Poking the Beehive from Space: K2 Rotation Periods for Praesepe

We continue to use the Praesepe membership catalog presented in Paper II, which includes 1130 cluster members with membership probabilities ${P}_{\mathrm{mem}}\geqslant 50 \%$ as calculated by Kraus & Hillenbrand (2007) and 39 previously identified members too bright to be included by those authors in their catalog for the cluster. We assign these bright stars ${P}_{\mathrm{mem}}=100$%. We also continue to use the photometry and stellar masses presented in Table 5 of Paper II. For most of our analysis, as in that work, we include only the 1099 stars with ${P}_{\mathrm{mem}}\geqslant 70 \%$.

In Papers I and II, we combined P_rot measurements from PTF data with P_rot measurements from Scholz & Eislöffel (2007), Scholz et al. (2011), and Delorme et al. (2011) to produce a catalog of 135 known rotators in Praesepe.⁴ Of these stars, 83 have a Kraus & Hillenbrand (2007) ${P}_{\mathrm{mem}}\gt 95 \%$.

To this catalog, we now add 180 P_rot measurements from Kovács et al. (2014); 174 of these stars have ${P}_{\mathrm{mem}}\geqslant 70 \%$. Forty-four stars have previous P_rot measurements by other authors: the majority of these measurements are consistent to within 0.5 days, but 13 stars have significantly discrepant P_rot measurements (see Table 1). In all 13 cases, Kovács et al. (2014) measure the P_rot to be at least twice as long as previous authors. This discrepancy undermines the validity of the other Kovács et al. (2014) P_rot values, and we therefore retain the previous literature P_rot wherever possible.

In total, we add 136 rotators with non-K2 P_rot to our Praesepe catalog, including 131 with ${P}_{\mathrm{mem}}\gt 70 \%$. The mass–period data for Praesepe members with existing P_rot measurements is shown in Figure 1.

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The pointing in K2 is held in an unstable equilibrium against solar pressure by the two functioning reaction wheels. The spacecraft rolls about the boresight by up to 1 pixel at the edge of the focal plane. To correct for this, thrusters can be fired every 6 hr (if needed) to return the spacecraft to its original position. This drift causes stars to move in arcs on the focal plane, inducing a sawtooth-like signal in the 75-day light curve for each star (Van Cleve et al. 2016).

Several groups have developed methods for extracting photometry and removing the effect of the pointing drift from the raw light curve. We tested the light curves produced using several detrending methods (Vanderburg & Johnson 2014; Aigrain et al. 2016; Luger et al. 2016), as well as our own (Paper III). We chose to use the light curves generated by the K2 Systematics Correction method (K2SC; Aigrain et al. 2016) for our analysis, as this approach does the best job of removing systematics and long-period trends, which can bias period measurements or completely wash out periodic signals.

Aigrain et al. (2016) use a semi-parametric Gaussian process model to correct for the spacecraft motion. These authors begin with the light curves and centroid positions produced by the Kepler Science Operations Center pipeline. They then simultaneously model the position-dependent, time-dependent, and white-noise components of the light curve. The time-dependent component should describe the intrinsic variability of the star, and the position-dependent component should describe the instrumental signal resulting from the spacecraft roll. In cases where a significant period between 0.05 and 20 days is detected in the raw light curve, Aigrain et al. (2016) use a quasi-periodic kernel to describe the time-dependent trend; otherwise these authors use a squared-exponential kernel.

Since we wish to measure stellar variability, we remove only the position-dependent trend. The provided light-curve files include the position-dependent, time-dependent, and white-noise components in separate columns for both the simple aperture photometry (SAP) and pre-search data conditioning (PDC) pipeline light curves (Van Cleve et al. 2016). We use the PDC light curves, and compute the final light curve for our analysis by adding the white noise and time-dependent components, and then subtracting the median of the time-dependent component.⁵

We use the Press & Rybicki (1989) FFT-based Lomb–Scargle algorithm⁶ to measure rotation periods. We compute the Lomb–Scargle periodogram power for 3 × 10⁴ periods ranging from 0.1 to 70 days (approximately the length of the Campaign).

