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# The Rotation-activity Correlations in K and M Dwarfs. II. New Constraints on the Dynamo Mechanisms in Late-K and M Dwarfs before and at the Transition to Complete Convection*

E. R. Houdebine<sup>1,2,3</sup>, D. J. Mullan<sup>4</sup>, B. Bercu<sup>5</sup>, F. Paletou<sup>2,3</sup>, and M. Gebran<sup>6</sup>

<sup>1</sup> Armagh Observatory, College Hill, BT61 9DG Armagh, UK; eric_houdebine@yahoo.fr

<sup>2</sup> Université de Toulouse, UPS-Observatoire Midi-Pyrénées, IRAP, Toulouse, France

<sup>3</sup> CNRS, Institut de Recherche en Astrophysique et Planétologie, 14 av. E. Belin, F-31400 Toulouse, France

<sup>4</sup> Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA; mullan@udel.edu

<sup>5</sup> Université de Bordeaux, Institut de Mathématiques, UMR 5251, 351 cours de la libération, F-33405 Talence cedex, France; Bernard.Bercu@math.u-bordeaux1.fr

<sup>6</sup> Department of Physics & Astronomy, Notre Dame University-Louaize, PO Box 72, Zouk Mikaël, Lebanon

Received 2015 May 4; revised 2017 January 7; accepted 2017 January 25; published 2017 March 7

## Abstract

We study the rotation-activity correlations (RACs) in a sample of stars from spectral type dK4 to dM4. We study RACs using chromospheric data and coronal data. We study the Ca II line surface fluxes-$P$/sin$i$ RACs. We fit the RACs with linear homoscedastic and heteroscedastic regression models. We find that these RACs differ substantially from one spectral sub-type to another. For dM3 and dM4 stars, we find that the RACs cannot be described by a simple model, but instead that there may exist two distinct RAC behaviors for the low-activity and the high-activity stellar sub-samples, respectively. Although these results are preliminary and will need confirmation, the data suggest that these distinct RACs may be associated with different dynamo regimes. We also study $R'_{HK}$ as a function of the Rossby number $R_0$. We find (i) for dK4 stars, $R'_{HK}$ as a function of $R_0$ agrees well with previous results for F-G-K stars and (ii) in dK6, dM2, dM3, and dM4 stars, at a given $R_0$, the values of $R'_{HK}$ lie at a factor of 3, 10, 20, and 90, respectively, below the F-G-K RAC. Our results suggest a significant decrease in the efficiency of the dynamo mechanism(s) as regards chromospheric heating before and at dM3, i.e., before and at the transition to complete convection. We also show that the ratio of coronal heating to chromospheric heating $L_X/L_{HK}$ increases by a factor of 100 between dK4 and dM4 stars.

**Key words:** stars: activity – stars: chromospheres – stars: coronae – stars: late-type – stars: low-mass – stars: magnetic field

*Supporting material:* machine-readable tables

## 1. Introduction

In the present paper, we have two principal goals. First, we present data that have a bearing on the existence of correlations between rotation and activity (RACs) in our sample of stars. Second, we ask if the empirical correlations can be interpreted in the context of stellar dynamo theories (which attempt to predict the properties of the magnetic fields that are generated in a star with a prescribed structure and rotation). The second goal is admittedly a challenging one: the basic mechanisms underlying stellar dynamo theory are being continuously improved as computer resources permit inclusion of more realistic physical effects. Rather than attempting to provide a comprehensive discussion of these complexities, we simplify our discussion by restricting attention to certain broad classes of dynamos. Here, we briefly outline the distinction between these classes. Whatever the source of the magnetic field, the observational consequences of such a field in a star is expected to be the same: dissipation of mechanical energy associated with magnetic processes leads to enhanced emission (relative to what is generated by the photosphere) from the chromosphere (in spectral lines such as Ca II H and K or $H_α$) and from the corona (in X-ray continuum). In this paper, these enhanced emissions arising from magnetic effects in the stellar

atmosphere are considered to be generic indicators of what we refer to as “magnetic activity,” or more briefly, “activity.” We seek to quantify how (an observational quantity that is associated with) “activity” in low-mass dwarfs is correlated with (an observational quantity that is associated with) “rotation.” In order to set the stage for interpreting our result, we first need to define certain terms that are related to dynamo models.

### 1.1. Three Classes of Dynamo Models

Models of dynamos in the Sun and stars are typically based on “mean-field electrodynamics” (MFE): the fluid flow and the magnetic field are separated into mean $(\langle u \rangle, \langle B \rangle)$ and turbulent $(u', B')$ components (e.g., Racine et al. 2011). Although the mean values of $u'$ and $B'$ are by definition zero, the mean value of their cross-product does not reduce to zero, but instead produces a non-zero turbulent MFE, E. The essence of MFE is to express the (vector) E in terms of the large-scale (vector) magnetic field $\langle B \rangle$ by means of a tensor expansion in the mean field plus its gradients. The leading order term in this expansion $E_i = \alpha_{ij} \langle B_j \rangle$ is called the “$\alpha$-effect.” Parker (1955) first suggested that the $\alpha$-effect might arise because cyclonic convective turbulence can systematically twist a large-scale magnetic field, and in the process, regenerate a large-scale poloidal field. All of the dynamos we consider here rely on the $\alpha$-effect, as well as on (at least) one more factor.

<footnote>* Based on observations available at Observatoire de Haute Provence and the European Southern Observatory databases and on Hipparcos parallax measurements.</footnote>
In order to construct a mean-field dynamo model, ⟨B⟩ is written as the sum of poloidal and toroidal components, and ⟨u⟩ is assumed to be directed purely in the azimuthal direction φ, and to be axisymmetric, i.e., the angular velocity Ω can be a function of r and θ, but not of φ. With these assumptions, the induction equation for the time-varying magnetic field can be separated into two equations, one for the poloidal field, the other for the toroidal field. Source terms (S<sub>p</sub>, S<sub>t</sub>) appear in both equations, and the dominant source terms determine the dynamo class. For the poloidal equation, S<sub>p</sub> is a single term, namely, the φ component of E, i.e., the α-effect. For the toroidal equation, S<sub>t</sub> contains two terms: (i) a spatial gradient of Ω and (ii) a spatial gradient of E.

Three classes of mean-field dynamos are defined as follows: (a) α-Ω dynamo, in which only term (i) is retained in the toroidal equation; (b) α² dynamo, in which only term (ii) is retained in the toroidal equation; (c) α²-Ω dynamo, in which terms (i) and (ii) are both retained in the toroidal equation.

## 1.2. Stellar Internal Structure as It Relates to Dynamo Activity

Models of dwarf stars of spectral type F, G, K, and early M possess a radiative core and a convective envelope. In such stars, magnetic fields can be generated by an α-Ω dynamo (also referred to as an interface dynamo, or shell dynamo, Parker 1975).<fnref n="7" /> The strongest toroidal magnetic fields in these stars are expected to be produced in the vicinity of the interface (“tachocline”) between the radiative core and the convective envelope, where the differential rotation is the strongest. In stars that are massive enough to contain such an interface, and where, therefore, the possibility of an “interface dynamo” (ID) exists, data pertaining to enhanced emission from such stars in chromospheric spectral lines and/or in coronal X-rays reveal clearly that there is a strong correlation between rotation and activity indicators (e.g., Pallavicini et al. 1981; Wright et al. 2011), i.e., the Rotation-Activity Correlation (RAC) has a slope with a numerical value that is definitely non-zero.

However, since Limber (1958) derived models of cool main-sequence stars with lower and lower masses, it has been widely believed by stellar evolution modelers that interfaces do not exist in all low-mass stars. An anonymous referee has pointed out that “there has been no definitive evidence validating this theoretical prediction.” In principle, asteroseismology of M dwarfs might eventually provide *p*-mode frequencies with enough precision that (by analogy with helioseismological data) signatures of the interface could be identified. But although theoretical models have been calculated for such stars (see Rodriguez-Lopez et al. 2014), no empirical data on reliable *p*-mode periods in M dwarfs are yet available. Despite the lack of definitive evidence at the present time, we shall adopt the widely held belief that stars on the main sequence undergo a transition to complete convection (TTCC) at a certain spectral type. According to models, main-sequence stars with masses that are less than a critical value of about 0.30–0.35 M<sub>☉</sub> are completely convective. Such stars should have no tachocline whatsoever on the main-sequence. And yet the empirical evidence shows that stars with masses less than 0.30–0.35 M<sub>☉</sub> also show enhanced emission in chromospheric lines and in X-ray continuum. Since we have attributed such enhanced emission in warmer stars to magnetic activity, then it seems

natural also to ascribe the enhanced emissions in completely convective stars to magnetic activity. Such activity must rely on a non-interface type of dynamo, possibly an α² dynamo (distributive dynamo; DD), maintained by convective turbulence alone (e.g., Roberts & Stix 1972; Rosner 1980; Chabrier & Kuker 2006; Dobler et al. 2006).

It is important to note that the existence of complete convection in a star does not mean that the star is necessarily completely convective at all stages of its main-sequence phase of evolution. For example, Feiden & Dotter (2013) report that a 0.3 M<sub>☉</sub> model arrives on the main sequence with a radiative zone sandwiched between two convective zones: the models suggest that such a “sandwich” structure exists for several gigayears before the model becomes completely convective (see also Rodriguez-Lopez et al. 2014). However, at lower masses, 0.25 M<sub>☉</sub>, stellar models are completely convective at all ages (see Rodriguez-Lopez et al. 2014). Thus, when we use the label “completely convective” for stars of masses in the intermediate range (0.25–0.3) M<sub>☉</sub>, the label involves not merely a question of the mass of the model, but also of the age of the model. This complication may cause some ambiguity when we attempt to interpret RACs in the vicinity of the TTCC.

For simplicity, it would be convenient if a given star could be assigned to having either an ID or a DD. But we recognize that such a simplified approach cannot be the whole story: in cool stars where an interface exists, the inevitable presence of a deep convective envelope gives rise to the possibility that ID and DD may both be operating (e.g., Brandenburg & Subramanian 2005; Brown et al. 2010).

In order to distinguish between the observational properties of ID and DD, Durney et al. (1993) pointed out some differences between ID and DD. First, the magnetic field created by an ID should depend strongly on rotation whereas the magnetic field created by a DD should not. In terms of the notation to be used in the present paper, this would lead one to expect that in an ID star, the RAC should have a slope that is definitely non-zero, whereas in a DD star, the RAC slope should be essentially zero. Second, although IDs can produce activity cycles in the large-scale field it was not clear (in 1993) how DD could ever give rise to a cycle.

Since 1993, the second feature has been called into question. For example, Stefani & Gerberth (2003) have demonstrated that cyclic behavior is possible in α² dynamo models provided that the following condition is satisfied: the α-effect needs to have radial gradients that are sufficiently steep, including changes in the algebraic sign.

Moreover, recent modeling suggests that the third class of dynamo models (α²-Ω) might be a better description of the solar dynamo (where there is certainly an interface) than an α-Ω dynamo (e.g., Lawson et al. 2015). From the perspective of a change from an α-Ω dynamo to an α²-Ω dynamo, perhaps in completely convective stars, we might also encounter a change from an α² dynamo to an α²-Ω dynamo. If this happens, then we might find that rather than seeing a zero slope for the RAC in a completely convective star, the inclusion of Ω in the dynamo model could lead to a non-zero RAC slope.

## 1.3. Observational Evidence for the TTCC?

A long-standing problem has been the search for an observational signature of the putative transition between different types of dynamos at the TTCC. There have been

<footnote n="7">⁷ For a quantitative evaluation of such a dynamo in mid-K to early-M stars with known rotation periods, see Mullan et al. (2015).</footnote>
several attempts to detect a change in magnetic activity diagnostics (e.g., Mullan & MacDonald 2001) or magnetic field topologies (e.g., Donati et al. 2008; Morin et al. 2008, 2010; Phan-Bao et al. 2009; Stassun et al. 2011) in the vicinity of the TTCC. None of these studies have identified unambiguous and definitive signatures of significant changes in magnetic activity at the TTCC. However, in a recent paper, West et al. (2015) have reported on a study of RACs in two groups of dwarfs: M1–M4 and M5–M8. West et al. (2015) found that the RAC in M1–M4 stars has a somewhat different slope than the RAC in M5–M8 stars (see Section 4 for a more detailed discussion). One interpretation of this behavior is that something changes as regards the dynamo between M4 and M5. We will return to a discussion of this interpretation (as well as an alternative interpretation) in Section 4.

In order to address the TTCC-dynamo-transition topic meaningfully, we first need to identify at what spectral type the TTCC occurs. Limber (1958) states that “in that part of the main sequence where the inner radiative region ... is becoming vanishingly small,” the corresponding spectral type is M3V-M4V. Subsequently, Dorman et al. (1989) placed the TTCC at $M \sim 0.25 M_\odot$, which corresponds to the spectral subtype dM4. Even more recently, Chabrier & Baraffe (1997) predict that the TTCC occurs at $M \sim 0.35 M_\odot$, which corresponds to the spectral subtype dM2. Therefore, it appears that the TTCC may lie somewhere in the range between subtypes dM2 and dM4, corresponding to masses of $0.25-0.4 M_\odot$ (Stassun et al. 2011). The theoretical mass limit at the TTCC has been found to shift toward slightly smaller masses if different boundary conditions are used for the stellar models (Mullan et al. 2015). (Even larger shifts of the TTCC toward lower masses were, at one time, proposed by Mullan & MacDonald (2001) if interior magnetic fields were to be as large as $10^{-8}$ G. If fields as large as that were to exist inside stars, the definition of a “fully convective star” could become more ambiguous: the onset of complete convection would then depend not only on the mass and age of a star, but would also depend on how strong its magnetic field is. However, we may not in fact need to worry about this ambiguity: Browning et al. (2016) have recently argued that such strong fields would be unstable.

In standard main-sequence models, the radius of a model scales almost linearly with mass: therefore, the TTCC is expected to lie in the radius range of roughly $0.25-0.4 R_\odot$. According to the radius-$T_{\text{eff}}$ calibration of Houdebine et al. (2017) this radius range yields an effective temperature range of 3200–3500 K for the TTCC.

In this paper, we examine the topic of a possible dynamo transition at the TTCC using a more extensive and more fine-grained data set than has previously been available for study.

## 1.4. Dynamos: Unsaturated and Saturated

Empirically, evidence for rotationally driven dynamos in cool stars first emerged when researchers plotted the strength of chromospheric emission versus stellar rotation (e.g., see Kraft 1967, Vaughan et al. 1981, Soderblom 1982, Vogt et al. 1983, Noyes et al. 1984, Marcy & Chen 1992, Patten & Simon 1996, Fekel 1997, Delfosse et al. 1998, Jeffries et al. 2000, Pizzolato et al. 2003, Mohanty & Basri 2003, Browning et al. 2010, Wright et al. 2011, Rebassa-Mansergas et al. 2013, West et al. 2015).

The key signature is the following: emission in chromospheric spectral lines is observed to be stronger in stars with

faster rotation speed $v_r$ (or shorter rotation period $P$; e.g., Vaughan et al. 1981). This is defined as a “RAC.” The existence of an RAC is consistent with the expectations that (i) the faster the rotation is, the stronger are the magnetic fields that can be generated (e.g., Mullan et al. 2015), and (ii) stronger fields are associated with stronger chromospheric heating (Skumanich et al. 1975) and with stronger coronal heating (e.g., Mullan 2009). Thus, if chromospheric emission intensity is plotted as a function of $P$, it is found that over a certain range of periods, the RAC has a clearly negative slope.

However, as more data are accumulated, it emerges that the negative slope of the RAC does not extend indefinitely to shorter and shorter $P$. Instead, when $P$ becomes shorter than a certain value $P_c$, no further increase in emission occurs: for $P \le P_c$, the RAC becomes flat (e.g., Vilhu 1984), with a slope of zero. By definition, in the flattened portion of the RAC, increasing rotation does not result in increased chromospheric/coronal emission. For solar type stars, the transition to a flat curve occurs for $P \approx 3$ days. Vilhu suggested the term “saturated” to refer to such conditions. Similarly, Wright et al. (2011) found that the saturation regime occurs at about $R_0 \approx 0.8$. Reiners et al. (2009) found that saturation occurs at the critical Rossby number $R_0 \approx 0.1$. We shall compare these values to those we obtain for our stellar samples below. Pizzolato et al. (2003) found that saturation occurs from a period of about $\approx 2$ days in solar type stars to about $\approx 10$ days in M dwarfs. Rebassa-Mansergas et al. (2013) found that stars with $v \sin i \gtrsim 5$ km s$^{-1}$ are all in the saturated regime. This yields a rotation period of $\approx 3$ days for stars at the spectral type dM3 (see Paper I). Therefore, we shall consider that saturation occurs for periods of about $\approx 3$ days or $R_0 \approx 0.5$. Since in our stellar samples we have stars with shorter rotation periods or $R_0$, we shall consider below that these stars likely lie in the saturated regime.

Vilhu suggested that “saturation” might be due to a complete coverage of the star’s surface by magnetic fields. Another explanation of “saturation” was offered by Mullan (1984) in terms of the maximum possible flux of mechanical energy that can be generated by convection. In the present paper, we shall not attempt to identify the physical process that leads to saturation. Instead, we shall adopt a purely empirical approach, and we shall refer to the flat portion of an RAC as “saturated.” In the same vein, we shall refer to an RAC with a statistically significant negative slope as “unsaturated.” To the extent that an RAC owes its existence to the operation of a dynamo of some kind, we can say that the dynamo reveals itself in two regimes: saturated and unsaturated. The properties of the two different dynamo regimes we find below (see Section 3.8.1) for the low and high-activity sub-samples, respectively, cannot be due to effects of the chromospheric response to non-thermal heating mechanisms. In fact, there does not seem to be any definitive evidence that the influence of a magnetic field in a stellar atmosphere ever attains a saturated level. In support of this claim, we note that the response of stellar chromospheres to non-thermal heating mechanisms is continuous and monotonically increasing from basal chromospheres to flaring chromospheres (e.g., Houdebine 1992; Houdebine & Doyle 1994a, 1994b; Houdebine et al. 1995; Houdebine & Stempels 1997; Houdebine 2009a, 2009b, 2010a, 2010b). Therefore, the two different types of RACs we find below for the low and high-activity stars, respectively, highlight the different properties of the dynamo mechanisms in these two types of stars.
It seems to us that, in order to study the properties of stellar dynamos most profitably, it would be preferable to concentrate as much as possible on stars in the unsaturated regime. The reason for this claim is that, when conditions are saturated, extra factors come into play that may obscure some physical properties that are directly associated with dynamo action. In view of Vilhu's identification of a critical period $P_c$, it seems that the slowest rotators have the best chance of being in the unsaturated regime. For that reason, we consider it worthwhile to push the spectroscopic measurements of stellar rotation toward the smallest possible values of $v \sin i$ that can be reliably measured.

In general, since slow rotation means less chromospheric heating, we expect that the slowest rotators in our data set will be low-activity stars that are classified as dK and dM (i.e., those that by definition show no emission in the Balmer lines), while the fastest rotators will be highly active stars that are classified as dKe and dMe (i.e., those where, by definition, the Balmer lines have an emission core). In what follows, we shall be especially interested in determining the (negative) slope of the RAC for low-activity stars (e.g., see Figure 13 below).

## 1.5. Extending our Previous Work

One reason for the present study has emerged from recent work by Houdebine & Mullan (2015; hereafter HM): they found that another diagnostic of magnetic fields, namely, the efficiency of magnetic braking (which manifests itself in the rotational velocity), undergoes a detectable change at spectral sub-type dM3. Basing their analysis on a new data set of precise rotational velocities, HM found that the mean rotation period of M3 stars is abnormally large compared to those of the adjoining spectral types dM2 and dM4. This indicates that the dM3 stars have been slowed down more than the stars in the immediately adjacent sub-types dM2 and dM4. This excess slowing at dM3 may be associated with a change in magnetic properties of the stars at dM3. Specifically, HM suggested that the change might be associated with an earlier report (Mullan et al. 2006) that the lengths of flaring magnetic loops undergo a significant increase at spectral type dM3.

In the present paper, we extend the work of HM in two distinct ways. First, we expand the database of precise rotational properties of K and M dwarfs of various sub-types. Second, we expand and analyze a separate database, which deals with the second physical parameter that enters into the RAC: the radiative properties that are associated with magnetic “activity” in our sample of rotating stars. Our goal here is to use the activity data to construct RACs. Moreover, we quantify the RACs in two different parts of the stellar atmosphere: the chromosphere (using the Ca II lines) and the corona (using $L_X$). The present study is based on a larger sample of stars, and a finer grid of spectral sub-types, than have been used previously in constructing RACs for K and M stars. Our goal is to explore whether the unusual rotational signature reported by HM at dM3 is accompanied by unusual behavior in the activity indicators of either chromosphere or corona or both, either as regards the intensity of the radiation, or as regards the slopes of the RACs.

A key physical factor, which is known to be well correlated with chromospheric emission and coronal emission, has to do with magnetic fields on the stellar surface (e.g., Skumanich et al. 1975; Schrijver et al. 1989). The existence of RACs may be interpreted as an indication that the surface magnetic field

intensities are correlated with the stellar rotation rate. This relationship is to be expected on the basis of standard dynamo theory (e.g., Parker 1979; Krause & Radler 1980; Mullan et al. 2015). However, most of the observational investigations cited above suffer from two inadequacies: (i) they included only a few stars that are rotating slowly enough to be in the unsaturated regime and (ii) the targets included stars that were spread out over a broad range of spectral types. In this paper, we attempt to remedy both of these inadequacies.

## 1.6. Aspects of the Data Used in the Present Study

In an effort to extend the RACs to slow rotators among late-type dwarfs, we have been reporting, over the past several years, improved spectroscopic measurements of rotational broadening $v \sin i$ in stars of spectral sub-types dK4 (Houdebine 2011a, Paper XVI thereafter), dK6 (Houdebine et al. 2016, Paper I), dM2 (Houdebine 2008, Paper VIII; Houdebine 2010a, Paper XIV), dM3 (HM), and dM4 (Houdebine 2012a, Paper XVII; Paper I).

Combining our (previous) rotational measures with our measures of the Ca II line equivalent widths (EWs), we have already reported RACs for dK4 (Houdebine 2011a, Paper XVI), dM2 (Houdebine 2011b, Paper XV), and dM4 stars (Houdebine 2012b, Paper XVIII), for slow and rapid rotators alike. In those earlier papers, we proposed empirical RACs that included large samples of slow rotators. We found that, for later spectral types, it is crucial to examine a fine grid of spectral sub-types. Specifically, we found that the RACs vary significantly among dK4–dM4 stars: the RACs were found to have different gradients and different saturation levels (Papers XV, XVIII, and the present study). The previous data, combined with the present study, now provide us with large enough data sets in each spectral sub-type that we can investigate with improved confidence the differences (if any) between the RACs in five different spectral sub-types. In view of the results reported in HM, it is notable that the present study enables us to study the RACs in the vicinity of the TTCC.

When we consider slow rotators, our data confirm that the mean rotation periods of stars in the range dK4–dM4, in general, decrease with decreasing effective temperature (see HM and Paper I). However, we also find that something unusual happens in the rotation rates between dM2 and dM4. The overall trend toward decreasing rotation period as we go from dK4 to dM4 is interrupted at spectral sub-type dM3: at that sub-type, the mean rotation period increases to a local peak, such that the mean rotational period at dM3 is longer than the overall trend between dK4 and dM4 would have predicted (HM). But when we extend our investigation to include fast rotators among the dK4–dM4 stars, we find the following overall trend: the mean rotation period tends to increase slightly from dK4 to dM4. But once again, at dM3, we find an exception: the mean rotation period of fast rotators at dM3 is locally significantly longer than the overall trend would have predicted (HM). HM interpret these abnormally long rotation periods at sub-type dM3 as possibly being associated with the occurrence of increasing coronal loop lengths (previously reported by Mullan et al. 2006 in a study of flare stars). The mean rotation period of the slow rotators is an important constraint on the temporal history of the dynamo mechanisms and magnetic braking mechanisms (HM, Paper I).
## 1.7. Studying RACs in Various Formats

An important aspect of the present paper is that we wish to investigate the RACs in various formats. In the first place, we construct RACs separately for the chromosphere and the corona. Moreover, we explore correlations between various observations of the “activity” and various aspects of “rotation.” As an example, we will examine, for the H and K lines of Ca II, a plot of the quantity $R'_{HK}$ as a function of the Rossby number $R_0$. We also plot the Ca II surface flux as a function of $P/\sin i$. We use these different formats in an attempt to improve our chances of identifying changes (if any) in the dynamo regime in the vicinity of the TTCC. We shall find that changes near the TTCC are more readily detectable in some RACs than in others. This may explain why the empirical detection of the TTCC has been elusive in the past.

## 2. Selection of Spectroscopic Data

Stepień (1989, 1993, 1994) has reported that RACs exhibit certain differences at different spectral types. As an extension of this finding, we have found, in previous studies (Paper XVIII, HM) and also in the present study, that it is important in constructing RACs to select samples of stars with $T_{\text{eff}}$ values that are confined within a narrow range. There are two principal reasons for this, one related to the choice of an optimal set of photospheric absorption lines, and the second related to our analysis of chromospheric emission lines. As regards the choice of photospheric lines, we have already discussed the first of the above reasons at length in Paper I in the context of optimizing the measurement of rotational velocities at each spectral subtype between dK4 and dM4. In the present paper, we turn now to the second reason.

