ROTATION AND MAGNETIC ACTIVITY IN A SAMPLE OF M-DWARFS

Across a broad swath of the Hertzsprung–Russell diagram, rotation and magnetic activity appear to be intimately linked. In stars of spectral types from late F to mid-M, coronal, and chromospheric emission—thought partly to trace magnetic heating of the stellar atmosphere—are correlated with rotation (e.g., Noyes et al. 1984, hereafter N84; Delfosse et al. 1998; Pizzolato et al. 2003). Emission in chromospheric Ca ii H and K and Hα tends to rise with increasing rotational velocity, saturating at a threshold velocity that depends upon stellar mass; emission declines somewhat in the most rapid rotators (James et al. 2000). The relationship between rotation and these emission measures is tightened when the rotational influence is expressed using the Rossby number Ro ∼ P/τ_c, with P being the stellar rotation period and τ_c a typical convective overturning time (e.g., N84). This rotation–activity connection is generally thought to reflect the underlying rotational dependence of the magnetic dynamo realized in Sun-like stars.

In such stars, with masses between about 0.35 and 1.3 solar masses, the global dynamo is believed to be seated at the interface layer between the convective envelope and the inner stable region (e.g., Ossendrijver 2003; Parker 1993; Charbonneau & MacGregor 1997). The identification of this region as the likely host of global dynamo action was spurred partly by helioseismology, which revealed that in the Sun this interface is a site of very strong radial shear, called the tachocline (e.g., Thompson et al. 2003). This shear, plus the stable stratification that allows fields to be built up there without quickly becoming susceptible to magnetic buoyancy instabilities (Parker 1975), makes the tachocline the most plausible site where strong toroidal magnetic fields could be amplified and organized, ultimately to emerge at the surface as sunspots. Although the rotational dependence of this process is not yet fully understood, some guidance is perhaps provided by mean field dynamo theory. Such theories parameterize the field generation in solar-like stars as an "α–Ω-dynamo," with the Ω-effect representing the generation of toroidal fields from poloidal by differential rotation, and the α-effect referring to the twisting of fields by helical convection (e.g., Moffatt 1978; Steenbeck et al. 1966). In mean field theory, both these effects are sensitive to rotation—the α-effect because it depends upon the helicity of the convection which itself senses the overall rotation rate, and the Ω-effect because more rapidly rotating stars are expected to possess stronger internal angular velocity contrasts (Brown et al. 2008). Thus the rotation–activity correlation among stars like the Sun is at least qualitatively consistent with our understanding of the dynamo process.

Toward the low-mass end of the main sequence, our observational and theoretical understanding is far less clear. Mohanty & Basri (2003) argued that a sample of M5–M9 dwarfs exhibited a common "saturation-type" rotation–activity relationship, with the observed chromospheric emission roughly independent of rotation rate above a threshold value. But measuring the rotation rates of M-dwarfs in the "unsaturated" regime of the rotation–activity correlation is difficult, since the corresponding rotational velocities imply modest rotational line broadening that is typically indistinguishable from other effects. Thus it is unclear whether activity in such stars increases gradually with rotation rate, or instead changes more suddenly than in massive stars. Ample observational evidence also indicates that surface magnetic activity is common in the mid-late M-dwarfs, with the fraction of stars that show chromospheric Hα emission reaching a maximum around M7 (West et al. 2004).

Very recently, the connection between such emission measurements and the underlying magnetic field has begun to be probed directly, using both magnetically sensitive FeH molecular line ratios (Reiners & Basri 2007) and Zeeman Dopper imaging of very rapidly rotating M-stars (Donati et al. 2006; Morin et al. 2008). These observations reinforce the view that strong magnetic fields can be built in low-mass stars, and that the fields can possess significant large-scale structure (Donati et al. 2006). Theoretically, a general expectation has been that stars of spectral types later than about M3.5 (with masses less than about 0.35 M_☉) might harbor global magnetic dynamos very different from those in more massive stars. Such low-mass stars are convectively unstable throughout their interiors, and so cannot operate an "interface dynamo" precisely like that presumed to act in solar-like stars. Helical convection may still be able to build large-scale magnetic fields, as for instance in the "α²" mean field models of Chabrier & Kuker (2006); others have argued that in the absence of organizing shear and a stable layer, the magnetic fields realized in such stars should be mostly on small scales, reflecting the typical size of convective eddies (Durney et al. 1993). Numerical simulations have suggested that fully convective stars can build magnetic fields on a variety of spatial scales, and that rotation may still influence the field strengths that are realized (Dobler et al. 2006; Browning 2008). Although these models are suggestive, a thorough understanding of how the strength and geometry of dynamo-generated magnetic fields might vary with rotation rate remains elusive.

Part of the problem lies in the difficulty of obtaining reliable estimates of magnetic activity in the M-dwarfs. Most previous studies have employed the chromspheric Hα line as a proxy for magnetic activity, but measurements of that line are fraught with uncertainties (e.g., Cram & Mullan 1979). With increasing activity, Hα first appears in absorption, then fills in the absorption with an emission core, then finally appears in emission. Thus an absence of observable Hα emission can reflect either no activity, or a moderate amount of activity. Chromospheric Ca ii emission in the H and K lines does not share this particular problem, but has proven difficult to probe—it lies blueward of the spectral region where most CCDs have high sensitivity, so has been very difficult to detect in already-faint M-dwarfs. The few M-dwarf studies that have observed the H and K lines have generally done so at low dispersion, since the faintness of these stars would otherwise result in very low signal-noise ratios. Only very recently, with the advent of both 10 m class telescopes and high-resolution spectrographs that are sensitive in the blue, has it become possible to examine the quiescent H and K emission from low-activity M-dwarfs at high spectral resolution, as done by Rauscher & Marcy (2006). Below, we draw extensively upon their data. Such high-resolution data are helpful in discerning weak Ca ii emission embedded in deep absorption cores (see, e.g., Walkowicz & Hawley 2009).

Additional information about the magnetic fields in M-dwarfs, and their variation with mass and rotation rate, has come from studies of stellar spindown. Stars are born rapidly rotating, arriving on the main sequence with velocities that vary from less than 3 to more than 100 km s⁻¹ (Bouvier et al. 1997). They slow over time through angular momentum loss via a magnetized stellar wind (e.g., Skumanich 1972; MacGregor & Brenner 1991). Theoretically, an expectation is that this rotational braking should depend upon the strength and geometry of the stellar magnetic field and hence also upon rotation rate (e.g., Weber & Davis 1967; Matt & Pudritz 2008). Observationally, it appears that the time needed for stars to spin down is a strong function of stellar mass (e.g., Barnes 2003). Among the low-mass M-dwarfs, several authors have concluded that the spindown timescale increases with decreasing mass (e.g., Delfosse et al. 1998, Mohanty & Basri 2003). Indeed, most observably rotating M-dwarfs are of spectral types later than M3, suggesting that more massive stars have already spun down to below a detection threshold. The interpretation of these results has been slightly clouded by the small number of stars for which rotation has been measured, and by a systematic bias towards younger stars with decreasing stellar mass (e.g., West et al. 2006). Nonetheless, it has been tempting to ascribe the apparent mass dependence of spindown times to systematic changes in the magnetic field strength or morphology (e.g., Barnes 2003; Durney et al. 1993). In a related vein, West et al. (2008) recently used Sloan Digital Sky Survey (SDSS) measurements of Hα emission, in conjunction with a model for galactic dynamical heating of a stellar population, to argue that the "activity lifetime"—the period during which stars exhibit detectable emission—increases significantly between spectral classes M3 and M5. This finding is a further hint that some aspect of the field generation process, or the atmospheric heating mechanism, may change at approximately the mass where stars become fully convective.