The periodogram power, which is normalized so that the maximum possible power is 1.0, is the first measurement of detection quality. The normalized power, P_LS, is related to the ratio of ${\chi }^{2}$ for the sinusoidal model to ${\chi }_{0}^{2}$ for a pure noise model (Ivezić et al. 2013):

$$\begin{eqnarray}&&{P}_{\mathrm{LS}}=1-\displaystyle \frac{{\chi }^{2}}{{\chi }_{0}^{2}}.\end{eqnarray} \tag{ 1 }$$

A higher P_LS indicates that the signal is more likely sinusoidal, and a lower P_LS indicates that it is more likely noise. Therefore, P_LS gives some information about the relative contributions of noise and periodic modulation to the light curve. The distribution of P_LS for all stars in our sample is shown in Figure 4. We do not impose a global minimum value for P_LS. Instead, we compute a minimum significance threshold for each light curve.

We identify periodogram peaks using the scipy.signal.argrelextrema function, and define a peak as any point in the periodogram higher than at least 100 of the neighboring points. This value was chosen after some trial and error, and has the benefit of automatically rejecting most long-period trends, because the periodogram is logarithmically sampled and has fewer points at long periods. Long-period trends appear as a peak near 60–70 days with a series of harmonic peaks; these are generally rejected by argrelextrema. When there is a sinusoidal stellar signal in the light curve, it dominates the periodogram above any trends and is detected by argrelextrema.

We determine minimum significance thresholds for the periodogram peaks using bootstrap re-sampling, as in Paper III. We hold the observation epochs fixed and randomly redraw and replace the flux values to produce new scrambled light curves. We then compute a Lomb–Scargle periodogram for the scrambled light curve, and record the maximum periodogram power. We repeat this process 1000 times, and take the 99.9th percentile of peak powers as our minimum significance threshold for that light curve. A peak in our original light curve is significant if its power is higher than this minimum threshold, which is listed in Table 3. We take the highest significant peak as our default P_rot value; 23 of our targets show no significant periodogram peaks.

We combine automated and by-eye quality checks to validate the P_rot. The automated check comes from the peak periodogram power along with the number of, and power in, periodogram peaks beyond the first. Following Covey et al. (2016), we label a periodogram as clean if there are no peaks with more than 60% of the primary peak's power. The presence of such peaks may indicate that the P_rot measurement is incorrect. The clean flag is included in Table 3; only 46 K2 detections are not clean.

In addition, since instrumental signals can occasionally be detected at high significance, we inspect the periodograms and phase-folded light curves by eye to confirm detections. Clearly spurious detections are flagged as Q = 2, and questionable detections as Q = 1. This is similar to the approach used in Paper III, but we are more generous here and try to identify only the most obvious bad detections. In total, we remove 94 light curves. Additionally, a Q = 3 flag indicates that there were no significant periodogram peaks; as noted earlier, this occurred for 23 stars.

The Q flag is separate from the clean/not-clean classification, and we do not change the Q value based on the clean/not-clean classification. We consider P_rot measurements with a clean periodogram and Q = 0 to be high-quality detections. In cases where we measure a K2 P_rot for a star with a P_rot in the literature, the agreement is generally excellent (see Section 5.1). This indicates that our methods produce reasonable and valid P_rot measurements.

Following McQuillan et al. (2013), we plot the full light curve and, for ${P}_{\mathrm{rot}}\gt 2$ days, vertical dashed lines at intervals of the detected period. We check that light-curve features repeat over several intervals. We identify six cases where the phased light curve looks reasonable, but the pattern identified by eye does not match that detected in the periodogram (see Figure 5 and the top panel of Figure 3), and we flag these with Q = 2.

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We also identify 13 light curves where the dominant periodogram peak is likely for half of the true period and there is double-dip structure in the light curve (see second panel, Figure 3). There is typically a periodogram peak at this longer period that is weaker than the dominant peak. This feature is common in stellar light curves and usually attributed to symmetrical spot configurations and/or an evolving spot pattern on the stellar surface (McQuillan et al. 2013).

In most Q = 2 cases, the phase-folded light curve does not look sinusoidal (third panel, Figure 3), and the light curve is likely just noise. We also remove three stars where the saturation strip from a nearby star crosses the target pixel stamp, and one where the target is extended and likely a galaxy based on its Sloan Digital Sky Survey Data Release 12 (SDSS DR12; Alam et al. 2015) image.

As part of our visual inspection, we also note cases where two or more periods are detected, i.e., due to multiple stars being present in the aperture (fourth panel, Figure 3), or where we find evidence for spot evolution (second panel, Figure 3). We assign flags for targets with multiple periods and with spot evolution: "Y" for yes, "M" for maybe, and "N" for no.