In the context of chromospheric analysis, it is important to deal with stars with closely similar $T_{\text{eff}}$ when we are attempting to quantify the EW of the chromospheric lines (Ca II resonance doublet and $H_\alpha$) with as much precision as possible. These lines inevitably include some contributions from the background photospheric continuum and from the temperature minimum region (e.g., Cram & Mullan 1979; Houdebine & Doyle 1994c; Houdebine & Stempels 1997; Paper XV). Initially, our samples of stars had been selected for the purpose of chromospheric modeling studies (e.g., Houdebine & Stempels 1997; Houdebine 2009b, Paper XII; Houdebine 2010b, Paper IX) and, in these studies, the selection of stars with closely similar spectral types was essential in order to develop reliable grids of semi-empirical model chromospheres, each of which would be superposed on a particular photospheric model.

Based on our previous papers, we have found that the most suitable initial selection parameter when we wish to identify a homogeneous sample of K or M dwarfs belonging to a specific sub-type is the $(R-I)$ color: this color is sensitive to $T_{\text{eff}}$, but less so to metallicity (e.g., Leggett 1992; Ramirez & Melendez 2005; Mann et al. 2015). Moreover, broad-band colors of high precision are widely available in the literature for many of the cool dwarfs that are of interest to us.

As an example of how we selected data for our RAC studies, we now describe how we gathered the relevant data for a sample of dK6 stars. We selected a sample of 419 late K dwarfs on the basis of $(R-I)$ measurements available in the literature. For example, our sample of dK6 stars contains stars with $(R-I)_C$ (i.e., $(R-I)$ color in the Cousins system) in the range [0.684; 0.816], which also corresponds to $(R-I)_K$ ($(R-I)$ in the Kron

system) in the range [0.503; 0.613] according to the transformation formulae of Leggett (1992) (see Leggett 1992 for more information on the Cousins and Kron photometric systems). According to Kenyon & Hartmann (1995), this range of colors is centered on $(R-I)_C = 0.75$, i.e., the spectral type dK7. However, when we compiled and derived effective temperatures (see Paper I) for this sample of late-K dwarfs, we found, on average, higher temperatures than what would be expected from the $(R-I)_C-T_{\text{eff}}$ tabulation of Kenyon & Hartmann (1995) (see Paper I). Our dK6 stellar sample contains stars that have similar $(R-I)_C$ colors and the same effective temperatures to within ±110 K (see Paper I). We refer to HM for a discussion of corresponding data for our sample of dM3 stars.

The literature provided us with a starting list of a large number (419) of late-K dwarfs. Searching through databases at the European Southern Observatory (ESO) and Observatoire de Haute Provence (OHP), we identified spectra of 112 different stars that are suitable for our purposes. The final list of 105 stars for which the available spectra would allow us to determine reliable rotational data (i.e., $v \sin i$ values) in our sample of late-K dwarfs has already been provided in Paper I. For the present paper, we found spectra that would allow us to make reliable measurements of the EWs of Ca II and $H_\alpha$ for a sub-sample of only 89 late-K dwarfs. It is the combination of reliable rotations (Paper I) and reliable EWs of chromospheric lines that enable us to undertake, in the present paper, the study of the RAC in our sub-sample of dK6 dwarfs.

The spectra that we use for determining the Ca II and $H_\alpha$ EWs in the present study of dK4-dM4 stars came from three different échelle spectrographs; HARPS (High Accuracy Radial velocity Planet Search, ESO), SOPHIE (OHP), and FEROS (The Fiber-fed Extended Range Optical Spectrograph). We included the SOPHIE observations obtained in the high efficiency mode in our sample. The modes in which the spectra were obtained are indicated in Table 1 together with the corresponding signal-to-noise ratio (S/N). For further details of the spectrographs, see HM.

As a second example of how we selected pertinent data for our spectral sub-samples, we consider here briefly our sample of dM3 stars. As described in HM, we started off with a list of 381 dM3 objects based on $(R-I)$ data in the literature. Searching through the same databases as above, we found suitable observations, which allowed us to determine reliable $v \sin i$ values for 86 different dM3 stars. A subsequent search of the available spectra in the above databases allowed us to obtain reliable measurements of the EWs of Ca II and $H_\alpha$ for a sub-sample of only 59 M3 dwarfs for our chromospheric RAC (excluding probable spectroscopic binaries and low $\sin i$ stars).

### 2.1. Biases in Our Stellar Samples

The stars in our samples include all stars from all observing programs that have been carried out with HARPS and SOPHIE for stars belonging to the following spectral sub-types: dK4, dK6, dM2, and dM3. For dM4 stars, we compiled all measurements of $v \sin i$ available in the literature (see Paper I). For the dK6 and dM3 samples, we also supplemented our own measurements with measurements available in the literature, notably for active stars (see Paper I).

In the HARPS and SOPHIE databases, many of the spectra were obtained in connection with planet-search programs. In such programs, observers tend to avoid stars with high levels of
<table><caption>**Table 1**

Parameters of our Homoscedastic and Heteroscedastic Regression Models</caption><thead><tr><th colspan="7">Homoscedastic Least Square Fits</th></tr><tr><th>Spectral Type</th><th>a</th><th>b</th><th>F<sub>Ca II</sub> = b × (P/ sin i)<sup>a</sup><br>Correlation Coefficient</th><th>χ<sup>2</sup></th><th>No. Stars</th><th>Statistical Significance</th></tr></thead><tbody><tr><td>dK4+dK4e</td><td>-0.756 ± 0.05</td><td>1.57 ± 0.34 10<sup>6</sup></td><td>0.884</td><td>0.032</td><td>34</td><td>>99.9%</td></tr><tr><td>dK6+dK6e</td><td>-0.81 ± 0.06</td><td>1.95 ± 0.34 10<sup>6</sup></td><td>0.876</td><td>0.021</td><td>55</td><td>>99.9%</td></tr><tr><td>dM2+dM2e</td><td>-1.481 ± 0.068</td><td>1.89 ± 0.85 10<sup>6</sup></td><td>0.949</td><td>0.018</td><td>66</td><td>>99.9%</td></tr><tr><td>dM3+dM3e</td><td>-2.08 ± 0.19</td><td>1.52 ± 1.00 10<sup>7</sup></td><td>0.735</td><td>0.118</td><td>59</td><td>>99.9%</td></tr><tr><td>dM4+dM4e</td><td>-2.526 ± 0.20</td><td>7.94 ± 2.05 10<sup>5</sup></td><td>0.861</td><td>0.090</td><td>58</td><td>>99.9%</td></tr><tr><th colspan="7">Heteroscedastic Least Square Fits</th></tr><tr><th>Spectral Type</th><th>a</th><th>b</th><th>F<sub>Ca II</sub> = b × (P/ sin i)<sup>a</sup><br>Correlation Coefficient</th><th>χ<sup>2</sup></th><th>No. Stars</th><th>Statistical Significance</th></tr><tr><td>dK4e</td><td>-1.880 ± 0.55</td><td>9.55 ± 5.18 10<sup>6</sup></td><td>0.924</td><td>0.014</td><td>4</td><td>99.7%</td></tr><tr><td>dK6e</td><td>-1.364 ± 0.16</td><td>3.97 ± 0.74 10<sup>6</sup></td><td>0.961</td><td>0.006</td><td>9</td><td>90%</td></tr><tr><td>dM2e</td><td>-1.819 ± 0.14</td><td>3.40 ± 0.51 10<sup>6</sup></td><td>0.959</td><td>0.009</td><td>11</td><td>>99.9%</td></tr><tr><td>dM3e</td><td>-1.165 ± 0.52</td><td>2.19 ± 1.58 10<sup>6</sup></td><td>0.675</td><td>0.060</td><td>8</td><td>99.5%</td></tr><tr><td>dM3e+Sat.</td><td>-0.37 ± 0.11</td><td>3.16 ± 0.59 10<sup>5</sup></td><td>0.704</td><td>0.050</td><td>16</td><td>99.7%</td></tr><tr><td>dM4e</td><td>-0.960 ± 0.30</td><td>2.30 ± 0.59 10<sup>5</sup></td><td>0.592</td><td>0.025</td><td>23</td><td>92%</td></tr><tr><td>dM4e+Sat.</td><td>-0.47 ± 0.08</td><td>1.26 ± 0.08 10<sup>5</sup></td><td>0.720</td><td>0.020</td><td>34</td><td>99.9%</td></tr><tr><th colspan="7">EW<sub>Ca II</sub> = a × EW<sub>Hα</sub> + b</th></tr><tr><th>Spectral Type</th><th>a</th><th>b</th><th>Correlation Coefficient</th><th>χ<sup>2</sup></th><th>No. Stars</th><th>Statistical Significance</th></tr><tr><td>dM3e</td><td>2.90 ± 0.05</td><td>1.14±0.125</td><td>0.9994</td><td>0.023</td><td>6</td><td>>99.9%</td></tr><tr><td>dM4e</td><td>1.909 ± 0.18</td><td>-1.035±0.58</td><td>0.877</td><td>2.2</td><td>35</td><td>>99.9%</td></tr><tr><th colspan="7">EW<sub>Ca II</sub> = b × S<sup>a</sup><sub>HK</sub></th></tr><tr><th>Spectral Type</th><th>a</th><th>b</th><th>Correlation Coefficient</th><th>χ<sup>2</sup></th><th>No. Stars</th><th>Statistical Significance</th></tr><tr><td>dM4+dM4e</td><td>1.20 ± 0.07</td><td>-0.31 ± 0.04</td><td>0.970</td><td>0.024</td><td>22</td><td>>99.9%</td></tr><tr><th colspan="7">L<sub>X</sub> = a × L<sub>Ca II</sub> + b</th></tr><tr><th>Spectral Type</th><th>a</th><th>b</th><th>Correlation Coefficient</th><th>χ<sup>2</sup></th><th>No. Stars</th><th>Statistical Significance</th></tr><tr><td>dK4+dK4e</td><td>1.54 ± 0.29</td><td>-15.34 ± 8.2</td><td>0.859</td><td>0.17</td><td>13</td><td>>99.9%</td></tr><tr><td>dK6+dK6e</td><td>1.57 ± 0.21</td><td>-16.1 ± 5.9</td><td>0.321</td><td>0.11</td><td>16</td><td>&lt;10%</td></tr><tr><td>dM2+dM2e</td><td>1.57 ± 0.17</td><td>-15.0 ± 4.6</td><td>0.878</td><td>0.24</td><td>28</td><td>>99.9%</td></tr><tr><td>dM3+dM3e</td><td>1.74 ± 0.11</td><td>-19.1 ± 3.0</td><td>0.952</td><td>0.09</td><td>28</td><td>>99.9%</td></tr><tr><td>dM4+dM4e</td><td>1.37 ± 0.12</td><td>-8.6 ± 3.2</td><td>0.882</td><td>0.18</td><td>42</td><td>>99.9%</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th rowspan="2">Spectral Type</th><th colspan="2">a</th><th colspan="2">b</th><th colspan="2">F<sub>Ca II</sub> = b × (P/ sin i)<sup>a</sup></th><th rowspan="2">No. Stars</th><th rowspan="2">Statistical Significance</th></tr><tr><th></th><th></th><th></th><th></th><th>Correlation Coefficient</th><th>χ<sup>2</sup></th></tr></thead><tbody><tr><td>dK4e</td><td>-1.877 ± 0.51</td><td>9.73 ± 5.07</td><td>10<sup>6</sup></td><td>...</td><td>0.038</td><td>4</td><td>99.7%</td></tr><tr><td>dK6e</td><td>-1.402 ± 0.43</td><td>4.20 ± 3.52</td><td>10<sup>6</sup></td><td>...</td><td>0.023</td><td>9</td><td>90%</td></tr><tr><td>dM2e</td><td>-1.793 ± 0.14</td><td>3.28 ± 0.54</td><td>10<sup>6</sup></td><td>...</td><td>0.014</td><td>11</td><td>>99.9%</td></tr><tr><td>dM3e</td><td>-1.041 ± 0.58</td><td>1.60 ± 1.20</td><td>10<sup>6</sup></td><td>...</td><td>0.133</td><td>8</td><td>99.5%</td></tr><tr><td>dM4e</td><td>-0.951 ± 0.31</td><td>2.27 ± 0.58</td><td>10<sup>5</sup></td><td>...</td><td>0.077</td><td>23</td><td>92%</td></tr></tbody></table>

magnetic activity. Therefore our spectral samples are likely to be biased in general toward low-activity stars, i.e., stars that are in the unsaturated portions of the RACs. For reasons outlined above (see Section 1.1), we consider this bias to be an advantage in the present study. The biases in our samples may contribute somewhat to the density of the sampling at different parts of the RAC, but this is not expected to cause significant discrepancies as regards the overall RACs. Note that, for the dM4 and dK6 samples, the sampling of the RACs should be more complete as regards the measures of P/sin i (because we included other measures of v sin i from the literature, see Paper I).

### 3. The RACs in Late-K and M Dwarfs

In this section, we first (Section 3.1) evaluate the surface fluxes in the continuum in the vicinity of the Ca II resonance doublet by using our estimates of T<sub>eff</sub> and also by using the synthetic spectra of de Laverny et al. (2012).

Then we construct the RACs of M and K dwarfs using a variety of approaches: some approaches may facilitate the extraction of information that is more difficult to extract by means of other approaches. In all of the approaches, we plot a quantity related to “activity” as the ordinate, and a quantity related to “rotation” as the abscissa.

Our first approach (Sections 3.2–3.6) specifies the “activity” of a star in terms of the surface fluxes of chromospheric lines. Combining these surface flux results with our results for P/sin i (Paper I), we construct a set of RACs for M and K dwarfs belonging to various spectral sub-types (see Figures 2, 3, 5, 8, and 12). We first analyze the RAC for K4 and K6 dwarfs. Then we re-visit the RAC in M2 dwarfs that was first investigated in Paper XV. We then analyze the RAC in M3 dwarfs for which v sin i measures were reported in HM and for which we present new measurements of the chromospheric line EWs here. We also revisit the RAC in M4 dwarfs that was studied in Paper XVIII with the new values of the stellar parameters determined in Paper I. Finally, we compare the RACs for five spectral K and M subtypes, dK4, dK6, dM2, dM3, and dM4, and draw conclusions about the differential variations of the RAC from mid-K dwarfs to M4 dwarfs.

We emphasize that our RACs are plotted as a function of the projected rotation period P/sin i and not the rotation period P. The average sin i is 0.6. Then, in order to recover the RACs as a function of P, one must multiply our RACs by a factor of 1.67. The scatter due to variable sin i is included in our correlations. Nevertheless, we show that one can obtain reasonably good empirical RACs in spite of these uncertainties and the uncertainties on our measures.

In Sections 3.7 and 3.8, we present an overview of the systematic properties of the RACs in all of our sub-types, first as regards the slopes of the RACs (Section 3.7), and then as

regards the absolute values of the chromospheric emission levels (Section 3.8).

In Section 3.9, we switch to a different approach to constructing RACs. As regards rotation, we switch to the Rossby number. As regards chromospheric activity, we switch to a quantity that expresses the output power in the H and K lines of Ca II as a fraction of the star’s output power (L<sub>bol</sub>) (Figure 15).

Switching our attention to the corona (Section 3.10), we construct a different type of RAC for our targets, this time referring to conditions in the corona, rather than the chromosphere. The coronal RAC is obtained by plotting L<sub>X</sub>/L<sub>bol</sub> as a function of the Rossby number (Figure 16). Comparing “activity” in chromosphere and corona (Section 3.11; Figures 17 and 18) can provide information as to how any given M dwarf star partitions its deposition of mechanical energy between chromosphere and corona. To quantify this partition, in Section 3.11 we examine the correlations between L<sub>X</sub> and L<sub>HK</sub> (see Figures 17 and 18).

In the final Section 3.12, we summarize the properties of the various RACs that our data have enabled us to construct.

#### 3.1. The Mean Fluxes in the Vicinity of the Ca II Resonance Doublet

In order to evaluate the efficiency of the dynamo mechanism(s) as a function of spectral type in M and K dwarfs, it seems preferable to inter-compare the RACs calibrated in terms of absolute energy fluxes, rather than confining our attention to the values of the EW. (Nevertheless, the EW have proven useful in terms of surface magnetic fields: see Mullan et al. 2015.) In order to convert from EW to energy fluxes, we calculated the surface fluxes in the continuum in the vicinity of the Ca II lines from the theoretical model atmospheres of de Laverny et al. (2012) and Palacios et al. (2010)<fnref n="8" /> for log(g) = 5.0, [M/H] = 0.0 and α = 0.0. We found that, in M2, M3, and M4 dwarfs, for our observations, the continuum at about 3950 Å represents a good evaluation of the background continuum flux for the Ca II lines. But there are significant discrepancies between the observations and the models of de Laverny et al. (2012). We show a spectrum of a dM2 star (G1 205) in Figure 1 together with the model of de Laverny et al. (2012) for an effective temperature of 3500 K. We normalized these two spectra at 1 for the continuum flux at 3950 Å. As one can see in this figure, there are large differences in the continuum fluxes at 3910 and 4000 Å between the model and the observation. We believe these differences are due to missing opacities in the models. We also show, in the lower panel of Figure 1, the spectrum of G1 570A together with the model for an effective temperature of

<footnote n="8">⁸ http://npollux.lupm.univ-montp2.fr/ or ftp://ftp.oca.eu/pub/laverny/DEPOT/AMBRE_Grid_Flux/</footnote>
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Upper panel: the observed spectrum of Gl 205 (dM2, solid line) together with the theoretical spectrum of de Laverny et al. (2012) for an effective temperature of 3500 K (dotted line). Note substantial disagreements between the observations and the model. Lower panel: the observed spectrum of Gl 570A (dK4, solid line) together with the model for an effective temperature of 4500 K (dotted line). Here again there are significant differences between the model and observations. All the spectra are normalized for the continuum flux at about 3950 Å, i.e., between the H & K line centers.</figcaption></figure>

4500 K. Here again we note important differences: in the spectrum of Gl 570A, the continuum flux at 3950 Å is not a good estimate of the continuum flux in the vicinity of the Ca II lines. In this case, one has to interpolate between the fluxes at 3910 and 4000 Å. Therefore, for our estimates of the theoretical surface fluxes in the vicinity of the Ca II lines for M2, M3, and M4 dwarfs, we took the average of the flux at 3950 Å and the value interpolated between 3910 and 4000 Å. For K6 and K4 dwarfs, we used the value interpolated between 3910 and 4000 Å.

Using this approach, we found that the mean surface fluxes in the continuum in the vicinity of the Ca II lines are: $5.51 \times 10^5$, $2.18 \times 10^5$, $4.53 \times 10^4$, $2.72 \times 10^4$, and $1.74 \times 10^4$ erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$ for dK4, dK6, dM2, dM3, and dM4 stars, respectively. The strong decline (factor of ~30) in surface flux in the violet with decreasing $T_{\text{eff}}$ from dK4 to dM4 is apparent in these numbers. Using these figures, we are now in a position to examine quantitatively how the surface fluxes in the Ca II lines behave as a function of $P/\sin i$. In the subsequent sections, the Ca II surface fluxes were computed for each star according to its effective temperature. We expect that the models give estimates of the continuum surface fluxes with a precision of the order of 40%. As mentioned above, this figure is far below the factor of ~30 in the continuum surface fluxes in the violet with decreasing $T_{\text{eff}}$ from dK4 to dM4. Therefore, the decline in the Ca II surface fluxes that we observe below (see Figure 14) in the RACs from dK4 to dM4 is highly significant.

## 3.2. The RAC in K4 Dwarfs

In this subsection, we re-investigate the RACs in our sample of dK4 stars, which were first investigated in Houdebine (2012b, Paper XVIII). We reiterate that the RACs are of fundamental importance in order to constrain an essential parameter of the dynamo mechanism(s): the role played by rotation. In our previous studies (e.g., Papers VII, XIV, XVI, XVII, HM), we reported on our results for $v \sin i$ and $P/\sin i$ for stars of low-activity level (i.e., slow rotators) for the spectral sub-types dK4, dM2, dM3, and dM4. In Paper I, we have reported similar rotational data for dK6 stars. Combining these rotational data with the surface fluxes of chromospheric lines in dK and dM stars provides a unique opportunity to investigate the RACs in a fine-grained sample of M and K dwarfs, which are sub-divided across five closely spaced, but distinct, spectral sub-types.

In Figure 2, we show the RAC that we have obtained for our dK4 stars. In previously published studies of RACs in cool dwarfs, the majority of the results were reported in terms of homoscedastic linear least square fits (LSF) in order to fit their observations. In the present study, we propose different approaches. First, we perform homoscedastic linear and quadratic least square fits to our samples of low+high-activity stars (dK+dKe or dM+dMe). Second, we perform heteroscedastic linear and quadratic least square fits (LSF that account for measurement errors, see the Appendix) to our samples of low+high-activity stars. Third, we also compute homoscedastic and heteroscedastic linear least square fits to two particular sub-sets of our data, namely, stars in which activity is at a low level (dK, dM) and at a high level (dKe, dMe), respectively. We inter-compare the results of these fits for various spectral sub-types in turn in this subsection and in the following subsections. As we shall see, these various fits allow us to account for the complexity of our data sets. In our least-square fits, we do not include in general the suspected low $\sin i$ stars, lower limits, or spectroscopic binaries. However, in the present sub-section, for dK4 stars, it appears that some of the lower limit measurements do correlate with the other measurements of dK4 stars. Therefore, we included some of these measurements in our correlations (we did not include obvious outliers).

The heteroscedastic linear least-squares fit to the data in Figure 2 is shown by the solid line. The equation for the solid
<figure><img src="image_4.png" /><figcaption>Figure 2. Correlation between the mean surface flux of the Ca II resonance doublet and log(P/sin i) for stars with spectral sub-type dK4. We overplot the heteroscedastic linear least square fit (LSF; solid line). We also plot the heteroscedastic linear LSF separately to the low activity and to the high-activity stellar sub-samples (see the two distinct straight dotted-dashed lines).</figcaption></figure>

line in Figure 2 is as follows:

$$F_{\text{Ca II}} = 1.916 \pm 0.32 \times 10^6 \times (P/\sin i)^{-0.8140 \pm 0.059}. \quad (1)$$

The χ² for the fit in Equation (1) is only 0.033 for our sample of 34 dK4+dK4e stars. The statistical significance of this fit is 99.9% (see Table 1). (Results of all least-squares fits for this and subsequent spectral sub-types can be found in Table 1.)

The homoscedastic linear least-squares fit yields similar results:

$$F_{\text{Ca II}} = 1.57 \pm 0.34 \times 10^6 \times (P/\sin i)^{-0.756 \pm 0.05}. \quad (2)$$

The χ² for this fit is 0.032 for 34 dK4+dK4e stars. The correlation coefficient for this fit is found to be 0.884 and the statistical significance of the correlation is at least 99.9% (see Table 1). Therefore, these two fits are highly statistically significant. The reason for studying heteroscedastic fits in addition to the homoscedastic fits is that our P/sin i measurements have large errors and these errors vary as a function of the values of P/sin i (rapid rotators have smaller errors than slow rotators). The presence of these variable errors may yield heteroscedastic fits that are in some regards different from the homoscedastic fits. We shall see examples of this statement in some of the following subsections.

Two aspects of the fits in Equations (1) and (2) will be referred to in the subsequent discussion. First, the exponent of P/sin i will be referred to as the “RAC slope” in a plot of log(F Ca II) versus log(P/sin i). Systematic changes in the

numerical value of the RAC slope will lead us to an important conclusion of the present paper. Second, the numerical coefficient closest to the equals sign is a measure of the absolute level of the chromospheric emission among the stars in the sample. Systematic changes in the numerical value of the coefficient will also be an important conclusion of this paper.

So far in this sub-section, we have done an analysis of all of our dK4 stars, i.e., we have combined both the dK4 stars and the dK4e stars into a single sample. Now we split our sample up into two groups (dK4 in one, dK4e in the second), and analyze each group separately. Specifically, we now apply heteroscedastic and homoscedastic linear LSF to the sub-samples of only the low-activity stars (dK4) and then repeat the exercise including only the high-activity stars (dK4e) (see Table 1). The reason for performing these separate fits is that we found that the linear fits did not reproduce both the low-activity and high-activity sub-samples well; this result led us to wonder if these two samples might contain different dynamo modes. If it turns out that indeed different dynamo modes are at work in slow rotators and fast rotators, then separate analyses of the data sets is warranted. We find that the linear heteroscedastic LSF to the dK4 low-activity stars yield:

$$F_{\text{Ca II}} = 7.66 \pm 1.89 \times 10^5 \times (P/\sin i)^{-0.568 \pm 0.084}. \quad (3)$$

The χ² for this fit is 0.034 for our sample of 30 dK4 stars and the statistical significance of the correlation is at least 99.9%
(see Table 1). This fit is shown as the straight dotted-dashed line in Figure 2. For the homoscedastic fit, we obtain:

$$F_{\text{Ca II}} = 9.33 \pm 3.16 \times 10^5 \times (P/\sin i)^{-0.624 \pm 0.11}. \quad (4)$$

The $\chi^2$ for this fit is only 0.031 for 30 dK4 stars, the correlation coefficient is 0.744 and the statistical significance is at least 99.9% (see Table 1). Therefore, both of these fits (homo and hetero) are highly statistically significant in spite of the scatter in the data. We note that the differences between the homoscedastic and the heteroscedastic fits fall within the uncertainties of the parameters of the fits.

The linear heteroscedastic LSF to the dK4e high-activity stars yield:

$$F_{\text{Ca II}} = 9.73 \pm 5.07 \times 10^6 \times (P/\sin i)^{-1.877 \pm 0.51}. \quad (5)$$

The $\chi^2$ for this fit is 0.038 for four dK4e stars and the statistical significance is 99.7% (see Table 1). This fit is shown as the upper dotted-dashed straight line in Figure 2. We must admit that our dK4e sample contains only four stars and that therefore the RAC for dK4e stars is not well constrained. However, as we shall see in the subsequent sub-sections, we find that this RAC is consistent with the trend of the high-activity star RACs at other spectral types.