These observational and theoretical findings raise a few key questions. One concerns the apparent rarity of measurable rotation in early-M field dwarfs. Stauffer & Hartmann (1987), for instance, found rotational broadening above their detection threshold (v sin i≈ 10 km s⁻¹) in only about 5% of the roughly 200 field M-dwarfs they examined. Likewise, Marcy & Chen (1992) found only five field M-stars (out of a sample of 47) rotating more rapidly than their detection threshold of roughly 3 km s⁻¹. Is this paucity of rotators a genuine feature of early-M dwarfs? That is, does it persist in larger samples with slightly lower detection thresholds? And how does the fraction of observed rotators vary with decreasing mass in the M-dwarfs? Another, related question is whether there is any sign of a marked transition in the rotational properties of low-mass stars at the onset of full interior convection—if fully convective stars rotate more rapidly, then this would imply larger spindown times, and so hint at changes in magnetic wind braking. It is already fairly clear that very low mass stars in the late M and L spectral types do tend to rotate more rapidly (Reiners & Basri 2008), but it is uncertain whether this transition to rapid rotation occurs near M3.5, where full convection sets in, or at somewhat lower masses. Quantifying this apparent transition could shed light on whether it is due to a change in the dynamo, or instead to decreasing surface ionization effects that might prevent effective mass and angular momentum loss. The former would probably imply a transition in rotation rates near M3.5, whereas the latter might be more likely in late-M stars, where the outer layers become increasingly cool and neutral (Mohanty et al. 2002). A final question is whether rotation continues to be linked to magnetic activity in M-dwarfs, as seen in more massive stars (e.g., N84), and in particular if there is any change in this linkage at the mass where stars become fully convective. These questions motivate the work described here.

In this paper, we attempt to measure rotation rates for a largely unbiased sample of 123 M-dwarfs, using Keck HIRES data from the California Planet Search Program. Using cross-correlation of the high-resolution stellar spectra with a slowly rotating template, we determine the fraction of stars that rotate more rapidly than about 2.5 km s⁻¹ in each spectral type. We analyze the correlation between our measured rotational velocities and chromospheric activity measurements of the same sample (Rauscher & Marcy 2006). Our work is most similar in aims and methods to that of Delfosse et al. (1998), who analyzed the rotation and activity of 118 field M-dwarfs. Indeed, owing to the still small number of low-mass stars accessible to high-resolution spectrographic observations, there is substantial overlap (48 stars) between our target stars and those of Delfosse. However, our sample includes more early-M stars and fewer late-M ones; thus our work can further quantify the apparent rarity of rapid rotators in the early-M dwarfs, and help elucidate the transition in rotation rates throughout the M spectral type. We describe the observational sample and the methods used to measure rotation rates in Section 2. In Section 3, we present our rotation analysis, and in Section 4 we relate these to estimates of chromospheric and coronal activity. Section 5 contains a discussion of the kinematics and ages of the stars in our sample. We close in Section 6 with brief comments on the implications of this work.

2.1. Observational Sample

2.2. Methodology

Measuring the rotation rates of field low-mass stars is challenging for several reasons. Most of these stars appear to rotate so slowly that rotational line broadening is typically small compared to instrumental broadening in moderate-resolution spectra. The threshold rotational velocity above which rotational broadening can be detected of course decreases with increasing spectral resolution, but obtaining high-resolution spectra of faint M-dwarfs is taxing even on the largest of telescopes. Faced with these difficulties, we attempt to extract as much information as possible from the HIRES data available as part of the Planet Search program, while recognizing that even higher resolution observations would likely reveal rotation in stars that to us appear rotationless.

To measure the rotation rates of stars in our sample, we cross correlate each stellar spectrum with a high-resolution spectrum of a slowly rotating comparison star. The cross-correlation function (XCF) of the spectra, as defined here, has a minimum at the shift (in pixels) where differences between the spectra are minimized (i.e., where a chi-squared goodness of fit test would give the best results). The actual value of the shift is immaterial to our rotation measurements, but the width of the XCF function at the minimum is a measure of the average width of features in the two spectra. In effect, the XCF measures the broadening of many different spectral features simultaneously, and so may be less affected by variations in individual lines from star to star. By comparing the widths of the XCF of many different stars versus a common template, we can determine which stars have the broadest lines, and thus are likely the most rapidly rotating. Below a certain threshold rotational velocity (described below), variations from star to star in the width of the XCF will be dominated by random noise, arising either from variations in the instrumental line broadening or from uncertainties in our determination of the XCF width. Note that the XCF used here, which depends on the difference between two spectra, differs slightly from the more typical approach of defining the XCF in terms of the product of the two spectra under consideration; we find that the approach used here is somewhat better for noisy spectra.

For this analysis, we selected a sample of slowly rotating "template" stellar spectra, identified as the stars in each spectral class with the narrowest auto-correlation function width. These template spectra were all observed using the "B1" decker on HIRES, yielding a resolution of R ≈ 60, 000; most of the remaining sample consists of observations with the "B5" decker (with R ≈ 45, 000). We then cross-correlated every observed spectrum with the template star of its spectral type; for spectral classes for which no "B1" observation was available, we used the template from a nearby spectral class. The template spectra used for each spectral class are identified in Table 1. Because the observations in our sample were taken from the Planet Search program, the stellar spectra are overlaid with lines from an iodine cell placed in the ray path in order to provide a precise RV standard (e.g., Butler et al. 1996). In order to minimize contamination of the XCF determination by the iodine lines, we perform the cross-correlation analysis in the I-band, where the iodine lines are weaker and less numerous than in the R-band. We also chose wavelength regions that are largely free of telluric lines, and which (in the M-dwarfs) are dominated by molecular features. These considerations led us to examine the I-band wavelength regions 6675–6765 Å, 7070–7160 Å, 7530–7565 Å, 7720–7780 Å, and 7870–7930 Å. These regions were divided into 15 Å segments, with a separate cross-correlation analysis performed for each segment; this step was necessary because we did not have an accurate absolute wavelength scale for each observation, so the dispersion varied across each order. Which segments are best for cross-correlation, and which are poor (either because they lack many individual molecular lines or because they are dominated by a single broad feature) varies with spectral type; not all of the above wavelength regions are useful at any one type.

