone, the numerical dissipation introduced by the MPDATA scheme plays the role of the SGS contribution to the large scale motions and magnetic field.

We consider a spherical shell, $0 \le \varphi \le 2 \pi$, $0 \le \theta \le \pi$, radially spaning from $r=0.6R$ to $r=0.96R$, and a mesh resolution of $128\times 64 \times 48$ grid points in $\varphi$, $\theta$ and $r$, respectively. The full set of equations are described in [@G+16]. The boundary conditions for the velocity field are impearmeable and stress free at both boundaries. For the potential temperature perturbations we consider zero radial derivative of the radial convective flux at the bottom, and zero convective radial flux at the top. The magnetic boundary conditions correspond to a perfect conductor at the bottom and pseudo-vacuum at top. The simulations start with white noise perturbations in the potential temperature, the radial velocity and the azimuthal magnetic field. Since the main input parameter of the model is the stellar stratification below we present details of its setup.

As described in [@G+16; @C+17] two states are necessary in the formulation of convective simulations with EULAG-MHD: the reference, isentropic, and the ambient states. The later allows adiabatic displacements of fluid parcels in the super-adiabatic layers, and inhibits convective motions where the ambient is sub-adiabatic. Both states are described by the hydrostatic equilibrium equations,

$$\begin{aligned}
    \centering
    \frac{\partial T_{\rm i}}{\partial r} &=& -\frac{g}{R_g (m_{\rm i}+1)}\;,
    \label{equ:equilT}
    \\[10pt]
    \frac{\partial \rho_{\rm i}}{\partial r} &=& -\frac{\rho_{\rm i}}{T_{\rm i}}
     \left(\frac{g}{R_g} - \frac{\partial T_{\rm i}}{\partial r} \right)\;,
     \label{equ:equilrho}\end{aligned}$$

where the index $i$ stands either by $s$, for the reference state, or by $e$, for the ambient state. In the reference state the polytropic index $m_s=1.5$. The ambient stratification is a bi-polytropic fit to the mixing length (ML) stratification model obtained with the evolutionary code TGEC as described by .

Our fit takes into account the recent results of [@G+19a], indicating that the interface between the radiative and convective zones might play a relevant role in the dynamo generating mechanism. Their results also show that the magnetic field is stored in the stable layer where it is prone to magnetoshear instabilities [see, e.g., @Miesch+07], in turn developing helical non-axisymmetric motions and currents (source of magnetic field via the so called $\alpha$-effect). As demonstrated in [@G+19b], these instabilities are highly dependent on the stratification of the atmosphere. For this reason we calibrate the ambient state to reproduce the Brunt–Väisälä frequency profile of TGEC in the stable stratified layer. Since the motions in the convection zone define the development of the differential rotation, in this region we adjust the polytropic index allowing for a radial profile of the turbulent velocity compatible with that of the TGEC model (see Fig. \[fig:turbvel\]). Thus, the quantity $m_e$ in Eq. (\[equ:equilT\]) is given by

![Radial profile of the turbulent velocity for simulations P29 (red), P25 (yellow) and P21 (green). The blue line shows the TGEC stratification model.[]{data-label="fig:turbvel"}](figures/conv_velocity.pdf){width="\linewidth"}

$$\begin{aligned}
    \centering
    m_e(r) = m_{rz} + \frac{\Delta m}{2} \left[1 + {\rm erf} \left( \frac{r - r_{tac}}{w_t} \right) \right]\end{aligned}$$

where $m_{rz} = 2.0$ is the polytropic index of the radiative zone, $\Delta m = m_{cz} - m_{rz}$, with $m_{cz} = 1.49989$ the polytropic index in the convective zone, and $w_t = 0.1 R$ the width of transition between both regions. The radiative zone extends until $r_{tac}=0.76R$, the convective zone corresponds to the upper envelope. The profiles of temperature and density are shown in Fig. \[fig:strat\]. The yellow solid and dashed lines depict the ML stratification from TGEC model, the blue and red solid (dashed) lines correspond to the fitted density (temperature) profile for the ambient and reference states, respectively.