An important feature of our results emerges when we compare Equations (1) and (3), and when we compare Equations (2), (4), and (5): whether we consider homo- or heteroscedastic results, we find that the gradient of the RAC for the low-activity stars alone (−0.57, −0.64) is shallower in magnitude than that of the combined sample of low+high-activity stars (−0.81, −0.76), and that the gradient of the RAC for the high-activity stars alone (−1.88, −1.877) is larger in magnitude than that of the combined sample of low+high-activity stars (−0.81, −0.76). We shall find that this occurs systematically between the low-activity sub-samples and the full samples for all five of our spectral sub-types (see Table 1 and Sections 3.3–3.6). The effect is more pronounced as the spectral type increases (see also Section 3.7). For the high-activity sub-samples, we shall find that the slope is steeper than that of the combined samples for dK4, dK6, and dM2 stars, but that it reverses for the dM3 and dM4 stars (at the TTCC and beyond) and becomes shallower than that of the combined samples. As a consequence, the linear fits to the combined samples of low+high-activity stars tend to overestimate the slope in the low-activity sub-samples, and underestimate the slopes in the high-activity stars sub-samples for dK4, dK6, and dM2 stars. Therefore, we also performed homoscedastic and heteroscedastic quadratic LSF to our samples of low+high-activity stars for our five spectral sub-types. The quadratic fits allow us to reproduce both the shallower gradient of the linear LSF among the low-activity stars as well as the higher fluxes among the high-activity stars. This shows that the quadratic fit may give a better description of the data than the linear fit for the combined samples of low+high-activity stars. However, according to an anonymous referee comment, the quadratic fits may not be significantly different from the results of the linear fits. To argue on this point, the referee performed a simulation on sub-samples of dM and dMe stars with random errors typical of those we find in this study. The referee found also a shallower slope among his sub-sample of slow rotators. Given the evidence at hand, the signatures identified as possible evidences supporting the case for a quadratic fits are most probably due to random errors working in combination with a fairly narrow $log(P/\sin i)$ domain as compared to those errors.

Therefore, although the slopes of the high-activity sub-samples in dK4, dK6, and dM2 are in favor of a quadratic description of the data, we cannot yet conclude that quadratic fits represent definitely a better representation of the full data sets. We can also note that the $\chi^2$s are comparable between linear and quadratic fits. More data will be required to conclude.

As an anonymous referee rightly pointed to, if the data are localized to a domain that is only two to three times as large as a typical error bar, random measurement errors can easily randomize the data and weaken the slope of any true correlation. However, this point is not statistically correct. If the distribution of the errors is Gaussian (which we assume here), then the correct parameter to compare to the RAC domain $R$ is $\delta = \frac{\langle\text{error}\rangle}{\sqrt{n}}$ where $n$ is the number of measures. The value of $\frac{R}{\delta}$ is 17.2 for our low-activity dK4 star sample, which is far larger than the values of 2–3 mentioned above. It appears that $\frac{R}{\delta}$ lies in the range 17–40 for our low-activity and high-activity RACs. The confidence levels of a given LSF depend not only on the mean error but also on the number of measures. Therefore, our LSFs should be established to a fairly high level of confidence, which is confirmed by the high statistical significances that we obtained for our least square fits (see Table 1). Nevertheless, the simulation of the referee is of interest to us. We take this important point into account and we emphasize here the preliminary character of our results. Undoubtedly, these interesting results should be confirmed with additional data, obtained preferably with a higher resolution spectrograph for the slow rotators such as ESPRESSO (ESO, $R = 220,000$). Therefore, in the following sub-sections and Section 3.7, our results on the separate fits to the sub-samples of low and high-activity stars should be considered with caution and are only preliminary. They should be confirmed with larger stellar samples.

### 3.3. The RAC in K6 Dwarfs

In this subsection, we discuss the results for our sample of dK6 stars. For the sake of consistency with previous measurements in this series of papers, and to avoid duplication in describing the method, we refer the reader to Papers VI and XV to see how we evaluate the EW for the Ca II resonance doublet and the $H_\alpha$ line. In Table 2, we list the EW we have obtained for the Ca II resonance doublet and for the $H_\alpha$ line for our dK6 stars.

We note that our dK6 stellar sample contains only one star with $H_\alpha$ definitely in emission (i.e., a dK6e star): Gl 517 (EQ Vir), with the fastest rotation ($v \sin i = 9.77$ km s$^{-1}$, $P/\sin i = 3.05$ days, see Paper I). Our dK6 sample also includes one star of “intermediate” activity with $H_\alpha$ neither in emission nor absorption (i.e., a dK6(e) star): Gl 208 (Table 2). As regards EQ Vir, this star has been classified in the literature as a dK5 star. In fact, Gl 517 is a BY Dra star with colors that vary with time (depending on spot coverage).

Our observations contain many slow rotators but very few fast rotators (dKe stars). Therefore, in order to complete our sample and have a better defined RAC for dK6 stars, in Paper I we compiled $v \sin i$ and $P \sin i$ measures from the literature. We found 43 additional $v \sin i$ and $P/\sin i$ measures, which brings our total compilation of measures to 150.

We also retrieved some FEROS spectra from the ESO Archive for Gl 142, Gl 885A, Gl 900, and HIP 113597. The measures of the Ca II EW for these stars are given in Table 2.
<table><caption>Table 2
Equivalent Width (EW, in Units of Å) of the Ca II Resonance Doublet and Hα Line for Our Sample of dK6 Stars</caption><thead><tr><th rowspan="2">Star</th><th rowspan="2">No. meas.</th><th colspan="3">HARPS</th><th colspan="3">SOPHIE</th><th rowspan="2">K/H Ratio</th><th rowspan="2">(Ca II) EW (Å)</th><th rowspan="2">Ca II Flux (10<sup>5</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th rowspan="2">Ca II Flux<sup>a</sup> (10<sup>5</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th rowspan="2">No. meas.</th><th rowspan="2">S/N HARPS @5000</th><th rowspan="2">S/N SOPHIE @5000</th></tr><tr><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>Hα EW (Å)</th><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>Hα EW (Å)</th></tr></thead><tbody><tr><td>GJ 1056</td><td>9</td><td>-0.61 ± 0.01</td><td>-0.68 ± 0.01</td><td>0.643 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-0.645</td><td>2.052</td><td></td><td>9</td><td>151</td><td>...</td></tr><tr><td>GJ 1066</td><td>3</td><td>-0.91 ± 0.10</td><td>-1.01 ± 0.10</td><td>0.605 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-0.96</td><td>3.295</td><td></td><td>3</td><td>64</td><td>...</td></tr><tr><td>GJ 1067</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.79 ± 0.10</td><td>-1.11 ± 0.10</td><td>0.554 ± 0.020</td><td>1.41</td><td>-0.95</td><td>3.171</td><td></td><td>3</td><td>...</td><td>119</td></tr><tr><td>GJ 1177A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.919</td><td>4.525</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1248</td><td>3</td><td>...</td><td>...</td><td>0.239 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td><td>3</td><td>29</td><td>...</td></tr><tr><td>GJ 1267</td><td>1</td><td>-1.01 ± 0.15</td><td>-1.08 ± 0.15</td><td>0.518 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.07</td><td>-1.045</td><td>1.755</td><td>2.944</td><td>1</td><td>38</td><td>...</td></tr><tr><td>GJ 1279</td><td>2</td><td>-0.79 ± 0.01</td><td>-0.87 ± 0.01</td><td>0.638 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.10</td><td>-0.83</td><td>2.849</td><td></td><td>2</td><td>177</td><td>...</td></tr><tr><td>GJ 3072</td><td>3</td><td>-0.92 ± 0.03</td><td>-1.05 ± 0.03</td><td>0.585 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-0.985</td><td>1.501</td><td>3.132</td><td>3</td><td>66</td><td>...</td></tr><tr><td>GJ 3411</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.463</td><td>2.598</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3494</td><td>3</td><td>-0.97 ± 0.05</td><td>-1.27 ± 0.05</td><td>0.517 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.31</td><td>-1.12</td><td>2.905</td><td></td><td>3</td><td>64</td><td>...</td></tr><tr><td>GJ 3551</td><td>5</td><td>-0.19 ± 0.03</td><td>-0.17 ± 0.03</td><td>0.659 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>0.89</td><td>-0.18</td><td>0.402</td><td>0.553</td><td>5</td><td>91</td><td>...</td></tr><tr><td>GJ 3996</td><td>6</td><td>-1.51 ± 0.03</td><td>-1.65 ± 0.03</td><td>0.498 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-1.58</td><td>2.527</td><td>3.491</td><td>6</td><td>113</td><td>...</td></tr><tr><td>GJ 4140</td><td>3</td><td>-0.94 ± 0.03</td><td>-1.07 ± 0.03</td><td>0.544 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-1.005</td><td>1.840</td><td>7.050</td><td>3</td><td>78</td><td>...</td></tr><tr><td>GJ 9250</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.212</td><td>2.934</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 9299</td><td>8</td><td>-0.82 ± 0.01</td><td>-0.91 ± 0.01</td><td>0.618 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-0.865</td><td>2.033</td><td></td><td>8</td><td>134</td><td>...</td></tr><tr><td>GJ 9667</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.55 ± 0.10</td><td>-0.77 ± 0.10</td><td>0.625 ± 0.020</td><td>1.40</td><td>-0.66</td><td>1.394</td><td>3.788</td><td>1</td><td>...</td><td>59HE</td></tr><tr><td>GJ 9714</td><td>8</td><td>-0.76 ± 0.01</td><td>-0.84 ± 0.01</td><td>0.612 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-0.80</td><td>2.345</td><td></td><td>8</td><td>137</td><td>...</td></tr><tr><td>GJ 9827</td><td>2</td><td>-0.50 ± 0.03</td><td>-0.63 ± 0.03</td><td>0.647 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.26</td><td>-0.565</td><td>1.217</td><td>2.097</td><td>2</td><td>54</td><td>...</td></tr><tr><td>Gl 14</td><td>2</td><td>...</td><td>...</td><td>...</td><td>-1.00 ± 0.10</td><td>-1.61 ± 0.10</td><td>0.534 ± 0.020</td><td>1.61</td><td>-1.305</td><td>2.564</td><td>3.747</td><td>2</td><td>...</td><td>76</td></tr><tr><td>Gl 17.1</td><td>10</td><td>-0.86 ± 0.03</td><td>-0.98 ± 0.03</td><td>0.553 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-0.92</td><td>0.601</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 40A</td><td>2</td><td>-0.83 ± 0.10</td><td>-0.92 ± 0.10</td><td>0.583 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-0.875</td><td>2.228</td><td></td><td>2</td><td>59</td><td>...</td></tr><tr><td>Gl 45</td><td>4</td><td>-0.68 ± 0.02</td><td>-0.79 ± 0.02</td><td>0.633 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.16</td><td>-0.735</td><td>1.687</td><td>3.999</td><td>4</td><td>100</td><td>...</td></tr><tr><td>Gl 50</td><td>4</td><td>-0.31 ± 0.03</td><td>-0.37 ± 0.03</td><td>0.648 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.19</td><td>-0.34</td><td>0.761</td><td>1.299</td><td>4</td><td>79</td><td>...</td></tr><tr><td>Gl 52</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.31 ± 0.10</td><td>-0.49 ± 0.10</td><td>0.597 ± 0.020</td><td>1.58</td><td>-0.40</td><td>0.820</td><td>1.555</td><td>3</td><td>...</td><td>89</td></tr><tr><td>Gl 57</td><td>5</td><td>-0.32 ± 0.05</td><td>-0.35 ± 0.05</td><td>0.454 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-0.335</td><td>0.491</td><td>16.04</td><td>5</td><td>100</td><td>...</td></tr><tr><td>Gl 105.5</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.437</td><td>0.996</td><td></td><td>1</td><td>...</td><td>65HE</td></tr><tr><td>Gl 112.1</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.627</td><td>3.532</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 116</td><td>1</td><td>...</td><td>...</td><td>...</td><td>-0.37 ± 0.05</td><td>-0.44 ± 0.05</td><td>0.474 ± 0.020</td><td>1.19</td><td>-0.355</td><td>0.586</td><td>7.377</td><td>6</td><td>...</td><td>124</td></tr><tr><td>Gl 142<sup>b</sup></td><td>1</td><td>-1.16 ± 0.05</td><td>-1.53 ± 0.05</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.32</td><td>-1.345</td><td>2.372</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 143.1</td><td>6</td><td>-1.41 ± 0.05</td><td>-1.57 ± 0.05</td><td>0.534 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-1.49</td><td>2.271</td><td>3.319</td><td>6</td><td>109</td><td>...</td></tr><tr><td>Gl 146</td><td>46</td><td>-0.67 ± 0.01</td><td>-0.75 ± 0.01</td><td>0.638 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.12</td><td>-0.71</td><td>1.529</td><td>2.418</td><td>46</td><td>279</td><td>...</td></tr><tr><td>Gl 153A</td><td>2</td><td>...</td><td>...</td><td>...</td><td>-0.43 ± 0.10</td><td>-0.56 ± 0.10</td><td>0.606 ± 0.020</td><td>1.30</td><td>-0.495</td><td>1.118</td><td>3.277</td><td>2</td><td>...</td><td>72</td></tr><tr><td>Gl 156</td><td>2</td><td>-0.83 ± 0.10</td><td>-0.99 ± 0.10</td><td>0.629 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.19</td><td>-0.91</td><td>1.635</td><td>2.430</td><td>2</td><td>52</td><td>...</td></tr><tr><td>Gl 162.2</td><td>14</td><td>-0.88 ± 0.03</td><td>-0.96 ± 0.03</td><td>0.605 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-0.92</td><td>3.014</td><td></td><td>14</td><td>153</td><td>...</td></tr><tr><td>Gl 182</td><td>1</td><td>-7.65 ± 0.05</td><td>-8.33 ± 0.05</td><td>-1.26 ± 0.030</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-7.99</td><td>20.3</td><td>...</td><td>1</td><td>115<sup>3</sup></td><td>...</td></tr><tr><td>Gl 186AB</td><td>3</td><td>-0.27 ± 0.05</td><td>-0.38 ± 0.05</td><td>0.651 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.41</td><td>-0.325</td><td>0.733</td><td>1.486</td><td>3</td><td>84</td><td>...</td></tr><tr><td>Gl 191</td><td>30</td><td>-0.10 ± 0.01</td><td>-0.12 ± 0.01</td><td>0.258 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.20</td><td>-0.11</td><td>0.033</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 208</td><td>19,19</td><td>-2.49 ± 0.05</td><td>-3.03 ± 0.05</td><td>0.082 ± 0.020</td><td>-4.29 ± 0.05</td><td>-6.24 ± 0.02</td><td>-0.291 ± 0.020</td><td>1.22</td><td>-4.01</td><td>4.859</td><td>11.52</td><td>19, 19</td><td>135</td><td>309</td></tr><tr><td>Gl 221</td><td>65</td><td>-0.60 ± 0.05</td><td>-0.71 ± 0.05</td><td>0.637 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.18</td><td>-0.655</td><td>1.174</td><td>1.953</td><td>65</td><td>337</td><td>...</td></tr><tr><td>Gl 256</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.296</td><td>4.083</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 322</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.96 ± 0.15</td><td>-1.42 ± 0.15</td><td>0.519 ± 0.020</td><td>1.48</td><td>-1.19</td><td>2.806</td><td></td><td>3</td><td>...</td><td>83</td></tr><tr><td>Gl 369</td><td>2</td><td>-0.48 ± 0.05</td><td>-0.57 ± 0.05</td><td>0.417 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.19</td><td>-0.525</td><td>0.385</td><td></td><td>2</td><td>45</td><td>...</td></tr><tr><td>Gl 389.1</td><td>11</td><td>-0.39 ± 0.01</td><td>-0.42 ± 0.01</td><td>0.668 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.08</td><td>-0.405</td><td>0.872</td><td>1.284</td><td>11</td><td>147</td><td>...</td></tr><tr><td>Gl 397</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.005</td><td>1.827</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 401</td><td>7</td><td>-0.55 ± 0.01</td><td>-0.72 ± 0.01</td><td>0.422 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.31</td><td>-0.635</td><td>0.415</td><td></td><td>7</td><td>90</td><td>...</td></tr><tr><td>Gl 412.3</td><td>6</td><td>-0.39 ± 0.03</td><td>-0.47 ± 0.03</td><td>0.649 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.21</td><td>-0.43</td><td>1.024</td><td></td><td>6</td><td>146</td><td>...</td></tr><tr><td>Gl 414A</td><td>8</td><td>...</td><td>...</td><td>...</td><td>-0.66 ± 0.10</td><td>-0.98 ± 0.10</td><td>0.619 ± 0.020</td><td>1.48</td><td>-0.82</td><td>1.646</td><td>2.463</td><td>8</td><td>...</td><td>302</td></tr><tr><td>Gl 416</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td><td>1</td><td>...</td><td>32</td></tr><tr><td>Gl 421B</td><td>7</td><td>-0.69 ± 0.03</td><td>-0.75 ± 0.03</td><td>0.625 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-0.72</td><td>0.976</td><td>7.235</td><td>7</td><td>116</td><td>...</td></tr><tr><td>Gl 425B</td><td>1</td><td>-1.55 ± 0.05</td><td>-2.00 ± 0.05</td><td>0.334 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.29</td><td>-1.775</td><td>3.876</td><td>4.802</td><td>1</td><td>67</td><td>...</td></tr><tr><td>Gl 455.1</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-5.777</td><td>11.28</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 466</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td><td>2</td><td>...</td><td>34</td></tr><tr><td>Gl 496.1</td><td>47</td><td>-0.76 ± 0.01</td><td>-0.85 ± 0.01</td><td>0.653 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.12</td><td>-0.805</td><td>1.125</td><td>1.497</td><td>47</td><td>365</td><td>...</td></tr><tr><td>Gl 509.1</td><td>1</td><td>...</td><td>...</td><td>...</td><td>-1.00 ± 0.10</td><td>-1.60 ± 0.10</td><td>0.326 ± 0.020</td><td>1.60</td><td>-1.30</td><td>3.219</td><td></td><td>1</td><td>...</td><td>66</td></tr></tbody></table>
<table><caption>Table 2
(Continued)</caption><thead><tr><th rowspan="2">Star</th><th rowspan="2">No. meas.</th><th colspan="3">HARPS</th><th colspan="3">SOPHIE</th><th rowspan="2">K/H Ratio</th><th rowspan="2">(Ca II) EW (Å)</th><th rowspan="2">Ca II Flux (10<sup>5</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th rowspan="2">Ca II Flux<sup>a</sup> (10<sup>5</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th rowspan="2">No. meas.</th><th rowspan="2">S/N HARPS @5000</th><th rowspan="2">S/N SOPHIE @5000</th></tr><tr><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>H<sub>α</sub> EW (Å)</th><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>H<sub>α</sub> EW (Å)</th></tr></thead><tbody><tr><td>Gl 517</td><td>5</td><td>-2.36 ± 0.05</td><td>-2.83 ± 0.05</td><td>-0.611 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.20</td><td>-2.595</td><td>11.33</td><td></td><td>5</td><td>141</td><td>...</td></tr><tr><td>Gl 522</td><td>3</td><td>-0.56 ± 0.05</td><td>-0.64 ± 0.05</td><td>0.662 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-0.60</td><td>1.292</td><td>1.911</td><td>3</td><td>75</td><td>...</td></tr><tr><td>Gl 524.1</td><td>6</td><td>-0.94 ± 0.03</td><td>-0.99 ± 0.03</td><td>0.577 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.05</td><td>-0.965</td><td>2.389</td><td></td><td>6</td><td>130</td><td>...</td></tr><tr><td>Gl 542.2</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.231</td><td>3.251</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 546</td><td>9</td><td>...</td><td>...</td><td>...</td><td>-0.75 ± 0.10</td><td>-1.10 ± 0.10</td><td>0.591 ± 0.020</td><td>1.47</td><td>-0.925</td><td>2.617</td><td></td><td>9</td><td>...</td><td>146</td></tr><tr><td>Gl 558</td><td>2</td><td>...</td><td>...</td><td>...</td><td>-0.52 ± 0.10</td><td>-0.86 ± 0.10</td><td>0.572 ± 0.020</td><td>1.65</td><td>-0.69</td><td>1.573</td><td>3.379</td><td>2</td><td>...</td><td>80</td></tr><tr><td>Gl 562</td><td>6</td><td>...</td><td>...</td><td>...</td><td>-0.83 ± 0.10</td><td>-1.19 ± 0.10</td><td>0.558 ± 0.020</td><td>1.43</td><td>-1.01</td><td>2.580</td><td></td><td>6</td><td>...</td><td>126</td></tr><tr><td>Gl 571</td><td>4</td><td>-1.01 ± 0.05</td><td>-1.08 ± 0.05</td><td>0.525 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.07</td><td>-1.045</td><td>2.031</td><td>4.872</td><td>4</td><td>99</td><td>...</td></tr><tr><td>Gl 571.1</td><td>4</td><td>-1.10 ± 0.03</td><td>-1.24 ± 0.03</td><td>0.597 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.13</td><td>-1.17</td><td>1.813</td><td>2.412</td><td>4</td><td>123</td><td>...</td></tr><tr><td>Gl 583</td><td>4</td><td>-1.01 ± 0.03</td><td>-1.17 ± 0.03</td><td>0.579 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.16</td><td>-1.09</td><td>2.316</td><td>4.884</td><td>4</td><td>102</td><td>...</td></tr><tr><td>Gl 619</td><td>6</td><td>...</td><td>...</td><td>...</td><td>-0.38 ± 0.10</td><td>-0.66 ± 0.10</td><td>0.615 ± 0.020</td><td>1.74</td><td>-0.52</td><td>1.087</td><td>1.731</td><td>6</td><td>...</td><td>129</td></tr><tr><td>Gl 629.1</td><td>5</td><td>-0.86 ± 0.03</td><td>-0.97 ± 0.03</td><td>0.612 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.13</td><td>-0.915</td><td>1.587</td><td>2.191</td><td>5</td><td>95</td><td>...</td></tr><tr><td>Gl 659B</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.278</td><td>4.195</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 673</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.632</td><td>2.541</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 707</td><td>9</td><td>-0.77 ± 0.01</td><td>-0.87 ± 0.01</td><td>0.617 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.13</td><td>-0.82</td><td>1.979</td><td></td><td>9</td><td>226</td><td>...</td></tr><tr><td>Gl 710</td><td>3</td><td>-1.01 ± 0.10</td><td>-1.02 ± 0.10</td><td>0.588 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.01</td><td>-1.015</td><td>1.760</td><td>2.955</td><td>3</td><td>77</td><td>...</td></tr><tr><td>Gl 726</td><td>5</td><td>-0.79 ± 0.03</td><td>-0.88 ± 0.03</td><td>0.574 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-0.835</td><td>2.035</td><td></td><td>5</td><td>126</td><td>...</td></tr><tr><td>Gl 728</td><td>18</td><td>...</td><td>...</td><td>...</td><td>-0.43 ± 0.10</td><td>-0.60 ± 0.10</td><td>0.612 ± 0.020</td><td>1.40</td><td>-0.515</td><td>0.850</td><td>1.160</td><td>18</td><td>...</td><td>373</td></tr><tr><td>Gl 747.1</td><td>1</td><td>...</td><td>...</td><td>0.393 ± 0.050</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 747.3AB</td><td>7</td><td>-0.58 ± 0.03</td><td>-0.64 ± 0.03</td><td>0.624 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.10</td><td>-0.61</td><td>1.270</td><td>1.753</td><td>7</td><td>161</td><td>...</td></tr><tr><td>Gl 757</td><td>2</td><td>-0.76 ± 0.10</td><td>-0.86 ± 0.10</td><td>0.545 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.13</td><td>-0.81</td><td>1.231</td><td>3.367</td><td>2</td><td>49</td><td>...</td></tr><tr><td>Gl 763</td><td>15</td><td>...</td><td>...</td><td>...</td><td>-1.07 ± 0.05</td><td>-1.47 ± 0.05</td><td>0.486 ± 0.020</td><td>1.37</td><td>-1.271</td><td>1.910</td><td>4.386</td><td>1</td><td>...</td><td>91HE</td></tr><tr><td>Gl 773</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.903</td><td>2.328</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 782</td><td>2</td><td>-0.84 ± 0.01</td><td>-0.92 ± 0.01</td><td>0.592 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.10</td><td>-0.88</td><td>2.572</td><td></td><td>15</td><td>206</td><td>...</td></tr><tr><td>Gl 786.1</td><td>1</td><td>...</td><td>...</td><td>...</td><td>-0.73 ± 0.10</td><td>-1.11 ± 0.10</td><td>0.619 ± 0.020</td><td>1.52</td><td>-0.92</td><td>1.673</td><td>2.573</td><td>2</td><td>...</td><td>73</td></tr><tr><td>Gl 795AB</td><td>1</td><td>...</td><td>...</td><td>...</td><td>-0.45 ± 0.05</td><td>-0.53 ± 0.05</td><td>0.600 ± 0.020</td><td>1.18</td><td>-0.49</td><td>0.994</td><td>3.030</td><td>1</td><td>...</td><td>105HE</td></tr><tr><td>Gl 798</td><td>22</td><td>-0.51 ± 0.01</td><td>-0.58 ± 0.01</td><td>0.603 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-0.545</td><td>0.979</td><td>2.943</td><td>22</td><td>380</td><td>...</td></tr><tr><td>Gl 801</td><td>4</td><td>-0.36 ± 0.05</td><td>-0.36 ± 0.05</td><td>0.667 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.00</td><td>-0.36</td><td>0.750</td><td>1.151</td><td>4</td><td>76</td><td>...</td></tr><tr><td>Gl 818</td><td>12, 1</td><td>-0.43 ± 0.01</td><td>-0.50 ± 0.01</td><td>0.699 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.16</td><td>-0.465</td><td>1.396</td><td></td><td>12, 1</td><td>362</td><td>82HE</td></tr><tr><td>Gl 820B</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.63</td><td>2.51</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 826.1</td><td>1</td><td>-1.28 ± 0.10</td><td>-1.33 ± 0.10</td><td>0.571 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.04</td><td>-1.305</td><td>2.937</td><td></td><td>1</td><td>40</td><td>...</td></tr><tr><td>Gl 830</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td><td>1</td><td>33</td><td>...</td></tr><tr><td>Gl 842</td><td>7</td><td>-1.18 ± 0.01</td><td>-1.31 ± 0.01</td><td>0.482 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-1.245</td><td>0.615</td><td></td><td>7</td><td>173</td><td>...</td></tr><tr><td>Gl 847A</td><td>14</td><td>-0.84 ± 0.05</td><td>-0.92 ± 0.05</td><td>0.568 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.10</td><td>-0.88</td><td>1.489</td><td>5.806</td><td>14</td><td>131</td><td>...</td></tr><tr><td>Gl 855</td><td>10</td><td>-1.34 ± 0.03</td><td>-1.49 ± 0.03</td><td>0.491 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-1.415</td><td>0.965</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 857.1</td><td>2</td><td>...</td><td>...</td><td>...</td><td>-0.58 ± 0.05</td><td>-0.84 ± 0.05</td><td>0.623 ± 0.020</td><td>1.45</td><td>-0.71</td><td>1.897</td><td></td><td>2</td><td>...</td><td>241HE</td></tr><tr><td>Gl 884</td><td>1</td><td>-1.33 ± 0.10</td><td>-1.39 ± 0.10</td><td>0.527 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.05</td><td>-1.36</td><td>2.033</td><td>3.207</td><td>1</td><td>38</td><td>...</td></tr><tr><td>Gl 885A<sup>b</sup></td><td>2</td><td>-2.650 ± 0.05</td><td>-2.974 ± 0.05</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.12</td><td>-2.812</td><td>5.774</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 889A</td><td>1</td><td>-1.21 ± 0.10</td><td>-1.19 ± 0.10</td><td>0.547 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>0.98</td><td>-1.20</td><td>2.691</td><td>4.954</td><td>1</td><td>32</td><td>...</td></tr><tr><td>Gl 891</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.876</td><td>0.932</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 894</td><td>3</td><td>-0.89 ± 0.05</td><td>-0.94 ± 0.05</td><td>0.610 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.06</td><td>-0.915</td><td>2.048</td><td>2.953</td><td>3</td><td>69</td><td>...</td></tr><tr><td>Gl 895.3</td><td>5</td><td>-1.17 ± 0.03</td><td>-1.29 ± 0.03</td><td>0.563 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.10</td><td>-1.23</td><td>3.045</td><td></td><td>5</td><td>87</td><td>...</td></tr><tr><td>Gl 898</td><td>40</td><td>-1.30 ± 0.03</td><td>-1.44 ± 0.03</td><td>0.516 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-1.37</td><td>3.210</td><td></td><td>40</td><td>343</td><td>...</td></tr><tr><td>Gl 900<sup>b</sup></td><td>1</td><td>-3.247 ± 0.03</td><td>-3.581 ± 0.03</td><td>...</td><td>-2.44 ± 0.10</td><td>-3.29 ± 0.10</td><td>0.110 ± 0.020</td><td>1.35</td><td>-3.414</td><td>6.565</td><td></td><td>1</td><td>...</td><td>71HE</td></tr><tr><td>Gl 900</td><td>2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-5.638</td><td>10.84</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 906</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.66 ± 0.10</td><td>-0.91 ± 0.10</td><td>0.579 ± 0.020</td><td>1.38</td><td>-0.785</td><td>1.180</td><td>2.735</td><td>3</td><td>...</td><td>84</td></tr><tr><td>Gl 907.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-6.222</td><td>8.698</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 14593A</td><td>3</td><td>-0.49 ± 0.03</td><td>-0.51 ± 0.03</td><td>0.660 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.04</td><td>-0.50</td><td>1.179</td><td></td><td>3</td><td>69</td><td>...</td></tr><tr><td>HIP 19410ABC</td><td>4</td><td>-0.62 ± 0.03</td><td>-0.75 ± 0.03</td><td>0.670 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.21</td><td>-0.685</td><td>1.841</td><td></td><td>4</td><td>70</td><td>...</td></tr><tr><td>HIP 21865</td><td>4</td><td>-0.34 ± 0.03</td><td>-0.42 ± 0.03</td><td>0.703 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.24</td><td>-0.38</td><td>1.292</td><td></td><td>4</td><td>91</td><td>...</td></tr><tr><td>HIP 40170</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.31</td><td>3.398</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 42108AB</td><td>2</td><td>-0.056 ± 0.01</td><td>-0.122 ± 0.01</td><td>0.683 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>2.18</td><td>-0.089</td><td>0.169</td><td>0.491</td><td>2</td><td>62</td><td>...</td></tr><tr><td>HIP 42910</td><td>1</td><td>-0.70 ± 0.15</td><td>-0.76 ± 0.15</td><td>0.667 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-0.73</td><td>1.945</td><td></td><td>1</td><td>30</td><td>...</td></tr><tr><td>HIP 50773</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.88 ± 0.20</td><td>-1.37 ± 0.20</td><td>0.535 ± 0.020</td><td>1.56</td><td>-1.125</td><td>1.856</td><td>3.116</td><td>3</td><td>...</td><td>65</td></tr></tbody></table>
<table><caption>Table 2
(Continued)</caption><thead><tr><th rowspan="2">Star</th><th rowspan="2">No. meas.</th><th colspan="3">HARPS</th><th colspan="3">SOPHIE</th><th rowspan="2">K/H Ratio</th><th rowspan="2">(Ca II) EW (Å)</th><th rowspan="2">Ca II Flux (10<sup>5</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th rowspan="2">Ca II Flux<sup>a</sup> (10<sup>5</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th rowspan="2">No. meas.</th><th rowspan="2">S/N HARPS @5000</th><th rowspan="2">S/N SOPHIE @5000</th></tr><tr><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>H<sub>α</sub> EW (Å)</th><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>H<sub>α</sub> EW (Å)</th></tr></thead><tbody><tr><td>HIP 51073</td><td>4</td><td>-0.43 ± 0.05</td><td>-0.44 ± 0.05</td><td>0.655 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.02</td><td>-0.435</td><td>1.111</td><td></td><td>4</td><td>79</td><td>...</td></tr><tr><td>HIP 51263</td><td>1</td><td>&gt;-0.05</td><td>&gt;-0.05</td><td>0.700 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>&lt;0.142</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 53175</td><td>7</td><td>...</td><td>...</td><td>...</td><td>-0.54 ± 0.20</td><td>-0.71 ± 0.20</td><td>0.472 ± 0.020</td><td>1.31</td><td>-0.625</td><td>1.425</td><td>1.913</td><td>7</td><td>...</td><td>84</td></tr><tr><td>HIP 56838</td><td>3</td><td>-0.35 ± 0.03</td><td>-0.40 ± 0.03</td><td>0.678 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-0.375</td><td>0.839</td><td>1.137</td><td>3</td><td>...</td><td>...</td></tr><tr><td>HIP 58945</td><td>1</td><td>...</td><td>...</td><td>...</td><td>-0.84 ± 0.20</td><td>-1.24 ± 0.20</td><td>0.579 ± 0.020</td><td>1.48</td><td>-1.04</td><td>2.412</td><td>4.359</td><td>1</td><td>...</td><td>39</td></tr><tr><td>HIP 59247</td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.591 ± 0.020</td><td>...</td><td>...</td><td>...</td><td></td><td>4</td><td>...</td><td>58</td></tr><tr><td>HIP 60438A<sup>b</sup></td><td>1</td><td>-5.400 ± 0.03</td><td>-5.613 ± 0.03</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.04</td><td>-5.507</td><td>8.971</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 60438B<sup>b</sup></td><td>1</td><td>-2.419 ± 0.03</td><td>-2.827 ± 0.03</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.17</td><td>-2.623</td><td>4.273</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 60501</td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.659 ± 0.020</td><td>...</td><td>...</td><td>...</td><td></td><td>4</td><td>...</td><td>51</td></tr><tr><td>HIP 60661</td><td>2</td><td>...</td><td>...</td><td>...</td><td>-0.39 ± 0.25</td><td>-0.76 ± 0.25</td><td>0.592 ± 0.020</td><td>1.95</td><td>-0.575</td><td>...</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 72044</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.39 ± 0.05</td><td>-0.57 ± 0.05</td><td>0.610 ± 0.020</td><td>1.46</td><td>-0.48</td><td>1.678</td><td></td><td>3</td><td>...</td><td>102</td></tr><tr><td>HIP 76550</td><td>3</td><td>-0.28 ± 0.05</td><td>-0.27 ± 0.05</td><td>0.571 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>0.96</td><td>-0.275</td><td>0.448</td><td>5.785</td><td>3</td><td>52</td><td>...</td></tr><tr><td>HIP 78395</td><td>11</td><td>-0.68 ± 0.03</td><td>-0.72 ± 0.03</td><td>0.601 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.06</td><td>-0.70</td><td>1.991</td><td></td><td>11</td><td>131</td><td>...</td></tr><tr><td>HIP 80083</td><td>3</td><td>-0.28 ± 0.05</td><td>-0.30 ± 0.05</td><td>0.624 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.07</td><td>-0.29</td><td>1.281</td><td></td><td>3</td><td>50</td><td>...</td></tr><tr><td>HIP 103150</td><td>1</td><td>-0.46 ± 0.15</td><td>-0.58 ± 0.15</td><td>0.579 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.26</td><td>-0.52</td><td>0.965</td><td>2.802</td><td>1</td><td>33</td><td>...</td></tr><tr><td>HIP 110245</td><td>3</td><td>-0.82 ± 0.05</td><td>-0.93 ± 0.05</td><td>0.599 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.13</td><td>-0.875</td><td>2.935</td><td></td><td>3</td><td>73</td><td>...</td></tr><tr><td>HIP 110714</td><td>2</td><td>-0.97 ± 0.05</td><td>-1.11 ± 0.05</td><td>0.615 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-1.04</td><td>1.786</td><td>2.534</td><td>2</td><td>58</td><td>...</td></tr><tr><td>HIP 113597<sup>b</sup></td><td>2</td><td>-2.758 ± 0.03</td><td>-3.019 ± 0.03</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-2.889</td><td>5.810</td><td></td><td>...</td><td>...</td><td>...</td></tr><tr><td>HIP 116011</td><td>7</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.576 ± 0.020</td><td>...</td><td>...</td><td>...</td><td></td><td>7</td><td>...</td><td>55</td></tr></tbody></table>