Table 1. Stars Used as Slowly Rotating Templates

Type	Gl	HIP/Other
M0	353	46769
M1.5	414b	HD97101b
M2	411	54035
M2.5	436	57087
M3	1097	36338
M3.5	849	109388
M4	876	113020
M6	406	...

Notes. For cross-correlation of stars in spectral classes with no template listed, we used the template of a nearby spectral class. Because Gl 876 had the narrowest auto-correlation function of any star considered here, we used it to determine our quoted vsin i for all stars M4 or later, but cross-correlation against Gl 406 gave similar results.

Download table as:  ASCIITypeset image

Having calculated the XCF for each of these 15 Å regions, we fit its minimum with a Gaussian profile. Although the choice of a Gaussian profile is arbitrary, an excellent fit to the primary minimum in the XCF can usually be obtained. To automate this process over the ∼500 spectra considered here, we follow a simple procedure for identifying the feature to be fitted: we climb from the minimum of the XCF in both directions until we reach a threshold value that is 95% of a local XCF "continuum." We find that this procedure reliably distinguishes the global minimum of the XCF from many smaller local minima, which arise due to the fairly regular spacing of the molecular lines that dominate M-dwarf spectra. We then store the full-width at half-maximum (FWHM) of the Gaussian fit for each XCF. Outlying FWHMs, which correspond to XCFs for which the Gaussian fit fails, are thrown out; a mean XCF FWHM is calculated for each observation using the remaining FWHM measurements. Finally, we calculate a mean XCF FWHM for each star by averaging over the individual observations. The standard deviation of our individual determinations of the FWHM is typically ≈0.3 pixels across the ∼20 different XCFs calculated for each observation; mean FWHM determinations for multiple observations of the same star typically have a standard deviation of 0.1–0.2 pixels. A typical dispersion in the spectra is about 1.3 km s⁻¹ pixel⁻¹, so that 0.3 pixels corresponds to 0.4 km s⁻¹.

The measured widths of the XCF for each star are related to the typical width of spectral lines in that star, and hence to its rotational broadening. To calibrate the connection between the XCF widths and rotational velocity, we constructed "spun-up" spectra of our slowly rotating templates, using a standard analytical expression for the rotational broadening (Gray 1992) with a limb-darkening coefficient of 0.6. As a small further correction, we convolved these broadened B1-decker observations with a Gaussian profile chosen to mimic the degraded resolution of the B5-decker observations that constitute the bulk of the sample. The width of the Gaussian convolution profile was fixed by comparing ThAr calibration spectra taken using both B1 and B5 deckers, so that B1 decker calibration spectra convolved with the Gaussian had the same auto-correlation FWHM as B5 spectra. We then cross-correlated these rotationally broadened templates with the original spectrum (presumed slowly rotating on the basis of its narrow auto-correlation function), and calculated the typical FWHM of the XCF, as described above, as a function of rotational broadening velocity. The resulting calibration of FWHM versus vsin i is shown in Figure 1. This figure indicates that we are not reliably sensitive to rotational velocities less than about 2.5 km s⁻¹: the broadening of the XCF induced by smaller rotational velocities is not significantly greater than the uncertainty in our determination of the FWHMs. Above this threshold, an error of 0.2 pixels in the FWHM determination implies a 1σ uncertainty of well under 1 km s⁻¹ in the vsin i measurement.

Zoom In Zoom Out Reset image size

Also striking in Figure 1 is the larger FWHM of the M0 template with respect to an M4 template at the same rotation rate. Our other slowly rotating template observations, with spectral classes between M0.5 and M3.5, have typical FWHMs intermediate between these two. This variation in typical line width likely reflects changes in the turbulent broadening as a function of mass: decreasing luminosity implies declining convective velocities (with v_c ∝ L^1/3_c, with v_c being the typical convective velocity and L_c the luminosity that must be carried by convection), so this translates to less broadening in the least massive stars. A similar decrease in the typical width of cross-correlation profiles with decreasing mass was noted by Delfosse et al. (1998). Likewise, Valenti & Fischer (2005) noted a nearly linear decline in the turbulent macrovelocity with decreasing effective temperature, with v_turb declining by about 1 km s⁻¹ per 650 K for effective temperatures below 5770 K. This is somewhat steeper than would be expected from the variation in luminosity alone; that scaling would predict, for instance, a drop of about 0.6 km s⁻¹ in going from T_eff of 5770 to 5120 K (since v_c ∝ T^4/3_eff). Other effects may also play a role in influencing the XCF FWHM—e.g., the thermal broadening of the molecular lines declines with decreasing temperature, and for these stars (with spectra dominated by molecular lines) this effect would also tend to lead to a decrease in typical line width.

By spectral type M4, the broadening in our observations appears to be almost entirely instrumental: applying our cross-correlation technique to ThAr spectra with the B1 and B5 HIRES deckers gave XCF FWHM ∼5 pixels, comparable to the width of the XCF in our slowly rotating template stars at that spectral class. This is broadly consistent with extrapolating the Valenti & Fischer (2005) relation for macroturbulent velocities into the low-T_eff range probed here; their model would predict zero turbulent broadening below T_eff ≈ 3200 K.

An exception to the trend toward decreasing line width occurs in the very lowest mass stars in our sample. All six of the M4.5–M6 stars have slightly broader lines (as measured, e.g., by the FWHM of the auto-correlation function) than the slowly rotating M4 template. This may indicate that a small amount of unresolved rotational broadening is present in all the stars later than M4, including those we list here as non-detections (and discuss below).

Our rotation measurements are collected in Table 2. There we indicate our estimates of vsin i in km s⁻¹, together with identifying information for each star. The most striking feature that emerges from our measurements is how rare measurable rotation is: of the 122 M-dwarfs included in our sample, we detect significant rotational broadening in only seven. Another six stars have XCF FWHMs that are ≈2σ larger than the typical FWHM (with σ defined here as the typical standard deviation among determinations of the FWHM for multiple observations of one star), but manual inspection suggests that these are still not readily distinguishable from the slowly rotating template; these stars are noted as "marginal outliers" in Table 2. In practice, we are sensitive to rotational velocities only above ≈2.5 km s⁻¹. We have indicated upper limits to vsin i for the remaining stars in Table 2; these are stars whose XCF is indistinguishable from that of the slowly rotating template at the ≈2σ level.