We performed three simulations with the same stratification and different rotational period. They are labeled as P29, P25 and P21 for 29, 25 and 21 days, respectively. The rotation periods cover the uncertainties in the determination of $P_{\mathrm{rot}}$ from the analysis presented in §\[sec.sr\]. In Fig. \[fig:turbvel\], we compare the convective velocity from TGEC model with the turbulent velocity, $u_{\rm RMS}$ obtained in the three global simulations performed with EULAG-MHD. Since rotation constrains the convective motions, the amplitude of $u_{\rm RMS}$ decreases with the decrease of $P_{\mathrm{rot}}$. All three velocity profiles exhibit, however, the same trend. The values of $u_{\rm RMS}$ volume averaged in the convection zone are presented in Table \[tab:2\]. The Rossby number presented in the same table is computed as in [@gilman80], with $\tau_c$ computed from the pressure scale height and the resulting $u_{\rm RMS}$ one pressure scale height above the star’s tachocline. At this radial point the turbulent velocity in simulations P25 and P21 is smaller than the one predicted by the mixing length model. Thus, Ro is smaller than the one obtained from the TGEC model for the same rotational period, but still within the determination error of Ro$^T$.

Also in contrast with TGEC, the turbulent velocity in the simulations is different from zero in the radiative zone. These motions have two origins, overshooting of the downward flows coming from the convection zone, and non-axisymmetric motions developed in the stable layer due to MHD instabilities. As will be presented below, the realistic stratification of HD43587 captured in the ambient state allows us to reproduce some aspects of the magnetism of this star which are compatible with the observational results obtained in the previous sections.

Simulation results 
-------------------

Figure \[fig:omega\] shows the differential rotation profiles resulting from the simulations (a) P29 to (c) P21. The mean angular velocity, $\bar{\Omega}$, is an average on longitude and time (along the last two magnetic cycles of each simulation). All the cases show a clear formation of a radial shear layer at the base of the convection zone. For the three rotational periods studied, the resulting differential rotation is solar like, with slow poles and a fast equator. This kind of differential rotation appears when the time-scale of the convective flow is similar to the rotational one, i.e., rotationally constrained motions. In this case the transport of the angular momentum is directed towards the equator [see e.g., @FM2015; @Guerrero20].

In the simulations P29-P21, we notice that the latitudinal gradient of angular velocity diminishes as the rotational period decreases (Fig. \[fig:omega\] (a) to (c)) This is quantitatively verified in the third column of Table \[tab:2\] showing the relative differential rotation, $\Delta \Omega/\Omega_{eq}$, where $\Delta \Omega = \Omega_{eq} - \Omega_{45}$. In all the cases this value is smaller than the solar one, $\sim 0.1$, and about one order of magnitude smaller than in other stars where it was inferred by astroseismology [@Benomar+18].

![Mean differential rotation in the meridional plane from simulations (a) P29, (b) P25 and (c) P21. The color scale shows the angular velocity with respect to the rotation frame. The dashed lines depict the tachocline location.[]{data-label="fig:omega"}](figures/omega3.pdf){width="\linewidth"}

In Fig. \[fig:butterfly\], the colored contours depict the radial magnetic field averaged over longitude, $\overline{B}_r$, and the solid (dashed) contour lines represent the positive (negative) mean toroidal magnetic field, $\overline{B}_{\varphi}$. In general, the simulations result in oscillatory magnetic fields with polarity reversals. Like in [@G+19a], the results suggest the existence of a $\alpha^2-\Omega$ dynamo operating inside HD43587. The magnetic fields are anti-symmetric across the equator. For slow rotation the field seems further and further concentrated near the poles. The averages in volume and time of the toroidal and radial fields are also presented in Table \[tab:2\]. We have choosen to present $\langle \overline{B}_r \rangle$ instead of the poloidal field because the radial field is directly observed in the Sun.