Finally, we note any other interesting light curve features, typically transits or eclipses (see the Appendix for a discussion of the latter light curves). An example set of our inspection plots is shown in Figure 5, and the plots for all of our objects are available as an electronic figure set.

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View figure set (794 panels)

Tarball of figure set images

We measure the amplitude of variability for a given star using the 10th and 90th percentiles (P₁₀ and P₉₀) of the light curve in counts. We calculate the amplitude in magnitudes as

$$\begin{eqnarray}&&2.5\times \displaystyle \frac{[{{\rm{log}}}_{10}({P}_{90})-{{\rm{log}}}_{10}({P}_{10})]}{2}.\end{eqnarray} \tag{ 2 }$$

This number may be slightly misleading, however, in cases where the median flux level varies over the course of the Campaign (a minor example is shown in the second panel of Figure 3). Therefore, we also calculate a smoothed version of the phase-folded light curve, and measure the amplitude as the percent difference between the maximum and minimum values of the smoothed light curve. This method, already used in Paper III, tends to underpredict the amplitude of very fast rotators. We list both amplitudes in Table 3, but use the amplitude calculated using Equation (2) for all analyses below. Our results do not change significantly when using the amplitudes calculated by either method.

We examine a co-added K2 image, a Digital Sky Survey (DSS) red image, and an SDSS (Alam et al. 2015) r-band image of each target to look for neighboring stars (see Figure 5). We use a flag of "Y" for yes, "M" for maybe, and "N" for no to indicate whether the target and a neighbor have blended PSFs on the K2 chip. Stars flagged as "Y" are labeled candidate binaries; we find 159 such targets, or 23% of stars with K2 P_rot.

To determine the likelihood that these are chance alignments, we offset the cluster positions by 15° in both R.A. and decl. and search for neighbors in the SDSS DR12. We restrict this search to objects with $g\leqslant 22$ mag, the SDSS 95% completeness limit. We find an SDSS object within 10'' (20'') of 8% (13%) of these offset positions. This suggests that at least 10% of Praesepe members have a very wide but bound companion, with separations on the order of 10³–10⁴ au at Praesepe's distance (181.5 ± 6 pc; van Leeuwen 2009). The other neighboring stars are likely background stars that could still contribute flux to the K2 light curve. Lacking the observations to confirm which neighboring stars are actually bound companions, we consider all of these stars to be candidate binaries in our analysis.

As in previous work, we identify candidate unresolved binaries that are overluminous for their color (see Figure 6). We identify a binary main-sequence (MS) offset by 0.75 mag for a given color from that of single stars (as in Steele & Jameson 1995). We then label as candidate binary systems stars with $(r^{\prime} -K)\lt 4$ that lie above the midpoint between the single-star and binary MSs (Hodgkin et al. 1999). This method is biased toward binaries with equal masses; thus, we are certainly missing candidate binaries with lower mass ratios. Indeed, confirmed multiples appear at all distances from the putatively single MS (as shown in Figure 6; also see Figure 3 in Paper III for a similar analysis in the better-surveyed Hyades). Further observations and analyses are required to confirm the binary status of all cluster members.

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We only apply this method to stars with $(r^{\prime} -K)\lt 4$ because the single-star MS is less apparent for stars redder than this value. The observed spread in magnitudes could be due to binary systems at a variety of mass ratios, or to increased photometric uncertainties for these faint red stars. Identifying candidate binaries in this regime requires more information than just photometry.

Surveys for multiple systems in Praesepe have been undertaken using lunar occulations (Peterson & White 1984; Peterson et al. 1989), spectroscopy (Treanor 1960; Mermilliod et al. 1990; Bolte 1991; Abt & Willmarth 1999; Mermilliod & Mayor 1999; Halbwachs et al. 2003), speckle imaging (Mason et al. 1993; Patience et al. 2002), adaptive optics imaging (Bouvier et al. 2001), and time-domain photometry (e.g., Pepper et al. 2008). Spectroscopic binaries in Praesepe have also been identified through larger radial velocity (RV) surveys (Wilson & Joy 1950; Pourbaix et al. 2004; Mermilliod et al. 2009). Several of these surveys also note RV-variable or candidate binary systems. Bolte (1991) and Hodgkin et al. (1999) identify candidate binary systems by their position above the cluster main sequence (similar to our method above).

