A CATALOG OF ROTATION AND ACTIVITY IN EARLY-M STARS

The goal of our project is to construct a statistically meaningful, yet neither complete nor unbiased, sample of early-M field dwarf spectra. Literatures available at the time of observations containing considerable samples of high spectral resolution analysis of field-M dwarf rotation velocities were Marcy & Chen (1992) and Delfosse et al. (1998). During the course of our project, Jenkins et al. (2009) and Browning et al. (2010) added more stars to this list, and a few other rotational velocities were presented in Reiners (2007) and Reiners & Basri (2007). We did not exclude young or halo stars from the sample in order to achieve a representative picture of the stars in the solar neighborhood.

Observations for this project were carried out at the spectrographs FOCES (CAHA, Calar Alto) in 2005 and FEROS (ESO, La Silla) in 2006 using time allocated through the Max-Planck Institute for Astronomy (MPIA) at both observatories. In total, we have obtained spectra of 239 M0–M4.5 stars.

The FOCES échelle spectrograph at the 2.2 m Telescope at CAHA, Calar Alto, was operated at a spectral resolving power of approximately 40,000. Observations carried out with FEROS at the ESO/MPG 2.2 m Telescope at La Silla have a spectral resolution of approximately 48,000. Typical exposure times are between a few minutes and one hour per star for FEROS and up to two hours with FOCES. Individual exposures are always shorter than 30 minutes to avoid crowding with cosmic rays. The resulting signal-to-noise ratios (S/Ns) are typically around 50; further analysis uncertainties are discussed in Section 4.

Data reduction followed standard procedures including bias subtraction, flat fielding, and wavelength calibration provided by ThAr or ThArNe lamps. For the FEROS spectra, reduction was done with the dedicated pipeline based on the MIDAS context, the FEROS Data Reduction System.

FOCES data were reduced using standard reduction procedures implemented in ESO-MIDAS. To avoid over- or underexposure for different échelle orders, three flat fields were taken with different exposure times, which is the usual procedure for FOCES data reduction. Orders were grouped in three sets each corresponding to different flat-field exposure times, with the longest time corresponding to red and the shortest time to blue regions in the spectrum. All three flat fields were merged to create a master flat field. The raw object spectra were freed from cosmic-ray contamination and, after bias subtraction, flat fielded with the master flat field. The wavelength scale was calibrated for each night using ThAr lamp calibrations. Scattered light was removed using a standard background subtraction routine in ESO-MIDAS. Since FOCES is a fiber-fed spectrograph, the object spectra were first extracted and then divided by the order-extracted master flat field.

Although the methodologies determining v sin i employed in the different studies are basically identical, there are some differences in their implementation. The basic idea is to compare the spectrum of a known, slowly rotating star (the template spectrum) with a spectrum of the science target in which the value of vsin i is to be determined. The template spectrum is convolved with rotational broadening profiles according to a set of different velocities using the scheme described in Gray (2005).

There are in principle two methods used to determine vsin i. The first method is to directly compare a chosen set of individual spectral lines to artificially broadened template spectra. The value of vsin i is the one providing the best fit. This method is prone to systematic uncertainties induced by a mismatch between the template and science objects' spectra. The line profiles for comparison must be selected very carefully and systematic differences may occur if different sets of lines are used. This method determining vsin i was employed by Marcy & Chen (1992), Reiners (2007), Reiners & Basri (2007), and Jenkins et al. (2009). While Marcy & Chen (1992) and Jenkins et al. (2009) used atomic lines at optical wavelength ranges where blending is a serious issue, Reiners (2007) and Reiners & Basri (2007) employed lines of molecular FeH that are relatively free of blends and embedded in a rather well-defined continuum.

The second method to derive values of vsin i is the so-called cross-correlation technique (Tonry & Davis 1979; Basri et al. 2000). Here, a slowly rotating template star is identified and its spectrum is cross-correlated with a series of the same template spectrum that is artificially broadened according to different rotational velocities. The widths of the correlation functions provide a calibrated measure of the value of vsin i. Then, the target spectrum is cross-correlated with the non-broadened template spectrum and the width of the resulting correlation peak is converted into vsin i according to the calibration. Spectra are usually divided into several sections (e.g., spectral orders), and cross-correlation functions of individual orders may be averaged, or the median of individually derived vsin i values from different orders can be used. The latter procedure may also provide an estimate of the uncertainty. This cross-correlation method was employed in the analysis (see Section 4.2) of our new observations and in the work of Delfosse et al. (1998) and Browning et al. (2010).