![Correlation between $P_{\rm cyc}$ and $P_{\rm rot}$ for the simulations presented in Table \[tab:2\], green points. The blue and red points correspond to F, G, and K stars data as presented in [@Brandenburg2017]. The black star shows the position of HD 43587, and the yellow point marks the position of the Sun. []{data-label="fig:prpc"}](figures/prpc_proc2.pdf){width="\linewidth"}

The magnetic cycle periods, $P_{\rm cyc}$, were computed by Fourier analysis (see Table \[tab:2\]). The values, between 6.42 and 8.59 years are slightly smaller than the obtained in the chromospheric activity analysis of 10.44 years. Nevertheless, the cases for 21 and 25 days are still within the $1\sigma$ error of 3.3 years. In Figure \[fig:prpc\] we show the results of the simulations in the $P_{\rm rot}$ vs. $P_{\rm cyc}$ plane. The figure contains the observational points of F, G, and K stars as presented in [@Brandenburg2017]. The rotational and magnetic cycles of HD43587 are presented with a black star. The periods of the three simulations fall in the inactive branch of activity closer to the position of the Sun. The agreement between observation and simulation is encouraging, and indicates that the simulations are capturing the physics of the interior of the star.

In the recent work of [@Olspert2018], the results of novel analysis of the MW data suggest that the active branch presented in Fig. \[fig:prpc\] may not exist. Instead, they found a branch merging the active and transitional branches. Although this might be the case, we consider that there is not yet sufficient stars observed with fiduciary determination of their periods. Thus, while the activity-rotation problem remains in discussion [@BoroSaikia2018], we decided to present our results together with the canonical A and I branches.

![image](figures/butter3.pdf){width="\textwidth"}

The amplitude of the toroidal magnetic field in the generating radial shear layer is about $0.4$ Tesla and is larger in the simulations with larger period. The radial field component, on the other hand, is about $0.01$ Tesla and is larger in the simulations with shorter period (fast rotating). Figure \[fig:butterfly\] indicates that the magnetic field migrates upwards, however, at the upper boundary of the model, $r=0.95R_{\star}$, both, the toroidal and the radial field components decay about one order of magnitude. Furthermore, the simulation results do not exhibit relevant dynamo action nearby the upper boundary. Thus, it is expected that it will decrease further when reaching the star surface which is not considered in the model.

In summary, the simulations are able to reproduce cyclic dynamo action, including field reversals, with a differential rotation smaller than in the Sun, and a magnetic field which at the top boundary is similar to the radial field observed at the solar surface ($\sim 10^{-3}$ Tesla). The 10% difference observed in $\langle S_{MW} \rangle$ between the Sun and HD43587 might be correlated to a diminished differential rotation which in the Sun is at least twice larger than in simulations P25 and P21. These cases are the ones that better reproduce the cycle period of HD43587 inferred from the spectroscopic chromospheric activity.

Conclusions and summary
=======================

The spectral analysis is fundamental to understand astrophysical processes and stellar evolution. Currently, the stellar magnetic activity is one of the most important observables in the context of stellar interiors, and it might impact exoplanets detection as shown by @aigrain. In this work, from the high-resolution spectra archive and already published data, we investigated the chromospheric activity evolution of the solar analog star HD43587.