**Notes.** Results were obtained using spectra from HARPS, SOPHIE, and FEROS. For each star, we give the S/N ratio for the sum of all available spectra, as well as the number of spectra (no. meas.) that we used to compute the equivalent widths.

a Corrected or metallicity effects.

b FEROS spectra.

(This table is available in machine-readable form.)
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Correlation between the logarithm of the mean surface flux of the Ca II resonance doublet and $log(P/\sin i)$ for stars with spectral sub-type dK6. We overplot the heteroscedastic linear LSF (solid line). We also plot the heteroscedastic linear LSF to the low-activity stellar sub-sample (dotted-dashed straight line between $log(P/\sin i) = 1.6$ and 0.9) and the heteroscedastic linear LSF to the high-activity stellar sub-sample (a separate dotted-dashed straight line between $log(P/\sin i) = 0.85$ and 0.25). In the top left of the figure, we show an estimate of the error on the Ca II mean fluxes due to metallicity effects.</figcaption></figure>

We also searched the literature for other measurements of the Ca II EW. We found only a few measurements of the Ca II EW or flux for our dK6 stars (Marilli et al. 1986; Rutten 1987; Duncan et al. 1991; Browning et al. 2010). These measures are also listed in Table 2.

We saw in Paper XV (see also Section 3.4) that correction for metallicity effects in dM2 stars is essential in order to obtain a good correlation between the Ca II EW and $P/\sin i$. This is due to the fact that the Ca II line formation depends sensitively on the Ca abundance (E. R. Houdebine & P. M. Panagi 2017, in preparation). Here the same applies to dK6 stars and to the Ca II surface fluxes. In Paper I, we compiled [M/H] measures for our dK6 stellar sample and here we computed the Ca II surface fluxes corrected for metallicity effects, assuming a proportionality between [M/H] and Ca II surface fluxes.

In Figure 3, we show the RAC that we have obtained for our dK6 stars. The heteroscedastic linear LSF to the data in Figure 3 is shown by the solid line. One of the stars in our sample, Gl 208 (a dM6(e) star with $P/\sin i = 8.94$ days), is probably also a fast rotator: but its value of $v\sin i$ is too small (and its value of $P/\sin i$ is too large) to be entirely consistent with its observed activity level. It is possible that we are viewing this star close to its rotation axis, i.e., $\sin i$ may be atypically low. We also report on two relatively active stars with a rather slow rotation: Gl 455.1 and Gl 907.1. These stars depart noticeably from the main correlation. We believe that

these stars also have $\sin i$. Most of the other stars follow the solid-line correlation fairly well with a few exceptions: a group of stars that lie below the main correlation and are low-activity fast rotators. One of them is a subdwarf: their internal structure may cause their dynamos to operate differently from those in main-sequence K stars. For the few other stars, similar discrepancies from the RAC have been found in dM2 stars (Paper XV and Section 3.4); they will also be noted among dM3 stars (see Section 3.5). Reiners et al. (2012) also observed a few stars that are discrepant compared to the global trend: Reiners et al. disregarded such objects as being due to measurement errors. However, the repetition with which they appear in our samples, and the fact that they are relatively rapidly rotating, lead us to believe that these low-activity relatively fast rotators may indeed exist as a possibly significant sub-set of late K and M dwarfs. The best explanation we have is that, in a star such as Gl 412.3AB, which exhibits a slight asymmetry when its photospheric spectral lines are subjected to rotational analysis (see Paper I), these stars may be unresolved spectroscopic binaries: as such, they could yield abnormally broadened cross-correlation profiles (see also Gl 186AB in Paper I). Therefore these stars are probably spectroscopic binaries that are unresolved at the time the observations were made. This is quite plausible since binarity is a common characteristic of late-type dwarfs. We label these stars as spectroscopic binaries in Figure 3.
Including the known binaries in our dK6 sample, the number of probable spectroscopic binaries yield a value of 20% binaries in our full sample. This is consistent with the recent finding of Ward-Duong et al. (2015) that binary stars constitute about 25% of their stellar sample for $M_* \sim 0.7 M_⊙$. In the subsequent subsections, we will see that the proportion of binaries amount to about 28%, 37%, and 48% for our dM2, dM3, and dM4 stellar samples, respectively. Our dM2 and dM3 results are again compatible with the fraction of about 32–37% derived by Ward-Duong et al. (2015) for these stars. For our dM4 sample, our figure of 48% is somewhat larger than the 37% fraction expected at this spectral type. This disagreement may find a simple explanation in the bias in detecting parallaxes for nearby M dwarfs. Indeed, faint single M4 dwarfs are much more difficult to detect than the companions to nearby brighter M and K dwarfs. Therefore, our dM4 sample, which is largely based on parallax surveys of nearby dwarfs, is biased toward the detection of faint companions to nearby brighter M and K dwarfs. This should explain why we have a larger binary fraction for the M4 dwarf sample.

We should like to emphasize four points regarding the plot in Figure 3. (i) Metallicity differences from star to star contribute to the scatter about the dK6 RAC in Figure 3. By analogy, in Paper XV we reported that metallicity differences among dM2 stars are responsible for the greater part of the scatter in the RAC. (ii) Another contribution to the scatter is that our statistics on the Ca II line EW are poor; we have very few measures of the Ca II line EW for most of our dK6 stars compared to what we have for the stars in our dM2 sample, and the Ca II line EW is known to vary with rotation and with the phase of the activity cycle (e.g., Baliunas et al. 1995). However, we have little or no information as to which part of the cycle our observations happened to “catch” for any particular star. (iii) The value of $\sin i$ may vary from 1 to 0 and certainly contributes to the scatter in Figure 3. (iv) The range of effective temperatures in our dK6 sample is somewhat larger than in our samples of dM2, dM3, and dM4 stars (e.g., only $\pm 70$ K in dM3 stars, where $T_{eff}$ ranges from 3210 to 3350 K).

We would like to emphasize, as well, that a few of the active stars in our dK6 stellar sample are candidate young stars that may not yet have contracted to the main sequence (MS). A reliable way to identify the pre-main-sequence (PMS) stars in our samples is the stellar radius: PMS stars that have not yet contracted to the MS have abnormally large radii. We identified three such stars in our dK6 sample: GJ 1177A, GJ 182, and GJ 425B are possible PMS stars (see Paper I). However, GJ 425B is a rather low-activity star. As such, the abnormally large radius for this star is probably due to binarity. Nevertheless, we find that these stars do correlate well with the MS stars. In our M2 sample (see Section 3.4), we have also identified two PMS stars: GJ 1264 and GJ 803. But again, as we shall see, these PMS stars correlate very well with the other MS stars. In our M3 sample, GJ 277A is a possible PMS star. In our M4 sample, GJ 2069A, GJ 3322, GJ 669A, GJ 695B, and GJ 812A are possible PMS stars according to their radii. We emphasize that all these stars do not rotate especially fast. There are many MS stars that rotate faster. These PMS stars are actually expected to spin up as they contract to the MS, and young MS stars are expected to be the fastest rotators (e.g., Barnes 2003; for a theoretical model of this process, see e.g., Figure 2 in Mullan & MacDonald 2003). In fact, that is what we observe in our samples of stars: young MS stars are the fastest rotators (e.g.,

among our M4 sample, GJ 3631, GJ 3789, GJ 4020B, GJ 4338B, GJ 431, GJ 630.1, GJ 791.2A) with $\sin i$ in excess of 15 km s$^{-1}$ and up to 56 km s$^{-1}$.

All of our RACs demonstrate that PMS stars do not stand out as significantly different from the main correlations of the MS stars (see subsequent sections). Numerous previous studies also found similar results (e.g., Mamajek & Hillebrand 2008, Browning et al. 2010, Christian et al. 2011, West et al. 2015). This finding is rather intriguing since PMS stars may have internal structures that differ from those of MS stars: in fact, some PMS stars may even be fully convective. More investigation is required to confirm this result, but so far, in our samples, we found no definitive evidence that PMS stars obey different RACs from those of MS stars.

The heteroscedastic linear LSF to the RAC in Figure 3 gives the following:

$$F_{CaII} = 3.402 \pm 0.021 \times 10^6 \times (P/\sin i)^{-1.047 \pm 0.042}. \quad (6)$$

The parameters of this LSF are given in Table 1. The $\chi^2$ is 0.093 and the statistical significance is 99.9% for 55 dK6 stars. The homoscedastic linear LSF to the same data set gives:

$$F_{CaII} = 1.95 \pm 0.34 \times 10^6 \times (P/\sin i)^{-0.81 \pm 0.06}. \quad (7)$$

The correlation coefficient for this fit is 0.876, the statistical significance is 99.9% and the $\chi^2$ is only 0.021 for 55 dK6 stars. We find, in this case, that the heteroscedastic and the homoscedastic linear fits give slightly different results for the slope and for the flux amplitude: This is due to the fact that high-activity stars (fast rotators) have smaller uncertainties in $P/\sin i$ compared to low-activity stars, and this gives them much higher weights in the least square fit. As a result, the heteroscedastic linear solution fits the high-activity stars better, whereas the homoscedastic linear solution clearly underestimates the fluxes among high-activity stars. Therefore, in this case, the heteroscedastic fit gives more sensible results.

For future reference, we note that, in Figure 3, the correlation in Equation (6) spans the entire range of $P/\sin i$ values for which we have dK6 data. In particular, it is important to notice that the dK6 data exhibit no evidence for a flattening (or “saturation”) of the RAC at the shortest periods ($P/\sin i = 1.8$ days). This is consistent with previous results for the earlier spectral type (dK4), which also showed no signs of saturation (see Section 3.2). Neither is there evidence for saturation among dM2 stars (see Figure 5).

Having analyzed the combined samples of slow and fast rotators, we now turn to performing heteroscedastic and homoscedastic linear LSF on the sub-samples of low-activity stars (dK6) and high-activity stars (dK6e) separately. The heteroscedastic linear LSF to the dK6 low-activity stars yields:

$$F_{CaII} = 8.23 \pm 2.23 \times 10^5 \times (P/\sin i)^{-0.531 \pm 0.12}. \quad (8)$$

The $\chi^2$ for this fit is 0.017 and the statistical significance is 99.9% for 41 dK6 stars (see Table 1). This fit is shown as the straight dotted-dashed line in the lower left portion of Figure 3. The homoscedastic linear LSF to this sub-sample gives:

$$F_{CaII} = 1.29 \pm 0.36 \times 10^6 \times (P/\sin i)^{-0.637 \pm 0.11}. \quad (9)$$

The correlation coefficient for this fit is 0.706, the statistical significance is 99.9% and the $\chi^2$ is only 0.014 for 41 dK6 stars. Therefore, these two fits are highly statistically significant at a confidence level greater than 99.9%.
<figure><img src="image_6.png" /><figcaption>Figure 4. Correlation between the logarithm of the mean surface flux of the Ca II resonance doublet and log(P/ sin i) for stars with spectral sub-type dM2. In this plot, no corrections have been made for metallicity differences.</figcaption></figure>

We also fitted the high-activity stellar sub-sample. The heteroscedastic linear LSF to the dK6 high-activity stars yields:

$$F_{CaII} = 4.20 \pm 3.52 \times 10^6 \times (P/\sin i)^{-1.402 \pm 0.43}. \quad (10)$$

The $\chi^2$ for this fit is 0.023 and the statistical significance is 99.9% for nine dK6e stars (see Table 1). This fit is shown as the straight dotted-dashed line in the upper right portion of Figure 3.

We again find that the gradient of the RAC for the low-activity stars alone (-0.86, -0.64) seems shallower than that of the low+high-activity star sample (-1.05, -0.81) and that the gradient of the RAC for the high-activity stars alone (-1.402, -1.364) seems steeper than that of the low+high-activity star sample (-1.05, -0.81). We find that this difference in RAC steepness occurs systematically between the low-activity sub-samples and the full samples for all five of our spectral sub-types (see Table 1 and Sections 3.4–3.6). The parameter $R/\delta$ is 17.20 for the high-activity star sub-sample and 39.68 for the low-activity star sub-sample. Therefore, the parameter $\delta = \langle\langle\text{error}\rangle\rangle/\sqrt{n}$, which is a normalized estimate of the mean error on the measurements, is much smaller than the period domains of the RACs in both cases. Hence, these LSFs should be relatively well established, which corresponds to the high statistical significances we obtain (Table 1). However, we consider these results as still preliminary because

the domains of the RACs are relatively small compared to the typical uncertainties on individual measures.

If this difference between the RACs of the low and high-activity sub-samples is confirmed, this may represent a discovery of interest. Indeed, to the extent that the slope of an RAC is related in some way to an underlying dynamo process, the difference that we found between the slopes of the low and high-activity sub-samples suggests that there may exist two different dynamo regimes for these two sub-samples of stars. We find that this difference between low and high-activity regimes persists for all our samples of stars with different spectral sub-types. However, we observe a difference for stars that are more massive than the TTCC and stars that are less massive than the TTCC. For the former, the slope among high-activity stars is steeper than the slope among the low+high-activity stars, whereas for the latter, we find that the slope among high-activity stars is shallower than the slope among the low+high-activity stars. This preliminary result could be of interest for the dynamo mechanisms. More data with a higher resolution spectrograph will be needed to confirm these results.