We collected measurements of projected rotational velocities from the literature. Unfortunately, not all literature also provide activity measurements together with rotation, or they provide only information on whether Hα emission is detected or not, but not the value of equivalent width.

Delfosse et al. (1998) and Browning et al. (2010) included values of log L_Hα/L_bol in their tables, and we included their results in our catalog. Marcy & Chen (1992) did not provide information on Hα emission. Jenkins et al. (2009) provided only information on whether Hα was detected, and whether they found Hα in absorption in their spectra. If no Hα was detected at all, we calculated upper limits for the stars assuming similar detection thresholds as for our data (see Section 4.1) because the data quality is similar. In cases where Jenkins et al. (2009) found Hα in absorption, we treated these stars as inactive but mark them in our table. Although the existence of Hα in absorption may be an indicator of weak activity (Cram & Mullan 1979), we classified these stars as inactive because all other works only consider Hα emission as an indicator for activity. In those cases where Jenkins et al. (2009) detected Hα in emission, we do not provide any value in our table. The stars can easily be identified in the catalog as those stars from Jenkins et al. (2009) that have no value of log Hα/L_bol in our catalog.

Normalized luminosities or upper limits of it are available for 244 stars (73% of our total sample of 334).

Chromospheric activity was measured from Hα emission in our spectra. We estimated the continuum around Hα taking the median of two different regions on either side of Hα line, namely, 6545–6559 Å and 6567–6580 Å. This value was used to normalize the spectra. We integrated the equivalent width of Hα emission in the spectral range 6552–6572 Å.

Many stars of our sample do not show significant line emission at Hα. We conservatively estimated the detection limit of our spectra to 0.2 Å, which is consistent with the approach of Cayrel (1988) assuming a 3σ detection limit with

$$\begin{equation} \sigma _{\mathrm{EqW}} = 1.5 \frac{\sqrt{\mathrm{FWHM}_{{\rm line}} \delta x_{{\rm line}}}}{\mathrm{S}/\mathrm{N}} \end{equation} \tag{ 1 }$$

and a typical S/N of 50. In this equation, FWHM_line is the full width at half-maximum of the expected line and δx_line is the size of a resolution element.

Equivalent width is not a suitable indicator of stellar activity because the continuum flux is a steep function of effective temperature. We used PHOENIX model atmospheres (assuming log g = 5.0; Hauschildt et al. 1999) to transform equivalent widths to Hα flux, F_Hα, and to determine the flux ratio F_Hα/F_bol using F_bol = σT⁴. Effective temperatures were derived from spectral type using the conversion given by Kenyon & Hartmann (1995). Finally, we used the identity F_Hα/F_bol = L_Hα/L_bol to determine the ratio between Hα and bolometric luminosity, i.e., normalized Hα luminosity.

In order to derive projected rotational velocities, vsin i, from our spectra, we used the cross-correlation method as mentioned above. As a first step, the spectrum of a slowly rotating star was chosen as a template spectrum and artificially broadened according to a set of different velocities (Gray 2005). We chose velocities in the range [1, 40] km s⁻¹ and limb darkening was set to 0.6 (see Browning et al. 2010). The cross-correlation functions between the unbroadened template spectrum and the set of broadened spectra were calculated and the FWHMs were determined as a function of vsin i. To derive vsin i in a target spectrum, the cross-correlation function between the object spectrum and the template was calculated; an example is shown in Figure 2. The FWHMs were measured and converted into vsin i according to the calibration established from the broadened template spectra.

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The two spectrographs used for our observations have somewhat different spectral resolving power. In order to avoid systematic differences between the two data sets, we employed different template spectra, i.e., one for each instrument. For the FOCES sample, we used a spectrum of Gl 2 as a template star, and for the FEROS sample, Gl 84 was used. Both objects are of spectral type M2. We tried to use a few different template stars but found no systematic differences.