The distribution of observed rotation rates varies significantly with spectral type. Of the 69 stars in the M0–M3 spectral classes, only 2 (3%) had measurable rotation: hip 63510 (with v sin i ≈ 9.5 km s⁻¹) and AU Mic (with an upper limit to vsin i of 8.5 km s⁻¹); two other stars, HD 95650 and Gl 191, show marginally significant excess broadening, but are still limited to vsin i ≲3.0 km s⁻¹. (Gl 272 also shows evidence of rotational broadening, but our measurements of its rotation velocity were inconclusive.) The fraction of measurable rotators appears to increase around M3: of the remaining 54 M-dwarfs with spectral types M3 and later in our sample, 5 (9%) had measurable rotation rates, with another 2 showing marginally significant broadening. There are only 6 stars in our sample in the spectral classes M4.5-M6, but 2 of them (hip 37766 and Gl 1245b) are significant detections, and a third (Gl 83.1) shows a small amount of excess broadening. The fraction of stars observed to rotate above our detection threshold of ≈2.5 km s⁻¹ is displayed as a function of spectral type in Figure 2. The error bars shown there are purely statistical, and correspond to 1.5σ errors drawing from a binomial distribution in each mass bin. Because only 17 stars later than M3.5 are included in our sample, the error bars on the rotation fractions in that mass range are large. The conclusion that rotation more rapid than 2.5 km s⁻¹ is highly uncommon in the M0–M2.5 spectral types, as previously suggested by Marcy & Chen (1992), appears to us to be robust. If we instead calculated the error in the rotation fraction by augmenting or subtracting all measured FWHMs by 1σ (with σ again defined not as the standard deviation of the sample, but as a typical standard deviation of the FWHM determinations for multiple observations of the same star), and adopted an arbitrary 2.5 km s⁻¹ threshold for detection, the error bars in Figure 2 would be smaller—typically only one or two stars lie within one standard deviation of this cutoff.

Zoom In Zoom Out Reset image size

The underlying distribution of XCF FWHMs for stars in each spectral class, upon which our velocity measurements are based, is shown in Figure 3. The dashed vertical lines in each panel indicate the FWHM of the narrow-lined template in each class, rotationally broadened to vsin i = 2.5 km s⁻¹. The detected rotators are generally clear outliers in this distribution. Furthermore, throughout the M1–M3 spectral classes, no stars are particularly close to the ∼2.5 km s⁻¹ detection limit. There is an overall shift of the average FWHM toward smaller values with decreasing mass, as also realized in Figure 1. This shift is consistent with the overall decline in convective luminosities in going toward decreasing mass.

Zoom In Zoom Out Reset image size

We believe that the higher fraction of detected rotators in the M3.5–M6 spectral classes, relative to the earlier M subtypes, is not simply a result of the larger FWHM in the early-M slowly rotating template stars. We also performed cross-correlation of the early-M types with one of the late-M template stars, which had the narrowest auto-correlation FWHM of any star in our sample, in order to determine whether cross-correlation with the broader M0 template was "hiding" some slow rotators. No new rotation detections in the early-M subtypes emerged from this analysis. In general, we found that cross-correlation with a template of a different spectral type often led to slightly larger values of the XCF FWHM, owing to small mismatches between the two spectra (and to XCFs that are not fitted as well by a simple Gaussian). Typically, the "error" associated with this cross-correlation across different spectral types is less than about 0.5 km s⁻¹. Similarly, we also examined whether our rotational detection threshold in stars later than M4 might be somewhat higher than in M3–M4 stars, owing to the slightly broader lines of the XCF template used for the very lowest mass stars. Cross-correlating all the M4.5–M6 stars with our M4 template did not significantly change our determinations of rotational velocity.

Our rotation measurements are generally consistent with prior estimates. Our six clear rotation detections are HIP 63510 (vsin i ≈9.7 km s⁻¹), HIP 92403 (GJ 729, 4.0 km s⁻¹), HIP 112460 (GJ 873, 3.5 km s⁻¹), AD Leo (Gl 388, 2.7 km s⁻¹), HIP 37766 (GJ 285, 4.6 km s⁻¹), and Gl 1245b (7.0 km s⁻¹). Johns-Krull & Valenti (1996) used R = 120, 000 spectra to obtain a rotational velocity for GJ 729 (v sin i = 3.5 ±  0.5 km s⁻¹) comparable to our own measurement. Several of the detected stars were also observed by Reiners & Basri (2007) using HIRES in a slightly lower resolution setting; they estimated a vsin i of 4 km s⁻¹ for GJ 729, in agreement with our measurement, and find vsin i ≈3 km s⁻¹ for AD Leo, also in reasonable accord with our findings. They did not detect rotation in GJ 873, which we find exhibits modest rotational broadening. Delfosse et al. (1998), on the other hand, estimated vsin i ≈6.9 km s⁻¹ for the same star; they also found a vsin i of 6.2 km s⁻¹ for AD Leo, likewise in appreciable disagreement with our estimate. Although the reasons for this discrepancy are not clear, it may partly stem from the slightly different strategies employed here and in Delfosse et al. (1998) for choosing a slowly rotating template spectrum: they adopted a common K0 template for comparison with almost all of their stars, whereas we choose a different template for each spectral class. Trial calculations in which we instead used a single M0 template for all spectral classes tended to yield larger velocity estimates, but were still consistent with vsin i <4 km s⁻¹ for GJ 873. Vogt et al. (1983) also measured rotation in AD Leo, finding vsin i of 5 km s⁻¹ using spectra with R ≈ 50, 000. Our measurement of the rotational velocity of Gl 1245b is also identical to that of Reiners & Basri (2007), despite the somewhat different auto-correlation strategies employed here; Delfosse et al. (1998) also found v sin i = 6.8 km s⁻¹, consistent with our estimate. HIP 37766 (GJ 285) is the well-known flare star YZ CMi; both Reiners & Basri (2007) and Delfosse et al. (1998) also published rotation estimates for it, with our measurement consistent with the former (who suggested a value of 5 km s⁻¹) and somewhat smaller than the latter (6.5 km s⁻¹). Finally, the sole star earlier than M3.5 (other than AU Mic) for which we detected measurable rotation, HIP 63510 (Gl 494) was also discussed by Beuzit et al. (2004); they quote a measurement of v sin i = 9.6 km s⁻¹ (determined using the ELODIE spectrograph), in excellent agreement with our own determination.

The near-absence of measurable rotation in the early-M spectral classes suggests that magnetic braking is extremely effective in that mass range. Stars in this mass range are initially rapid rotators, arriving on the main sequence with vsin i as high as 80 km s⁻¹; even the slowest are typically initially rotating at ∼10 km s⁻¹ (Stauffer & Hartmann 1987). Thus the stars in our sample have slowed by at least a factor of 5. A detection vsin i threshold of ≈2.5 km s⁻¹ corresponds to rotation periods of about 13 days at M0 and about 8 days at M3.5 (where we have employed typical mass–radius relationships from, e.g., Ribas 2006).

The near-absence of measurable rotation between M0 and M3 is also consistent with the recent estimates of West et al. (2008), who derived "activity lifetimes" for a large sample of M-dwarfs observed as part of the SDSS. If rotation is correlated with magnetic activity in this mass range, as suggested by, e.g., Delfosse et al. (1998), then the fraction of stars showing magnetic activity should serve as a proxy for the fraction that are rotating more rapidly than some threshold value needed for measurable activity. West et al. (2008) estimate activity lifetimes of about 1 Gyr at spectral type M0, 0.5 Gyr at M1, 2 Gyr at M3, 4 Gyr at M4, and 7 Gyr at M5, suggesting that we should expect to see rotation in 20%–40% of stars in the M3–M4 mass range, and in less than 10% of stars from M0–M3. This is very roughly the case: the fraction of stars with measurable rotation is about 10% at M3/M3.5, and 30% in the M4.5–M6 grouping; earlier types show essentially no rotation. This close correspondence provides support for the view that rotation and magnetic activity are intimately linked throughout the mass range probed here. Furthermore, it suggests that the threshold velocity needed for magnetic activity detectable by West et al. (2008) is similar to the minimum velocity measurable in our sample. In the following section, we examine the relation of rotation and activity in more detail.