### 3.4. The RAC in M2 Dwarfs

We have re-investigated the RAC in the sample of dM2 stars of Paper XV. We have given the new stellar parameters and new P/ sin i values in Paper I. We show the raw data in
<table><caption>**Table 3**
Values of $v \sin i$, $P/\sin i$ as well as [M/H] from Paper I for Our dM2 Stars</caption><thead><tr><th>Star</th><th>v sin i (km s<sup>-1</sup>)</th><th>P/ sin i (days)</th><th>[M/H] (dex)</th><th>Ca II EW (Å)</th><th>F<sub>HK</sub> (10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th>F<sub>HK</sub><sup>a</sup> (10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th></tr></thead><tbody><tr><td>GJ 1010A</td><td>...</td><td>...</td><td>-0.098</td><td>-0.605</td><td>2.621</td><td>10.555</td></tr><tr><td>GJ 1062</td><td>9.8</td><td>1.92-0.53+0.65</td><td>-0.584</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1114</td><td>3.8</td><td>5.73-1.50+2.07</td><td>-0.339</td><td>-0.117</td><td>0.507</td><td>0.664</td></tr><tr><td>GJ 1264</td><td>6.46</td><td>6.82-1.51+2.06</td><td>+0.378</td><td>-4.45</td><td>20.85</td><td>8.732</td></tr><tr><td>GJ 2085</td><td>2.46</td><td>10.80-3.23+5.31</td><td>+0.000</td><td>-0.77</td><td>4.071</td><td>4.071</td></tr><tr><td>GJ 3084</td><td>3.16</td><td>9.74-4.07+5.98</td><td>+0.023</td><td>-0.841</td><td>4.373</td><td>4.147</td></tr><tr><td>GJ 3098</td><td>2.60</td><td>8.97-2.49+3.98</td><td>-0.200</td><td>-0.86</td><td>4.215</td><td>6.680</td></tr><tr><td>GJ 3207</td><td>...</td><td>...</td><td>-0.346</td><td>-0.19</td><td>0.801</td><td>1.777</td></tr><tr><td>GJ 3340</td><td>2.67</td><td>9.00-2.91+4.59</td><td>-0.048</td><td>-0.83</td><td>4.256</td><td>4.753</td></tr><tr><td>GJ 3215</td><td>...</td><td>...</td><td>-0.314</td><td>-0.45</td><td>1.950</td><td>4.018</td></tr><tr><td>GJ 3440</td><td>2.71</td><td>8.95-3.56+5.59</td><td>-0.165</td><td>-0.6</td><td>2.599</td><td>3.800</td></tr><tr><td>GJ 3759</td><td>2.71</td><td>9.10-2.52+3.95</td><td>-0.121</td><td>-0.645</td><td>3.148</td><td>4.159</td></tr><tr><td>GJ 3778</td><td>1.72</td><td>13.86-6.49+13.4</td><td>+0.002</td><td>-0.635</td><td>2.515</td><td>2.503</td></tr><tr><td>GJ 3915</td><td>1.88</td><td>22.51-12.4+24.0</td><td>+0.413</td><td>-0.60</td><td>2.694</td><td>1.041</td></tr><tr><td>GJ 4155</td><td>...</td><td>...</td><td>-0.231</td><td>-0.755</td><td>3.005</td><td>5.115</td></tr><tr><td>GJ 9381</td><td>2.19</td><td>14.01-4.81+8.44</td><td>+0.100</td><td>-0.90</td><td>3.899</td><td>3.097</td></tr><tr><td>GL 2</td><td>2.76</td><td>9.74-2.48+3.85</td><td>-0.018</td><td>-1.316</td><td>6.166</td><td>6.427</td></tr><tr><td>Gl 15A</td><td>1.43</td><td>13.87-4.92+12.0</td><td>-0.386</td><td>-0.303</td><td>1.366</td><td>3.322</td></tr><tr><td>Gl 16</td><td>2.34</td><td>11.23-3.27+5.53</td><td>-0.105</td><td>-1.084</td><td>4.696</td><td>5.980</td></tr><tr><td>GL 27.1</td><td>2.63</td><td>10.01-3.03+4.83</td><td>-0.016</td><td>-1.080</td><td>5.413</td><td>5.616</td></tr><tr><td>Gl 29.1A</td><td>10.6</td><td>3.16-0.64+0.77</td><td>+0.101</td><td>-9.69</td><td>43.12</td><td>34.17</td></tr><tr><td>Gl 29.1B</td><td>9.5</td><td>3.52-0.74+0.91</td><td>+0.101</td><td>-9.81</td><td>43.65</td><td>34.59</td></tr><tr><td>Gl 49</td><td>2.49</td><td>11.83-3.13+5.12</td><td>+0.026</td><td>-1.797</td><td>7.785</td><td>7.333</td></tr><tr><td>Gl 63</td><td>&lt;1</td><td>&gt;20.90</td><td>-0.309</td><td>-0.159</td><td>0.762</td><td>1.552</td></tr><tr><td>Gl 87</td><td>3.99</td><td>5.86-1.24+1.68</td><td>-0.027</td><td>-0.294</td><td>1.377</td><td>1.465</td></tr><tr><td>Gl 91</td><td>1.68</td><td>15.82-5.34+11.3</td><td>-0.090</td><td>-0.968</td><td>3.938</td><td>4.845</td></tr><tr><td>GL 114.1A</td><td>1.82</td><td>12.68-4.11+8.14</td><td>-0.242</td><td>-0.466</td><td>1.868</td><td>3.261</td></tr><tr><td>Gl 130</td><td>1.27</td><td>13.59-5.98+16.7</td><td>-0.766</td><td>-0.332</td><td>1.399</td><td>8.162</td></tr><tr><td>Gl 133</td><td>&lt;1</td><td>&gt;20.70</td><td>-0.466</td><td>-0.28</td><td>1.051</td><td>3.073</td></tr><tr><td>Gl 134</td><td>2.53</td><td>13.90-3.93+6.37</td><td>+0.128</td><td>-1.596</td><td>7.290</td><td>5.429</td></tr><tr><td>Gl 140AB</td><td>9.4</td><td>2.93-0.74+0.92</td><td>-0.002</td><td>-2.175</td><td>9.167</td><td>9.209</td></tr><tr><td>Gl 150.1B</td><td>3.7</td><td>7.26-1.74+2.42</td><td>-0.084</td><td>-1.063</td><td>4.980</td><td>6.043</td></tr><tr><td>Gl 153B</td><td>1.4</td><td>20.97-8.24+20.6</td><td>+0.017</td><td>&gt;0.25</td><td>&lt;1.054</td><td>&lt;1.014</td></tr><tr><td>Gl 155.1</td><td>2.93</td><td>7.67-2.05+3.10</td><td>+0.175</td><td>-0.355</td><td>1.851</td><td>1.237</td></tr><tr><td>Gl 162</td><td>2.35</td><td>11.61-3.48+5.86</td><td>+0.019</td><td>-1.115</td><td>5.621</td><td>5.380</td></tr><tr><td>Gl 173</td><td>2.35</td><td>10.27-2.91+4.91</td><td>-0.225</td><td>-0.430</td><td>1.661</td><td>2.788</td></tr><tr><td>Gl 191</td><td>9.15</td><td>1.55-0.26+0.33</td><td>-1.024</td><td>-0.164</td><td>0.867</td><td>9.163</td></tr><tr><td>Gl 205</td><td>2.73</td><td>11.92-2.94+4.59</td><td>+0.101</td><td>-1.555</td><td>7.453</td><td>5.907</td></tr><tr><td>Gl 212</td><td>1.98</td><td>14.85-4.55+8.51</td><td>+0.067</td><td>-1.872</td><td>10.03</td><td>8.596</td></tr><tr><td>Gl 218</td><td>1.66</td><td>15.49-5.21+11.1</td><td>-0.099</td><td>-0.643</td><td>2.767</td><td>3.475</td></tr><tr><td>Gl 229</td><td>2.63</td><td>11.22-2.80+4.46</td><td>+0.060</td><td>-1.124</td><td>5.845</td><td>5.091</td></tr><tr><td>Gl 250B</td><td>...</td><td>...</td><td>-0.247</td><td>-0.400</td><td>1.294</td><td>2.285</td></tr><tr><td>Gl 275.1</td><td>2.7</td><td>13.87-4.14+6.51</td><td>+0.172</td><td>-0.16</td><td>0.807</td><td>0.543</td></tr><tr><td>Gl 289</td><td>&lt;1</td><td>&gt;18.37</td><td>-0.741</td><td>-0.16</td><td>0.693</td><td>3.817</td></tr><tr><td>Gl 330</td><td>3.05</td><td>9.62-2.57+3.82</td><td>+0.023</td><td>-0.302</td><td>1.447</td><td>1.372</td></tr><tr><td>Gl 361</td><td>1.95</td><td>12.54-3.90+7.37</td><td>-0.212</td><td>-0.926</td><td>3.476</td><td>5.663</td></tr><tr><td>Gl 366</td><td>&lt;1</td><td>&gt;27.69</td><td>-0.014</td><td>-0.284</td><td>1.331</td><td>1.375</td></tr><tr><td>Gl 378</td><td>2.25</td><td>17.30-5.20+8.99</td><td>+0.096</td><td>-0.755</td><td>2.524</td><td>2.023</td></tr><tr><td>Gl 382</td><td>2.9</td><td>9.49-2.37+3.60</td><td>-0.059</td><td>-1.398</td><td>5.563</td><td>6.372</td></tr><tr><td>Gl 390</td><td>2.46</td><td>10.78-2.97+4.89</td><td>-0.030</td><td>-1.231</td><td>5.767</td><td>6.179</td></tr><tr><td>Gl 411</td><td>0.61</td><td>32.77-11.9+30.8</td><td>-0.442</td><td>-0.199</td><td>0.792</td><td>2.191</td></tr><tr><td>Gl 412A</td><td>1.6</td><td>11.55-3.82+8.40</td><td>-0.435</td><td>-0.180</td><td>0.907</td><td>2.469</td></tr><tr><td>Gl 430.1</td><td>0.46</td><td>65.7-28.3+108</td><td>+0.089</td><td>-0.43</td><td>2.261</td><td>1.842</td></tr><tr><td>Gl 433</td><td>&lt;1</td><td>&gt;23.54</td><td>-0.199</td><td>-0.439</td><td>1.850</td><td>2.925</td></tr><tr><td>Gl 450</td><td>2.47</td><td>9.14-2.41+3.96</td><td>-0.247</td><td>-1.054</td><td>4.690</td><td>8.283</td></tr><tr><td>Gl 477</td><td>1.95</td><td>14.79-5.07+9.57</td><td>-0.057</td><td>-1.063</td><td>4.168</td><td>4.753</td></tr><tr><td>Gl 490A</td><td>8.4</td><td>3.87-0.91+1.15</td><td>+0.078</td><td>-6.03</td><td>29.55</td><td>24.69</td></tr><tr><td>Gl 494AB</td><td>9.75</td><td>2.17-0.39+0.47</td><td>-0.339</td><td>-7.222</td><td>33.84</td><td>73.86</td></tr><tr><td>Gl 507.1</td><td>2.32</td><td>13.39-3.83+6.49</td><td>+0.065</td><td>-1.036</td><td>4.367</td><td>3.760</td></tr><tr><td>Gl 508.2</td><td>1.78</td><td>15.01-5.01+10.1</td><td>-0.002</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 510</td><td>2.49</td><td>10.33-3.02+4.93</td><td>-0.162</td><td>-1.77</td><td>7.252</td><td>10.53</td></tr><tr><td>Gl 514</td><td>2.07</td><td>12.62-3.63+6.59</td><td>-0.030</td><td>-0.788</td><td>3.972</td><td>4.256</td></tr><tr><td>Gl 521</td><td>0.85</td><td>30.84-9.69+18.7</td><td>-0.017</td><td>-0.505</td><td>2.420</td><td>2.517</td></tr></tbody></table>
<table><caption>**Table 3**
(Continued)</caption><thead><tr><th>Star</th><th>v sin i<br>(km s<sup>-1</sup>)</th><th>P/ sin i<br>(days)</th><th>[M/H]<br>(dex)</th><th>Ca II EW<br>(Å)</th><th>F<sub>HK</sub><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th><th>F<sub>HK</sub><sup>a</sup><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th></tr></thead><tbody><tr><td>Gl 526</td><td>1.00</td><td>25.41–6.99+12.1</td><td>-0.086</td><td>-0.498</td><td>2.275</td><td>2.773</td></tr><tr><td>Gl 536</td><td>&lt;1</td><td>&gt;26.27</td><td>-0.032</td><td>-0.689</td><td>3.228</td><td>3.475</td></tr><tr><td>Gl 537AB</td><td>3.6</td><td>7.11–1.82+2.55</td><td>-0.046</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 540</td><td>2.09</td><td>14.97–4.52+8.16</td><td>+0.073</td><td>-0.915</td><td>4.613</td><td>3.899</td></tr><tr><td>Gl 552</td><td>0.58</td><td>45.55–17.9+49.1</td><td>-0.100</td><td>-0.734</td><td>2.756</td><td>3.470</td></tr><tr><td>Gl 563.2A</td><td>2.24</td><td>2.06–0.72+1.24</td><td>-1.742</td><td>-0.355</td><td>1.621</td><td>89.49</td></tr><tr><td>Gl 563.2B</td><td>1.82</td><td>2.42–0.93+1.85</td><td>-1.754</td><td>-0.33</td><td>1.274</td><td>72.31</td></tr><tr><td>Gl 618.4</td><td>1.45</td><td>17.56–7.21+17.4</td><td>-0.119</td><td>-0.515</td><td>2.191</td><td>2.822</td></tr><tr><td>Gl 634</td><td>&lt;0.6</td><td>&gt;37.12</td><td>-0.303</td><td>-0.345</td><td>1.373</td><td>2.758</td></tr><tr><td>Gl 637</td><td>2.19</td><td>8.94–2.68+4.71</td><td>-0.415</td><td>-0.33</td><td>1.569</td><td>4.080</td></tr><tr><td>Gl 645</td><td>2.80</td><td>7.75–2.64+4.09</td><td>-0.257</td><td>-0.91</td><td>4.174</td><td>7.543</td></tr><tr><td>Gl 649</td><td>2.27</td><td>11.77–3.29+5.65</td><td>-0.020</td><td>-1.036</td><td>5.077</td><td>5.316</td></tr><tr><td>Gl 654AB</td><td>5.7</td><td>4.40–1.02+1.46</td><td>-0.090</td><td>-0.262</td><td>1.073</td><td>1.320</td></tr><tr><td>Gl 672.1</td><td>2.99</td><td>8.70–2.75+4.13</td><td>-0.086</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 686</td><td>2.49</td><td>8.94–2.35+3.85</td><td>-0.245</td><td>-0.412</td><td>1.930</td><td>3.393</td></tr><tr><td>Gl 701</td><td>1.6</td><td>14.65–4.91+10.8</td><td>-0.169</td><td>-0.592</td><td>2.837</td><td>4.187</td></tr><tr><td>Gl 724</td><td>2.97</td><td>9.17–2.37+3.58</td><td>-0.005</td><td>-1.21</td><td>5.930</td><td>5.999</td></tr><tr><td>Gl 737B</td><td>...</td><td>...</td><td>+0.134</td><td>-1.79</td><td>8.685</td><td>6.379</td></tr><tr><td>Gl 745A</td><td>3.0</td><td>5.03–1.25+1.89</td><td>-1.03</td><td>-0.136</td><td>0.525</td><td>5.625</td></tr><tr><td>Gl 745B</td><td>2.8</td><td>5.55–1.44+2.22</td><td>-1.07</td><td>-0.109</td><td>0.421</td><td>4.946</td></tr><tr><td>Gl 767A</td><td>2.9</td><td>12.30–3.29+5.01</td><td>+0.126</td><td>-1.08</td><td>4.679</td><td>3.501</td></tr><tr><td>Gl 781</td><td>12.7</td><td>1.08–0.22+0.25</td><td>-1.10</td><td>-2.136</td><td>11.11</td><td>139.87</td></tr><tr><td>Gl 800A</td><td>2.34</td><td>12.33–3.68+6.22</td><td>-0.113</td><td>-0.738</td><td>3.537</td><td>4.588</td></tr><tr><td>Gl 803</td><td>9.68</td><td>4.03–0.68+0.84</td><td>+0.154</td><td>-8.364</td><td>40.089</td><td>28.12</td></tr><tr><td>Gl 806</td><td>0.46</td><td>47.86–20.9+80.5</td><td>-0.336</td><td>-0.624</td><td>2.410</td><td>5.224</td></tr><tr><td>Gl 808</td><td>1.27</td><td>14.94–7.23+20.1</td><td>-0.882</td><td>-0.16</td><td>0.637</td><td>4.854</td></tr><tr><td>Gl 809</td><td>2.66</td><td>11.04–2.76+4.36</td><td>+0.067</td><td>-0.956</td><td>4.971</td><td>4.260</td></tr><tr><td>Gl 815AB</td><td>7.61</td><td>3.35–0.73+0.95</td><td>-0.167</td><td>-8.38</td><td>33.35</td><td>48.99</td></tr><tr><td>Gl 821</td><td>2.63</td><td>7.10–1.89+3.02</td><td>-0.484</td><td>-0.185</td><td>0.867</td><td>2.643</td></tr><tr><td>Gl 832</td><td>...</td><td>...</td><td>-0.268</td><td>-0.569</td><td>2.197</td><td>4.072</td></tr><tr><td>Gl 842</td><td>2.80</td><td>10.05–2.56+3.96</td><td>+0.050</td><td>-1.15</td><td>6.362</td><td>5.670</td></tr><tr><td>Gl 855</td><td>2.99</td><td>9.97–2.61+3.92</td><td>+0.049</td><td>-1.275</td><td>6.833</td><td>6.104</td></tr><tr><td>Gl 863</td><td>2.64</td><td>8.90–2.44+3.88</td><td>-0.132</td><td>-0.699</td><td>3.524</td><td>4.776</td></tr><tr><td>Gl 867A</td><td>7.02</td><td>4.81–0.98+1.31</td><td>+0.080</td><td>-7.272</td><td>28.08</td><td>23.36</td></tr><tr><td>Gl 880</td><td>2.07</td><td>14.25–4.09+7.43</td><td>+0.040</td><td>-1.165</td><td>5.321</td><td>4.853</td></tr><tr><td>Gl 887</td><td>...</td><td>...</td><td>-0.092</td><td>-0.670</td><td>3.284</td><td>4.059</td></tr><tr><td>Gl 895</td><td>0.52</td><td>56.16–22.6+71.4</td><td>+0.049</td><td>-1.675</td><td>7.848</td><td>7.011</td></tr><tr><td>Gl 908</td><td>2.25</td><td>9.54–2.59+4.48</td><td>-0.271</td><td>-0.270</td><td>1.294</td><td>2.415</td></tr><tr><td>St 497</td><td>6.5</td><td>3.95–1.45+1.98</td><td>-0.027</td><td>-6.20</td><td>29.05</td><td>30.91</td></tr><tr><td>St 928</td><td>2.1</td><td>11.16–4.86+8.74</td><td>-0.176</td><td>-0.94</td><td>4.183</td><td>6.273</td></tr><tr><td>G192-11A</td><td>2.6</td><td>10.14–2.71+4.33</td><td>-0.009</td><td>-2.405</td><td>12.12</td><td>12.37</td></tr><tr><td>MCC 354A</td><td>2.47</td><td>12.99–3.82+6.27</td><td>+0.109</td><td>-1.80</td><td>7.163</td><td>5.573</td></tr><tr><td>MCC 452</td><td>...</td><td>...</td><td>+0.360</td><td>-0.96</td><td>4.992</td><td>2.179</td></tr><tr><td>MCC 488</td><td>3.2</td><td>10.60–3.03+4.43</td><td>+0.113</td><td>-1.87</td><td>9.427</td><td>7.267</td></tr><tr><td>LHS 1155</td><td>...</td><td>...</td><td>...</td><td>-0.88</td><td>3.399</td><td>...</td></tr></tbody></table>

Note. We also give the Ca II EW (from the compilation of Paper XVIII), surface fluxes $F_{HK}$, and the surface fluxes corrected for the metallicity effects.

a Surface flux corrected from metallicity effects.

(This table is available in machine-readable form.)

Figure 4. We can see in this figure that there is a large scatter among M2 dwarf low-activity stars. Most of the subdwarfs also lie significantly apart from most of the M2 dwarfs. Most of this large scatter is due to metallicity effects on the Ca II line formation.

We used the empirical correlation found by Houdebine (2008, Paper VII) to determine the metallicity for each star as a function of its radius (see Paper I). We use these values here to correct the Ca II surface fluxes for metallicity effects, assuming a proportionality between surface flux and metallicity (optically

thin case). The Ca II line EW, surface fluxes and surface fluxes corrected for metallicity are all listed in Table 3.

We show in Figure 5 the Ca II surface fluxes (after the data have been corrected for metallicity effects) as a function of $P/\sin i$. The improvement of the correlation between this figure and Figure 4 is striking: it shows how important it is to correct the RAC for metallicity effects in M dwarfs. In Figure 5, the scatter has been much reduced among the slow rotators, and also among the fast rotators. Moreover, even subdwarfs correlate with normal metallicity dwarfs.
<figure><img src="image_7.png" /><figcaption>Figure 5. Correlation between the logarithm of the mean surface flux of the Ca II resonance doublet corrected for metallicity effects and log(P/ sin i) for stars with spectral sub-type dM2. The comparison between this correlation and that of Figure 4 is striking: it shows how important it is to correct for metallicity effects on the RAC in M dwarfs. In this figure, even subdwarfs now correlate with normal metallicity dwarfs. The straight solid line shows the heteroscedastic linear LSF to the data. The dotted-dashed lines show the heteroscedastic linear LSF to the low-activity star and high-activity star sub-samples. We show in the upper-left corner of the figure estimates of the uncertainties on the Ca II mean surface fluxes due to the corrections for metallicity effects, for dwarfs and subdwarfs.</figcaption></figure>

In Figure 5, we plot the uncertainties on P/ sin i as dotted lines for each star. For these uncertainties, we assumed an uncertainty of ±0.14 km s⁻¹ for measures of v sin i below 1 km s⁻¹, ±0.30 km s⁻¹ for v sin i between 1 km s⁻¹ and 6 km s⁻¹, and ±0.50 km s⁻¹ for v sin i above 6 km s⁻¹ (see Paper I). Uncertainties in the values of P/ sin i have already been included in Figures 2 and 3, following the same prescription as we describe here. Similar uncertainties will be included in Figures 8 and 12 in connection with dM3 and dM4 stars, respectively.

In Figure 5, the straight solid line shows the heteroscedastic linear LSF to the data. There is little difference between this fit and the linear LSF that was obtained in Paper XV. There is a moderate shift between the two correlations, and the gradient is almost unchanged (see Table 2). We identified some low-activity, relatively fast rotators in this M2 dwarf sample. As mentioned in the previous section, these stars are probably unresolved spectroscopic binaries, so we have labelled these stars “spectroscopic binaries” in Figure 5. There are also a few stars that have rotation periods that are too long for their surface fluxes. We believe these stars are a sub-group of stars with low sin i. These stars are shown as squares in Figure 5.

The heteroscedastic linear LSF to the RAC in Figure 5 gives the following:

$$F_{\text{Ca II}} = 2.312 \pm 0.276 \times 10^6 \times (P/\sin i)^{-1.575 \pm 0.058}. \quad (11)$$

The parameters of this LSF are given in Table 1. The χ² is 0.028 for 66 dM2 stars.

The homoscedastic linear LSF to the RAC in Figure 5 gives the following:

$$F_{\text{Ca II}} = 1.89 \pm 0.85 \times 10^6 \times (P/\sin i)^{-1.481 \pm 0.068}. \quad (12)$$