We estimated the detection limit for rotational broadening from our procedure to determine vsin i. If we artificially broadened the spectra to simulate slow rotation (vsin i  ≲  3 km s⁻¹), the FWHM of the cross-correlation profile only marginally differed from the autocorrelation function of the template. In our case, the spectral resolving power was not high enough to fully resolve the lines of slow rotators so that the threshold at which significant broadening becomes visible is determined by the spectral resolving power of the instrument. We found that from the FOCES spectra (R = 40, 000) we can determine values of vsin i in excess of 4 km s⁻¹. For the FEROS spectra (R = 48, 000), the detection limit is at vsin i = 3 km s⁻¹.

For the cross-correlation procedure, we used only selected spectral regions that are virtually free of telluric absorption and emission lines. We calculated correlation functions in 13 different spectral regions, each covering approximately 20 Å. The spectral regions are not identical in the FEROS and FOCES samples, which is due to the different performances of the instruments and their coverage of échelle orders. The projected rotational velocity was calculated for each spectral region; the adopted final rotational velocity is the mean of the individual values. The standard deviation for stars with vsin i  <  20 km s⁻¹ is typically below 1 km s⁻¹. Note that this value is much smaller than the detection limit because it is the typical scatter in vsin i between individual orders while the detection limit is the lowest value of vsin i at which rotation can be distinguished against other broadening agents like temperature and instrumental broadening. The uncertainty of our vsin i measurements is also not the same as the intra-order scatter; it is the accuracy at which we can distinguish rotation from other broadening agents; we estimated the final uncertainty to be ∼3 km s⁻¹ for slow rotators but not less than 10% (see Reiners 2007; Reiners & Basri 2007).

Complete information on binarity in stars is difficult to achieve. Binarity may be detected using high spatial resolution imaging, but in many cases binary components are too close to each other and cannot be spatially resolved. In such a case, the observed spectrum consists of light from all components weighted according to their luminosity. If both components are similar in luminosity, both spectra appear in the spectrum with a separation according to the difference in radial velocities at the time of observation. For the search for spectroscopic binaries, the cross-correlation profile is very useful. Three cases can be distinguished. (1) The separation is larger than the typical line width. In such a case, two systems of spectral lines are visible in the spectrum and the cross-correlation profile shows two separated maxima. (2) The separation is on the order of the typical line width. Here, the cross-correlation function is broader than individual correlation maxima due to single stars. The profile may appear asymmetric depending on the luminosity difference of the components and their radial velocity difference. Figure 3 shows a typical cross-correlation profile with significant asymmetry that is attributed to a spectroscopic binary. (3) The separation is marginally different from zero. The profile appears only slightly wider than the typical single profile and may be asymmetric.

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If a system is a multiple system instead of a single star, the determination of rotation becomes meaningless unless individual components can be disentangled from each other. We found that 20 stars of our originally observed sample are spectroscopic binaries that could be unambiguously identified. The stars are listed in Table 1 and are not used for further analysis and the catalog. For completeness, Table 1 also includes spectroscopic binaries of spectral types M0.0–M4.5 reported in the literature.

Our cross-correlation analysis revealed that 13 other stars have asymmetric cross-correlation profiles. The degree of asymmetry is small but justifies the assumption that the stars are no regular single objects. They may be spectroscopic binaries with long periods or observed at very similar radial velocities. We excluded these objects from our sample analysis. The marginal outliers are listed in Table 2; further observations at a different epoch may clarify whether these stars are in fact binaries.

In Figure 4, we show normalized Hα activity as a function of spectral type. In total, our catalog contains 244 stars with Hα measurements, 95 of them (39%) are active. It is well established that activity lifetimes are substantially longer at later spectral types (e.g., Hawley et al. 1996; Gizis et al. 2002; Silvestri et al. 2005; West et al. 2008). Stars on the cool side of the boundary to complete convection appear active much longer, leading to the observation that many more fully convective stars show activity. On the hot side of that boundary, where stars are believed to still harbor a tachocline and hence may drive a Sun-like large-scale dynamo, only very few active stars are known in the field. Virtually all active early-M stars are members of young associations; several examples can be found in Torres et al. (2006). Partially convective early-M stars in the field, however, that are believed to be older, in general do not possess significant activity. Within the literature concerned for our catalog, there is no early-M type star (<M3) with significant activity that is not a known member of a young association. Thus, any single early-M type active star is probably young, which means not older than a few 100 Myr. In fully convective stars, however, activity can persist for several Gyr and we expect to find many more active stars of spectral type M3 and later. Note that rapid rotation and hence enhanced activity may be maintained in tidally locked binaries.