We turn now to estimates of the surface magnetic activity for the stars in our sample, and to the question of whether that activity is linked to the rotation rates we have so far measured. We focus here on the chromospheric emission in the Ca ii H and K lines, which have the advantage over Hα that they are much more easily observable in emission if there is any non-radiative heating. This arises partly because these lines are closer to LTE and partly because of the high contrast of the Planck function in the blue. The Ca ii H and K emission for our sample has previously been reported by Rauscher & Marcy (2006; hereafter RM06), who published analyses of the H and K equivalent widths (EWs) for all but four of the stars considered here. All the stars we detect as rotators are among the most active emitters in the RM06 compilation; more specifically, all the detected rotators are in the largest 10% of EW measurements, or are noted (in the case of AD Leo and GJ 873) as very active stars whose Ca emission could not be fitted to a Gaussian in the manner used by RM06. Thus all stars rotating above our detection threshold of approximately 2.5 km s⁻¹ appear to be active. The converse statement—that all active stars are detectably rotating—is, however, untrue, with the non-rotators in our sample spanning a wide range of Ca ii H and K EWs.

To analyze more quantitatively the connection between rotation and magnetic activity in our sample, we have constructed EW measurements analogous to those of RM06, and converted these to values of L_CaH/L_bol and L_CaK/L_bol—the luminosity in each line, normalized by the bolometric luminosity. Unlike the automated EW measurements of RM06, we fit a double Gaussian (to account for the absorption core) to each emission line in each spectrum by hand. Our EW values are computed using the same continuum regions as RM06. The individual line values can be found in A. A. West et al. (2010, in preparation). The conversion to L_Ca/L_bol facilitates comparison of activity across different M subtypes: comparing the EW measurement alone can be misleading, since the continuum strength varies by a factor of 50–100 from M0 to M5.5. To construct the L_Ca/L_bol values, we followed a procedure analogous to that of Walkowicz et al. (2004) and West et al. (2005), using fluxed spectra with known distances to calibrate the conversions. We measured the absolute flux in the continuum regions used to compute the EWs in the spectra of 21 nearby M-dwarfs. Using the bolometric corrections from Leggett et al. (1996, 2001), we computed L_bol and the Walkowicz χ (continuum/L_bol) for each star. L_Ca/L_bol values were calculated by multiplying the EW of each star by the average χ value at that spectral type (West & Hawley 2008).

The resulting estimates of L_Ca/L_bol are plotted versus vsin i in Figure 4. The H and K luminosity (as a fraction of the bolometric luminosity) is roughly constant across all the detected rotators. The strongest emission observed is L_CaH/L_bol ≈ 9.0 × 10⁻⁵ and L_CaK/L_bol ≈ 5.8 × 10⁻⁵, realized in the star Gl 494a (Hip 63510) rotating at vsin i ≈9.7 km s⁻¹. The stars without detectable rotation span two orders of magnitude in L_Ca/L_bol, with the strongest non-rotating emitters possessing normalized Ca luminosities comparable to those of the most rapidly rotating active stars in our sample. Thus measurable rotation (with vsin i >2.5 km s⁻¹) appears to be a sufficient condition for high activity, but not a necessary one.

Zoom In Zoom Out Reset image size

We presume that some of the highly active stars without measurable rotation might simply be at low inclination angles relative to our line of sight. Of the nine stars in our sample with "saturated" Ca ii emission (which we define as log  L_CaH/L_bol > − 4.6, approximately the activity level of the least active rotator), three are not detectably rotating. To estimate roughly how likely it is that these three stars happen to lie at fairly low inclination angles, we turn to the binomial function

$$\begin{equation} P(\mathit {k \ \mbox{out}\ \mbox{of}\ n}) = \frac{n!}{k! (n-k)!}\ p^k q^{n-k}, \end{equation} \tag{ 1 }$$

which gives the probability of observing k events in n trials, if the probability of an event on each trial is p (and q = 1 − p is the probability of the event not occurring in each trial). For our purposes here, n = 9 is the number of active stars observed, and p is the probability that a randomly chosen star will have an inclination angle less than or equal to a chosen angle i; on geometrical grounds, p = 1 − cos i. The binomial formula indicates that we would expect, for instance, to observe three out of nine stars at i < 45° about half the time—i.e., the probability of 3 such events occurring in 9 trials is about P = 0.50. Smaller inclination angles occur less frequently, so we can constrain the maximum saturation velocity from our observation that one-third of the active stars have projected velocities below our detection threshold. For example, if all active stars rotated at 6 km s⁻¹, then the likelihood of 3 or more non-detections (given a detection threshold of 2.5 km s⁻¹) would only be about 3%. On the other hand, if all active stars rotated at 4 km s⁻¹, then the likelihood that 3 or more stars would be at sufficiently low inclination to escape detection rises to 31%. From this we conclude that the true saturation velocity (in the mass range studied here) is probably less than 6 km s⁻¹; it could be substantially below that.

This relation between rotation and magnetic activity is qualitatively similar to that noted by prior authors who used Hα or X-ray emission as proxies for magnetic activity. Mohanty & Basri (2003) and Delfosse et al. (1998) both found that, in the M-dwarfs, chromospheric emission was consistent with a "saturation-type" rotation–activity correlation, with activity roughly independent of rotation rate above a threshold value; below that threshold, their samples—like ours—showed a wide range of activity levels. Pizzolato et al. (2003), who examined coronal X-ray emission, also found activity in the M-dwarfs to be "saturated" at a value L_x/L_bol ≈ 10^−3.3 in all stars with rotation periods shorter than about 10 days—roughly corresponding to our minimum detectable rotational velocity at M3.5. Despite this low threshold for detection, we have likewise found no measurable rotators with particularly low levels of activity, suggesting (as argued above) that any threshold rotational velocity needed for dynamo action must be small indeed.

On a more detailed level, the saturated emission level seen here—with log (L_Ca/L_bol) ≈−4.5—differs appreciably from the saturation plateau as observed in X-rays and, to a lesser extent, Hα (see Table 3 and discussion below). These tracers are both more luminous (relative to the bolometric luminosity) than the Ca ii emission discussed here, typically saturating at log (L_x/L_bol) ≈−3 and log (L_Hα/L_bol) between −3.5 and −4. The highest L_Ca/L_bol achieved in our sample are comparable to the largest values previously reported (using lower dispersion spectra) for the flux in the H and K bandpasses, normalized to the bolometric flux—e.g., by Noyes et al. (1984), who quote values of log (R'_hk) up to about −4.2 for a sample of more massive stars. The lower level of saturated Ca emission, relative to Hα and X-rays, presumably reflects the fact that less cooling occurs through the Ca ii channel than through the Balmer lines, and that the X-ray emission occurs in the corona rather than the chromosphere.