Three planets have been detected from RV observations of two Praesepe members (Quinn et al. 2012; Malavolta et al. 2016), including one hot Jupiter in each system. One confirmed and eight candidate transiting planets have also been discovered from the K2 data for the cluster (Barros et al. 2016; Libralato et al. 2016; Mann et al. 2016; Obermeier et al. 2016; Pope et al. 2016).

No eclipsing binaries in Praesepe have been published from the K2 data so far, but we identify four likely eclipsing binaries and two single-transit events by eye; see the Appendix for details. One of these candidate eclipsing binaries was previously identified from PTF data, and has been confirmed with RVs (A. L. Kraus et al. 2017, in preparation). We consider the other three eclipsing binaries to be candidate binaries until we can confirm that the eclipses are not from a background system.

We also consider stars with multiple periods visible in the K2 light curve to be candidate binaries if the two peaks are separated by at least 20% of the primary period. In other words, if

$$\begin{eqnarray}&&\displaystyle \frac{| {P}_{\mathrm{rot},1}-{P}_{\mathrm{rot},2}| }{{P}_{\mathrm{rot},1}}\gt 0.2,\end{eqnarray} \tag{ 3 }$$

we consider the target to be a candidate binary. This threshold is based on the maximum period separation for differentially rotating spot groups on the Sun (see Rebull et al. 2016). Fifty-eight K2 targets have a second period detected in their periodogram, and nine more have a second period identified by eye only, giving 67 (10%) multiperiodic targets out of the 677 K2 targets with measured P_rot.

In total, we find 82 confirmed binaries or triples, 92 candidate systems from our photometric analysis and the literature, and 170 additional candidate systems identified from our K2 analysis. Table 2 lists the binary members and their relevant properties, and they are also flagged in Table 3. Aside from the M-dwarf eclipsing binary noted above, however, confirmed binaries in Praesepe are only found above $0.72\,{M}_{\odot }$, which limits our ability to analyze the impact of binaries on rotation and activity in low-mass Praesepe members.

Table 3.  P_rot Measurements for Praesepe Stars Targeted in K2

Namea	EPIC	${P}_{\mathrm{rot},1}$	Power1	Q1	Clean	Threshold	${P}_{\mathrm{rot},2}$	Power2	Q2	Multi	Spot	Blend	Bin.	Ampl.(mag)
JC201	211930461	14.59	0.83910	0	Y	0.00816	⋯	⋯	⋯	N	Y	Y	Conf	0.00934
⋯	212094548	6.60	0.00890	1	N	0.00521	⋯	⋯	⋯	N	N	N	⋯	0.04953
⋯	211907293	⋯	⋯	2	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0	⋯
KW222	211988287	3.29	0.20730	0	N	0.00861	⋯	⋯	⋯	M	Y	N	⋯	0.00281
KW238	211971871	2.96	0.66740	0	Y	0.00747	⋯	⋯	⋯	N	Y	N	⋯	0.01633
KW239	211992776	1.18	0.30260	0	Y	0.00791	⋯	⋯	⋯	M	Y	N	⋯	0.00109
KW282	211990908	2.56	0.25900	0	Y	0.00802	⋯	⋯	⋯	Y	Y	Y	Conf	0.00497
AD 2305	212100611	1.34	0.44670	0	Y	0.00776	1.8001	0.18730	0	Y	N	M	⋯	0.03630
AD 2482	211795467	15.49	0.23690	0	Y	0.00794	⋯	⋯	⋯	N	Y	N	⋯	0.01088
JS352	211913532	16.33	0.09270	0	Y	0.00723	2.8027	0.01060	0	Y	Y	N	Cand	0.01570

Note.

^aLiterature name given in Kraus & Hillenbrand (2007). All are standard SIMBAD identifiers, except AD####, which correspond to stars in Adams et al. (2002).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

There are 207 Praesepe members with P_rot measured both from K2 data and from at least one ground-based survey. Another 51 members have P_rot measured by multiple ground-based surveys, but not by K2. The 43 stars with P_rot from at least two studies that differ by >10% are listed in Table 1. Overall, the agreement between K2 and literature P_rot measurements is excellent: half of our K2 measurements are consistent with previous measurements to within 2%, and >75% are consistent to within 5%.