The parameters of this LSF are given in Table 2. The correlation coefficient is 0.949 and the χ² is 0.018 for 66 dM2 stars. The gradient of this correlation (−1.481) is very close to that of the correlation found in Paper XV (−1.53). In fact there are few differences between the two results: the correlation has only been shifted slightly in P/ sin i. Both the homoscedastic and the heteroscedastic fit give significant correlations with a statistical significance better than 99.9%. The heteroscedastic linear fit yields a somewhat slightly larger slope than the homoscedastic linear fit. This is because the errors on the dM2e stars are significantly smaller than those of the dM2 stars, and therefore the weights of the dM2e stars in the correlations are larger than those of dM2 stars in the heteroscedastic fit. As a consequence, the heteroscedastic fit goes through the
<table><caption>**Table 4**
Equivalent Width (EW, in Units of Å) of the Ca II Resonance Doublet and Hα Line for Our Sample of dM3 Stars (see HM)</caption><thead><tr><th rowspan="2">Star</th><th rowspan="2">No. of meas.</th><th colspan="3">HARPS</th><th colspan="3">SOPHIE</th><th rowspan="2">K/H ratio</th><th rowspan="2">⟨Ca ii⟩</th></tr><tr><th>Ca II H EW (A)</th><th>Ca II K EW (A)</th><th>Hα EW (A)</th><th>Ca II H EW (A)</th><th>Ca II K EW (A)</th><th>Hα EW (A)</th></tr></thead><tbody><tr><td>GJ 1046</td><td>6</td><td>-0.370 ± 0.01</td><td>-0.470 ± 0.01</td><td>0.329 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.27</td><td>-0.420</td></tr><tr><td>GJ 1050</td><td>9</td><td>-0.101 ± 0.01</td><td>-0.125 ± 0.01</td><td>0.261 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.24</td><td>-0.113</td></tr><tr><td>GJ 1097</td><td>7</td><td>-0.431 ± 0.01</td><td>-0.506 ± 0.01</td><td>0.306 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.17</td><td>-0.469</td></tr><tr><td>GJ 1125</td><td>8, 8</td><td>-0.273 ± 0.01</td><td>-0.365 ± 0.01</td><td>0.248 ± 0.020</td><td>-0.29 ± 0.10</td><td>-0.48 ± 0.10</td><td>0.235 ± 0.020</td><td>1.34</td><td>-0.319</td></tr><tr><td>GJ 1203</td><td>8</td><td>-0.248 ± 0.01</td><td>-0.317 ± 0.01</td><td>0.298 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.28</td><td>-0.283</td></tr><tr><td>GJ 1212AB</td><td>1</td><td>...</td><td>...</td><td>0.249 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1271</td><td>8</td><td>...</td><td>...</td><td>...</td><td>-0.61 ± 0.05</td><td>-0.91 ± 0.05</td><td>0.332 ± 0.020</td><td>1.49</td><td>-0.76</td></tr><tr><td>GJ 2121</td><td>1</td><td>...</td><td>...</td><td>0.286 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.33</td><td>-0.560</td></tr><tr><td>GJ 3139</td><td>2</td><td>-0.48 ± 0.03</td><td>-0.64 ± 0.03</td><td>0.365 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3160AB</td><td>1</td><td>...</td><td>...</td><td>0.276 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3189</td><td>9</td><td>-0.065 ± 0.05</td><td>-0.127 ± 0.05</td><td>0.224 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.95</td><td>-0.096</td></tr><tr><td>GJ 3279</td><td>10</td><td>-0.446 ± 0.01</td><td>-0.546 ± 0.03</td><td>0.362 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.22</td><td>-0.496</td></tr><tr><td>GJ 3293</td><td>19</td><td>-0.736 ± 0.01</td><td>-0.824 ± 0.01</td><td>0.339 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.12</td><td>-0.780</td></tr><tr><td>GJ 3404A</td><td>19</td><td>-0.303 ± 0.01</td><td>-0.336 ± 0.01</td><td>0.268 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-0.320</td></tr><tr><td>GJ 3412AB</td><td>1</td><td>...</td><td>...</td><td>...</td><td>-0.29 ± 0.10</td><td>-0.43 ± 0.20</td><td>0.255 ± 0.020</td><td>1.48</td><td>-0.36</td></tr><tr><td>GJ 3459</td><td>7</td><td>-0.428 ± 0.02</td><td>-0.523 ± 0.02</td><td>0.269 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.22</td><td>-0.476</td></tr><tr><td>GJ 3528</td><td>6</td><td>-0.693 ± 0.02</td><td>-0.843 ± 0.02</td><td>0.342 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.22</td><td>-0.768</td></tr><tr><td>GJ 3563</td><td>11</td><td>-0.320 ± 0.01</td><td>-0.388 ± 0.01</td><td>0.315 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.21</td><td>-0.354</td></tr><tr><td>GJ 3598</td><td>1</td><td>...</td><td>...</td><td>0.316 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3634</td><td>49</td><td>-0.542 ± 0.01</td><td>-0.625 ± 0.01</td><td>0.376 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.15</td><td>-0.584</td></tr><tr><td>GJ 3643</td><td>1</td><td>...</td><td>...</td><td>0.293 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3708A</td><td>8</td><td>-0.381 ± 0.02</td><td>-0.430 ± 0.02</td><td>0.275 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.13</td><td>-0.406</td></tr><tr><td>GJ 3846</td><td>6</td><td>-0.483 ± 0.05</td><td>-0.496 ± 0.05</td><td>0.260 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.03</td><td>-0.490</td></tr><tr><td>GJ 3892</td><td>8</td><td>-0.481 ± 0.01</td><td>-0.561 ± 0.01</td><td>0.334 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.17</td><td>-0.521</td></tr><tr><td>GJ 3916AB</td><td>3</td><td>-0.357 ± 0.05</td><td>-0.418 ± 0.05</td><td>0.327 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.17</td><td>-0.388</td></tr><tr><td>GJ 4004</td><td>2</td><td>...</td><td>...</td><td>0.220 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4231</td><td>4</td><td>-13.31 ± 0.5</td><td>-12.79 ± 0.5</td><td>-4.064 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>0.96</td><td>-13.05<sup>a</sup></td></tr><tr><td>GJ 4129</td><td>1</td><td>...</td><td>...</td><td>0.242 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 12</td><td>6, 2</td><td>-0.252 ± 0.1</td><td>-0.214 ± 0.1</td><td>0.255 ± 0.020</td><td>...</td><td>...</td><td>0.270 ± 0.020</td><td>0.85</td><td>-0.233</td></tr><tr><td>Gl 70</td><td>7, 7</td><td>-0.538 ± 0.01</td><td>-0.659 ± 0.01</td><td>0.282 ± 0.020</td><td>-0.73 ± 0.05</td><td>-0.92 ± 0.05</td><td>0.251 ± 0.020</td><td>1.22</td><td>-0.701</td></tr><tr><td>Gl 84</td><td>2</td><td>-0.749 ± 0.05</td><td>-0.926 ± 0.05</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.24</td><td>-0.838<sup>a</sup></td></tr><tr><td>Gl 109</td><td>28</td><td>...</td><td>...</td><td>...</td><td>-0.54 ± 0.01</td><td>-0.77 ± 0.01</td><td>0.225 ± 0.020</td><td>1.43</td><td>-0.655</td></tr><tr><td>Gl 145</td><td>6, 2</td><td>-0.757 ± 0.02</td><td>-0.853 ± 0.02</td><td>0.236 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.13</td><td>-0.805</td></tr><tr><td>Gl 163</td><td>56</td><td>-0.369 ± 0.01</td><td>-0.431 ± 0.01</td><td>0.304 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.17</td><td>-0.400</td></tr><tr><td>Gl 204.2</td><td>10</td><td>-0.532 ± 0.02</td><td>-0.586 ± 0.02</td><td>0.302 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.10</td><td>-0.559</td></tr><tr><td>Gl 207.1</td><td>1, 4</td><td>-9.266 ± 0.2</td><td>-10.01 ± 0.2</td><td>-2.99 ± 0.020</td><td>-16.1 ± 1</td><td>-24.5 ± 2</td><td>-3.22 ± 0.20</td><td>1.08</td><td>-9.638</td></tr><tr><td>Gl 226</td><td>9</td><td>...</td><td>...</td><td>...</td><td>-0.40 ± 0.05</td><td>-0.68 ± 0.05</td><td>0.344 ± 0.020</td><td>1.70</td><td>-0.54</td></tr><tr><td>Gl 238</td><td>2</td><td>-0.409 ± 0.1</td><td>-0.381 ± 0.1</td><td>0.349 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>0.93</td><td>-0.395</td></tr><tr><td>Gl 251</td><td>23</td><td>...</td><td>...</td><td>...</td><td>-0.25 ± 0.03</td><td>-0.38 ± 0.03</td><td>0.240 ± 0.020</td><td>1.52</td><td>-0.315</td></tr><tr><td>Gl 298</td><td>6</td><td>-0.498 ± 0.02</td><td>-0.619 ± 0.02</td><td>0.312 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.24</td><td>-0.559</td></tr><tr><td>Gl 352</td><td>1</td><td>-0.261 ± 0.05</td><td>-0.326 ± 0.05</td><td>0.304 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.25</td><td>-0.294</td></tr><tr><td>Gl 357</td><td>6</td><td>-0.186 ± 0.01</td><td>-0.217 ± 0.01</td><td>0.266 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.17</td><td>-0.202</td></tr><tr><td>Gl 358</td><td>28</td><td>-2.45 ± 0.05</td><td>-2.72 ± 0.05</td><td>-0.532 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.11</td><td>-2.585</td></tr><tr><td>Gl 377</td><td>8</td><td>-0.964 ± 0.01</td><td>-1.10 ± 0.01</td><td>0.319 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-1.032</td></tr><tr><td>Gl 386</td><td>6</td><td>-0.509 ± 0.01</td><td>-0.571 ± 0.01</td><td>0.323 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.12</td><td>-0.540</td></tr><tr><td>Gl 388</td><td>40</td><td>-7.90 ± 0.05</td><td>-8.61 ± 0.05</td><td>-2.48 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.09</td><td>-8.255</td></tr><tr><td>Gl 399</td><td>7</td><td>-0.443 ± 0.02</td><td>-0.529 ± 0.02</td><td>0.326 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.19</td><td>-0.486</td></tr><tr><td>Gl 408</td><td>?</td><td>...</td><td>...</td><td>...</td><td>-0.54 ± 0.02</td><td>-0.77 ± 0.02</td><td>0.314 ± 0.020</td><td>1.43</td><td>-0.655</td></tr><tr><td>Gl 422</td><td>27</td><td>-0.212 ± 0.01</td><td>-0.246 ± 0.01</td><td>0.286 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.16</td><td>-0.229</td></tr><tr><td>Gl 436</td><td>141, 19</td><td>-0.325 ± 0.01</td><td>-0.376 ± 0.01</td><td>0.300 ± 0.020</td><td>-0.27 ± 0.02</td><td>-0.45 ± 0.02</td><td>0.259 ± 0.020</td><td>1.16</td><td>-0.345</td></tr><tr><td>Gl 443</td><td>13</td><td>-0.898 ± 0.02</td><td>-1.093 ± 0.01</td><td>0.320 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.22</td><td>-0.996</td></tr><tr><td>Gl 452A</td><td>11</td><td>-0.483 ± 0.03</td><td>-0.624 ± 0.03</td><td>0.357 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.29</td><td>-0.554</td></tr><tr><td>Gl 452.1</td><td>11</td><td>-3.430 ± 0.05</td><td>-3.783 ± 0.05</td><td>-0.799 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.10</td><td>-3.607</td></tr><tr><td>Gl 463</td><td>10</td><td>...</td><td>...</td><td>...</td><td>-0.51 ± 0.02</td><td>-0.72 ± 0.02</td><td>0.341 ± 0.020</td><td>1.41</td><td>-0.615</td></tr><tr><td>Gl 479</td><td>48</td><td>-1.608 ± 0.03</td><td>-1.847 ± 0.05</td><td>0.098 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.15</td><td>-1.728</td></tr><tr><td>Gl 480</td><td>5, 21</td><td>-0.640 ± 0.05</td><td>-0.736 ± 0.05</td><td>0.314 ± 0.020</td><td>-0.75 ± 0.05</td><td>-1.17 ± 0.05</td><td>0.300 ± 0.020</td><td>1.15</td><td>-0.799</td></tr><tr><td>Gl 513</td><td>7</td><td>-0.365 ± 0.03</td><td>-0.425 ± 0.03</td><td>0.327 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.16</td><td>-0.395</td></tr><tr><td>Gl 581</td><td>258</td><td>-0.190 ± 0.01</td><td>-0.224 ± 0.01</td><td>0.193 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.18</td><td>-0.207</td></tr><tr><td>Gl 588</td><td>21</td><td>-0.592 ± 0.01</td><td>-0.704 ± 0.01</td><td>0.318 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.19</td><td>-0.648</td></tr><tr><td>Gl 595</td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.254 ± 0.020</td><td>...</td><td>...</td></tr><tr><td>Gl 617B</td><td>9</td><td>...</td><td>...</td><td>...</td><td>-0.84 ± 0.03</td><td>-1.30 ± 0.03</td><td>0.309 ± 0.020</td><td>1.55</td><td>-1.07</td></tr></tbody></table>
<table><caption>**Table 4**
(Continued)</caption><thead><tr><th rowspan="2">Star</th><th rowspan="2">No. of meas.</th><th colspan="3">HARPS</th><th colspan="3">SOPHIE</th><th rowspan="2">K/H ratio</th><th rowspan="2">⟨Ca ii⟩</th></tr><tr><th>Ca II H EW (A)</th><th>Ca II K EW (A)</th><th>H<sub>α</sub> EW (A)</th><th>Ca II H EW (A)</th><th>Ca II K EW (A)</th><th>H<sub>α</sub> EW (A)</th></tr></thead><tbody><tr><td>Gl 618A</td><td>19</td><td>-0.448 ± 0.01</td><td>-0.517 ± 0.01</td><td>0.312 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.15</td><td>-0.483</td></tr><tr><td>Gl 623</td><td>3</td><td>...</td><td>...</td><td>...</td><td>-0.27 ± 0.10</td><td>-0.36 ± 0.10</td><td>0.248 ± 0.020</td><td>1.33</td><td>-0.315</td></tr><tr><td>Gl 634</td><td>9</td><td>-0.318 ± 0.03</td><td>-0.377 ± 0.03</td><td>0.353 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.19</td><td>-0.348</td></tr><tr><td>Gl 644AB</td><td>1</td><td>-2.625 ± 0.05</td><td>-2.941 ± 0.05</td><td>-1.292 ± 0.20</td><td>...</td><td>...</td><td>...</td><td>1.12</td><td>-2.783</td></tr><tr><td>Gl 655</td><td>9</td><td>...</td><td>...</td><td>...</td><td>-0.45 ± 0.10</td><td>-0.80 ± 0.10</td><td>0.250 ± 0.020</td><td>1.78</td><td>-0.625</td></tr><tr><td>Gl 660A</td><td>1</td><td>-0.423 ± 0.1</td><td>-0.531 ± 0.1</td><td>0.262 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.26</td><td>-0.477</td></tr><tr><td>Gl 671</td><td>7</td><td>...</td><td>...</td><td>...</td><td>-0.34 ± 0.02</td><td>-0.42 ± 0.02</td><td>0.279 ± 0.020</td><td>1.24</td><td>-0.38</td></tr><tr><td>Gl 674</td><td>44</td><td>-1.153 ± 0.01</td><td>-1.335 ± 0.01</td><td>0.125 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.16</td><td>-1.244</td></tr><tr><td>Gl 687AB</td><td>16</td><td>...</td><td>...</td><td>...</td><td>-0.36 ± 0.01</td><td>-0.58 ± 0.02</td><td>0.292 ± 0.020</td><td>1.64</td><td>-0.47</td></tr><tr><td>Gl 693</td><td>7</td><td>-0.391 ± 0.03</td><td>-0.447 ± 0.03</td><td>0.271 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.14</td><td>-0.419</td></tr><tr><td>Gl 694</td><td>11</td><td>...</td><td>...</td><td>...</td><td>-0.58 ± 0.01</td><td>-0.90 ± 0.01</td><td>0.317 ± 0.020</td><td>1.55</td><td>-0.74</td></tr><tr><td>Gl 725A</td><td>19</td><td>...</td><td>...</td><td>...</td><td>-0.22 ± 0.01</td><td>-0.34 ± 0.01</td><td>0.233 ± 0.020</td><td>1.55</td><td>-0.28</td></tr><tr><td>Gl 735A</td><td>1, 2</td><td>-8.286 ± 0.3</td><td>-8.743 ± 0.3</td><td>-1.99 ± 0.20</td><td>...</td><td>...</td><td>-1.71 ± 0.20</td><td>1.06</td><td>-8.515</td></tr><tr><td>Gl 735B</td><td>1, 2</td><td>-5.360 ± 0.3</td><td>-5.623 ± 0.3</td><td>-1.99 ± 0.20</td><td>...</td><td>...</td><td>-1.71 ± 0.20</td><td>1.05</td><td>-5.492</td></tr><tr><td>Gl 739</td><td>2</td><td>-0.636 ± 0.1</td><td>-0.788 ± 0.1</td><td>0.292 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.24</td><td>-0.712</td></tr><tr><td>Gl 752A</td><td>13, 7</td><td>-0.784 ± 0.01</td><td>-0.910 ± 0.01</td><td>0.338 ± 0.020</td><td>-0.72 ± 0.02</td><td>-1.16 ± 0.02</td><td>0.375 ± 0.020</td><td>1.16</td><td>-0.828</td></tr><tr><td>Gl 781.1A</td><td>3</td><td>>-0.30</td><td>>-0.30</td><td>0.206 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>...</td><td>>-0.30</td></tr><tr><td>Gl 793</td><td>8</td><td>...</td><td>...</td><td>...</td><td>-1.12 ± 0.04</td><td>-1.78 ± 0.04</td><td>0.111 ± 0.020</td><td>1.59</td><td>-1.45</td></tr><tr><td>Gl 844</td><td>16</td><td>...</td><td>...</td><td>...</td><td>-1.32 ± 0.05</td><td>-2.08 ± 0.05</td><td>0.236 ± 0.020</td><td>1.58</td><td>-1.70</td></tr><tr><td>Gl 849</td><td>36</td><td>-0.580 ± 0.01</td><td>-0.694 ± 0.01</td><td>0.355 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.20</td><td>-0.637</td></tr><tr><td>Gl 877</td><td>41</td><td>-0.454 ± 0.01</td><td>-0.528 ± 0.01</td><td>0.308 ± 0.020</td><td>...</td><td>...</td><td>...</td><td>1.16</td><td>-0.491</td></tr><tr><td>Gl 896A</td><td>1</td><td>...</td><td>...</td><td>...</td><td>-8.98 ± 0.20</td><td>-12.5 ± 0.20</td><td>-3.80 ± 0.20</td><td>1.39</td><td>-10.74</td></tr></tbody></table>

**Notes.** Results were obtained using spectra from HARPS, SOPHIE, and FEROS.

ª FEROS spectrum.

(This table is available in machine-readable form.)

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Correlation between the mean EW of the Ca II resonance doublet and the Hα EW for stars with spectral sub-type dM3e. The solid line is the least square fit to the data.</figcaption></figure>

sub-sample of dM2e stars whereas the homoscedastic fit underestimates clearly the higher fluxes of the dM2e stars. Therefore the heteroscedastic fit provides globally a better evaluation of the slope for the dM2+dM2e stars.

We also performed linear LSF to the sub-samples of only the dM2 low-activity and the high-activity dM2e stars (see Table 1). The heteroscedastic linear LSF to the dM2 low-activity stars sub-sample yields:

$$F_{\text{Ca II}} = 2.68 \pm 0.77 \times 10^5 \times (P/\sin i)^{-0.709 \pm 0.14}. \quad (13)$$

Therefore, the gradient of the RAC for the low-activity dM2 stars is again significantly shallower than that of the combined sample of low+high-activity stars.

For the homoscedastic linear LSF to the dM2 low-activity stars, we obtain:

$$F_{\text{Ca II}} = 4.17 \pm 1.15 \times 10^5 \times (P/\sin i)^{-0.891 \pm 0.12}. \quad (14)$$

Both the heteroscedastic and the homoscedastic fits show that the gradient of the RAC for the low-activity dM2 stars is again (as for dK4 and dK6 stars) significantly shallower than those of the combined sample of low+high-activity stars. The χ² for these fits are 0.046 and 0.013, respectively. These fits are statistically significant at a confidence level better than 99.9%. We note that the slope of the heteroscedastic fit is slightly shallower than that of the homoscedastic fit and is also shallower than the value found for the heteroscedastic fit for the dK6 sample. However, if we take into account errors (see Table 1), there are in fact no significant differences between these values.

The heteroscedastic linear LSF to the dM2e high-activity star sub-sample yields:

$$F_{\text{Ca II}} = 3.28 \pm 0.54 \times 10^6 \times (P/\sin i)^{-1.793 \pm 0.14}. \quad (15)$$

Therefore, the slope of the RAC for the high-activity dM2e stars is again steeper than that of the combined sample of low
<figure><img src="image_9.png" /><figcaption>Figure 7. Correlation between the logarithm of the mean surface fluxes of the Ca II resonance doublet and log(P/ sin i) for stars with spectral sub-type dM3. No corrections for metallicity were applied here. One can see that (as for the case of dM2 stars) there is a large scatter among the data when it is not corrected for metallicity effects.</figcaption></figure>

+high-activity stars. We note that this slope gets closer to that of the combined sample as we move from dK4 to dM2. The differences between these two slopes were larger for dK4 and dK6 stars.

The parameter R δ is 23.45 for the high-activity star sub-sample and 34.77 for the low-activity star sub-sample. Therefore, the parameter δ = (error) / √n, which is a normalized estimate of the mean error on the measurements, is much smaller than the period domains of the RACs in both cases. Hence, these LSFs should be relatively well established, which corresponds to the high statistical significances we obtain (Table 1). However, we consider again these results as still preliminary because the domains of the RACs are relatively small compared to the typical uncertainties on individual measures.

3.5. The RAC in M3 Dwarfs

In HM, we have already listed the results for the rotational parameters v sin i and P / sin i of our dM3 stellar sample. However, we re-computed the stellar parameters for our sample of dM3 stars in Paper I according to the new results of Mann et al. (2015). The revised values of the parameters for the dM3 sample can be found in Paper I. In the present paper, we report on chromospheric line data for dM3 stars: these data were not a part of HM. In Table 4, we list the EW we have obtained for

the Ca II resonance doublet and for the H α line for our dM3 stars.

In order to supplement our sample of v sin i and Ca II measurements, and in order to have an unbiased sample of measurements, we searched the literature for additional v sin i and Ca II measurements. We found several additional stars with v sin i measurements (see Paper I), but we found only a few measure- ments of the Ca II lines for these stars (Stauffer & Hartmann 1986; Rutten 1987; Giampapa et al. 1989; Rutten et al. 1989). Instead, we found several measures of the H α EW (Soderblom et al. 1991; Hawley et al. 1996; Kamper et al. 1997; Christian & Mathioudakis 2002; Gizis et al. 2002; Pace 2013). Fortunately, there is a relatively tight correlation between the mean Ca II line mean EW and H α EW in dM3 and dM4 stars. Therefore, we decided to infer the Ca II line mean EW from the measures of the H α EW. We show in Figure 6 the relationship between the Ca II line mean EW and the H α EW for our measurements of dM3 stars (see Table 4). As one can see, the correlation between these two parameters is very good for all stars except Gl 644AB, which lies slightly below. The homoscedastic linear least square fit to this data except Gl 644AB gives:

$$
\mathrm{EW}_{\mathrm{Ca}\,\mathrm{II}} = 2.90 \pm 0.05 \times \mathrm{EW}_{\mathrm{H}\alpha} + 1.14 \pm 0.125. \quad (16)
$$