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The total catalog contains 7 out of 129 (5%) early-M type stars exhibiting significant activity as Hα emission. In contrast, 47 out of 115 (41%) stars between M3 and M4.5 are active. The fraction of active stars as a function of spectral type is shown in Figure 5; it is in general higher than reported in West et al. (2008) from the Sloan Digital Sky Survey sample, which is consistent with our sample being younger than the sample used there (observed away from the Galactic plane), and our results are consistent with earlier work on the activity fraction of field stars (e.g., Hawley et al. 1996; Gizis et al. 2002; Silvestri et al. 2005). The general trend, however, is very well reproduced. A relatively sharp transition from a low activity fraction smaller than 10% to a significant fraction above 50% occurs around spectral type M3. From activity information alone, we cannot determine the reason for this dramatic increase. West et al. (2008) speculated that longer activity lifetimes may be explained by a transition from a rotationally dependent solar-like dynamo to a rotationally independent turbulent dynamo in which magnetic fields can survive much longer even if the stars are rotating slower. We come back to this point when we discuss the rotation of these stars in Section 5.3.

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Field stars are believed to be relatively old (∼Gyr) and early-M dwarfs (<M3) are in general not observed to be active in the field. In our survey, we found or confirmed activity in 7 out of 129 (5%) early-M targets with Hα measurements. The reason for their activity is probably youth since it is known that early-M dwarfs can be very active at young ages. Young stars are rotating much more rapidly and therefore can generate sufficient magnetism to generate magnetic activity (Pizzolato et al. 2003). A potential reason why a field early-M dwarf can generate activity is the prevention of angular momentum loss because of binarity. Another explanation is that the star is indeed young and entered our survey because it is relatively nearby compared to other young objects. Finally, some stars may be misclassified in spectral type so that they are actually within the regime of M3 or later.

The seven active early-M dwarfs found in our survey are presented in Table 4. Spectral types are between M0 and M2; misclassification may be an issue for the latest targets but is unlikely for all of them. Some of the stars have companions but we did not find evidence for binarity in any of the stars that would sufficiently influence the rotation of the star on Gyr timescales. Most of the objects are probably young objects in the solar neighborhood. We discuss the seven stars individually in the following.

Gl 182 is a young star known as V1005 Ori. The star is contained in the SACY sample (Torres et al. 2006; da Silva et al. 2009) and classified as a member of the β Pictoris young association (10 Myr). It shows substantial activity (log L_Hα/L_bol  =   −  4.11) and rapid rotation (vsin i  =  10.4 km s⁻¹).

Gl 494A has a companion of spectral type M7 at a separation of 0475 (Beuzit et al. 2004), and a planetary candidate companion of spectral type T8–9 at 102'' (Goldman et al. 2010). The secondary component is too faint to influence the rotational profile and our measurement of vsin i is most likely the one of the M0.5 primary alone. Beuzit et al. (2004) determined vsin i = 9.6 km s⁻¹ consistent with our observations. They concluded that the object is not a short-period locked binary but a rapidly rotating, young early-M star (see also Burgasser et al. 2010; Burningham et al. 2011).

Steph 546A (GJ 3331A) is also contained in the SACY survey under the name BD 21 1074A. It is classified as a member of the β Pic association (da Silva et al. 2009) and has a companion pair, BD 21 1074BC.

Wo 9520 (GJ 9520) was observed by Daemgen et al. (2007) using the Altair AO System at Gemini North Observatory. No companion could be detected. Shkolnik et al. (2009) observed the star at two different epochs and found no evidence for RV variability, providing evidence that Wo 9520 is not a binary system. Shkolnik et al. (2009) estimated an age of 15–150 Myr.