Discrepant measurements are typically $\tfrac{1}{2}\times$ or $2\times$ harmonics of each other (Figure 8). All but two stars with discrepant P_rot show evidence of evolving spot configurations: either a double-dip light-curve structure or a varying amplitude of modulation over the course of the campaign. This signature is usually better resolved in the K2 light curves, allowing us to measure the correct period even if it is not the highest periodogram peak.

Additionally, four stars with discrepant P_rot values show evidence for two periods in the light curve. The PTF and K2 periods for EPIC 211937872 and EPIC 211971354 are ≈1 days apart; both K2 light curves show a second ≈1-day period superimposed on the primary period, in addition to evidence of spot evolution. Two periods are detected in the K2 data for EPIC 212013132: ${P}_{\mathrm{rot},1}=2.13$ days, half of the SWASP ${P}_{\mathrm{rot}}=4.27$ days, and ${P}_{\mathrm{rot},2}=12.32$ days, consistent with the Kovács et al. (2014) ${P}_{\mathrm{rot}}=12.78$ days. Finally, two periods are also detected in the K2 light curve of EPIC 211734093: ${P}_{\mathrm{rot},1}=18.22$ days and ${P}_{\mathrm{rot},2}=7.74$ days. The latter of these is half of the Kovács et al. (2014) ${P}_{\mathrm{rot}}=15.87$ days. In all four cases, the second period in the light curve, possibly with additional spot evolution effects, accounts for the discrepancy between our K2 P_rot values and those in the literature.

In three cases, the P_rot measured by Scholz & Eislöffel (2007) and Scholz et al. (2011) is potentially a 1-day alias of the K2 period. Scholz & Eislöffel (2007) surveyed Praesepe over three observing runs lasting three to five nights each, and Scholz et al. (2011) surveyed the cluster again for nine nights. Measurements over such short baselines are more prone to aliasing, particularly when the periods are so close to 0.5 or 1 days.

We find no strong evidence that the Praesepe stars with literature P_rot have larger photometric amplitudes (Figure 9), which has often been invoked to explain low P_rot yields from ground-based surveys. The only partial exception to this are the PTF data: in Paper I, we could only measure P_rot for stars with amplitudes $\gtrsim 0.02$ mag (≳1%) for ${K}_{p}\gt 16$. Aside from this handful of PTF stars, small photometric amplitude—i.e., less contrast between starspots and the stellar photosphere—does not explain the incompleteness of ground-based surveys.

Overall, 86% of Praesepe K2 targets have detectable P_rot, suggesting that non-detections in ground-based surveys are due primarily to limitations of those surveys rather than to inclination effects or spot coverage. Our Praesepe and Hyades K2 P_rot are nearly all consistent with previous measurements (Figure 8 and Paper III). We conclude that ground-based P_rot measurements are reasonably reliable, and that these surveys are merely limited by trade-offs between photometric precision, cadence, baseline, and number of targets; interruptions due to daylight and weather; and variable spot patterns on the stars themselves. Further comparisons of the K2 data with ground-based light curves and survey techniques are needed to determine why previous surveys did not detect the rotators with new P_rot measured here.

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Nonetheless, the overall agreement between the K2 measurements and those of previous surveys indicates that our P_rot measurement procedures provide accurate results. It also bodes well for future ground-based surveys: while K2's superior precision allows us to resolve detailed light-curve features, it appears that, in general, ground-based surveys produce valid and reproducible P_rot measurements.

In Paper III, we found that nearly all the rapid rotators in the Hyades with ${M}_{* }\ \gtrsim 0.3\,{M}_{\odot }$ were confirmed or candidate binary systems. Of the three remaining rapid rotators, none had been surveyed for companions. The Hyades as a whole has been extensively surveyed for companions: >30% of all Hyads are confirmed binaries, including ≈45% of Hyads with measured P_rot. This suggested that all single stars with ${M}_{* }\ \gtrsim 0.3\,{M}_{\odot }$ have converged onto the slow-rotator sequence by ≈650 Myr.

For Praesepe stars, we define the cutoff between the slow-rotator sequence and more rapid rotators by computing the 75th percentile of periods for stars with $1.1\ \gtrsim {M}_{* }\ \gtrsim 0.3\,{M}_{\odot }$, and then lowering this threshold by 30%. This produces the orange line shown in Figure 10. We find that half of all rapidly rotating Praesepe stars are confirmed or candidate binaries.