Further information about rotation and activity can be gleaned from Figure 5, which examines the H and K luminosities of our sample as a function of spectral type. The stars for which we have detected rotation are shown there in green. Three main insights emerge from this plot. First, across much of the mass range considered here, there is a clear separation in L_Ca/L_bol between highly active stars, most of which are detected as rotators, and a significantly less active population of non-rotators. At spectral type M3.5, for instance, the two detectably rotating stars have L_CaK/L_bol and L_CaH/L_bol nearly a factor of 10 greater than the most active stars with no measurable rotation. This separation is reminiscent of the "Vaughan–Preston gap" identified in more massive stars (e.g., Vaughan & Preston 1980; Noyes et al. 1984). It could plausibly reflect either two distinct modes of dynamo action, or a range of magnetic fluxes in which magnetic spindown is especially rapid (so that few stars linger in that flux regime for long). Alternatively, the gap could simply be an artifact of the relatively small number of stars in each spectral type bin.

Zoom In Zoom Out Reset image size

Second, the mean level of emission among relatively "inactive" stars declines with decreasing mass. The lower L_Ca/L_bol realized at late types may reflect a decreasing ability of the Ca lines to cool the chromosphere. Interestingly, similar analysis of Hα activity as a function of spectral type in field M-dwarfs (West et al. 2004) has shown no decrease in L_Hα/L_bol at these spectral types: instead, the mean Hα emission remained roughly constant until spectral types M5 or later. This may suggest that the Hα measurements trace primarily the "active" population defined above; in that population, L_Ca/L_bol also remains fairly constant across the mass range studied here. To further examine this issue, we measured Hα EWs for all the stars in our sample that showed detectable Hα emission; those measurements are listed in Table 3. (A few additional targets showed Hα absorption, but interpreting that absorption correctly would require more sophisticated modeling than employed here; we defer analysis of those targets to subsequent papers.) We converted the measured Hα EWs to L_Hα/L_bol values using the spectral-type-dependent χ values from Walkowicz et al. (2004). The resulting L_Hα/L_bol values, which are also listed in Table 3, show no systematic trends with spectral type. Our measurements of log (L_Hα/L_bol) almost all lie between −3.5 and −4, consistent with prior studies (West et al. 2004). All of the stars with detectable rotation also show Hα emission; furthermore, the Hα emitters that are not detectable rotators are also fairly strong in Ca H and K. The Hα emission as measured by L_Hα/L_bol is between 4 and 11 times larger than L_CaH/L_bol for all of these stars. These findings reinforce the view that Hα emission measurements trace the "active" population, which shows little or no dependence of emission (normalized to bolometric luminosity) on rotation rate or spectral type.

Thirdly, although the most active stars are generally detectably rotating, a few are not; some of these stick out from the larger population of inactive non-rotators, and may be good candidates for photometric period determinations or follow-up observations at even higher resolution. At spectral class M2, for instance, the star Gl 569a has L_CaK/L_bol and L_CaH/L_bol about three times larger than the next most active star of that class, yet is not detected as a rotator. After performing this analysis, we found that Gl 569a was claimed by Marcy & Chen (1992) to rotate at 4.0 km s⁻¹; this is inconsistent with our measurement of the XCF FWHM. Very recently, Kiraga & Stepien (2007) found a photometric rotation period (with marginal significance) for this star of 13.7 days, which would yield a vsin i of about 1.4 km s⁻¹ if viewed edge-on. Such a low velocity is striking given its high level of activity. One possibility is that the true saturation velocity is actually that low (or even lower). This would be consistent with our observations, which only provide an upper limit to the saturation velocity. Other stars that have remarkably high activity but no measurable rotation include Gl 362 (spectral type M3) and Gl 406 (M6). As noted above, the existence of a few active stars without measurable rotation is consistent with the expectation that a small number of stars in a random sample will happen to lie nearly pole-on with respect to our line of sight.

There are a few other stars which are common to both our sample and that of Kiraga & Stepien (2007). The only such star with a detected rotation velocity (Gl 494) has a short period (2.9 days) and high activity. The stars with near-solar periods—Gl 205 (33.5 days) and Gl 382 (21.6 days)–were non-detections in rotation velocity and, in Ca ii emission, both lie near the top of the inactive group for their spectral class (M1.5). Finally, Gl 411 (M2, 48 days) and Gl 699 (M4, 130 days) lie near the bottom of their groups in activity.

All the stars at spectral class M4.5 have fairly high Ca luminosities, and so might also be good candidates for follow-up observation. As noted in Section 2, we may have no truly non-rotating template star in that mass range to compare with, so our detection threshold in vsin i may be slightly higher than in earlier spectral types. Still, any star rotating more rapidly than 3.0 km s⁻¹ in that spectral type should easily have been detected; cross-correlation with narrow-lined templates from other spectral classes did not reveal any such detections. Finally, the most active star in the M0 class was Gl 410 (HD 95650); its XCF FWHM was the largest of any star in that subclass, but was still smaller than that expected for a star rotating at 2.5 km s⁻¹, and so we had identified it as a "marginal outlier" in Table 1. It, too, would be an excellent candidate for follow-up observations.

Stars spin down as they age (e.g., Skumanich 1972; Barnes 2003), so for stars of a given spectral type, it is natural to assume that those which are measurably rotating are generally younger than those which are not rotating. When comparing stars of different spectral types, however, the situation is less clear: because the rate at which stars spin down may be a function of stellar mass (e.g., Wolff & Simon 1997; Terndrup et al. 2000; Barnes 2003), a young star of one type may still rotate more slowly than a somewhat older star of another type. Thus, the higher frequency of rotation in late-M dwarfs (relative to early-M dwarfs) in our sample could result if either (a) the spindown times in the late-M population are systematically longer or (b) the late-M population is systematically younger.

To explore which of these possibilities is more likely, we attempt here to constrain roughly the ages of the stars in our sample. In particular, we aim to determine whether our M0–M2 population, which is almost universally non-rotating, is systematically older than the population of types M3 or later. Determining the ages of main-sequence stars is notoriously difficult (e.g., Mamajek et al. 2008), and no method has yet been devised that yields precise age estimates in the mass range studied here. But as a crude means of constraining the age of a sample of stars, many authors have turned to analyses of kinematic data (see, e.g., Leggett 1992; Eggen et al. 1962; West et al. 2006). The idea behind a kinematic age estimate is straightforward: over time, stars are dynamically heated through interactions with molecular clouds and other gas in the galactic disk (Spitzer & Schwarzschild 1951, 1953), leading to an increase in their velocity dispersions with age (Wielen 1977). Thus, older stars should have higher velocity dispersions on average than younger stars; by extension, stars with high velocity dispersions are likely to be older than low-velocity stars. More specifically, the (U,V,W) space motions of stars are approximately correlated with their metallicity and, by inference, their age: e.g., stars in certain regions of the (U,V) plane are likely to belong to the "young disk" population, whereas "old disk" or halo stars tend to lie elsewhere on that plane (e.g., Eggen et al. 1962; Carney et al. 1990; Leggett 1992). Below, we follow the procedure described by Leggett (1992) for assigning the stars in our sample to different dynamical populations, and hence constraining their ages.