[c c c c c c c c c c c]{}\
Model & $P_{\mathrm{rot}}$ & ${\rm min}(Ra)$ & ${\Delta \Omega}/{\Omega_{eq}}$ & $\langle u_{RMS} \rangle$ & $Ro$ & $P_{cyc}$ & $\langle \overline{B}_{\phi} \rangle_{NSL}$ & $\langle \overline{B}_{r} \rangle_{NSL}$ & $\langle \overline{B}_{\phi} \rangle_{TAC}$ & $\langle \overline{B}_{r} \rangle_{TAC}$\
& \[days\] & & \[m s$^{-1}$\] & & \[years\] & \[T\] & \[T\] & \[T\] & \[T\]\
\
P29 & 29 & $4.5\cdot10^6$ & 0.087 & 87.06 & 2.13 & 6.42 & 0.0422 & 0.0012 & 0.4559 & 0.0113\
P25 & 25 & $4.5\cdot10^6$ & 0.047 & 80.13 & 1.64 & 6.87 & 0.0403 & 0.0013 & 0.4372 & 0.0147\
P21 & 21 & $4.5\cdot10^6$ & 0.023 & 79.35 & 1.38 & 8.59 & 0.0392 & 0.0016 & 0.4027 & 0.0174\

A minimum Rayleigh number for the simulations may be estimated as ${\rm min}(Ra) = \frac{g_0 r_{\rm cz}^2}{\nu \alpha \Theta_e} \frac{d \Theta_e}{d r}$, where $\Theta_e$ is the ambient state potential temperature, $\alpha=9.64\times10^{-9}$ s$^{-1}$, is the inverse of the Newtonian cooling time limiting the growing of perturbations of $\Theta$, $r_{\rm cz} = 0.2R_{\star}$ is the depth of the convection zone and $\nu = 10^8$ m$^2$ s$^{-1}$, is the maximum value of the numerical viscosity estimated by [@SBBCMS16], for EULAG-MHD simulations with similar resolution.

Differential rotation parameter, $\Delta \Omega /\Omega_{eq} = 
        (\Omega_{eq} - \Omega_{45})/\Omega_{eq}$, where $\Omega_{eq}$ and $\Omega_{45}$ are the angular velocities at $0^{\circ}$ and $45^{\circ}$ latitude, respectively.

$\langle u_{RMS} \rangle$ is the volume average of the turbulent velocity in the entire convection zone.

Rossby number is defined as ${\rm Ro}=P_{\mathrm{rot}}/\tau_c$ and computed with $\tau_c = \alpha H_p / u_{RMS}$, where $H_p$ is the pressure scale height, and it is measured at one pressure height scale above the bottom of the convective zone [@gilman80].

Toroidal and radial magnetic field averaged over longitude and time. The subscripts $NSL$ and $TAC$ correspond to $r=0.95R_{\star}$ and $r=0.76R_{\star}$, respectively.

We built the larger $S_{MW}$ time series already published for this star, using data spanning 50 years from 1966 until 2016. The derived $\braket{S_{MW}} = 0.154$ and $\log \braket{R'_{\mathrm{HK}}} = -4.97$ indicate a low chromospheric activity, beyond the Vaughan-Preston gap. From the GLS analysis of the $S_{MW}$ time series, we found a reliable 10.44 yrs cycle period (see Fig. \[fig:HD43587Sindex\]), which is similar to the solar value. The indirect determinations of a rotation period give $\overline{P}_{\mathrm{rot}} = 22.6 \pm 1.9$ d. These values of $P_{\rm cyc}$ and $\overline{P}_{\mathrm{rot}}$, are consistent for a solar analog star. The inferred Rossby number, Ro$^{N}$ = $2.05\pm0.05$ from the calibrations prescribed by @noyes84a, and Ro$^{T}$ = $1.96\pm0.38$ from TGEC modeling, point out a rotation period about twice larger than the convective turnover time which might represent a diminished differential rotation [@Metcalfe2016].

Based on the inferred rotational period and the internal structure of HD43587 we performed global simulations with the EULAG-MHD code. We considered three different rotational periods around the value obtained in §\[sec.sr\]. For all the cases the angular velocity is solar-like, i.e., fast equator and slow poles, with relative differential rotation smaller than the solar value. In principle, the smaller shear might be responsible for the low levels of activity of the simulated stars. The radial field at the top of the simulations (at $r=0.96\,R_{\star}$) is about $10^{-2}$ T. This value is similar to the radial field at the solar photosphere.