The parameters of this fit are given in Table 1. The fit is very good with a correlation coefficient of 0.9994. We give the Ca II line mean EW computed from the H α EW in Table 4.
<table><caption>**Table 5**
The Values of $v \sin i$, $P/\sin i$ and [M/H] from Paper I for Our Targets in Our List of dM3 Stars</caption><thead><tr><th>Star</th><th><i>v</i> sin<i>i</i><br>(km s<sup>-1</sup>)</th><th><i>P</i>/sin<i>i</i><br>(days)</th><th>[M/H]<br>(dex)</th><th>Ca II EW<br>(Å)</th><th><i>F</i><sub>HK</sub></th><th><i>F</i><sub>HK</sub><sup>a</sup><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th></tr></thead><tbody><tr><td>GJ 1046</td><td>2.63</td><td>7.41–1.1+1.2</td><td>-0.078</td><td>-0.420</td><td>1.276</td><td>1.527</td></tr><tr><td>GJ 1050</td><td>0.59</td><td>34.4–18+310</td><td>-0.035</td><td>-0.113</td><td>0.324</td><td>0.351</td></tr><tr><td>GJ 1054B</td><td>32.13</td><td>0.45–0.07+0.08</td><td>-0.323</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1097</td><td>&lt;0.5</td><td>&gt;41.91</td><td>-0.033</td><td>-0.469</td><td>1.178</td><td>1.271</td></tr><tr><td>GJ 1125</td><td>0.99</td><td>10.3–2.9+8.3</td><td>-0.691</td><td>-0.319</td><td>1.454</td><td>7.138</td></tr><tr><td>GJ 1203</td><td>0.82</td><td>23.5–7.1+18</td><td>-0.087</td><td>-0.283</td><td>0.772</td><td>0.943</td></tr><tr><td>GJ 1212A</td><td>0.69</td><td>25.3–9.6+57</td><td>-0.170</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1212B</td><td>0.90</td><td>19.3–7.1+13</td><td>-0.170</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 1271</td><td>1.39</td><td>20.0–5.1+7.7</td><td>+0.127</td><td>-0.76</td><td>2.265</td><td>1.691</td></tr><tr><td>GJ 2121</td><td>2.06</td><td>12.5–3.1+4.2</td><td>+0.059</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3139</td><td>0.90</td><td>29.4–7.8+14</td><td>+0.084</td><td>-0.560</td><td>1.577</td><td>1.300</td></tr><tr><td>GJ 3160A</td><td>0.95</td><td>22.0–6.2+9.9</td><td>-0.033</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3160B</td><td>1.06</td><td>19.8–5.3+7.4</td><td>-0.033</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3189</td><td>1.42</td><td>6.43–1.8+1.6</td><td>-0.801</td><td>-0.096</td><td>0.278</td><td>1.758</td></tr><tr><td>GJ 3279</td><td>0.85</td><td>23.7–13+26</td><td>-0.045</td><td>-0.496</td><td>1.417</td><td>1.572</td></tr><tr><td>GJ 3293</td><td>0.85</td><td>25.0–12+25</td><td>-0.031</td><td>-0.780</td><td>2.287</td><td>2.456</td></tr><tr><td>GJ 3404A</td><td>0.90</td><td>26.5–9.8+17</td><td>-0.005</td><td>-0.320</td><td>0.782</td><td>0.791</td></tr><tr><td>GJ 3412A</td><td>...</td><td>...</td><td>-0.280</td><td>-0.36</td><td>0.914</td><td>1.742</td></tr><tr><td>GJ 3412B</td><td>...</td><td>...</td><td>-0.280</td><td>-0.36</td><td>0.914</td><td>1.742</td></tr><tr><td>GJ 3459</td><td>0.69</td><td>23.1–7.7+48</td><td>-0.238</td><td>-0.476</td><td>1.215</td><td>2.102</td></tr><tr><td>GJ 3528</td><td>0.90</td><td>28.4–8.1+15</td><td>+0.057</td><td>-0.768</td><td>2.283</td><td>2.002</td></tr><tr><td>GJ 3563</td><td>0.85</td><td>22.2–6.3+14</td><td>-0.104</td><td>-0.354</td><td>1.033</td><td>1.313</td></tr><tr><td>GJ 3598</td><td>0.69</td><td>30.0–15+80</td><td>-0.035</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3634</td><td>0.85</td><td>24.6–11+22</td><td>-0.033</td><td>-0.584</td><td>1.780</td><td>1.921</td></tr><tr><td>GJ 3643</td><td>1.00</td><td>20.4–5.7+9.0</td><td>-0.036</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 3708A</td><td>0.50</td><td>37.0–17+∞</td><td>-0.122</td><td>-0.406</td><td>1.064</td><td>1.409</td></tr><tr><td>GJ 3846</td><td>0.82</td><td>20.5–6.4+16</td><td>-0.189</td><td>-0.490</td><td>1.230</td><td>1.901</td></tr><tr><td>GJ 3892</td><td>0.82</td><td>27.4–7.6+20</td><td>-0.026</td><td>-0.521</td><td>1.453</td><td>1.543</td></tr><tr><td>GJ 3916A</td><td>1.24</td><td>14.1–3.1+3.8</td><td>-0.168</td><td>-0.388</td><td>1.143</td><td>1.683</td></tr><tr><td>GJ 3916B</td><td>3.51</td><td>4.97–0.7+0.7</td><td>-0.168</td><td>-0.388</td><td>1.143</td><td>1.683</td></tr><tr><td>GJ 4004</td><td>0.59</td><td>28.6–18+290</td><td>-0.186</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4129</td><td>0.50</td><td>37.9–18+∞</td><td>-0.104</td><td>...</td><td>...</td><td>...</td></tr><tr><td>GJ 4231</td><td>80.00</td><td>0.24–0.08+0.09</td><td>-0.080</td><td>-13.05</td><td>33.48</td><td>40.25</td></tr><tr><td>GJ 4282AB</td><td>15.00</td><td>1.25–0.48+0.54</td><td>-0.115</td><td>-9.115</td><td>22.82</td><td>29.74</td></tr><tr><td>Gl 12</td><td>0.95</td><td>12.8–4.4+6.9</td><td>-0.508</td><td>-0.233</td><td>0.564</td><td>1.817</td></tr><tr><td>Gl 48</td><td>2.45</td><td>10.7–2.9+4.9</td><td>+0.071</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 70</td><td>1.01</td><td>20.8–5.4+10</td><td>-0.033</td><td>-0.701</td><td>2.233</td><td>2.409</td></tr><tr><td>Gl 84</td><td>...</td><td>...</td><td>-0.094</td><td>-0.838</td><td>2.555</td><td>...</td></tr><tr><td>Gl 109</td><td>1.34</td><td>13.9–3.2+4.8</td><td>-0.113</td><td>-0.655</td><td>1.716</td><td>2.226</td></tr><tr><td>Gl 119B</td><td>4.00</td><td>7.72–2.2+3.0</td><td>+0.217</td><td>-1.840</td><td>5.358</td><td>3.251</td></tr><tr><td>Gl 140C</td><td>6.70</td><td>2.99–1.0+1.3</td><td>-0.052</td><td>-1.36</td><td>4.173</td><td>4.704</td></tr><tr><td>Gl 145</td><td>0.85</td><td>19.6–5.1+12</td><td>-0.194</td><td>-0.805</td><td>2.246</td><td>3.511</td></tr><tr><td>Gl 163</td><td>0.85</td><td>26.0–6.8+16</td><td>-0.027</td><td>-0.400</td><td>0.966</td><td>1.028</td></tr><tr><td>Gl 204.2</td><td>0.69</td><td>36.1–14+83</td><td>+0.033</td><td>-0.559</td><td>1.552</td><td>1.438</td></tr><tr><td>Gl 207.1</td><td>9.52</td><td>2.51–0.4+0.4</td><td>+0.006</td><td>-9.638</td><td>27.08</td><td>26.71</td></tr><tr><td>Gl 226</td><td>1.12</td><td>19.6–5.1+9.0</td><td>-0.028</td><td>-0.54</td><td>1.583</td><td>1.688</td></tr><tr><td>Gl 238</td><td>0.85</td><td>28.0–7.3+17</td><td>+0.005</td><td>-0.395</td><td>1.118</td><td>1.105</td></tr><tr><td>Gl 251AB</td><td>1.14</td><td>16.9–4.4+7.5</td><td>-0.388</td><td>-0.315</td><td>0.746</td><td>1.823</td></tr><tr><td>Gl 277A</td><td>8.00</td><td>3.69–0.86+1.1</td><td>+0.175</td><td>-4.958</td><td>8.13</td><td>8.682</td></tr><tr><td>Gl 298</td><td>1.00</td><td>24.6–5.7+9.2</td><td>+0.023</td><td>-0.559</td><td>1.407</td><td>1.334</td></tr><tr><td>Gl 352</td><td>0.90</td><td>7.6–1.0+1.0</td><td>+0.178</td><td>-0.294</td><td>0.824</td><td>0.547</td></tr><tr><td>Gl 357</td><td>1.21</td><td>18.7–4.3+8.2</td><td>-0.189</td><td>-0.202</td><td>0.634</td><td>0.980</td></tr><tr><td>Gl 358</td><td>1.10</td><td>17.7–3.3+4.2</td><td>-0.030</td><td>-2.585</td><td>7.018</td><td>7.520</td></tr><tr><td>Gl 377</td><td>...</td><td>24.1–5.4+7.2</td><td>+0.084</td><td>-1.032</td><td>2.760</td><td>2.275</td></tr><tr><td>Gl 386</td><td>0.69</td><td>37.8–12+79</td><td>+0.068</td><td>-0.540</td><td>1.532</td><td>1.310</td></tr><tr><td>Gl 388</td><td>2.63</td><td>8.38–1.1+1.2</td><td>-0.028</td><td>-8.255</td><td>19.77</td><td>21.09</td></tr><tr><td>Gl 399</td><td>0.59</td><td>43.6–17+330</td><td>+0.064</td><td>-0.486</td><td>1.524</td><td>1.315</td></tr><tr><td>Gl 408</td><td>1.12</td><td>18.1–4.6+8.1</td><td>-0.045</td><td>-0.655</td><td>1.832</td><td>2.032</td></tr><tr><td>Gl 422</td><td>0.90</td><td>23.1–5.8+11</td><td>-0.033</td><td>-0.229</td><td>0.580</td><td>0.626</td></tr><tr><td>Gl 436</td><td>0.86</td><td>27.0–9.0+26</td><td>-0.019</td><td>-0.345</td><td>0.944</td><td>0.986</td></tr><tr><td>Gl 443</td><td>0.90</td><td>31.8–8.7+16</td><td>+0.150</td><td>-0.996</td><td>2.684</td><td>1.900</td></tr><tr><td>Gl 452A</td><td>0.95</td><td>27.1–7.3+12</td><td>+0.062</td><td>-0.554</td><td>1.485</td><td>1.287</td></tr><tr><td>Gl 452.1</td><td>1.60</td><td>6.66–1.9+2.2</td><td>-0.651</td><td>-3.607</td><td>8.541</td><td>38.24</td></tr></tbody></table>
<table><caption>**Table 5**
(Continued)</caption><thead><tr><th>Star</th><th>v sin i<br>(km s<sup>-1</sup>)</th><th>P/ sin i<br>(days)</th><th>[M/H]<br>(dex)</th><th>Ca II EW<br>(Å)</th><th>F<sub>HK</sub></th><th>F<sub>HK</sub><sup>a</sup><br>(10<sup>4</sup> erg s<sup>-1</sup> cm<sup>-2</sup>)</th></tr></thead><tbody><tr><td>Gl 463</td><td>1.17</td><td>23.1–6.4+10</td><td>+0.101</td><td>-0.615</td><td>1.720</td><td>1.363</td></tr><tr><td>Gl 479</td><td>1.06</td><td>21.1–4.4+6.3</td><td>-0.026</td><td>-1.728</td><td>4.679</td><td>4.968</td></tr><tr><td>Gl 480</td><td>0.73</td><td>34.0–12+53</td><td>+0.033</td><td>-0.799</td><td>1.897</td><td>1.758</td></tr><tr><td>Gl 513</td><td>0.77</td><td>29.0–11+39</td><td>-0.027</td><td>-0.395</td><td>0.965</td><td>1.027</td></tr><tr><td>Gl 581</td><td>...</td><td>...</td><td>-0.215</td><td>-0.207</td><td>0.508</td><td>0.833</td></tr><tr><td>Gl 588</td><td>0.77</td><td>31.8–8.3+35</td><td>+0.022</td><td>-0.648</td><td>1.847</td><td>1.756</td></tr><tr><td>Gl 595AB</td><td>1.62</td><td>9.25–2.1+2.8</td><td>-0.293</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 617B</td><td>1.12</td><td>22.6–6.0+11</td><td>+0.045</td><td>-1.07</td><td>2.803</td><td>2.527</td></tr><tr><td>Gl 618A</td><td>0.82</td><td>22.8–6.9+18</td><td>-0.115</td><td>-0.483</td><td>1.278</td><td>1.665</td></tr><tr><td>Gl 623</td><td>0.94</td><td>22.3–6.4+16</td><td>-0.034</td><td>-0.315</td><td>0.937</td><td>1.013</td></tr><tr><td>Gl 634</td><td>1.00</td><td>17.1–5.0+7.8</td><td>-0.182</td><td>-0.348</td><td>0.912</td><td>1.387</td></tr><tr><td>Gl 644A</td><td>1.79</td><td>12.2–2.0+2.2</td><td>-0.029</td><td>-2.783</td><td>7.366</td><td>7.875</td></tr><tr><td>Gl 644B</td><td>2.78</td><td>7.88–1.1+1.1</td><td>-0.029</td><td>-2.783</td><td>7.366</td><td>7.875</td></tr><tr><td>Gl 655</td><td>&lt;0.5</td><td>&gt;39.88</td><td>-0.057</td><td>-0.625</td><td>1.637</td><td>1.867</td></tr><tr><td>Gl 660A</td><td>0.82</td><td>22.6–8.3+20</td><td>-0.120</td><td>-0.477</td><td>1.130</td><td>1.490</td></tr><tr><td>Gl 671</td><td>0.94</td><td>21.2–6.3+15</td><td>-0.061</td><td>-0.38</td><td>1.083</td><td>1.246</td></tr><tr><td>Gl 674</td><td>0.90</td><td>20.0–4.5+8.6</td><td>-0.143</td><td>-1.244</td><td>3.521</td><td>4.894</td></tr><tr><td>Gl 687AB</td><td>0.73</td><td>30.3–11+130</td><td>-0.028</td><td>-0.47</td><td>1.151</td><td>1.228</td></tr><tr><td>Gl 693</td><td>0.82</td><td>16.8–4.3+12</td><td>-0.372</td><td>-0.419</td><td>1.024</td><td>2.412</td></tr><tr><td>Gl 694</td><td>1.29</td><td>17.7–4.2+6.2</td><td>-0.024</td><td>-0.74</td><td>2.155</td><td>2.277</td></tr><tr><td>Gl 725AB</td><td>0.85</td><td>22.5–7.0+18.4</td><td>-0.372</td><td>-0.28</td><td>0.714</td><td>1.682</td></tr><tr><td>Gl 735A</td><td>1.86</td><td>13.0–2.0+2.4</td><td>+0.012</td><td>-8.515</td><td>22.60</td><td>21.98</td></tr><tr><td>Gl 735B</td><td>1.83</td><td>13.2–2.1+3.1</td><td>+0.012</td><td>-5.492</td><td>14.57</td><td>14.17</td></tr><tr><td>Gl 739</td><td>3.19</td><td>7.59–1.0+1.0</td><td>+0.012</td><td>-0.712</td><td>2.122</td><td>2.064</td></tr><tr><td>Gl 752A</td><td>1.19</td><td>21.2–4.3+6.7</td><td>+0.045</td><td>-0.828</td><td>2.549</td><td>2.298</td></tr><tr><td>Gl 781.1A</td><td>9.20</td><td>1.93–0.50+0.62</td><td>-0.157</td><td>...</td><td>...</td><td>...</td></tr><tr><td>Gl 793</td><td>1.26</td><td>16.2–4.0+5.9</td><td>-0.036</td><td>-1.45</td><td>3.720</td><td>4.042</td></tr><tr><td>Gl 844</td><td>3.46</td><td>9.92–1.4+1.4</td><td>+0.324</td><td>-1.70</td><td>5.216</td><td>2.474</td></tr><tr><td>Gl 849</td><td>0.85</td><td>30.8–7.8+18</td><td>+0.076</td><td>-0.637</td><td>1.539</td><td>1.292</td></tr><tr><td>Gl 856AB</td><td>11.00</td><td>2.36–0.63+0.76</td><td>+0.067</td><td>-14.42</td><td>36.11</td><td>30.95</td></tr><tr><td>Gl 875.1</td><td>11.00</td><td>2.08–0.44+0.52</td><td>-0.024</td><td>-9.62</td><td>24.09</td><td>25.46</td></tr><tr><td>Gl 877</td><td>0.95</td><td>23.5–5.0+8.2</td><td>-0.026</td><td>-0.491</td><td>1.353</td><td>1.436</td></tr><tr><td>Gl 896A</td><td>14.03</td><td>1.41–0.14+0.14</td><td>-0.064</td><td>-10.74</td><td>25.43</td><td>29.47</td></tr><tr><td>Gl 897AB</td><td>7.10</td><td>3.18–0.97+1.29</td><td>-0.025</td><td>-7.250</td><td>21.06</td><td>22.31</td></tr></tbody></table>

Notes. We also give the Ca II EWs, surface fluxes $F_{HK}$, and the surface fluxes corrected for the metallicity effects.

a Surface flux corrected for metallicity effects.

(This table is available in machine-readable form.)

We show the RAC for dM3 stars for the raw data in Figure 7. In this figure, one can see that the scatter is very large among both dM3 and dM3e stars. The same kind of scatter was observed among the raw data of dM2 stars (Figure 4). Considering the good correction we had for the metallicity in the previous section for dM2 stars, we decided to compile all the metallicities published for our initial selection list of 381 dM3 stars, and try to obtain a metallicity–radius correlation for these stars, similarly to the dM2 stars in Paper VII. We found metallicities from the literature for 147 dM3 stars. This data and the radius–[M/H] relationship were reported in Paper I. Data for M3 stars corrected for metallicity can be found in Table 5 and Figure 8.

We show in Figure 8 the RAC corrected for metallicity effects for dM3 stars. The scatter is reduced compared to the raw data in Figure 7, but there remains a significant scatter among both dM3 and dM3e stars. The corrected correlation is not as good as for dM2 stars. We believe this is due partly to the poorer statistics we have on the Ca II line EWs. Indeed, for dM3 stars, we have very few measurements of the Ca II lines EWs (Table 4) whereas, for dM2 stars, we had several measurements for almost all stars (see Houdebine et al. 2012,

Paper XIX). We also observe a larger scatter in the radius–[M/H] relationship for dM3 stars (see Paper I) compared to dM2 stars. This may also lead to poorer corrections of the metallicity effects on the RAC.

Among our sample of dM3 stars, we observe a sub-sample of relatively fast rotating, low-activity stars. Again, as in the case of dM2 stars, we believe that these stars are unresolved spectroscopic binaries. We also found three stars with possibly low v sin i (see Figure 8). In the corrected RAC, most of the subdwarfs now follow the same correlation as normal dwarfs, indicating that our metallicity corrections are reasonably correct.

We find that for periods above 7 days or so, the gradient between the dM3 stars (open circles, the slowest rotators) and the dM3e stars (filled circles, the fastest rotators) is very steep. Because we have only a few dM3e stars compared to the larger group of dM3 stars, we do not obtain a good linear LSF to both the dM3 and dM3e sub-samples. This is due to the fact that the linear LSF to the dM3 sub-sample gives a shallower gradient (–0.90, Table 1) compared to the dM3+dM3e sample. Therefore, we gave higher weights to our dM3e data points: we find that a weight of 7 to dM3e stars and 1 to dM3
<figure><img src="image_10.png" /><figcaption>Figure 8. Correlation between the logarithm of the mean surface fluxes of the Ca II resonance doublet corrected for metallicity effects and log(P/ sin i) for stars with spectral sub-type dM3 (see Section 3.5). Note the possible feature of “saturation” at P/ sin i < 6 days. For “unsaturated” stars, i.e., those with P/ sin i > 7 days, the solid straight line is the heteroscedastic linear LSF to the combined data set of low+high-activity stars. The dotted-dashed curves represent the heteroscedastic linear LSFs to the low-activity star and high-activity star sub-samples, respectively (see the text). In the center-right hand of the figure, we show estimates of the uncertainties on the Ca II mean surface fluxes due to the corrections for metallicity effects, for dwarfs and subdwarfs.</figcaption></figure>

stars gives a good fit to both data sub-samples. For P/sin i ≥ 6.7 days, we obtain the following heteroscedastic linear LSF for our dM3+dM3e sample (Table 1):

$$F_{\text{Ca II}} = 1.169 \pm 0.476 \times 10^7 \times (P/\sin i)^{-2.020 \pm 0.11}. \quad (17)$$

The gradient in Equation (17) (−2.02 ± 0.11) is significantly steeper than the gradients that we determined for our dK4 and dK6 samples (−0.814 ± 0.059 and −1.047 ± 0.042, respectively; see Figures 2 and 3). But interestingly, the mean gradient in Figure 8 is intermediate between that of dM2 stars (−1.575 ± 0.058) and that of dM4 stars (−2.56 ± 0.19; Table 1). Once again, this is an encouraging sign that the physical parameters we have derived for dM3 stars are plausible when compared with stars that are slightly hotter and slightly cooler. We plot the linear LSF for stars with periods P/ sin i > 7 days (we refer to these as unsaturated stars; see below) in Figure 8 (solid line).

For the homoscedastic fit, we obtain the following linear LSF for our dM3+dM3e sample (Table 1):

$$F_{\text{Ca II}} = 1.52 \pm 1.00 \times 10^7 \times (P/\sin i)^{-2.075 \pm 0.19}. \quad (18)$$

The slopes of both the heteroscedastic and the homoscedastic linear fits are comparable. Note that the uncertainty on the slope is smaller for the heteroscedastic fit than for

the homoscedastic fit. Both fits confirm that the slope of the RAC is getting steeper as we move to later spectral types.

As we saw in Section 1.4, one would expect the RAC to flatten out for rotation periods shorter than 2–10 days. We have several objects in our M3 sample that have shorter rotation periods. We note that the RAC flattens out (“saturates”) for the fastest rotating dM3e stars for P/ sin i < 6 days. In this period range, we find seven fast rotators with about the same surface flux (~2.97 × 10⁵ erg s⁻¹ cm⁻²), which corresponds to the maximum observed in the unsaturated portion of the RAC. Although, in the present data set, we do not have enough data to clearly confirm the presence of a saturation plateau, one should expect saturation to occur among some of our sample stars according to previous investigations. We propose that saturation in our present dM3 sample occurs at about ~6 days, which agrees with the expectations of saturation occurring in the range of 2–10 days. In Figure 8, we therefore represent the fits of the RAC in the “unsaturated” portion, and draw a flat plateau in the expected “saturated” portion of the RAC.