GJ 2036A (CD-56 1032A) is part of a binary system with two active components. The system is classified as a member of the AB Doradus young association (70 Myr) by da Silva et al. (2009).

Gl 358 is identified as a possible member of the Carina-Near Stream by Zuckerman et al. (2006), which would imply an age of ∼200 Myr.

Gl 569A is accompanied by the brown-dwarf pair Gl 569Bab (Simon et al. 2006; Femenía et al. 2011) for which orbital parameters are well determined. Comparison of the colors of Gl 569Bab to theoretical isochrones and color mass diagrams suggest an age of 100–125 Myr, which is probably the same as for Gl 569A. The orbital inclination of Gl 569Bab is 324 ± 13 (sin i = 0.54; Simon et al. 2006).

Five out of the seven active early-M stars show detectable rotation in vsin i. The two other stars, in which no rotation was detected, Gl 358 and Gl 569A, may be observed under high inclination. Both stars show relatively low activity compared to the other stars in Table 4 so that their equatorial velocities are probably comparably low, i.e., only a few km s⁻¹. Inclination angles below ∼60° may be sufficient to push vsin i below the detection limit, which renders this scenario rather likely. In particular, this is consistent with the assumption of spin–orbit alignment in the Gl 569 multiple system (i ≈ 30°).

For each of the seven active early-M stars, Kiraga & Stepien (2007) provided a photometric period. This may be a hint that spot configurations in active early-M dwarfs are rather stable at least after a few 10 Myr. However, periodicity may not in all cases be due to rotation. We discuss photometric periods in Section 6.

We provide measurements or upper limits of the projected rotational velocity, vsin i, for all 334 stars of our catalog. As explained above, detection thresholds for our new measurements with the FOCES and FEROS spectrographs are estimated at 4 and 3 km s⁻¹, respectively, according to the spectrographs' different resolving powers. Detection limits of the collected data from the literature also vary. Browning et al. (2010) estimated a detection limit of vsin i_lim  =  2.5 km s⁻¹ (R  =  45, 000–60, 000), Reiners & Basri (2007) used vsin i_lim  =  3 km s⁻¹ (R  =  31, 000), and Reiners (2007) used extremely high-resolution data (R ≈ 200, 000), estimating vsin i_lim  =  1 km s⁻¹. Furthermore, we adopt vsin i_lim  =  3 km s⁻¹ for results from Marcy & Chen (1992, R  =  40, 000) and vsin i_lim  =  2 km s⁻¹ for those from Delfosse et al. (1998, R  =  42, 000) as written in those publications. Jenkins et al. (2009) did not quote a general detection threshold in vsin i. Using spectra at a spectral resolving power of R  =  37, 000, they determined individual upper limits as well as many detections at the vsin i  =  3 km s⁻¹ level. We adopt these values as they are given in the original literature but show below that their detection threshold is likely higher, around vsin i_lim  =  4 km s⁻¹.

Several stars are contained in more than one of the considered works. Individual measurements may differ because spectral appearance can change with chromospheric variability or different authors used different spectral lines in their analysis that are sensitive to chromospheric activity. We provide a list of all stars with more than one vsin i measurement in Table 5. In general, all measurements from different data are consistent within the uncertainties; the only outliers are two measurements of vsin i in Gl 388 and Gl 873 by Delfosse et al. (1998). For the inactive stars Gl 369, G 244-047.01, and GJ 1119, rotational broadening is reported in one paper but not detected in one or more other works. These stars are probably rotating very slowly as we discuss below. Gl 388 (AD Leo), Gl 729 (V1216 Sgr), and Gl 873 (EV Lac) are very active stars for that different analyses report vsin i values close to the detection limits. These stars are probably rotating on the few km s⁻¹ level.

For our catalog, we adopted vsin i values according to the following strategy. Preference was given to the vsin i data from the work done with the highest spectral resolution. The highest priority was given to the results from Reiners (2007) because they were derived from the highest resolution spectra. The second highest priority was given to the results from Browning et al. (2010). If both were not available and several other works provided measurements of vsin i, we chose to use the one from our new measurements. In Figure 6, we show a comparison between vsin i measured in this work and data from the literature. For all stars, values are consistent within the uncertainties and detection limits.