The magnetic field is oscillatory with magnetic field reversals. The magnetic cycle periods obtained in the simulations range from $6.4$ yrs, for the simulations with rotational period of 29 days, to $8.6$ yrs for the simulation with $P_{\rm rot} = 21 $ days. This cycle period is compatible with the one inferred in §\[sec:ts\].

It is worth mentioning that given the difficulties of finding the rotation period from the light curves, the indirect determination of the $P_{\rm rot}$ is the best approach to to characterize the star HD43587. The fact that the inferred rotation period fits with a solar analog encourages us to presume that it might be correct. Furthermore, ab-initio simulations of the star with initial parameters given by the stellar structure (the same as used to determine the rotational period) as well as the inferred $P_{\rm rot}$, result in activity cycles close enough to the independently determined from the $S$-index. These numerical results substantiate the observational ones.

Even though the possibility that HD43587 is under a Maunder minimum cannot be ruled out, all our results seem to indicate that it is a solar analog star, older than the Sun, and undergoing a “natural” decrease of its magnetic activity. Nevertheless, some authors argue that a transition in the stellar differential rotation might be the cause of these low levels of activity [@MS17]. The characterization of a Maunder minimum would only be valid if we could observe the star coming in and out of this activity minimum [@hall]. There is no current observation indicating this pattern.

The low activity level observed in HD43587 might be due to a small differential rotation in the convective zone. We have also found evidences that HD43587 could be at the minimum of a longer activity cycle larger than 50 years. However, in the context of observations of magnetic cycles in solar-like stars, the existence of long-activity cycles for old objects is rather controversial. The results presented by [@Brandenburg2017] show that, besides the Sun, all confirmed objects that have co-existing long and short cycles are younger than 2.3 Gyr. As a matter of fact, the reality of double periodicities in the observed stellar activity cycles is still questioned within the community [@BoroSaikia2018; @Olspert2018]. [****]{}

The results presented above demonstrate the relevance of studying HD43587 in the context of stellar evolution. Several authors have investigated its features and evolutionary status, as well as its magnetic activity evolution. Its similarity with the Sun, whether evolutive, spectroscopic or magnetic, gives a sight of the future evolution of the Sun and makes it a good candidate for being an earth-like exoplanet host. Its low activity level provides an excellent opportunity to attempt a signal of an earth-like planet without the interference of the magnetic activity [@Korhonen2015]. In this way, we estimate it is fundamental to keep monitoring the activity of HD43587 to confirm the co-existence of a short and long period cycle, as well as to increase the accuracy of the cycle and rotation periods determination. We also consider that it should be a key target for the ESA PLATO mission.

We would like to thank the anonymous referee for useful comments and suggestions to improve this paper. Research activities of the Ge$^3$[^6] stellar team at the Federal University of Rio Grande do Norte are supported by continuous grants from Brazilian scientific promotion agencies CAPES and CNPq. R.R.F. acknowledges CAPES graduate fellowships. J.D.N. and M.C. acknowledge support from CNPq ([*Bolsa de Produtividade*]{}). We also thank T.Morel for providing the HARPS spectra used in this paper.

[^1]: Available at Royal Observatory of Belgium: [<http://www.sidc.be/silso/datafiles>]{}

[^2]: Located at Mount Wilson, at 1742 m of altitude in [*San Gabriel Mountains, LA, California*]{}, the observatory has two telescopes: the Hale telescope (1,5 m), built in 1908, and the Hooker telescope (2,5 m), built in 1917.

[^3]: <http://polarbase.irap.omp.eu>

[^4]: [<https://www.nso.edu/data/historical-data/mount-wilson-observatory-hk-project/>]{}

[^5]: <http://polarbase.irap.omp.eu/>

[^6]: <http://ge3.dfte.ufrn.br>