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Despite the far more extensive P_rot catalog in Praesepe relative to the Hyades, however, we are currently unable to confirm our result from Paper III because Praesepe lacks a similarly rich binary catalog. Only 7% of all cluster members are confirmed binaries, and (with the exception of one eclipsing M-dwarf binary) these are restricted to ${M}_{* }\ \gtrsim 0.72\,{M}_{\odot }$. Our identification of candidate systems is also likely incomplete. Confident analysis of the impact of binaries on the mass–period plane requires additional binary searches in Praesepe.

Many stars on the slow-rotator sequence are also candidate binaries. This might suggest that companions have minimal impact on angular-momentum evolution. It could also indicate that different subsets of binaries undergo different rotational evolution.

The rapidly and slowly rotating binaries likely have different separation distributions, due to the impact of disk disruption on their initial angular-momentum content. Single stars experience braking due to their protoplanetary disks (Rebull et al. 2004). Binaries wider than 40 au are unlikely to disrupt each others' protoplanetary disks (Cieza et al. 2009; Kraus et al. 2016) and are far too wide to be affected by tides—these systems will therefore evolve as (two) single stars. Binaries closer than 40 au, on the other hand, are far more likely to have disrupted disks, which would allow the component stars to spin up without losing angular momentum to their disks. These systems will arrive on the MS spinning more rapidly, and eventually spin down to converge with single stars. We expect that future studies of Praesepe will find that binaries with slowly rotating components are wider than 40 au (≈02 at ≈180 pc), while the rapidly rotating stars have companions at closer separations.

In Paper III, we found that the Reiners & Mohanty (2012) and Matt et al. (2015) models for angular-momentum evolution predicted faster rotation than that observed for 0.9−0.3 ${M}_{\odot }$ stars. However, this comparison was limited by the number of Hyads with P_rot. We therefore compare our far richer Praesepe sample with the models of Matt et al. (2015) and Brown (2014), which were generously provided by these authors (S. Matt 2015, private communication; T. Brown 2017, private communication).

5.3.1. Matt et al. (2015)

Matt et al. (2015) derive a model for the angular-momentum evolution of a rotating solid sphere due to magnetic braking. These authors' initial conditions approximate the distribution of P_rot observed for 2−5 Myr old stars, but are not drawn directly from observations. Matt et al. (2015) allow the stellar radius to evolve according to model evolutionary tracks. Their prescription for the angular momentum lost via stellar winds is based on the Kawaler (1988) and Matt et al. (2012) solar-wind models, and the angular-momentum loss scales with stellar mass and radius. Matt et al. (2015) also use explicitly different spin-down rates for stars in the saturated and unsaturated regime.

The Matt et al. (2015) model accurately predicts the mass dependence of the slow-rotator sequence for Hyades and Praesepe stars with ${M}_{* }\ \gtrsim 0.8\ {M}_{\odot }$, with the exception of a handful of binary stars (see Figures 11–12). This indicates that, as in our comparison to the Hyades alone, the stellar-wind prescription used by Matt et al. (2015) is correct for solar-type stars.

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The lower envelope of P_rot predicted by Matt et al. (2015) approximates that observed in Praesepe, though the distribution of rapidly rotating stars with ${M}_{* }\,\lesssim \,0.8\,{M}_{\odot }$ is much more sparse than predicted by the model. Using the division between the slow sequence and faster rotators defined in Section 5.2, we observe that 26% of stars with masses 0.3−0.8 ${M}_{\odot }$ are rapidly rotating, relative to 77% of model stars. In Figure 13, we have binned the model and data P_rot distributions by mass to allow for easier comparisons of the period distribution. Below ≈0.8 ${M}_{\odot }$, the Matt et al. (2015) model predicts a broader distribution of periods than is observed, while the observed P_rot are more concentrated at slow periods with a tail of fast rotators. This suggests that, although the Matt et al. (2015) model may work for some 0.8−0.3 ${M}_{\odot }$ stars, it fails to predict the efficiency with which <50% of stars in this mass range spin down.

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The most obvious discrepancy between the Matt et al. (2015) models and our data occurs for slowly rotating early M stars with masses ≈0.6−0.3 ${M}_{\odot }$, as was previously noted in Matt et al. (2015) and Paper III. In our observations, more than half of the 0.6−0.3 ${M}_{\odot }$ stars have converged to the slow-rotator sequence, which extends fairly smoothly from ≈1−0.3 ${M}_{\odot }$ (see Figure 12), and more than half of the remaining rapid rotators are binaries (Figure 11).