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Projected rotational velocities, vsin i, for our catalog are plotted as a function of spectral type in Figure 7. The situation appears very similar to the one in Figure 4 where activity is shown as a function of spectral type. Again, we see only a few early-M type stars with significant rotation (shown as open circles in Figure 7). These stars are listed in Table 4 and discussed individually in Section 5.2. At later spectral types (⩾M3), significant rotation appears to be more frequent just as activity is more frequent in this spectral range. In total, 51 of our 334 stars (15%) show rotational broadening of vsin i ⩾ 3 km s⁻¹.

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Figure 8 shows the fraction of rapid rotators, i.e., stars with detected rotational broadening vsin i ⩾ 3 km s⁻¹, as a function of spectral type. The figure appears to be very similar to the fraction of active stars discussed above (Figure 5). While among early-M type stars (<M3), the fraction of rapid rotators is below 5%; it rises rapidly to approximately 45% at spectral type M4.

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After discussing activity and rotation in the sample catalog, we now turn to the relation between the two across the spectral range M0.0–M4.5, i.e., from partially convective Sun-like stars to fully convective stars. The two distributions in Figures 5 and 8 are strikingly similar. Within statistical uncertainties, the distributions of active and rapidly rotating stars are consistent with the assumption that active stars and rapid rotators are both drawn from the same underlying population. We tested the probability that both distributions are drawn from the same distribution following Kolmogorov–Smirnov statistics (Press et al. 1992). In numbers, the probability that the sample of active stars and the sample of rapidly rotating stars are drawn from independent distributions is lower than 4  ×  10⁻⁴. This means that rotation and activity are highly correlated and that the active stars in our sample are likely the same stars as the rotating ones. While this does not prove a causal relation between rotation and activity, it provides evidence that both occur in the same stars.

It is worth emphasizing that evidence for a correlation between activity and rotation exists over the entire sample including both partially and fully convective stars. In other words, before we discuss the relation between rotation and activity on the basis of individual stars, the distribution of rotation and activity in M0–M4 stars already provides strong evidence for the validity of the rotation–activity connection across the boundary to complete convection.

We plot the normalized Hα luminosity, log L_Hα/L_bol, against the projected rotational velocity, vsin i, in Figure 9. In the figure, we further discriminate between partially convective stars (<M3) and likely fully convective stars (M3 and later). The boundary between the two groups is likely not sharp and spectral type uncertainties on the order of 0.5–1 spectral subtypes further softens the location of this transition. As a first result, we can confirm the rotation–activity relation in the sense that low activity (log L_Hα/L_bol  <  − 4.5) only occurs at slow rotation. There are a handful of inactive stars (stars with low activity) for that detection of rotational line broadening on the order of 4–5 km s⁻¹ is detected. We discuss these stars in Section 5.5. Furthermore, we can conclude that the correlation between rotation and activity is valid at both sides of the convection boundary, i.e., for both early- and later-M type stars (open and solid symbols in Figure 9).

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Active stars are found at virtually all rotation rates. In our sample, we found 48 very active stars with log L_Hα/L_bol  >  − 4.5. Among them, 33 stars are rapid rotators (vsin i  >  3 km s⁻¹). This means that 15 out of 48 active stars (31%) have rotation velocities below our detection limit, or are observed under low inclination angles so that high rotation rates are not detected. In the latter scenario, the most active stars at low vsin i values are interpreted as stars observed under very low angles i. We can test the assumption whether a tight relation between rotation and activity is valid among all our sample stars. If so, all stars with very high Hα emission, say log L_Hα/L_bol  >  − 4.0, would be rotating at approximately vsin i ≳ 5 km s⁻¹, which means that at a typical detection threshold of vsin i_lim  =  3 km s⁻¹, inclination must be such that sin i  <  0.6 (i  <  37°). In our sample, 7 out of 36 (19%) stars with log L_Hα/L_bol  >  − 4.0 show no detectable rotation. In a sample of stars with randomly oriented rotation axes, a fraction of 19% will be observed at inclination angles smaller than 36°, i.e., sin i  <  0.59. Thus, we can conclude that the distribution of measurements in vsin i and activity is consistent with the assumption of a well-defined relation between rotation and activity that is spread out in Figure 9 due to projection effects from observing the stars under statistically distributed orientations.