The fact that certain stars in our samples probably lie in a saturated regime, while others are found to lie in an unsaturated regime, has a bearing on a methodological point that was made in Section 1.1: a study of stellar dynamos is best done (we believe) by focusing on stars in an unsaturated regime.
<table><caption>**Table 6**
Equivalent Width (EW, in Units of Å) of the Ca II Resonance Doublet and Hα Line for Our Sample of dM4 Stars</caption><thead><tr><th rowspan="2">Star</th><th rowspan="2">No. meas.</th><th colspan="3">FEROS</th><th rowspan="2">Mean Ca II EW (Å)</th><th rowspan="2">S<sub>HK</sub></th><th rowspan="2">Mean Ca II EW<sup>a</sup> (Å)</th><th rowspan="2">H<sub>α</sub> EW (Å)</th><th rowspan="2">Mean Ca II EW (Å)</th></tr><tr><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>H<sub>α</sub> EW (Å)</th></tr></thead><tbody><tr><td>GJ 1001A</td><td>1</td><td>-0.20 ± 0.04</td><td>-0.09 ± 0.07</td><td>0.215 ± 0.02</td><td>-0.235(2)</td><td>...</td><td>...</td><td>0.218(15)</td><td>-0.235(2)</td></tr><tr><td>GJ 1005</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.224</td><td>...</td></tr><tr><td>GJ 1006A</td><td>2</td><td>-4.09 ± 0.04</td><td>-4.72 ± 0.04</td><td>-3.14 ± 0.20</td><td>-4.405(2)</td><td>...</td><td>...</td><td>-3.14(2)</td><td>-4.405(2)</td></tr><tr><td>GJ 1006B</td><td>2</td><td>-7.05 ± 0.05</td><td>-8.55 ± 0.05</td><td>-2.98 ± 0.20</td><td>-7.80(2)</td><td>...</td><td>...</td><td>-2.98(2)</td><td>-7.80(2)</td></tr><tr><td>GJ 1065</td><td>3</td><td>-0.26 ± 0.08</td><td>-0.24 ± 0.08</td><td>0.167 ± 0.02</td><td>-0.25(3)</td><td>...</td><td>...</td><td>0.189(4)</td><td>-0.25(3)</td></tr><tr><td>GJ 1105</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.62(1)</td><td>...</td><td>...</td><td>0.203(3)</td><td>-0.62(1)</td></tr><tr><td>GJ 1129</td><td>3</td><td>-0.302 ± 0.03</td><td>-0.353 ± 0.03</td><td>0.190 ± 0.02</td><td>-0.328(3)</td><td>...</td><td>...</td><td>0.178(4)</td><td>-0.328(3)</td></tr><tr><td>GJ 1134</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.095(1)</td><td>...</td><td>...</td><td>0.181(3)</td><td>...</td></tr><tr><td>GJ 1138</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.180(4)</td><td>...</td></tr><tr><td>GJ 1207</td><td>8</td><td>-6.529 ± 0.08</td><td>-7.801 ± 0.08</td><td>-2.55 ± 0.20</td><td>-7.098(11)</td><td>...</td><td>...</td><td>-2.830(11)</td><td>-7.098(11)</td></tr><tr><td>GJ 1254</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.133</td><td>...</td></tr><tr><td>GJ 1289</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.680(1)</td><td>...</td><td>...</td><td>-0.432(18)</td><td>-1.680(1)</td></tr><tr><td>GJ 2036B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-14.49(2)</td><td>-8.135(2)</td><td>-14.49(2)</td></tr><tr><td>GJ 2069A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-16.91(1)</td><td>...</td><td>...</td><td>-4.548(5)</td><td>-16.91(1)</td></tr><tr><td>GJ 3149B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-5.647(1)</td><td>-3.50</td><td>-5.647(1)</td></tr><tr><td>GJ 3283B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-6.372(1)</td><td>-3.88</td><td>-6.372(1)</td></tr><tr><td>GJ 3322</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-10.58(2)</td><td>-6.082(2)</td><td>-10.58(2)</td></tr><tr><td>GJ 3522</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-7.10(1)</td><td>...</td><td>...</td><td>...</td><td>-7.10(1)</td></tr><tr><td>GJ 3631</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-19.29(2)</td><td>-10.65(2)</td><td>-19.29(2)</td></tr><tr><td>GJ 3666</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.21(1)</td><td>...</td><td>...</td><td>...</td><td>-0.21(1)</td></tr><tr><td>GJ 3707</td><td>2</td><td>-0.287 ± 0.08</td><td>-0.365 ± 0.08</td><td>0.200 ± 0.02</td><td>-0.341(9)</td><td>...</td><td>...</td><td>0.226(9)</td><td>-0.341(9)</td></tr><tr><td>GJ 3789</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-19.31(3)</td><td>-10.66(3)</td><td>-19.31(3)</td></tr><tr><td>GJ 3801</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.212</td><td>...</td></tr><tr><td>GJ 3804</td><td>4</td><td>-0.251 ± 0.05</td><td>-0.366 ± 0.05</td><td>0.193 ± 0.02</td><td>-0.302(9)</td><td>0.648(4)</td><td>-0.291(4)</td><td>0.211(5)</td><td>-0.299(13)</td></tr><tr><td>GJ 3873</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.465(3)</td><td>...</td><td>...</td><td>0.217(4)</td><td>-0.465(3)</td></tr><tr><td>GJ 3900</td><td>1</td><td>-0.401 ± 0.05</td><td>-0.504 ± 0.05</td><td>0.221 ± 0.02</td><td>-0.453(1)</td><td>...</td><td>...</td><td>0.240(3)</td><td>-0.453(1)</td></tr><tr><td>GJ 4020B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-12.94(1)</td><td>-7.32(1)</td><td>-12.94(1)</td></tr><tr><td>GJ 4063</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.380(9)</td><td>1.235(14)</td><td>-0.631(14)</td><td>0.232(2)</td><td>-0.533(23)</td></tr><tr><td>GJ 4333</td><td>2</td><td>-0.495 ± 0.08</td><td>-0.572 ± 0.08</td><td>0.179 ± 0.02</td><td>-0.587(15)</td><td>1.060(4)</td><td>-0.525(4)</td><td>0.180(5)</td><td>-0.574(19)</td></tr><tr><td>GJ 4338B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-10.95(2)</td><td>-6.277(2)</td><td>-10.95(2)</td></tr><tr><td>GJ 4378A</td><td>2</td><td>-2.674 ± 0.1</td><td>-3.004 ± 0.1</td><td>-1.486 ± 0.20</td><td>-2.839(2)</td><td>...</td><td>...</td><td>-1.808(3)</td><td>-2.839(2)</td></tr><tr><td>GJ 4378B</td><td>2</td><td>-7.509 ± 0.1</td><td>-9.684 ± 0.1</td><td>-4.434 ± 0.20</td><td>-8.597(2)</td><td>...</td><td>...</td><td>-4.434(2)</td><td>-8.597(2)</td></tr><tr><td>Gl 15B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.964(14)</td><td>1.162(18)</td><td>-0.888(18)</td><td>0.216(4)</td><td>-0.921(32)</td></tr><tr><td>Gl 46</td><td>2</td><td>-0.208 ± 0.02</td><td>-0.321 ± 0.02</td><td>0.231 ± 0.02</td><td>-0.295(4)</td><td>...</td><td>...</td><td>0.241(4)</td><td>-0.295(4)</td></tr><tr><td>Gl 54.1</td><td>6</td><td>-8.814 ± 0.2</td><td>-9.027 ± 0.2</td><td>-2.099 ± 0.20</td><td>-7.044(17)</td><td>4.816(4)</td><td>-3.230(4)</td><td>-2.010(8)</td><td>-6.318(21)</td></tr><tr><td>Gl 84.1B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.625(7)</td><td>...</td><td>...</td><td>0.549(7)</td><td>-0.625(7)</td></tr><tr><td>Gl 105B</td><td>1</td><td>-0.278 ± 0.05</td><td>-0.313 ± 0.05</td><td>0.204 ± 0.02</td><td>-0.319(4)</td><td>0.727(1)</td><td>-0.372(1)</td><td>0.188(3)</td><td>-0.330(5)</td></tr><tr><td>Gl 166C</td><td>1</td><td>-5.206 ± 0.2</td><td>-5.502 ± 0.2</td><td>-2.999 ± 0.20</td><td>-5.354(1)</td><td>6.588(2)</td><td>-4.705(2)</td><td>-3.227(8)</td><td>-4.921(3)</td></tr><tr><td>Gl 169.1A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.174(1)</td><td>...</td></tr><tr><td>Gl 179</td><td>2</td><td>-0.626 ± 0.03</td><td>-0.759 ± 0.03</td><td>0.228 ± 0.02</td><td>-0.676(34)</td><td>1.592(4)</td><td>-0.856(4)</td><td>0.230(21)</td><td>-0.695(38)</td></tr><tr><td>Gl 203</td><td>2</td><td>-0.119 ± 0.05</td><td>-0.137 ± 0.05</td><td>0.177 ± 0.02</td><td>-0.121(5)</td><td>...</td><td>...</td><td>0.203(13)</td><td>-0.121(5)</td></tr><tr><td>Gl 206AB</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-8.55(1)</td><td>...</td><td>...</td><td>-4.25(1)</td><td>-8.55(1)</td></tr><tr><td>Gl 206A</td><td>3</td><td>-7.348 ± 0.2</td><td>-8.802 ± 0.2</td><td>-3.630 ± 0.20</td><td>-8.154(4)</td><td>...</td><td>...</td><td>-3.585(4)</td><td>-8.154(4)</td></tr><tr><td>Gl 206B</td><td>3</td><td>-4.234 ± 0.2</td><td>-5.438 ± 0.2</td><td>-2.292 ± 0.20</td><td>-4.922(4)</td><td>...</td><td>...</td><td>-2.304(4)</td><td>-4.922(4)</td></tr><tr><td>Gl 213</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.315(8)</td><td>0.670(4)</td><td>-0.303(4)</td><td>0.215</td><td>-0.311(12)</td></tr><tr><td>Gl 232</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.36(4)</td><td>...</td><td>...</td><td>0.163(14)</td><td>-0.36(4)</td></tr><tr><td>Gl 234A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-5.88(1)</td><td>...</td><td>-4.797(32)</td><td>-3.055(32)</td><td>-4.830(33)</td></tr><tr><td>Gl 268AB</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-4.66(2)</td><td>...</td><td>...</td><td>-2.152(16)</td><td>-4.66(2)</td></tr><tr><td>Gl 268A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-5.95(1)</td><td>...</td><td>...</td><td>-2.38(1)</td><td>-5.95(1)</td></tr><tr><td>Gl 268B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-3.37(1)</td><td>...</td><td>...</td><td>-2.34(1)</td><td>-3.37(1)</td></tr><tr><td>Gl 273</td><td>5</td><td>-0.357 ± 0.03</td><td>-0.538 ± 0.03</td><td>0.207 ± 0.02</td><td>-0.380(21)</td><td>0.836(17)</td><td>-0.497(17)</td><td>0.189(18)</td><td>-0.432(38)</td></tr><tr><td>Gl 277B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-6.61(6)</td><td>6.149(2)</td><td>-4.331(2)</td><td>-1.977(7)</td><td>-6.040(8)</td></tr><tr><td>Gl 285</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-16.05(12)</td><td>28.48(4)</td><td>-27.26(4)</td><td>-7.003(15)</td><td>-16.04(12)</td></tr><tr><td>Gl 299</td><td>1</td><td>...</td><td>...</td><td>0.202 ± 0.02</td><td>-0.375(4)</td><td>...</td><td>...</td><td>0.185(15)</td><td>-0.375(4)</td></tr><tr><td>Gl 300</td><td>1</td><td>-0.592 ± 0.09</td><td>-0.784 ± 0.09</td><td>0.143 ± 0.02</td><td>-0.654(5)</td><td>...</td><td>...</td><td>0.167(2)</td><td>-0.654(5)</td></tr><tr><td>Gl 317</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>1.251(4)</td><td>-0.641(4)</td><td>0.281</td><td>-0.641(4)</td></tr><tr><td>Gl 319B</td><td>1</td><td>-0.181 ± 0.08</td><td>-0.292 ± 0.08</td><td>0.295 ± 0.02</td><td>-0.236(1)</td><td>...</td><td>...</td><td>0.295(1)</td><td>-0.236(1)</td></tr><tr><td>Gl 324B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.73(1)</td><td>...</td><td>...</td><td>0.249</td><td>-0.73(1)</td></tr><tr><td>Gl 375AB</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>2.560(2)</td><td>-1.513(2)</td><td>-4.75</td><td>-1.513(2)</td></tr><tr><td>Gl 375A</td><td>1</td><td>-9.234 ± 0.2</td><td>-8.184 ± 0.2</td><td>-3.722 ± 0.2</td><td>-8.709(1)</td><td>...</td><td>...</td><td>-3.722</td><td>-8.709(1)</td></tr><tr><td>Gl 375B</td><td>1</td><td>-7.352 ± 0.2</td><td>-8.578 ± 0.2</td><td>-3.746 ± 0.2</td><td>-7.965(1)</td><td>...</td><td>...</td><td>-3.746</td><td>-7.965(1)</td></tr></tbody></table>
<table><caption>**Table 6**
(Continued)</caption><thead><tr><th rowspan="2">Star</th><th rowspan="2">No. meas.</th><th colspan="3">FEROS</th><th rowspan="2">S<sub>HK</sub></th><th rowspan="2">Mean Ca II EW<sup>a</sup> (Å)</th><th rowspan="2">H<sub>α</sub> EW (Å)</th><th rowspan="2">Mean Ca II EW (Å)</th></tr><tr><th>Ca II H EW (Å)</th><th>Ca II K EW (Å)</th><th>H<sub>α</sub> EW (Å)</th></tr></thead><tbody><tr><td>Gl 398</td><td>7</td><td>-8.122 ± 0.2</td><td>-10.174 ± 0.2</td><td>-3.617 ± 0.2</td><td>-8.573(9)</td><td>...</td><td>...</td><td>-3.617(7)</td><td>-8.573(9)</td></tr><tr><td>Gl 402</td><td>5</td><td>-0.451 ± 0.05</td><td>-0.673 ± 0.05</td><td>0.209 ± 0.02</td><td>-0.699(15)</td><td>1.279(4)</td><td>-0.658(4)</td><td>0.175(21)</td><td>-0.690(19)</td></tr><tr><td>Gl 431</td><td>1</td><td>-7.620 ± 0.2</td><td>-8.782 ± 0.2</td><td>-4.017 ± 0.2</td><td>-8.790(3)</td><td>...</td><td>...</td><td>-4.519(3)</td><td>-8.790(3)</td></tr><tr><td>Gl 445</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.179(14)</td><td>0.566(4)</td><td>-0.195(4)</td><td>0.247(13)</td><td>-0.183(18)</td></tr><tr><td>Gl 447</td><td>1</td><td>-0.541 ± 0.05</td><td>-0.824 ± 0.05</td><td>0.187 ± 0.02</td><td>-0.452(13)</td><td>0.875(4)</td><td>-0.417(4)</td><td>0.176(10)</td><td>-0.444(17)</td></tr><tr><td>Gl 469</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.47(8)</td><td>...</td><td>...</td><td>0.189(7)</td><td>-0.47(8)</td></tr><tr><td>Gl 486</td><td>1</td><td>-0.248 ± 0.05</td><td>-0.314 ± 0.05</td><td>0.195 ± 0.02</td><td>-0.237(18)</td><td>0.648(5)</td><td>-0.284(5)</td><td>0.175(13)</td><td>-0.242(20)</td></tr><tr><td>Gl 487</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-1.125</td><td>...</td><td>...</td><td>-0.251(54)</td><td>-1.125</td></tr><tr><td>Gl 490B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-7.35(1)</td><td>-4.39(1)</td><td>-7.35(1)</td></tr><tr><td>Gl 512B</td><td>5</td><td>-0.415 ± 0.03</td><td>-0.526 ± 0.03</td><td>0.315 ± 0.02</td><td>-0.470(5)</td><td>...</td><td>...</td><td>0.315(5)</td><td>-0.470(5)</td></tr><tr><td>Gl 520C</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-7.65(1)</td><td>-4.55(1)</td><td>-7.65(1)</td></tr><tr><td>Gl 545</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.192(3)</td><td>...</td></tr><tr><td>Gl 553.1</td><td>3</td><td>-0.148 ± 0.03</td><td>-0.204 ± 0.03</td><td>0.186 ± 0.02</td><td>-0.178(15)</td><td>0.501(4)</td><td>-0.214(4)</td><td>0.174(11)</td><td>-0.186(19)</td></tr><tr><td>Gl 555</td><td>2</td><td>-0.384 ± 0.03</td><td>-0.573 ± 0.03</td><td>0.191 ± 0.02</td><td>-0.390(14)</td><td>0.904(8)</td><td>-0.576(8)</td><td>0.157(21)</td><td>-0.458(22)</td></tr><tr><td>Gl 568AB</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.203(1)</td><td>...</td></tr><tr><td>Gl 592</td><td>6</td><td>-0.190 ± 0.05</td><td>-0.234 ± 0.05</td><td>0.212 ± 0.02</td><td>-0.212(6)</td><td>...</td><td>...</td><td>0.212(6)</td><td>-0.212(6)</td></tr><tr><td>Gl 609</td><td>5</td><td>-0.269 ± 0.09</td><td>-0.401 ± 0.09</td><td>0.211 ± 0.02</td><td>-0.401(8)</td><td>...</td><td>...</td><td>0.206(11)</td><td>-0.401(8)</td></tr><tr><td>Gl 630.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-8.51(1)</td><td>-5.00(1)</td><td>-8.51(1)</td></tr><tr><td>Gl 643AB</td><td>2</td><td>-0.178 ± 0.05</td><td>-0.264 ± 0.05</td><td>0.210 ± 0.02</td><td>-0.197(3)</td><td>...</td><td>...</td><td>0.176(24)</td><td>-0.197(3)</td></tr><tr><td>Gl 644B</td><td>1</td><td>-0.213 ± 0.1</td><td>-0.419 ± 0.1</td><td>0.201 ± 0.02</td><td>-0.316(1)</td><td>...</td><td>...</td><td>0.201(1)</td><td>-0.316(1)</td></tr><tr><td>Gl 669A</td><td>2</td><td>-8.560 ± 0.2</td><td>-10.86 ± 0.2</td><td>-5.887 ± 0.2</td><td>-5.866(8)</td><td>4.431(2)</td><td>-2.923(2)</td><td>-2.696(6)</td><td>-5.277(10)</td></tr><tr><td>Gl 682</td><td>1</td><td>-0.399 ± 0.08</td><td>-0.552 ± 0.08</td><td>0.183 ± 0.02</td><td>-0.476(1)</td><td>...</td><td>...</td><td>0.183(1)</td><td>-0.476(1)</td></tr><tr><td>Gl 695B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>0.283</td><td>...</td></tr><tr><td>Gl 699</td><td>1</td><td>-0.445 ± 0.03</td><td>-0.370 ± 0.03</td><td>0.249 ± 0.02</td><td>-0.411(15)</td><td>0.761(7)</td><td>-0.353(71)</td><td>0.222(33)</td><td>-0.363(86)</td></tr><tr><td>Gl 720B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.29</td><td>...</td><td>...</td><td>0.253</td><td>-0.29</td></tr><tr><td>Gl 725B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.305(15)</td><td>0.651(22)</td><td>-0.292(22)</td><td>0.267</td><td>-0.298(37)</td></tr><tr><td>Gl 729</td><td>6</td><td>-4.375 ± 0.2</td><td>-5.110 ± 0.2</td><td>-1.685 ± 0.2</td><td>-4.743(6)</td><td>8.967(4)</td><td>-6.811(4)</td><td>-1.793(17)</td><td>-5.570(10)</td></tr><tr><td>Gl 732A</td><td>3</td><td>-3.120 ± 0.2</td><td>-3.803 ± 0.2</td><td>-1.141 ± 0.2</td><td>-3.462(3)</td><td>...</td><td>...</td><td>-1.141(3)</td><td>-3.462(3)</td></tr><tr><td>Gl 781.1B</td><td>1</td><td>-3.994 ± 0.2</td><td>-4.938 ± 0.2</td><td>-2.634 ± 0.2</td><td>-4.466(1)</td><td>...</td><td>...</td><td>-2.967(2)</td><td>-4.466(1)</td></tr><tr><td>Gl 791.2A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-7.775(3)</td><td>-4.615(3)</td><td>-7.775(3)</td></tr><tr><td>Gl 812A</td><td>4</td><td>-2.555 ± 0.1</td><td>-3.073 ± 0.1</td><td>-1.553 ± 0.2</td><td>-2.814(4)</td><td>...</td><td>...</td><td>-2.030(6)</td><td>-2.814(4)</td></tr><tr><td>Gl 860B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>3.304(2)</td><td>-2.055(2)</td><td>-2.098(3)</td><td>-2.055(2)</td></tr><tr><td>Gl 865AB</td><td>1</td><td>-7.308 ± 0.2</td><td>-8.846 ± 0.2</td><td>-2.688 ± 0.2</td><td>-6.816(2)</td><td>...</td><td>...</td><td>-2.369(2)</td><td>-6.816(2)</td></tr><tr><td>Gl 865A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-4.88(1)</td><td>-3.10(1)</td><td>-4.88(1)</td></tr><tr><td>Gl 865B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-8.70(1)</td><td>-5.10(1)</td><td>-8.70(1)</td></tr><tr><td>Gl 867B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-7.722(2)</td><td>-4.587(2)</td><td>-7.722(2)</td></tr><tr><td>Gl 873A</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-7.70(2)</td><td>12.525(4)</td><td>-10.171(4)</td><td>-3.784(17)</td><td>-9.347(6)</td></tr><tr><td>Gl 876A</td><td>5</td><td>-0.348 ± 0.01</td><td>-0.711 ± 0.01</td><td>0.200 ± 0.02</td><td>-0.448(137)</td><td>1.022(119)</td><td>-0.503(119)</td><td>0.197(52)</td><td>-0.473(256)</td></tr><tr><td>Gl 896AB</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-6.43(34)</td><td>-3.91(34)</td><td>-6.43(34)</td></tr><tr><td>Gl 896B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>5.490(2)</td><td>-9.156(9)</td><td>-6.143(7)</td><td>-9.156(9)</td></tr><tr><td>G 97-52B</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.126(1)</td><td>...</td></tr><tr><td>LHS 1723</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>-0.68(1)</td><td>-0.9(1)</td><td>-0.68(1)</td></tr></tbody></table>

Note. Results were obtained using FEROS spectra as well as a compilation of data published in the literature. When the Hα EW is known but with no observations of the Ca II EW, we derived the Ca II EW according to the Hα EW–Ca II EW correlation (see Figure 8).

a Inferred from the Hα EW–Ca II EW correlation.

(This table is available in machine-readable form.)

Therefore in what follows, the least squares fits that we will present refer only to stars in the unsaturated regime. In this regime, the observations show that there exist stars of both low (dM) and high (dMe) activity, so the sample provides access to a range of “dynamo strengths.” In contrast, in the saturated regime, only high-activity stars (dMe) are present.

In the regime of our unsaturated stars, we also performed heteroscedastic and homoscedastic linear LSFs to the separate sub-samples of low-activity dM3 and high-activity dM3e stars respectively (see Table 1). The heteroscedastic linear LSF to

the dM3 low-activity stars (all of which are in the unsaturated regime) yields:

$$F_{\text{Ca II}} = 2.33 \pm 1.10 \times 10^5 \times (P/\sin i)^{-0.837 \pm 0.20}. \quad (19)$$

Therefore, the gradient of the RAC for the low-activity stars is again much shallower than that of the combined sample of low+high-activity stars, and by more than 3σ. Therefore, the linear LSF to the combined sample may be somehow inadequate. We plot the linear LSF to the low-activity sub-sample also in Figure 8 (dotted-dashed line).
<figure><img src="image_11.png" /><figcaption>Figure 9. Correlation between the mean EW of the Ca II resonance doublet and the Hα EW for stars with spectral sub-type dM4e. The solid line is the linear LSF to the data.</figcaption></figure>

The homoscedastic linear LSF to the dM3 low-activity stars yields:

$$F_{\text{Ca II}} = 3.09 \pm 1.58 \times 10^5 \times (P/\sin i)^{-0.93 \pm 0.22}. \quad (20)$$

Both fits yield comparable results within errors. The $\chi^2$ for these fits are 0.039 and 0.047 respectively. Both fits are highly statistically significant at a 99.9% confidence level. Once again, we see that the slopes of the RAC among low-activity dM3 stars (−0.84, −0.93) are significantly shallower (by more than $3\sigma$) than the slopes of the RAC for the combined sample of high and low-activity dM3 stars (−2.02, −2.08).

Turning now to the high-activity dM3e stars in the unsaturated regime, we find that the heteroscedastic linear LSF yields:

$$F_{\text{Ca II}} = 1.60 \pm 1.20 \times 10^6 \times (P/\sin i)^{-1.041 \pm 0.58}. \quad (21)$$

In contrast to the results we obtained for the K4, K6, and M2 samples, this time for the dM3e sub-sample of unsaturated stars, the slope of the RAC is shallower than that of the combined sample of (unsaturated) low+high-activity stars. It seems that at the TTCC, the slope of the RAC for (unsaturated) high-activity stars is falling significantly to a value of about −1. Although there remains some uncertainty on our fit for dM3e stars (because we have only eight stars in the unsaturated regime, and with a significant scatter), we shall see in the next subsection that this decrease in the steepness of the slope is confirmed in our (unsaturated) dM4e sub-sample and is significant above the $3\sigma$ level (23 stars). Our M4 sample also confirms that, the slope of the RACs for the high-activity (unsaturated) sub-samples has fallen below the slope of the combined dM+dMe samples.

We have also performed LSFs to the full sub-sample of dM3e stars (unsaturated+saturated). We give the results of the fits in Table 1 (dM3e+sat). The slope for the full sub-sample has fallen to −0.39, which is considerably smaller than the value of −1.04 we found above. However, this presents us with the idea that at the TTCC, the slope of the RAC for dM3e stars

<figure><img src="image_12.png" /><figcaption>Figure 10. Correlation between the mean EW of the Ca II resonance doublet and the Ca II Mount Wilson S index for stars with spectral sub-type dM4. The solid line is the linear LSF to the data.</figcaption></figure>

is of the same order than that of the slope for the low-activity dM3 stars (see Section 3.7.6).

The relative invariance of the RAC slopes that we have obtained for the linear LSF to the low-activity stars in our five stellar samples gives an important degree of credibility to our results. It is important to note that all of our low-activity stars have been found to lie in the unsaturated regime of the RAC, i.e., in the regime where (we believe) dynamo theory can best be tested. To the extent that the slope of the RAC is determined (in the unsaturated regime) by the physical properties of a dynamo, our results suggest that there may exist only one dynamo regime for the low-activity stars from dK4 to dM4 within errors (see Section 3.7.2). The parameter $R/\delta$ is 18.28 for the high-activity star sub-sample and 21.77 for the low-activity star sub-sample. Therefore, the parameter $\delta = \langle\text{error}\rangle/\sqrt{n}$, which is a normalized estimate of the mean error on the measurements, is much smaller than the period domains of the RACs in both cases. Hence, these LSFs should be relatively well established, which corresponds to the high statistical significances we obtain (Table 1). However, these results may still be preliminary because the domains of the RACs are relatively small compared to the typical uncertainties on individual measures as highlighted by the referee.

There is the possibility that there exist two different dynamo regimes for dK, dM and dKe, dMe, respectively throughout the spectral range we investigate. Note that, in contrast to the low-activity dK and dM stars, the dynamo regimes in high-activity dKe and dMe stars vary with spectral type, especially at the TTCC. Might these changes be an indication that the dynamo mechanisms undergo a change at the TTCC, perhaps from an $\alpha-\Omega$ dynamo to an $\alpha^2$ type of dynamo?

There tends to be more and more high-activity, relatively slow rotators when we move to the spectral sub-types dM3 and dM4, which questions the validity of a single RAC for the combined samples at these spectral types. Given the consistency of the results we have obtained for the slopes of the RAC for the low-activity stars for our five spectral sub-types, it is
<figure><img src="image_13.png" /><figcaption>Figure 11. Correlation between the mean EW of the Ca II resonance doublet and log(P/ sin i) for stars with spectral sub-type dM4. Filled dots: dM4e. Hollow circles: dM4 stars.</figcaption></figure>

possible that a single dynamo mechanism may apply to all low-activity stars in our sample. However, we arrive at a different conclusion regarding the dynamo mechanism among the high-activity stars. At the TTCC and beyond, we propose that high-activity stars may switch to a different dynamo mechanism (see Section 3.7).

### 3.6. The RAC in M4 Dwarfs

Considering that, in our previous study of the RAC in dM4 stars, we had only a couple of dozen stars (Houdebine 2012a, Paper XVII), we decided to compile all available measures of v sin i published in the literature (see Paper I). With our own measures, we obtained v sin i measures for 106 dM4 stars (Paper I) from our initial list of 395 dM4 stars. Our dM4 stars in this study were selected according to their (R-I)C color, within the range [1.500;1.700]. This range is somewhat larger than the one used in Paper XVII in order to have more targets.

In order to obtain the RAC for dM4 stars, we also compiled the Ca II resonance doublet and the Hα line EW and fluxes from the literature. We give this compilation of data in Table 6. The data were collected from the following authors: Giampapa & Liebert (1986), Stauffer & Hartmann (1986), Fleming & Giampapa (1989), Herbst & Miller (1989), Young et al. (1989), Hawley et al. (1996), Delfosse et al. (1998), Gizis et al. (2002), Mohanty & Basri (2003), Wright et al. (2003), Rauscher & Marcy (2006), Morales et al. (2008), Walkowicz & Hawley (2009), Browning et al. (2010), Isaacson & Fischer (2010),

Houdebine (2012b), Reiners et al. (2012), and Pace (2013). In addition to this data, we also measured the Ca II and Hα EW from FEROS spectra from the ESO archive for 47 dM4 stars. We give the results in Table 6. Together with the measurements from Paper XVIII, this allows us to derive a correlation between the Ca II EW and the Hα EW for dM4e stars. We show this correlation in Figure 9. Although there is more scatter in this sample than in our dM3 star sample, we have a relatively good correlation with a correlation coefficient of 0.877 for 35 measures (see Table 1). This correlation is significant at a confidence level better than 99.8%. The LSF to this data gives:

$$ EW_{Ca\ II} = 1.909 \pm 0.18 \times EW_{H\alpha} - 1.035 \pm 0.58. \quad (22) $$

Equation (22) allows us to obtain an estimate of the Ca II EW when we have only the Hα line EW available for dM4e stars. Similarly, we compiled many measures of the Mount Wilson S index (Table 6). We show the empirical correlation between the Ca II EW and the S index in Figure 10. Again, we obtain a relatively good correlation between the two parameters, with a correlation coefficient of 0.970 for 22 data points (see Table 1). The relationship between the Ca II EW and the S index is:

$$ EW_{Ca\ II} = 0.4898 \pm 0.047 \times S^{1.20 \pm 0.07}. \quad (23) $$

All these compilations of data now allow us to compute the Ca II surface fluxes (Table 7). We show the Ca II surface fluxes as a function of P/ sin i for our dM4 stellar sample in
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Correlation between the logarithm of the mean surface fluxes of the Ca II resonance doublet corrected for metallicity effects and $log(P/\sin i)$ for stars with spectral sub-type dM4. Notice the onset of saturation for stars with $log(P/\sin i) \le 0.2$. LSFs have been obtained for the stars in the unsaturated regime as follows. Solid line, heteroscedastic linear LSF; separated dotted-dashed lines, the heteroscedastic linear LSF to the low-activity dM4 and to the high-activity dM4e sub-samples. We show in the center-right of the figure estimates of the uncertainties on the Ca II mean surface fluxes due to the corrections for metallicity effects, for dwarfs and subdwarfs.</figcaption></figure>

Figure 11. One can see in this diagram that the scatter is very large, and is similar to (or even worse than) the raw correlation for dM3 stars. It appears that the scatter increases from spectral sub-types dM2 to dM4. However, there is a parameter that plays a role in the scatter in Figure 11: the $(R-I)_C$ range for our dM4 sample ([1.500:1.700]) spans 0.2 dex, whereas for our dM2 and dM3 samples, it spans 0.132 dex. This larger range in our selected targets contributes to the larger scatter observed in Figures 11 and 12. In Figure 11, it is noteworthy that there is no overlap in Ca II fluxes between the low-activity dM4 stars (up to $log(F_{HK} = 4.4)$) and the high-activity dM4e stars ($log(F_{HK} \ge 4.6)$: there is a clear gap in the fluxes, although the two groups overlap in their rotation periods. The dM3 stars do not show this clean separation between dM and dMe (Figure 8). The dM2 stars also show some separation (Figure 5), but it is not as clean as for dM4 stars.

Considering the good correction we had for metallicity in dM2 stars, we decided to compile all the metallicities published for our initial selection list of 395 dM4 stars, and try to obtain a metallicity–radius correlation for these stars (see Paper I). We found metallicities from the literature for 179 dM4 stars. Data for M4 stars that have been corrected for metallicity can be found in Figure 12 and Table 7.

In Figure 12, we show the RAC corrected for metallicity effects for dM4 stars. The scatter is reduced compared to the raw data in Figure 11, but there remains a significant scatter among both dM4 and dM4e stars. The corrected correlation is not as good as for dM2 stars in spite of the relatively good statistics we have on the Ca II surface fluxes.

We also observe among our sample of dM4 stars a sub-sample of relatively fast rotating, low-activity stars. Again, as in the case of dM2 and dM3 stars, we believe that these stars are unresolved spectroscopic binaries. We also found a few stars with possibly low $\sin i$ (see Figure 12).

We find that when we apply the linear heteroscedastic and homoscedastic LSFs to the combined dM4+dM4e stars in the unsaturated regime, the gradients are very steep, steeper even than what we found for our dM2 and dM3 stellar samples. We also find that dM4 stars rotate much faster than dM2 and dM3 stars. The change in Ca II flux between dM4 and dM4e stars is very abrupt, as already noticed in Figure 11, a difference of only 0.1 in the $log(P/\sin i)$. Our data suggest that saturation in our dM4 sample occurs at about $P/\sin i \sim 1.8$ days. This agrees with the values found in previous studies. Specifically, for stars in the unsaturated regime, i.e., for $P/\sin i \ge 1.8$ days we obtain the following heteroscedastic linear LSF for the stars