By contrast, the model predicts an end to the slow-rotator sequence around 0.6 ${M}_{\odot }$, with the slowest rotators at lower masses being significantly faster than the slow rotators observed in our data. The median P_rot we observe for 0.6−0.3 ${M}_{\odot }$ stars is >75% slower than predicted (Figure 12). Furthermore, >60% of stars in this mass range rotate more slowly than the maximum P_rot predicted for their mass (Figure 13). It appears that real early M dwarfs brake far more efficiently than predicted by Matt et al. (2015).

This discrepancy suggests that most M dwarfs undergo enhanced angular-momentum loss relative to their higher mass counterparts. This could be due to a change in the structure of the magnetic field, i.e., a larger, less complex field with more open field lines near the star's equator that would allow the star to more efficiently shed angular momentum (i.e., Donati 2011; Garraffo et al. 2015). It could also indicate a departure from solid-body rotation, which is assumed by Matt et al. (2015), for early M dwarfs. We could be observing an effect of the deepening convective zone for M dwarfs, through a change in the moment of inertia or in the dynamo as the radiative core shrinks with decreasing mass.

Despite the failure of the Matt et al. (2015) model to predict the observed behavior of early M dwarfs, this model does reasonably well in reproducing the distribution of rapidly rotating, fully convective 0.1−0.2 ${M}_{\odot }$ stars. This suggests that whatever is to blame for the discrepancy with observed early M dwarfs, the physical assumptions of Matt et al. (2015) do apply to fully convective stars.

5.3.2. Brown (2014): The Metastable Dynamo Model

Brown (2014) derives an empirical model for the generation and evolution of the fast and slow-rotator sequences, called the Metastable Dynamo Model (MDM). He models all stars as solid bodies that are born with weak coupling between their dynamos and stellar winds, leading to minimal spin down. The stars then spontaneously and permanently switch into a strong coupling mode where they spin down rapidly. Brown (2014) does not employ a critical P_rot for this switch—it is purely stochastic, with a mass-dependent probability of switching by a given stellar age. Taking as its starting point the distribution of periods in the 13 Myr old cluster h Per, the MDM generates a bimodal distribution at older ages: a fast sequence and a slow sequence separated by a gap, similar to the distribution observed by Barnes (2003).

The Brown (2014) model approximately reproduces the overall morphology of the mass–period plane in Praesepe: there is a clear sequence of slowly rotating stars with some faster rotators. A more careful comparison, however, indicates that the model and data are discrepant. Specifically, the bimodality is not obvious in the data for Praesepe >0.5 ${M}_{\odot }$ stars, which is the mass regime covered by the MDM (Figure 14–15). The rapidly rotating Praesepe stars in this mass range are not strongly concentrated at any particular P_rot, nor is there an obvious gap at intermediate P_rot. Furthermore, using the division between slow and fast stars defined in Section 5.2, we find that 15% of observed 0.9−0.5 ${M}_{\odot }$ Praesepe stars are rapidly rotating, compared to only 7% in the model.

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We do observe stronger bimodality for 0.25−0.5 ${M}_{\odot }$ stars, below the mass range modeled by the Brown (2014). However, the observed morphology does not match the predictions for >0.5 ${M}_{\odot }$ stars: the rapid rotators extend to ${P}_{\mathrm{rot}}\approx \tfrac{1}{2}$ days and show a wider range of fast P_rot, in contrast to the clear lower limit of ≈1.5 days in the current model. Our observations of early M stars in Praesepe therefore support the Brown (2014) model's prediction of bimodality in the mass–period morphology, but adjustments are needed to extend the MDM to this mass range.

Finally, the predicted locations of the fastest and slowest rotators at a given mass do not match the observations. The slow-rotator sequence is too slow, while a handful of early M dwarfs with $0.5\ \lesssim {M}_{* }\ \lesssim 0.6\,{M}_{\odot }$ rotate faster than predicted by the MDM. The offset of the slow sequence is visible in Figure 6 of Brown (2014), who points out that more complicated physics is likely needed to explain the exact evolution of slow rotators. The too-fast rotators are not obvious in that figure; however, due to the use of a linear P_rot axis and the inclusion of only a few dozen P_rot from Delorme et al. (2011) and WEBDA, compared to the hundreds of P_rot included here.