To carry out this procedure, we required accurate (U,V,W) space motions for the stars in our sample. These in turn rely on measurements of the stars' proper motions, parallaxes, and RVs. We used proper motions and parallaxes from the Hipparchos catalog for all but a handful of stars; for those not included in the Hipparchos sample, we used the LSPM-North catalog (Lepine & Shara 2005), Tycho-2 (Hog et al. 2000), the Luyten Half-Second catalog (Salim & Gould 2003; Bakos et al. 2002), the USNO catalog (Harrington & Dahn 1980), or the Yale catalog of parallaxes (Jenkins 1963). The data are tabulated in Table 4. For all but 26 of the stars studied here, Nidever et al. (2002)—who also used Planet Search spectra—published mean barycentric RVs accurate to within about 0.4 km s⁻¹ in this mass range; we have used these RV measurements when possible. For the remaining 26 stars, we derived our own RV measurements. Rather than employing the complex spectral fitting procedure used in Nidever et al. (2002), we measured RVs by cross-correlating each star of unknown RV with a known-RV star observed on the same night. We corrected for Earth's motion between observations using the "baryvel" procedure in IDL. To test our RV measurements, we first confirmed that we could recover values close to those of Nidever et al. (2002); for each target star that was observed more than once, we also derived multiple independent RV measurements by cross-correlating against at least two distinct known-RV stars. These tests lead us to believe that our RV measurements are accurate to ± 1 km s⁻¹, with slightly greater errors possible for stars with very high RVs. The RV measurements are also listed in Table 4. In two cases (Gl 272 and Gl 803), our cross-correlation procedure gave inconsistent measurements of the RV, so for those two stars we have taken RV values from the literature (Reid et al. 1995; Wilson 1963).

With RV measurements in hand, we calculated the (U,V,W) space motions of all the stars in our sample. We used the "gal_uvw" IDL library procedure, which essentially follows the method of Johnson & Soderblom (1987), modified to use the Dehnen (1998) values for the solar motion. The resulting values are listed in Table 4, with U there taken to be positive away from the Galactic center. (Note that this convention differs from that used in, e.g., Delfosse et al. 1998.) To constrain the kinematic ages of the stars in our sample, we turned to the classification scheme of Leggett (1992). We used her cutoffs in (U,V,W) space to assign each star to either the young disk (YD), old disk (OD), halo (H), or young-old disk (YOD). Leggett (1992) also included an OD–H population, which we have subsumed into the OD group. The "young-old disk," which Leggett describes as lying "around the edge of the young disk UV ellipsoid" is arbitrarily defined here to include stars whose (U,V) values are within 10 km s⁻¹ of the cutoff values for the YD; this definition seems to be consistent with that employed in, e.g., Delfosse et al. (1998), though that paper does not explicitly quote cutoff (U,V,W) values for each population. Although these populations correspond only in a loose statistical sense to stellar ages, this analysis nonetheless provide some hints about whether the ages of stars in our sample vary systematically with mass.

Some links between rotation rate and kinematic age are evident from the results in Table 4. Of the seven detectably rotating stars, five are classified as "young disk," the youngest population; the others (Gl 873 and Gl 1245b) are classified here as YOD, but lie fairly close to the young disk (U,V,W) region identified by Leggett (1992). No OD star is detected as a rotator, even though 40% of our sample is classified as belonging to the OD. Being kinematically "young" (i.e., belonging to the YD or YOD disk) is, in our sample, a necessary but not sufficient condition for detectable rotation: most of the stars classified as "young disk" (34 out of 40) are non-rotators. This result suggests that even the "young disk" population includes stars that are old enough to have spun down below our detection limit.

The fraction of stars that are kinematically "young" varies slightly with mass. For instance, 33% of the stars with spectral types M0–M1 are classified as young disk (of the remainder, 24% are YOD and 42% OD), while 40% of the population with types M3.5 or later is classified as YD. But such modest variations with spectral type in the fraction of kinematically young stars do not appear to be sufficient to explain the low fraction of observed rotators in the early-M classes. For instance, suppose that the true "rotation fraction" was actually independent of spectral type, and given by the fraction of YD and YOD stars in the M0 to M2.5 types in which we detect rotation. This would imply that 5.4% of stars should be detectably rotating, independent of the spectral type (since we detect two rotators out of 37 YD or YOD M0–M2.5 stars). In the small fraction of our sample (six stars) later than M4, then, we would expect to see two or more rotators only 3.8% of the time, again presuming that this can be assessed using the binomial function discussed above. The fact that we see two rotators later than M4 is thus fairly compelling evidence that the rotation fraction in those spectral types differs intrinsically from the 5.4% we would infer from analysis of the early-Ms alone. Conversely, if we suppose that the true, type-independent rotation fraction is accurately given by the fraction of stars later than M4 that we detect as rotators, we would expect to see far more rotators in our early-M sample than are actually observed. Specifically, if the true probability of detectable rotation is 0.33 (the fraction of stars later than M4 that show rotation in our sample), the probability of randomly drawing two or fewer rotators from a sample of 37 stars (the number of YD and YOD stars in types M0 to M2.5) is less than 10⁻⁴.

We therefore conclude that systematic age differences among the spectral types in our sample cannot account for the low fraction of observed rotators in the early-M subsample, and are unlikely to account for the relatively high fraction of rotators in stars with types later than M4. The number of kinematically "young" stars (YD and YOD) in the early-M subsample is still large enough to render statistically significant constraints on the rotation fraction; indeed, we can rule out intrinsic rotation fractions in the early Ms higher than about 20% at the 99% confidence level. Our finding that the fraction of detectably rotating stars is significantly larger in the late-M subtypes is on statistically weaker ground because of the small number of stars involved. Still, we can also rule out very low true rotation fractions (<5%) in stars later than M4 at about 97% confidence—that is, there is a 97% chance that we would have detected fewer rotators in those spectral classes if the true rotation fraction was only 5%. More observations of kinematically young stars in those spectral types would allow tighter constraints on the true rotation fraction, and hence on the spindown of fully convective stars.

We have analyzed the rotation and chromospheric activity of 123 nearby M-dwarfs, using high-resolution spectra taken at Keck as part of the California Planet Search program. We found only seven stars rotating more rapidly than our vsin i detection threshold of approximately 2.5 km s⁻¹; in a further seven stars, we only able to place slightly weaker upper limits on vsin i, ranging from 2.5 to 6 km s⁻¹.