Active M stars are known to exhibit frequent flaring events that introduce substantial scatter in log L_Hα/L_bol and is difficult to quantify with just one observation. Kowalski et al. (2009) found that among a sample of 236 stars with flares, ∼3% show no Hα emission outside the flare. If we assume that inactive stars with occasional flaring are slow rotators, we can estimate that 1 star of our 33 slowly rotating flare stars is in fact an inactive, slow rotator. Furthermore, Hilton et al. (2010) determined the flare rates of stars at different spectral types; for stars of our sample the flare rates are ⩽1%. Although it is not trivial to compare the flare rates from Hilton et al. (2010) to our sample because of the inhomogeneous distribution of exposure times, we can exclude a significant influence of flaring on the occurrence of active stars for which no evidence of rotation could be detected.

In Figure 9, we have distinguished between early-M type stars (<M3, open symbols) and later M stars (solid symbols) expecting the transition from partially to fully convective stars at this spectral range. The subsample of early-M dwarfs is defined by a number of inactive, slowly rotating stars (with several upper limits in vsin i) and seven active stars, five of them with detected rotation. The early-M dwarfs exhibit a clear correlation between projected rotation rate and normalized Hα activity; all stars with vsin i  ≈  3 km⁻¹ and above are active, and the stars with vsin i ⩾ 5 km⁻¹ show higher activity than those at vsin i  ≈  3 km⁻¹. Interestingly, the scatter in activity in early-M stars at vsin i ⩾ 5 km⁻¹ is much smaller than the scatter in mid-M stars in our sample. From our small sample, it is not possible to decide whether this is an intrinsic effect or due to the small number of rapidly rotating early-M stars, but it is consistent with the conclusion of Gizis et al. (2002) and Lee et al. (2010) that the Hα variability is larger at later spectral types. In a fully convective star, the scatter of Hα activity is much larger than in early-M dwarfs, but as in early-M dwarfs all significantly rotating fully convective show significant Hα activity. In summary, the saturation-type rotation–activity relation is intact until spectral type M4.5, while the scatter is perhaps growing larger in lower-mass stars that generally tend to be more active in our sample.

The relation between rotation and activity may not be valid in all individual cases. In general, rotation generates magnetic activity, but we may still find activity in some stars that are observed at low projected rotation rates, or even in stars that are slowly rotating but active for reasons we have not understood. On the other hand, our assumption of magnetic dynamo operation triggered by rotation leads to the expectation that all rapidly rotating stars show significant values of magnetic activity. West & Basri (2009) reported the existence of three rapidly rotating but inactive stars at spectral types later than our sample (M6–M7). We searched for such stars in our sample of hotter M stars.

Table 6 lists nine stars in which no Hα emission was found but a detection of rotational broadening was reported. All nine stars are from the catalog of Jenkins et al. (2009), and all stars have values of vsin i between 3 and 4.5 km s⁻¹. Four of the nine stars are reported to show Hα in absorption, which is evidence for very low but non-zero activity.

Regardless of whether or not the stars with Hα absorption are indeed weakly active, it is striking that all nine stars are found in the same work. Our catalog contains 23 stars taken from Jenkins et al. (2009); 9 of them show no activity in the presence of rotation. Among the 311 stars taken from other sources, not a single star shows similar properties to another star. The spectral resolution of the data used by Jenkins et al. (2009) is R  =  37, 000. For such data, a detection limit between 3 and 5 km s⁻¹, which means at the same order as the measurements reported for the stars in Table 6, can be expected depending on S/N. We argue that the absence of rapidly rotating inactive stars in the rest of our sample provides ample evidence that all rapid rotators vsin i ≳ 3 km s⁻¹ show measurable Hα activity, and that the reported detections of rotation in inactive stars from Jenkins et al. (2009) are spurious and in fact upper limits to their real values of vsin i. Our conclusion is that substantial chromospheric emission is a fundamental consequence of rapid rotation in M0–M4.5 stars.