All but three of the detected rotators are of spectral type M3.5 or later, and are therefore probably fully convective, despite the fact that the bulk of our sample (72%) consists of stars in the M0–M3 classes. This finding suggests that rotation more rapid than about 2.5 km s⁻¹ is very rare in the early Ms, occurring in less than 10% of stars; it also suggests, though admittedly with still-poor statistics, that measurable rotation becomes more common with decreasing mass. This, in turn, bolsters the view that the process of rotational braking becomes less effective in fully convective objects—as also suggested, for instance, by observations of rapidly rotating L-dwarfs (e.g., Reiners & Basri 2008), and by studies of the "activity fraction" as a function of stellar mass (West et al. 2008).

We find that rotation is linked to magnetic activity in the following sense: all stars in which we detect rotation are active, but not all active stars are detectably rotating. It is plausible that all the active non-rotating stars happen to lie at low inclination angles relative to our line of sight. Examining the luminosity in the Ca ii H and K lines, normalized to the bolometric luminosity, reveals that over much of the mass range considered here, our sample can plausibly be separated into two fairly distinct populations: one of highly active stars, most of which are detectably rotating, and another of non-rotating stars that generally possess L_Ca/L_bol at least an order of magnitude smaller. The separation between these two groups is analogous to the Vaughan–Preston "gap" claimed in more massive stars (e.g., Vaughan & Preston 1980). The "active" population in our sample has L_CaH/L_bol and L_CaK/L_bol ≈ 10^−4.5, roughly independent of rotation rate, consistent with the "saturation-type" rotation–activity relationship that has previously been claimed in some fully convective stars (e.g., Mohanty & Basri 2003). The level at which L_Ca/L_bol "saturates" in our sample is roughly consistent with prior measurements of the R'_hk parameter in more massive stars (e.g., N84), and somewhat lower than the values at which L_x/L_bol or L_Hα/L_bol saturate (e.g., Pizzolato et al. 2003; Delfosse et al. 1998).

Our findings are actually fairly similar to those for more massive stars. Rotation is rare in the early M-dwarfs of our sample, but this may be more reflective of the decline in stellar radii with decreasing mass than of fundamentally more effective braking (relative to more massive stars) in that mass range. Our detection limits on vsin i, though representative of the best that can be done using cross-correlation of R ≈ 45000–60000 spectra on a 10 m telescope, still rule out only rather short rotation periods: M-dwarf radii are so small that a v sin i of 2.5 km s⁻¹ corresponds to a rotation period of about 8 days at M3.5. Thus, a star would have to have a rotation period considerably shorter than that of the Sun (whose period is about 28 days) to be detected here as a rotator. Our detection threshold in vsin i corresponds to periods of less than about 20 days for solar-type stars, but less than 12 days for all the low-mass stars considered here.

Our inferences about stellar activity in the M-dwarfs are also surprisingly in line with what has been obtained in more massive stars. The Ca ii H and K luminosities L_Ca/L_bol of active, rotating stars in our sample correspond well to the highest values of R'_HK noted in prior studies of Sun-like stars (e.g., N84). Furthermore, our threshold period for detection of rotation corresponds fairly well to the period separating young, active stars in the N84 survey from old, less active ones. Put another way, all our detected rotators would, if they were more massive but had the same rotation period, be part of the "active" population in N84's sample; our non-rotators would generally (if similarly converted to greater masses) fall into the solar-like low-activity population in that sample. Furthermore, our data are also consistent with the idea that magnetic activity is connected in some fashion to the Rossby number ∼P_rot/τ_c, with P_rot the rotation period and τ_c a typical convective overturning time (e.g., N84). Estimates of the overturning time are imprecise—in reality, the convective flows in stellar interiors are not well characterized by a single overturning time, since they consist of a vast range of motions with a variety of spatial scales, which in general vary significantly with depth (e.g., Browning 2008). But crude estimates of an average overturning time for energy-carrying eddies in the mass range considered here vary from about 30 days to about 100 (e.g., Saar 2001; Kiraga & Stepien 2007; Reiners et al. 2009). Our rotation detection threshold thus corresponds to Rossby numbers no greater than 0.4 for the most massive stars in the sample, and less than this (∼0.1) for the fully convective objects. The fact that activity does not seem to increase with rotation velocities above our detection threshold is thus broadly consistent with what we know of the rotation–activity correlation in more massive stars, for which activity is generally thought to "saturate" at Rossby numbers of about 0.1 (see Reiners et al. 2009). On the other hand, as noted above, we cannot strictly rule out true "saturation" velocities of less than about 6.0 km s⁻¹, which would in turn imply Rossby numbers closer to unity.

Probably the most striking difference between our activity measurements and those in more massive stars is the larger spread in L_Ca/L_bol values we observe. Some of the low-activity, non-rotating stars in our sample have L_Ca/L_bol as low as 10^−6.5, more than a factor of 10 lower than the smallest values of R'_HK reported in N84. It is unclear why the chromospheric luminosities of these least active M-dwarfs extend to much lower values than in more massive stars, particularly given that the L_Ca/L_bol achieved by the most active stars appears roughly independent of stellar mass. This may well simply reflect the decreasing efficiency of the Ca ii lines as a cooling channel, as noted above. It may also just reflect the fact that the N84 survey (taken through a 1 Å bandpass) may not have been sensitive to very low levels of Ca emission. Another possible explanation is that some of the stars in our sample are rotating much more slowly than any solar-like stars, and that activity continues to be related to rotation period in such slow rotators; testing this possibility will require photometric rotation period measurements for some of the inactive stars that we list as non-rotators.

The implications of our findings for dynamo theory are not entirely clear. On the one hand, there appear to be close analogs between dynamos in low-mass M-dwarfs (some of which are fully convective) and dynamos in more massive stars: both seem to obey some form of rotation–activity relation, both are capable of generating similar maximum chromospheric luminosities (L_Ca/L_bol), and both may possess fairly distinct active and inactive populations (separated by a "gap" in L_Ca/L_bol). Furthermore, the rotation period dividing the inactive, slowly rotating stars from their active cousins seems not to depend too strongly upon stellar mass. This latter point may reflect the central role of rotation in dynamo action, together with only modest variations in the convective turnover time τ_c with stellar mass. On the other hand, the fact that measurable rotation seems very rare in early-M stars in our sample—with only 2 M0–M2 stars in 70 showing vsin i in excess of 2.5 km s⁻¹—but fairly common in stars of spectral type M3.5 or later—where four stars in 17 are detectably rotating—indicates that something about stellar spindown changes near the onset of full convection in stars. Whether that change is due to variations in the strength of dynamo-generated fields, their geometry, their coupling to a stellar wind, or to some yet unforeseen effect, remains to be determined.

This work was supported in part by an NSF Astronomy and Astrophysics postdoctoral fellowship (AST 05-02413), and by research support from the Canadian Institute for Theoretical Astrophysics. Our observations were obtained from the W.M. Keck Observatory, which is operated as a scientific partnership by the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. We acknowledge the great cultural significance of Mauna Kea for native Hawaiians and express our gratitude for permission to observe from atop this mountain. G.B. acknowledges support from the NSF through grant AST-0606748.